development of a transient gradient enhanced non local continuum damage … 4.5 flowchart for the...

$: DEVELOPMENT OF A TRANSIENT GRADIENT ENHANCED NON LOCAL CONTINUUM DAMAGE … 4.5 Flowchart for the analysis of RVE ... damage growth rate which depends on the tensile fracture energy$
DEVELOPMENT OF A TRANSIENT GRADIENT

ENHANCED NON LOCAL CONTINUUM

DAMAGE MECHANICS MODEL FOR

MASONRY

By

Ali Jelvehpour

B.Sc., M.Sc.

A Thesis Submitted in Fulfilment of the

Requirements for the Degree of Doctor of Philosophy

(PhD)

School of Civil Engineering and Built Environment

Science and Engineering Faculty

Queensland University of Technology

2016

QUT Verified Signature

i Acknowledgments |

Acknowledgements

First and foremost, I wish to express my sincere gratitude to my principal supervisor Prof.

Manicka Dhanasekar for his continuous encouragement, support and guidance throughout my

PhD candidature. I would also like to thank Prof. Dhanasekar for providing me with a

supervisory scholarship. I appreciate all his time, efforts and ideas which lead me forward in

my PhD journey. He will always be an excellent example of a friend, a teacher and a

professor for me. I also wish to thank Dr. Xuemei Liu for her support whenever I needed.

Acknowledgment is due to Concrete Masonry Association of Australia (CMAA) for their

financial support of this research.

I like to thank School of Civil Engineering and Built Environment (CEBE) of Queensland

University of Technology for providing me with a tuition fee waiver scholarship an also for

providing me with excellent research facilities. I am grateful to QUT’s High Performance

Computing (HPC) team, especially Mr. Mark Barry, for their support throughout my

research.

I also would like to thank all my friends and fellow PhD candidates, especially Ashkan

Mohit, Idin Ravaz, Proshot Tehrani, Sanam Aghdamy, Thangarajah Janaraj, Julian Ajith

Thamboo, Shahid Nazir, Sarkar Noor-E-Khuda and Norrul Azmi Yahya for their continuous

support.

Finally, I like to thank my lovely wife, Hana, my amazing parents, Bagher and Shahnaz, my

caring siblings, Parisa and Hamid and my beautiful niece Proshot for their unconditional

support. None of this would have been possible without them.

ii Abstract |

Abstract Due to the advent of varied types of masonry systems a comprehensive failure mechanism of masonry essential for the understanding of its behaviour is impossible to be determined from experimental testing. As masonry is predominantly used in wall structures a biaxial stress state dominates its failure mechanism. Biaxial testing will therefore be necessary for each type of masonry, which is expensive and time consuming. A computational method would be advantageous; however masonry is complex to model which requires advanced computational modelling methods. This thesis has formulated a damage mechanics inspired modelling method and has shown that the method effectively determines the failure mechanisms and deformation characteristics of masonry under biaxial states of loading.

A continuum damage mechanics (CDM) model incorporating variable Poisson’s ratio which represents the evolution of microcracks has been formulated. The model is enhanced with a transient-gradient nonlocal formulation to account for the post peak softening of quasi-brittle materials without any mesh pathology. The enhanced model has been implemented for the development of representative volume elements (RVEs) for masonry. Two forms of RVEs have been developed and through application to simulate the test results of masonry under uniaxial stress states, it has been shown that the geometry of the RVE that incorporates a single unit surrounded by half thickness of binder layers is sufficient.

Through a series of simulation of biaxial tests on half-scale clay brick masonry panels it has been shown that the enhanced model provides an over-stiff predictions of masonry. The major reason for this over prediction is the inherent assumption of the perfect bond between the surfaces of the units and the mortar layers. To appropriately define the bond characteristics a nonlinear contact or other complex methods would have to be incorporated within the RVE, which could lead to computational complexity. An interfacial transition zone (ITZ) concept which only requires the basic formulations in the enhanced modelling method has been used to enrich the RVE. The ITZ enriched RVE has been used for the prediction of failure surfaces of conventional masonry systems. The predicted failure surfaces have been found to compare well with the experimental datasets where available.

The transient-gradient enhanced CDM model has been further developed to represent the behaviour of dry-stack masonry through a contact surface closure concept represented by initial damage parameter standing for surface imperfections. The model has been calibrated with uniaxial experimental datasets and extended for biaxial failure surface predictions. Failure envelopes in the non-dimensional zero-shear stress plane for all masonry systems fall in a zone with the hollow block dry-stack masonry as its lower bound and a combination of the conventional hollow block prisms and the modified hollow block dry-stack masonry as its upper bound envelopes.

The ITZ enriched CDM model has been used to simulate the uniaxial compression behaviour of masonry consisting of a range of units from unburnt clay bricks to dressed stone and two types of mortar involving 140 combinations. The results have been compared to the Australian masonry standard AS3700 provisions, which provided several conclusions.

iii Contents |

Contents

List of Symbols .................................................................................................................................... viii

List of Figures ...................................................................................................................................... xii

List of Tables ...................................................................................................................................... xxii

CHAPTER 1 ........................................................................................................................................... 1

Introduction............................................................................................................................................ 1

1.1 General Remarks .................................................................................................................... 1

1.2 Research Significance ............................................................................................................ 2

1.3 Aim .......................................................................................................................................... 3

1.4 Thesis Arrangement ............................................................................................................... 3

CHAPTER 2 ........................................................................................................................................... 5

Literature Review ................................................................................................................................... 5

2.1 Introduction ............................................................................................................................ 5

2.2 Experimental Research on Masonry ..................................................................................... 6

2.2.1 Uniaxial Compressive Behaviour ................................................................................ 6

2.2.2 Uniaxial Tensile Behaviour ......................................................................................... 8

2.2.3 Shear Behaviour .......................................................................................................... 9

2.2.4 Biaxial Behaviour ...................................................................................................... 10

2.3 Modelling Masonry .............................................................................................................. 13

2.3.1 Microscopic Models ................................................................................................... 14

2.3.2 Macroscopic Models.................................................................................................... 15

2.3.3 Homogenisation Models ............................................................................................ 15

2.4 Continuum Damage Mechanics .......................................................................................... 17

2.4.1 Damage Variable ........................................................................................................ 19

2.4.2 Principle of Strain-Equivalence ................................................................................ 21

2.4.3 Damage Evolution Law.............................................................................................. 23

2.5 Localisation and Mesh Sensitivity in Quasi-Brittle Material ............................................. 28

2.6 Non-Local Models ................................................................................................................ 29

2.6.1 Integral Non-local Model for Strain Softening Material ......................................... 30

2.6.2 Gradient Enhanced Non-local Model for Strain Softening Material ...................... 30

2.6.3 Finite Element Implementation of Gradient Enhanced Non-Local Model ............. 32

2.7 The Transient-Gradient Non-Local Model for Strain Softening Material ........................ 33

2.7.1 Nonlocal Damage Model with a Damage-Dependant Transient Length Scale ....... 34

2.7.2 Nonlocal Damage Model with a Strain-Dependant Transient Length Scale .......... 35

iv Contents |

2.8 First-Order Computational Homogenisation ...................................................................... 36

2.8.1 Choice of Homogenisation Boundary Condition ...................................................... 37

2.8.2 Periodic Boundary Conditions for Computational Homogenisation ....................... 38

2.9 Concluding Remarks ............................................................................................................ 39

CHAPTER 3 ......................................................................................................................................... 40

Development of a CDM Model Incorporating Variation of Poisson’s Ratio .................................... 40

3.1 Introduction .......................................................................................................................... 40

3.2 Uniaxial Behaviour of Quasi-brittle Material..................................................................... 41

3.3 Formulation ......................................................................................................................... 42

3.3.1 Refined Damage Evolution Law for Masonry Constituents ..................................... 42

3.3.2 Variable Poisson’s Ratio ............................................................................................ 44

3.4 Implementation .................................................................................................................... 47

3.5 Numerical Example ............................................................................................................. 50

3.5.1 Uniaxial Tension ....................................................................................................... 51

3.5.2 Uniaxial Compression ................................................................................................ 54

3.5.3 Pure Shear .................................................................................................................. 56

3.6 Parametric Study .................................................................................................................. 58

3.6.1 Uniaxial Tension Test ................................................................................................ 59

3.6.1.1 Effect of Damage Evolution Law Parameter α ............................................... 59

3.6.1.2 Effect of Damage Evolution Law Parameters β ............................................. 61

3.6.2 Uniaxial Compression Test ........................................................................................ 62

3.6.2.1 Effect of Equivalent Strain Parameter k ......................................................... 62

3.6.2.2 Effect of Damage Evolution Law Parameter ζ ................................................ 63

3.6.2.3 Effect of Damage Evolution Law Parameter 𝝎𝒄 ............................................. 64

3.6.2.4 Effect of the Volumetric Damage Parameter η ................................................. 66


CHAPTER 4 ......................................................................................................................................... 71

Enhancement of the CDM Model through a Non-Local Transient-Gradient Method ..................... 71

4.1 Introduction .......................................................................................................................... 71

4.2 Formulation of a Transient-Gradient Model ...................................................................... 72

4.3 Homogenisation Technique for Modelling Masonry ......................................................... 74

4.3.1 Choice of a Mesoscopic Representative Volume Element (RVE) ............................ 74

4.3.2 Boundary conditions for the Representative Volume Element (RVE) ..................... 76

4.4 Implementation .................................................................................................................... 78

4.5 Numerical Example ............................................................................................................. 81

4.5.1 Stress-strain Behaviour of Individual Constituents .................................................. 81

v Contents |

4.5.2 Numerical Analysis of the RVEs ............................................................................... 84

4.6 Parametric Study .................................................................................................................. 89

4.6.1 Effect of the Non-Local Length Scale Parameter c .................................................. 89


CHAPTER 5 ......................................................................................................................................... 93

Application of the Transient Enhanced RVE to Brick Masonry under Biaxial Loading ................. 93

5.1 Introduction .......................................................................................................................... 93

5.2 Problem definition ................................................................................................................ 94

5.2.1 Tensile and Compressive Behaviour of Individual Constituents (Brick and Mortar) 95

5.2.2 RVE Geometry, Discretisation and Loading Configuration ....................................... 98

5.2.3 Stress-Strain Behaviour of the RVE under Different Loading Configurations ....... 100

5.3 Observations ....................................................................................................................... 105

5.4 Concluding Remarks .......................................................................................................... 106

CHAPTER 6 ....................................................................................................................................... 107

Enrichment of the RVE with Interfacial Transition Zone (ITZ) ..................................................... 107

6.1 Introduction ........................................................................................................................ 107

6.2 Interfacial Transition Zone (ITZ) Concept ....................................................................... 108

6.2.1 Enrichment of the RVE with an Interfacial Transition Zone (ITZ) ...................... 109

6.2.2 Elastic Properties of the Layers of the Interfacial Transition Zone (ITZ) ............. 110

6.3 Parametric study ................................................................................................................ 111

6.3.1 Effect of Thickness of the Interfacial Transition Zone (ITZ) ................................ 113

6.3.2 Effect of the Interfacial Transition Zone (ITZ) Parameter 𝝀𝑬 .............................. 117

6.4 Validation of the Interfacial Transition Zone (ITZ) Enriched CDM Model ................... 122

6.4.1 Validation of Uniaxial Tests Conducted by Dhanasekar (1985) ............................ 122

6.4.2 Validation of Uniaxial Compression Tests Conducted by Barbosa & Hanai ......... 126


CHAPTER 7 ....................................................................................................................................... 134

Application of the ITZ Enriched CDM Model – 1: Constitutive Properties of Conventional

Masonry under Biaxial Stresses ........................................................................................................ 134

7.1 Introduction ........................................................................................................................ 134

7.2 Problem Definition ............................................................................................................. 134

7.3 Masonry RVEs, their Dimensions and Properties ............................................................ 135

7.3.1 RVE Considered to Simulate Biaxial Tests by Dhanasekar (1985) ........................ 136

7.3.2 RVE for the Simulation of Biaxial Tests on Hollow Concrete Masonry ................ 138

7.4 Numerical Modelling of Biaxial Testings ......................................................................... 139

7.4.1 Simulation of Clay Brick Masonry Experimental Tests . Error! Bookmark not defined.

vi Contents |

7.4.1.1 Bed Joint Angles of 𝜽 = 𝟎° and 𝜽 = 𝟗𝟎°(Zero-shear state) ......................... 142

7.4.1.2 Bed Joint Angles of 𝜽 = 𝟐𝟐. 𝟓° and 𝜽 = 𝟔𝟕. 𝟓° ............................................. 144

7.4.1.3 Bed Joint Angle of 𝜽 = 𝟒𝟓° ............................................................................. 147

7.4.1.4 Failure Envelope .............................................................................................. 148

7.4.1.5 Prediction of Modes of Failure ........................................................................ 153

7.4.2 Simulation of Concrete Block Masonry Biaxial Experiments ................................ 162

7.4.2.1 Analysing Case 1 (P1) ....................................................................................... 162

7.4.2.2 Analysing Case 2 (P2) ....................................................................................... 163

7.4.2.3 Analysing Case 3 (P3) ....................................................................................... 163

7.4.2.4 Analysing Case 4 (P4) ....................................................................................... 164

7.4.3 Comparison of Failure Envelopes for Conventional Masonry .............................. 165


CHAPTER 8 ....................................................................................................................................... 168

Application of the ITZ Enriched CDM Model – II: Constitutive Properties of Dry-stack Masonry

under Biaxial Stresses ........................................................................................................................ 168

8.1 Introduction ........................................................................................................................ 168

8.2 Dry-Stack Masonry ............................................................................................................ 168

8.3 Constitutive Modelling of Dry-Stack Masonry ................................................................. 169

8.3.1 Formulation of Damage Evolution Law for Dry-Stack Joint ................................ 171

8.3.2 Implementation ........................................................................................................ 172

8.4 Parametric Study ................................................................................................................ 174

8.4.1 Effect of the Full Contact Strain 𝒌𝒋𝒄 ....................................................................... 175

8.4.2 Effect of Initial Imperfection Parameter 𝑰𝟎 ............................................................ 177

8.4.3 Effect of Damage Slope Parameter 𝑺𝒋 ..................................................................... 178

8.5 Numerical Validation of the Model ................................................................................... 180

8.5.1 Simulation of Experimental Tests Conducted by Oh (1994) ................................... 180

8.5.2 Masonry Unit Properties ........................................................................................... 182

8.5.3 Analysis of the RVEs for Uniaxial Compression Test ............................................. 187

8.6 Prediction of Biaxial Failure Envelope for Dry-Stack Masonry ..................................... 188

8.6.1 Prediction of Biaxial Failure Envelope for the H-Block ......................................... 190

8.6.2 Prediction of Biaxial Failure Envelope for the Modified H-Block ......................... 191

8.6.3 Prediction of Biaxial Failure Envelope for the Conventional Hollow Block ......... 191

8.6.4 Comparison of Biaxial Failure Envelopes of Different Masonry Systems ............. 192


CHAPTER 9 ....................................................................................................................................... 196

Application of the Model for the Masonry Compressive Strength Prediction ................................. 196

vii Contents |

9.1 Introduction ........................................................................................................................ 196

9.2 Current AS 3700 (2011) Provisions for Compressive Strength of Masonry .................... 196

9.3 Effect of Different Parameters on Compressive Strength of Masonry ............................ 197

9.3.1 Effect of Characteristic Unconfined Compressive Strength of Masonry Unit 𝒇′𝒖𝒄 200

9.3.2 Effect of Masonry Unit Height to Mortar Bed Joint Thickness 𝒉𝒖/𝒕𝒋 ................... 202


CHAPTER 10 ..................................................................................................................................... 209

Conclusions and Recommendations ................................................................................................. 209

10.1 Summary............................................................................................................................. 209

10.2 Conclusion .......................................................................................................................... 210

10.3 Recommendations for Future Work .................................................................................. 211

References .......................................................................................................................................... 213

Appendix A ......................................................................................................................................... A-1

Appendix B ......................................................................................................................................... B-1

Appendix C ......................................................................................................................................... C-1

Appendix D ......................................................................................................................................... D-1

Appendix E ......................................................................................................................................... E-1

Appendix F ......................................................................................................................................... F-1

Appendix G ......................................................................................................................................... G-1

Appendix H......................................................................................................................................... H-1

viii List of Symbols |

List of Symbols

cA Model Parameter

tA Model Parameter

cB Model Parameter

tB Model Parameter

B Interpolation function

𝒄 = A function of length scale parameter

0c An arbitrary positive value that prevents non-local interaction

𝑪𝒊𝒋𝒌𝒍 = The elasticity matrix components

E Young’s modulus

0E Initial Young’s modulus

0( )E n The initial Young’s modulus of the nth

layer of the ITZ

ME Young’s modulus of the mortar layer

E Damaged Young’s modulus

mf ' Characteristic compressive strength of masonry

ucf ' Characteristic compressive strength of units

Fg fracture energy

tg Tension evolution function

cg Compression evolution function

G shear modulus

0G Initial shear modulus

H Height of the RVE or prism

uh height of unit

𝐼1 = the first invariant of strain tensor

0I The initial imperfection parameter

𝐽2 = the second invariant of deviatoric strain tensor

𝑘 = the equivalent strain tension to compression sensitivity parameter

K Bulk modulus

0K Initial Bulk modulus

hK joint thickness factor

cl Length scale parameter

L Length of the RVE or prism

ix List of Symbols |

N The total number of layers within the ITZ

𝑵 = interpolation function

TN Transpose of interpolation Matrix

n Vector normal to the surface

n Transient-gradient model parameter

jS Damage slope parameter

jt thickness of the mortar joint

ITZT thickness of the total ITZ

i u Displacement on boundary i

A

u Displacements of the controlling node A

B

u Displacements of the controlling node B

C

u Displacements of the controlling node C

𝒖𝒌 = the displacement with respect to Cartesian coordinate system

u x The strain-periodic displacement field

W Width of the RVE or prism

�⃗⃗⃗� (�⃗⃗� ) = a mesoscopic displacement fluctuation field

𝒙 = material point

�⃗⃗� = the position vector

,eqf

damage loading function

A parameter controlling the softening tail

t Tensile damage portion

c Compressive damage portion

β= A parameter controlling the damage growth rate

nu displacement increment of integration point n

n Strain increment of integration point n

i principal strains

𝜺𝒌𝒍 = The linear strains

eq The equivalent strain

eq non-local equivalent strain

,i eqε non-local equivalent strain of boundary i

f a model parameter controlling the initial slope of the softening curve

Maximum transient gradient strain threshold

M ε Macroscopic strain tensor

x List of Symbols |

𝝋𝒊 = test function

= A function of damage parameter

A model parameter

Damage evolution law parameter

damage threshold

𝜅𝑖 = The threshold for damage initiation

d The strain in which the material completely loses its integrity

c threshold strain beyond which damage grows rapidly

jc The strain at which the joint is fully closed

E A parameter controlling the stiffness degradation

A parameter controlling the decrease of Poisson’s ratio

η= Path parameter

𝝂 = Poisson’s ratio

0 Initial Poisson’s ratio

M The Poisson’s ratio of the mortar layer

0( )n The initial Poisson’s ratio of the nth

layer of the ITZ

𝜔 = Scalar damage parameter

T Tensile damage variables

c the threshold damage beyond which damage grows rapidly

C Compressive damage variables

Nonlocal damage parameter

K volumetric damage parameter

energy

𝜎1 = Stress in principal axes 1

𝜎2 = Stress in principal axes 2

𝝈𝒊𝒋 = The Cauchy stress

𝜎𝑛 = Stress normal to RVE bed joint

𝜎𝑝 = Stress parallel to RVE bed joint

𝜎𝑥𝑥 = Stress in direction x-x

𝜎𝑦𝑦 = Stress in direction y-y

𝜎𝑥𝑦 = Stress in direction x-y

𝜏 = Shear stress applied on the RVE

𝜃 = The orientation of the loading axes with respect to masonry bed joints

Bulk modulus damage portion

xi List of Symbols |

A homogenous and isotropic weight function

the relative distance from point 𝒙

( ) Variable length scale parameter as a function of damage parameter

( )eq Variable length scale parameter as a function of nonlocal equivalent strain

The original cross sectional area

volume of the element

c fully damaged region

d partially damaged region

Damage evolution law parameter

𝜵(𝒏) = the nth

order gradient

2 the Laplacian operator

. The McAuley brackets

𝕮𝟎 = Zero order continuous domain

𝕮𝟏 = First order continuous domain

xii List of Figures |

List of Figures

Figure 1.1 Varied structural masonry systems......................................................................... 1

Figure 2.1 Local state of stress in the constituents of masonry prisms under vertical

compression perpendicular to bed joint .................................................................................... 7

Figure 2.2 Modes of failure of solid clay masonry under uniaxial compression, from

(Dhanasekar, 1985) ................................................................................................................... 8

Figure 2.3 Modes of failure of solid clay masonry under uniaxial tension, from

(Dhanasekar, 1985) ................................................................................................................... 9

Figure 2.4 Modes of failure of solid clay masonry under uniaxial compression and uniaxial

tension considering different load orientations with respect to bed joint, from (Dhanasekar,

1985) ........................................................................................................................................ 10

Figure 2.5 Modes of failure of solid clay masonry under biaxial loads(Dhanasekar, 1985) . 11

Figure 2.6 Mode of failure of solid clay masonry under biaxial compression-compression,

from (Dhanasekar, 1985) ......................................................................................................... 11

Figure 2.7 Biaxial failure envelope of solid clay masonry (Page, 1981; Page, 1983) .......... 12

Figure 2.8 The periodic assemblage of masonry ................................................................... 14

Figure 2.9 Illustration of different scales in multi-scale methods.......................................... 17

Figure 2.10 Distribution of damage on an element and representation of microstructural

defects ...................................................................................................................................... 20

Figure 2.11 a) Damage growth plotted from Equation (2.12); b) Stress-strain curve

corresponding to Equation (2.12) ............................................................................................ 25


corresponding to Equation (2.13) ............................................................................................ 25

Figure 2.13 Exponential softening law from Jirásek, et al. (2004) ........................................ 27

Figure 3.1 Typical behaviour of quasi brittle material under uniaxial tension ..................... 41

Figure 3.2 Typical behaviour of quasi brittle material under uniaxial compression ............ 41

Figure 3.3 Qualitative damage evolution in terms of equivalent strain under uniaxial

compression ............................................................................................................................. 43

Figure 3.4 Implemented program’s flow chart ...................................................................... 48

Figure 3.5 The 8-noded element used for the numerical analysis ......................................... 50

Figure 3.6 The 8-noded element under uniaxial tension ........................................................ 52

Figure 3.7 Stress-strain behaviour of the material based on the proposed model in uniaxial

tension test ............................................................................................................................... 52

xiii List of Figures |

Figure 3.8 Evolution of Poisson’s ratio for the material based on the proposed model in

uniaxial tension test ................................................................................................................. 53

Figure 3.9 Damage evolution of the material based on the proposed model in uniaxial

tension test for η =1.0 .............................................................................................................. 53

Figure 3.10 The 8-noded element under uniaxial compression ............................................. 54


compression test ....................................................................................................................... 55


uniaxial compression test ......................................................................................................... 55


compression test for η =0.5 ..................................................................................................... 56

Figure 3.14 The 8-noded element under pure shear .............................................................. 57

Figure 3.15 Stress-strain behaviour of the material based on the proposed model in pure

shear test ............................................................................................................................ 57


pure shear test .......................................................................................................................... 58

Figure 3.17 Damage evolution of the material based on the proposed model in pure shear

test ............................................................................................................................................ 58

Figure 3.18 Influence of parameter α on the stress-strain behaviour of the proposed model

in uniaxial tension test ............................................................................................................. 60

Figure 3.19 Influence of parameter α on the damage evolution of the proposed model in


Figure 3.20 Influence of parameter β on the stress-strain behaviour of the proposed model

in uniaxial tension test ............................................................................................................. 61

Figure 3.21 Influence of parameter β on the damage evolution of the proposed model in


Figure 3.22 Influence of parameter k on the stress-strain behaviour of the proposed model in


Figure 3.23 Influence of parameter k on damage evolution of the model in uniaxial

compression test .................................................................................................................. 63

Figure 3.24 Influence of parameter ζ on the stress-strain behaviour of the proposed model in


Figure 3.25 Influence of parameter ζ on damage evolution of the model in uniaxial

compression test ................................................................................................................... 64

Figure 3.26 Influence of parameter ω_c on the stress-strain behaviour of the proposed

model in uniaxial compression test .......................................................................................... 65

Figure 3.27 Influence of parameter ω_c on damage evolution of the proposed model in


xiv List of Figures |

Figure 3.28 Influence of parameter η on the stress-strain behaviour of the proposed model

in uniaxial compression test..................................................................................................... 66

Figure 3.29 Influence of parameter η on the volumetric damage variable with respect to the

damage variable based on the proposed model in uniaxial compression test ........................ 67

Figure 3.30 Influence of parameter η on Poisson’s ratio with respect to strain based on the

proposed model in uniaxial compression test .......................................................................... 67

Figure 3.31 Influence of parameter η on Poisson’s ratio with respect to the damage variable

based on the proposed model in uniaxial compression test ..................................................... 68

Figure 3.32 Influence of parameter η on Poisson’s ratio with respect to the volumetric

damage variable based on the proposed model in uniaxial compression test ....................... 68

Figure 3.33 Influence of parameter η on Bulk’s modulus with respect to the damage variable

based on the proposed model in uniaxial compression test ..................................................... 69

Figure 4.1 Typical periodic RVEs used for masonry (Anthoine, 1995; Massart, 2003;

Lourenco, et al., 2007) ............................................................................................................. 75

Figure 4.2 Identified RVEs ..................................................................................................... 76

Figure 4.3 Controlling nodes and periodicity conditions on a typical masonry RVE ........... 77

Figure 4.4 Loading modes applied on the RVE ..................................................................... 78

Figure 4.5 Flowchart for the analysis of RVE ....................................................................... 79

Figure 4.6 Dimensions considered for the RVE ..................................................................... 82

Figure 4.7 Stress-strain behaviour of the RVE constituents (brick and mortar) under

uniaxial compression ............................................................................................................... 83

Figure 4.8 Stress-strain behaviour of the RVE constituents (brick and mortar) under

uniaxial tension ........................................................................................................................ 83

Figure 4.9 Discretisation of individual RVEs ........................................................................ 84

Figure 4.10 Loading and boundary conditions of RVE-1 under uniaxial tension

perpendicular to bed joint ........................................................................................................ 85

Figure 4.11 Loading and boundary conditions of RVE-1 under uniaxial tension parallel to

bed joint ............................................................................................................................... 85

Figure 4.12 Loading and boundary conditions of RVE-1 under pure shear.......................... 85

Figure 4.13 Stress-strain behaviour of both RVEs under uniaxial compression perpendicular

to bed joint ............................................................................................................................... 86

Figure 4.14 Stress-strain behaviour of both RVEs under uniaxial tension perpendicular to

bed joint .............................................................................................................................. 87

Figure 4.15 Stress-strain behaviour of both RVEs under uniaxial compression parallel to

bed joint ............................................................................................................................. 87

Figure 4.16 Stress-strain behaviour of both RVEs under uniaxial tension parallel to bed

joint .......................................................................................................................................... 88

xv List of Figures |

Figure 4.17 Stress-strain behaviour of both RVEs under pure shear .................................... 88

Figure 4.18 Influence of different length scale parameters on stress-strain behaviour of the

RVE under uniaxial compression perpendicular to bed joint .................................................. 90

Figure 4.19 Strength of the RVE in terms of the nonlocal length scale parameter c ............. 91

Figure 4.20 Stress distribution at interface between mortar and brick for c = 1.0 ............... 91

Figure 4.21 Stress distribution at interface between mortar and brick for c = 5.0 ............... 92

Figure 5.1 Dhanasekar’s biaxial load-control test setup (Dhanasekar, 1985) ..................... 94

Figure 5.2 The 8-noded element used for the numerical analysis ......................................... 95

Figure 5.3 Stress-strain behaviour of brick element under uniaxial compression ................ 96

Figure 5.4 Stress-strain behaviour of mortar element under uniaxial compression ............. 97



Figure 5.7 A typical discretisation of RVE ............................................................................. 98

Figure 5.8 Typical dimensions of the modelled RVE ............................................................. 99

Figure 5.9 Loading configuration of the panel ...................................................................... 99

Figure 5.10 Loading configuration and boundary conditions of the RVE ........................... 100

Figure 5.11 Stress-Strain behaviour of the RVE under uniaxial compression parallel to bed

joint ........................................................................................................................................ 101

Figure 5.12 Stress-Strain behaviour of the RVE under biaxial compression with 0 and

1 2/ 1 ............................................................................................................................... 102


1 2/ 2 .............................................................................................................................. 102


1 2/ 4 .............................................................................................................................. 103

Figure 5.15 Stress-Strain behaviour of the RVE under uniaxial compression perpendicular

to bed joint ............................................................................................................................ 104


1 2/ 2 .............................................................................................................................. 104


1 2/ 4 .............................................................................................................................. 105

Figure 6.1 Representation of the Interfacial Transition Zone (ITZ) for concrete ............... 108

Figure 6.2 Representation of the masonry RVE utilised with a 5-layered Interfacial

Transition Zone (ITZ) ............................................................................................................ 109

xvi List of Figures |

Figure 6.3 Representation of the masonry RVE thicknesses with an n-layered Interfacial

Transition Zone (ITZ) ............................................................................................................ 111

Figure 6.4 Finite Element discretisation of the RVE ........................................................... 113

Figure 6.5 Influence of ITZ thickness on stress-strain behaviour of the RVE under uniaxial

compression perpendicular to bed joint ................................................................................ 115


compression parallel to bed joint .......................................................................................... 115


tension perpendicular to bed joint ......................................................................................... 116


tension parallel to bed joint ................................................................................................... 116

Figure 6.9 Influence of ITZ thickness on stress-strain behaviour of the RVE under pure

shear loading ....................................................................................................................... 117

Figure 6.10 Influence of ITZ parameter E on stress-strain behaviour of masonry under

uniaxial compression perpendicular to bed joint .................................................................. 118


uniaxial compression parallel to bed joint ............................................................................. 119


uniaxial tension perpendicular to bed joint ........................................................................... 120


uniaxial tension parallel to bed joint ..................................................................................... 120


pure shear loading ................................................................................................................. 121

Figure 6.15 Dimensions of the modelled RVE and its ITZ for experiments conducted by

Dhanasekar (1985) ............................................................................................................... 123

Figure 6.16 Comparison of the stress-strain behaviour of the RVE under uniaxial

compression parallel to bed joint .......................................................................................... 124

Figure 6.17 Comparison of the stress-strain behaviour of the RVEs under uniaxial

compression perpendicular to bed joint ................................................................................ 124

Figure 6.18 Comparison of the stress-strain behaviour of the RVE under uniaxial tension

parallel to bed joint ................................................................................................................ 125

Figure 6.19 Comparison of the stress-strain behaviour of the RVEs under uniaxial tension

perpendicular to bed joint ...................................................................................................... 125

Figure 6.20 Hollow concrete block with dimensions from Barbosa, et al. (2010) .............. 126

Figure 6.21 Idealised dimensions of the modelled RVE....................................................... 127

xvii List of Figures |


Barbosa & Hanai (2009) ....................................................................................................... 130

Figure 6.23 Finite Element discretisation of the RVE ......................................................... 131

Figure 6.24 Comparison of the stress-strain behaviour for test P1 of the ITZ enhanced RVE

against the experimental data and numerical analysis by Barbosa, et al. (2010)................. 131







Figure 7.1 Representation of material and principal axes in masonry ................................ 135

Figure 7.2 Typical dimensions of the modelled RVE and its ITZ ......................................... 137

Figure 7.3 Finite Element discretisation of the RVE ........................................................... 138

Figure 7.4 Typical dimensions considered for the tested blocks.......................................... 139

Figure 7.5 Typical dimensions of the modelled RVE and its ITZ ......................................... 139

Figure 7.6 Loading configuration and boundary conditions of the RVE ............................. 140

Figure 7.7 Loading configuration and boundary conditions of the RVE for 6 load cases

considering θ = 90° ............................................................................................................... 140

Figure 7.8 Failure envelope of the RVE in terms of principle stresses for bed joint angle

0° ............................................................................................................................................ 143

Figure 7.9 Failure envelope of the RVE in terms of principle stresses for bed joint angle 90°

................................................................................................................................................ 144


22.5° ....................................................................................................................................... 145


67.5° ....................................................................................................................................... 145

Figure 7.12 Failure envelope of the RVE in terms of shear stress τ and stress normal to bed

joint for bed joint angles of 22.5° and 67.5° ........................................................................ 146

Figure 7.13 Failure envelope of the RVE in terms of shear stress τ and stress parallel to bed

joint for bed joint angles of 22.5° and 67.5° ........................................................................ 146


45° .......................................................................................................................................... 147


joint for bed joint angle of 45° ............................................................................................. 147


joint for bed joint angle of 45° ............................................................................................. 148

xviii List of Figures |

Figure 7.17 Failure envelope of the RVE in terms of principle stresses for bed joint angles

0°, 22.5°, 45°, 67.5° and 90° .............................................................................................. 149


joint for bed joint angles of 0°, 22.5°, 45°, 67.5° and 90° ................................................... 149


joint for bed joint angle of 0°, 22.5°, 45°, 67.5° and 90° .................................................... 150

Figure 7.20 3-D failure envelope of the RVE in terms of shear stress τ, stress normal to bed

joint and stress parallel to bed joint .................................................................................. 150

Figure 7.21 Failure envelope of the RVE in the normal-parallel stress plane ................... 151

Figure 7.22 Fitted ellipse and its corresponding failure cases for cases with the summation

of the normal and parallel stresses less than -1 ...................................................................... 152

Figure 7.23 Fitted ellipse and its corresponding failure cases for cases with the summation

of the normal and parallel stresses more than -1 .................................................................... 153

Figure 7.24 Failure mode and damage progression in the ITZ layers for uniaxial

compression parallel to bed joint in the initial loading stages .............................................. 154


compression parallel to bed joint in the final loading stages ............................................... 155


compression normal to bed joint in the initial loading stages ............................................... 156


compression normal to bed joint in the final loading stages ................................................. 157

Figure 7.28 Failure mode and damage progression in the ITZ layers for uniaxial tension

parallel to bed joint in the final loading stages ................................................................... 158


normal to bed joint in the final loading stages ...................................................................... 159

Figure 7.30 Failure mode and damage progression in the ITZ layers for biaxial

compression-compression (bed joint angle of 0°) in the final loading stages ....................... 160

Figure 7.31 Failure mode and damage progression in the ITZ layers for biaxial

compression-tension (bed joint angle of 45°) in the final loading stages.............................. 161

Figure 7.32 Failure envelope of the RVE in terms of principle stresses in zero-shear (P1) 162




Figure 7.36 Comparison of the failure envelopes of conventional masonry with different

strength and geometry............................................................................................................ 166

Figure 8.1 Representation of a dry joint in dry-stack masonry ........................................... 169

xix List of Figures |

Figure 8.2 Qualitative stress-strain and damage evolution of dry-stack masonry due to joint

closure .................................................................................................................................... 170

Figure 8.3 Eight-noded plane stress element used in the analysis (CPS8R) ....................... 174

Figure 8.4 Influence of different values of the full contact strain parameter on damage

evolution in uniaxial compression ......................................................................................... 176

Figure 8.5 Influence of different values of the full contact strain parameter on stress-strain

behaviour under uniaxial compression .................................................................................. 176

Figure 8.6 Influence of different values of the initial imperfection parameter on damage


Figure 8.7 Influence of different values of the initial imperfection parameter on stress-

strain behaviour under uniaxial compression ....................................................................... 178

Figure 8.8 Influence of different values of the damage slope parameter on damage


Figure 8.9 Influence of different values of the damage slope parameter on stress-strain

behaviour under uniaxial compression .................................................................................. 179

Figure 8.10 Measured dimensions of the H-block presented by Oh (1994) ........................ 181

Figure 8.11 Measured dimensions of the conventional hollow block presented by Oh

(1994) ..................................................................................................................................... 181

Figure 8.12 Idealised dimensions of the H-block and modified H-block and their RVE ..... 182

Figure 8.13 Idealised dimensions of the conventional hollow block and its RVE ............... 184

Figure 8.14 Stress-strain behaviour of H-block under uniaxial compression ..................... 185

Figure 8.15 Stress-strain behaviour of conventional hollow block under uniaxial

compression ........................................................................................................................... 185

Figure 8.16 Stress-strain behaviour of the fictitious joint of H-block under uniaxial

compression ........................................................................................................................... 186

Figure 8.17 Stress-strain behaviour of the fictitious joint of conventional hollow block under

uniaxial compression ............................................................................................................. 186

Figure 8.18 Stress-strain behaviour of conventional hollow block under uniaxial

compression .......................................................................................................................... 187

Figure 8.19 Stress-strain behaviour of the H-block under uniaxial compression ............... 187

Figure 8.20 Stress-strain behaviour of the modified H-block under uniaxial compression 188

Figure 8.21 Representation of material and principal axes in masonry ............................... 189

Figure 8.22 Loading configuration and boundary conditions of the RVE .......................... 189

Figure 8.23 Failure envelope of the H-block in terms of principal stresses ....................... 190

Figure 8.24 Failure envelope of the modified H-block in terms of principal stresses ......... 191

xx List of Figures |

Figure 8.25 Failure envelope of the conventional hollow block in terms of principal stresses

................................................................................................................................................ 192

Figure 8.26 Comparison of the failure envelopes of different masonry systems with various

strengths ................................................................................................................................. 193

Figure 8.27 Failure envelop zone of masonry systems with various strengths and geometries

................................................................................................................................................ 194

Figure 9.1 Representation of a dry joint in dry-stack masonry ........................................... 198

Figure 9.2 Dimensions of the RVEs used for different height to thickness ratios ................ 199

Figure 9.3 Comparison of the predicted compressive strength using the proposed model and

AS3700 for compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 7.6 for two mortar strengths .. 200


AS3700 for compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 19 for two mortar strengths ... 201

Figure 9.5 Comparison of the predicted compressive strength of masonry of using the

proposed model and AS3700 with unit characteristic compressive strength 𝑓′𝑢𝑐 = 3.0 𝑀𝑃𝑎

for two mortar strengths ........................................................................................................ 202





proposed model and AS3700 with unit characteristic compressive strength 𝑓′𝑢𝑐 = 10 𝑀𝑃𝑎

















xxi List of Figures |







Figure 10.1 An RVE incorporating grouting and rendering 3D view ................................. 212

Figure 10.2 An RVE incorporating grouting and rendering 2D view ................................. 212

xxii List of Tables |

List of Tables

Table 3.1 Algorithm for the implementation of the introduced damage model ...................... 49

Table 3.2 Input properties of the tested element...................................................................... 51

Table 4.1 Algorithm for the implementation of the introduced transient gradient model on the

Representative Volume Element (RVE) ................................................................................... 79

Table 4.2 Mechanical properties of the tested element ........................................................... 82

Table 4.3 Transient-gradient properties of the constituents ................................................... 86

Table 5.1 Mechanical properties of the tested materials ........................................................ 96

Table 5.2 Transient-gradient properties of RVE’s constituent .............................................. 100

Table 6.1 Properties of the constituent materials .................................................................. 112


Table 6.3 The Interfacial Transition Zone parameters for each test ..................................... 114

Table 6.4 Young’s modulus and Poisson’s ratio of each individual layer ............................ 114


Table 6.6 Young’s modulus and Poisson’s ratio of each individual layer ............................ 119

Table 6.7 Young’s modulus and Poisson’s ratio of each ITZ layer....................................... 122

Table 6.8 Material properties of the tested constituents ....................................................... 127

Table 6.9 Mechanical properties of the tested prism 1 (P1) ................................................. 128

Table 6.10 Mechanical properties of the tested prism 2 (P2) ............................................... 128



Table 7.1 Young’s modulus and Poisson’s ratio of each ITZ layer....................................... 136


Table 7.3 Load factors for each load case ............................................................................ 141

Table 7.4 Characteristic strength of units in different experiments ...................................... 166

Table 8.1 Algorithm for the implementation of the dry-surface damage model .................... 172

Table 8.2 Mechanical properties of the tested element ......................................................... 175

Table 8.3 Properties of the H-block and conventional block ................................................ 182

Table 8.4 Properties of interface layers ................................................................................ 182

Table 8.5 Load factors for each load case ............................................................................ 190

Table 8.6 Characteristic strength of units in different experiments ...................................... 192

Table 9.1 Compressive strengths considered for units .......................................................... 197

xxiii List of Tables |


Table 9.3 Masonry unit height to mortar bed joint thicknesses considered for units ........... 198

1 Introduction |

CHAPTER 1

Introduction

1.1 General Remarks

Masonry is one of the oldest building materials used since the early civilizations. Masonry is

still extensively used in buildings as structural and cladding elements due to its aesthetics, durability,

fire and heat resistance and low material costs. Example of masonry buildings form “Chogha

Zanbil” in Iran from 1250 BC and the Great Wall of China from 220 BC to European castles

and bridges from the industrial era and museums and buildings like the Sorø art museum in

Denmark and the old government house in Queensland University of Technology are shown

in Figure 1.1.

Figure 1.1 Varied structural masonry systems

Different assemblage of quasi-brittle masonry units, their various strengths and geometries

and their type of bonding leads to complex anisotropic behaviour and failure mechanisms for

masonry. The behaviour of quasi-brittle material is also rather complex by itself. In the past

decades, with the progress in the areas of composite material, continuum damage mechanics,

2 Introduction |

plasticity and also with the emergence of powerful computers, a lot of researchers have

focused on better understanding the behaviour and failure mechanism of masonry.

In this thesis, with a view to better understand the behaviour of such structures, a new

continuum damage mechanics constitutive model has been introduced which is capable of

reproducing the behaviour of different masonry systems. This model has been further

enhanced with a transient-gradient nonlocal model in order to eliminate the localization

problems associated with softening damage model. The advantage of this model is that it

would enable us to obtain the behaviour and failure mechanism of masonry using the

properties of its individual constituents i.e. mortar and unit without any need for testing of

masonry assemblages under biaxial loading, which is complex, time consuming and

expensive.

1.2 Research Significance Masonry is predominantly used in wall structures, the thickness of which is smaller than its

height and length. Therefore structural walls are often idealised as plane stress membranes

with the normal and shear stresses acting on the bed ( ),nσ τ and perpend ( ),pσ τ joints of

masonry significantly affecting their deformation and failure. These normal and shear stresses

expressed as a failure envelope in a three dimensional stress space ( ), ,n pσ σ τ . To define this

failure surface, biaxial testing of masonry wall panels under varying stress ratios is essential,

which is very expensive and time consuming. Considering the range of available products

for unit and mortar, their different geometries and strengths, it is impossible to undertake

experimental studies to obtain their behaviour and modes of failure. Therefore, introduction

of a constitutive model capable of computationally describe the behaviour of different types

of masonry is crucial. This thesis contains formulation and application of a damage

mechanics inspired computational modelling method capable of predicting the complete

(hardening and softening) monotonic stress-strain behaviour of masonry of any construct and

providing a reliable biaxial failure surfaces of structural walls.

3 Introduction |

1.3 Aim The aim of this study is to develop a nonlocal computational modelling method capable of

describing the behaviour of masonry from its constituents and their interaction. The aim also

involves demonstrating the capability of the developed model for rationally predicting the

deformation and failure surfaces of both the conventional and the dry-stack masonry systems.

The aim will be achieved through the following set of enabling objectives:

1. Developing a continuum damage mechanics (CDM) constitutive law which

incorporates variation of Poisson’s ratio for quasi-brittle material such as mortar, clay

brick and concrete block.

2. Enhancement of the constitutive law with a transient-gradient nonlocal model to

eliminate localisation concerns.

3. Formulating a conventional masonry representative volume element (RVE) through

application of the model and validate the RVE with existing uniaxial experimental

datasets.

4. Improving the RVE with an interfacial transition zone (ITZ) technique to account for

interfacial damages between the unit and binder layer surfaces.

5. Development of biaxial failure envelopes of the conventional and dry-stack masonry

through the ITZ enriched RVE.

6. Demonstration of further application of the ITZ enriched RVE for determining

strength properties of practical importance and mapping the properties with those

determined from the Australian Masonry Structures Standard AS3700 (2011).

1.4 Thesis Arrangement This thesis is organised in the following chapters:

In Chapter 1, an outline to the research is provided. The significance of this study, its aims

and objectives are also presented.

Chapter 2 reviews the literature on

• Experimental work on masonry

• Numerical Modelling of masonry

• Continuum Damage Mechanics (CDM)

4 Introduction |

• Gradient enhanced and transient-gradient nonlocal models

• Computational homogenisation

In Chapter 3, a new formulation of a CDM constitutive model incorporating variable

Poisson’s ratio is presented and its performance is demonstrated through numerical examples

and a parametric study.

In Chapter 4, formulation of a transient-gradient enhanced nonlocal model is described and

the model is shown to be of free of the localisation problems associated with softening

models. Numerical examples and parametric study are also provided.

Chapter 5 delves into the application of the model formulated in Chapter 4 to predict the

behaviour of brick masonry under biaxial loading conditions using a standard RVE.

In Chapter 6, reformulation of the RVE with an interfacial transition zone (ITZ) enrichment

is introduced so that the damage between the interfaces of the two constituents of masonry

could be accounted for appropriately. The model is validated through different experiments

on conventional masonry.

Chapter 7 illustrates the ability of the model to predict the biaxial failure envelope of

conventional masonry under different loading conditions. The results are validated with

available experimental data.

In Chapter 8 the proposed constitutive model is further improved to incorporate the joint

closure phenomena observed in dry-stack masonry. Parametric study of the new model is also

illustrated in this chapter. Finally the model is validated with experimental data on dry-stack

masonry and its prediction of dry-stack biaxial failure envelop is presented.

Chapter 9 presents some practical applications of the model for compressive strength

determination with comparison of the predicted strength with that of the Australian Masonry

Standard AS3700 (2011).

In Chapter 10, conclusions obtained from these studies and recommendations for future work

are presented.

5 Literature Review |

CHAPTER 2

Literature Review

2.1 Introduction

Masonry is one of the primary building materials. It is composed of two or more materials,

i.e. units (stone, clay bricks, or concrete blocks) and mortar. With these two constituents, a

large number of arrangements are generated using various methods of bonding methods (for

example, Stretcher/ English/ Flemish).

In spite of the simplicity associated with building in masonry, the analysis of the mechanical

behaviour of masonry materials is a very demanding task. Masonry is non-homogeneous due

to the different mechanical properties of its quasi-brittle constituents and their geometrical

bonding methods.

The complex behaviour of quasi-brittle material is affected by the composite nature of these

materials, which necessitates development of appropriate constitutive models. Although

recently there have been a large number of noteworthy contributions (Lourenço, et al., 1998;

Papa, et al., 2000; Addessi, et al., 2002; van Zijl, 2004; Chaimoon & Attard, 2007; da Porto,

et al., 2010; Petersen, et al., 2012; Kuutti & Kolari, 2012; Sousa, et al., 2013; Nazir &

Dhanasekar, 2013) with different levels of complexity and applicability, the complete

features of the material behaviour have not always been examined. Further development in

the constitutive modelling of masonry materials is, therefore, needed. With this motivation

and inspired by some key experimentally-observed features of the material behaviour, an

analytical modelling method capable of evaluating the macroscopic constitutive properties of

masonry is developed in this thesis.

Literature review covering the following key topics that are relevant to the aim of this thesis

is presented under the following sub-headings:

Experimental research on masonry


Modelling methods of masonry

Computational homogenisation and Multi-scale modelling

Continuum Damage Mechanics

Localisation and mesh sensitivity

Non-local and gradient enhanced damage models

Transient gradient non-local models

2.2 Experimental Research on Masonry

The structural behaviour of masonry is affected by the properties of its principal constituents,

units and mortar. The bond between the unit and the mortar also significantly affects the

masonry behaviour. Strength of masonry under different loadings is affected by the range of

principal stresses and their orientation to bed joint.

In the following sections, experimental tests, both uniaxial and biaxial, capable of providing a

comprehensive description of masonry, are reviewed.

2.2.1 Uniaxial Compressive Behaviour

The uniaxial compressive strength of masonry perpendicular to the bed joints has been

always considered one of its most important properties. The masonry compressive failure is

largely governed by the interaction between its constituents. Many researchers (Hilsdorf,

1996; Hendry, 1990) have derived relationship between the compressive strength of unit and

mortar and the compressive strength of masonry. Hilsdorf (1996) has also shown that the

difference in unit and mortar elastic properties (Poisson’s ratio and Young’s modulus) is an

important factor of failure as they induce inconsistent deformation of the constituent

materials as shown in Figure 2.1. In this figure p is the stress imposed on the prism and

superscripts b and m represent brick and mortar, respectively.


Figure 2.1 Local state of stress in the constituents of masonry prisms under vertical

compression perpendicular to bed joint

The tendency for incompatible deformation of the constituents, even a masonry prism under

uniaxial compression would generate a complex stress state of triaxial compression in the

mortar and uniaxial compression-biaxial tension in the unit as shown in Figure 2.1.

There has been considerably fewer research on uniaxial compression tests parallel to the bed

joints. Nevertheless, being an anisotropic material, the resistance of masonry to compressive

loads parallel to the bed joints occur due to the splitting of the bed joints as in Figure 2.2.


Figure 2.2 Modes of failure of solid clay masonry under uniaxial compression, from

(Dhanasekar, 1985)

2.2.2 Uniaxial Tensile Behaviour

Tensile strength of masonry is low and hence significantly contributes to the failure of the

masonry structures. Different types of tests have been used by researchers to determine the

tensile strength of masonry (Graubohm & Brameshuber, 2011) which can be classified into

(1) flexural bond strength tests (Marrocchino, et al., 2007; Nicholas, et al., 2008) and (2)

direct tensile bond strength tests (Jukes & Riddington, 1997). In the direct tensile test,

ensuring uniform stress distribution on the joint is not easy. In the flexural test the bond

strength at the edge of the mortar joint is calculated by imposing bending onto a masonry

wallet.

Debonding between the unit and bed joint is the main cause of failure under tension

perpendicular to the bed joints. However, tension failure usually depends on the type of

mortar and units. For the case of stronger mortar and weaker units the tension crack passes

along the mortar head joints and through the centre of the bricks. For the case of relatively

weaker mortar joints and stronger units, the tension crack passes along the mortar bed joints.

In this case, the masonry tensile strength can roughly be estimated by the tensile bond

strength of the joints.

Figure 2.3 shows different modes of failure observed by Page (1983) on solid clay units

masonry walls subjected to uniaxial tension.


Figure 2.3 Modes of failure of solid clay masonry under uniaxial tension, from (Dhanasekar,

1985)

2.2.3 Shear Behaviour

Understanding the behaviour of masonry in shear is important because it is the governing

mode of failure in masonry under lateral loading such as wind and earthquake. Determination

of the pure shear strength of masonry joints is a demanding task, since generating a uniform

state of stress in the joints requires very complex test set-ups. Hofmann and Stockl (1986),

Atkinson et al. (1989), Jukes & Riddington (1997), van der Pluijm (1998), Da Porto (2005)

and Morandi, et al. (2013) have proposed different setups for shear behaviour. It can be seen

that increasing the confining compressive stress, increase the shear strength which may be

attributed to the frictional behaviour of masonry once the adhesion is broken. The effect of

joint thickness has also been noted in masonry. Increase of the joint thickness reduces the

shear strength.

Dilatancy (defined as vertical motion under applied horizontal displacement) of the masonry

joint depends on the microscopic geometrical and mechanical features of unit and mortar

layers which makes its behaviour quite complex. The angle of dilatancy reduces under

increase of normal stress and also increase of relative tangential displacement (van Zijl,

2000).

Masonry shear mode of failure can be seen in Figure 2.6. Biaxial tension-compression

loading with a 45 degree bed joint orientation represents shear failure.


2.2.4 Biaxial Behaviour

Behaviour of masonry systems, their strength and the failure mode varies depending on the

orientation of loading with respect to bed joints (Page, 1983; Dhanasekar, 1985; Dhanasekar,

et al., 1985) due to their anisotropic nature (Figure 2.4).

Figure 2.4 Modes of failure of solid clay masonry under uniaxial compression and uniaxial

tension considering different load orientations with respect to bed joint, from (Dhanasekar,

1985)

Because of this anisotropic nature, strength envelope and behaviour of masonry is

significantly affected by the orientation of the principal stresses to the bed joints. Therefore,

the constitutive behaviour of masonry cannot be fully described under different biaxial stress

states, solely based on uniaxial loading conditions. Consequently, the biaxial failure envelope

is described in terms of both principal stresses and the orientation of the loading axes with

respect to masonry bed joints. A summary of these failure modes can be seen in Figures 2.5

and 2.6.


Figure 2.5 Modes of failure of solid clay masonry under biaxial loads, (Dhanasekar, 1985)

Figure 2.6 Mode of failure of solid clay masonry under biaxial compression-compression,

from (Dhanasekar, 1985)


In all biaxial compression stress states when the stress ratio is unity, regardless of the

orientation of principal stresses to the bed joints, failure occurs by splitting of the specimen in

a plane parallel to the free surface of the wallets, see Figure 2.6. However, when compression

in one direction is considerably greater than compression of the other direction, orientation of

the loading to the bed joints plays a significant role in failure strength. In such a case, a

combination of both joint failure and splitting occurs.

Figure 2.7 Biaxial failure envelope of solid clay masonry (Page, 1981; Page, 1983)

Figure 2.7 demonstrates the full set of experimental data by Page (1981; 1983) carried out

with half-scale solid clay under biaxial loading.

Only a few studies were performed to obtain the full experimental failure envelope for

masonry. Samarasinghe & Hendry (1980) tested a series of brickwork panels, under uniform

biaxial tension-compression. In an attempt to obtain a three dimensional failure surface, Page

(1981) tested a series of biaxial compression-compression tests and a series of biaxial

tension-compression tests (Page, 1983) for the brick masonry as a function of the principal


stresses and the orientation to bed joint. The failure surface was presented in terms of the two

normal stresses and the inplane shear stresses in the bed joints by Dhabnasekar (Dhanasekar,

1985; Dhanasekar, et al., 1985).

Hamid (1978) conducted prism compression and tension tests to study the anisotropic nature

of grouted and ungrouted concrete masonry. Naraine & Sinha (1992) carried out biaxial

compression tests on clay brick masonry and developed a failure envelop. Only two bed joint

orientations of 0° and 90° with varying biaxial stress ratios were examined. Khattab (1993),

Liu, et al. (2009) and Drysdale & Khattab (1995) did similar studies to that of Page on the

characteristics of grouted concrete masonry under a series of compression and compression-

tension tests. Senthivel & Uzoegho (2004) carried out biaxial cyclic compression tests on

calcium silicate brick masonry. Similar to Naraine & Sinha (1992), only two bed joint

orientations of 0° and 90° with varying biaxial stress ratios were tested. Vermeltfoort (2005)

carried on biaxial tests on thin layer mortared clay masonry panels. Badarloo et al. (2009)

developed biaxial compression failure envelop for a series of compression biaxial tests on

full-scale grouted unreinforced brick masonry (bed joint orientations of 0° and 90°).

It should be noted that the failure envelope illustrated in Figure 2.7 and also the ones obtained

from other tests like Naraine & Sinha (1992), Drysdale & Khattab (1995), Senthivel &

Uzoegho (2004), Badarloo, et al. (2009) and Liu, et al. (2009) have limited applicability for

other masonry types. Based on the type of unit and mortar, their shape and the type of

assemblage, different strength envelopes are likely to be obtained for different masonry

systems. Obtaining failure envelopes through experimental work for each type of masonry is

time consuming and expensive. Therefore, a range of analytical and computational methods

have been developed in the last three decades to describe the constitutive behaviour of

different types of masonry.

2.3 Modelling Masonry

Considering the range of available products for unit and mortar, their different geometries

and strengths, it is impossible to undertake experimental studies to obtain their behaviour and

modes of failure. Researchers have used numerical methods as valuable tools for modelling

masonry during the last decades, e.g. Lourenço (1996); Papa, et al. (2000); Giordano et al.

(2002); Berto et al. (2002); Pietruszczak and Ushaksarei (2003); Massart (2003); van Zijl

(2004); Lourenço et al. (2007); Chaimoon & Attard (2007); Zucchini and Lourenço (2009);


da Porto, et al. (2010); Petersen, et al. (2012); Sousa, et al. (2013); Nazir & Dhanasekar

(2013). Many of the available computational methods use significant simplifications, for

example assuming a purely brittle behaviour for the constituent materials, Luciano and Sacco

(1997) or the assumption of isotropy for masonry material, Hanganu et al. (2002).

A periodic mesostructure of masonry is shown in Figure 2.8. Typically the mortar layer is

relatively weaker than the units, which leads to stiffness degradation along the preferential

orientations due to the periodic arrangement of the structure. The initial and the crack-

induced anisotropy of the material are coupled to the mesostructure, mostly to the geometry,

the bonding method, and the material properties of the units, the mortar and the grout. The

mesostructure, therefore, has a significant effect on the structural failure mechanisms of the

macroscopic masonry walls. This suggests that for accurate analyses of masonry structures

the mesostructure could be taken into consideration (Massart, et al., 2007), instead of the

whole wall structure, which would save cost, time and computational effort.

The existing computational methods intended for modelling masonry structures can be

classified into three categories based on their purpose and the level of their microstructural

detail. These three categories are (1) Microscopic models, (2) Macroscopic models and (3)

Multi-scale models.

Figure 2.8 The periodic assemblage of masonry

2.3.1 Microscopic Models

The “microscopic models” makes use of the constituents’ material information. This

approach has been used by many authors including, Page (1978); Lourenço (1996);

Giambanco & De Gati (1997); Giambanco, et al. (2001). Lourenço and Rots (1997) proposed


a multi-surface interface model in which the interface elements were used as potential crack,

slip and crushing planes. Plasticity formulations with non-associated flow rules combined

with Coulomb friction laws were used in this model. Giambanco & De Gati (1997) used a

cohesive interface model to predict the shear behaviour of masonry joints in cyclic loading.

The down side of this approach is that it cannot be used for large scale structural modelling

because of its requirement of using very large number of elements that lead to exorbitant

computational cost. This weakness has been partially overcome by simplified micro models

(Gambarotta & Lagomarsino, 1997; Sutcliffe, et al., 2001). In case of microscopic modelling,

the properties of the constituents such as the brick or block, the mortar, as well as the

properties of the constituents interaction are required which are usually obtained from

uniaxial tests (Dhanasekar, 2010).

2.3.2 Macroscopic Models

By treating masonry as a pseudo homogenous material, the “macroscopic models” reduce the

computational cost problem by formulating a closed-form macroscopic law between the

average stress and the average strain on an equivalent homogenised continuum, see Lourenço

(1998), Papa & Nappi (1997), Berto, et al. (2002). These models have been widely used to

analyse the behaviour of complex structures like historical constructions (Roca, et al., 2010)

and masonry bridges (Pelà, et al., 2009). The problem with this kind of approach is that the

closed-form formulation should suitably account for the interaction between damage induced

and initial anisotropy (Massart, et al., 2007). Moreover, if the homogenised continuum based

on smeared crack approach has not accounted for the localisation of the failure planes,

spurious mesh pathology problems are encountered (Lourenco, 1996; DeJong, 2009).

2.3.3 Homogenisation Models

The multi-scale approach bridges between the microscopic and the macroscopic

representation of masonry. Homogenisation techniques have been developed to obtain

macroscopic constitutive laws for masonry from the microscopic constitutive laws of its

constituents (Lourenco, et al., 2007). This approach solves the identification problem of


finding a closed-form formulation to some extent. Afterwards, they perform all of the

computations in a macroscopic framework. This type of multi-scale approach can be found in

the work of Anthoine (1995; 1997), Pegon & Anthoine (1997) and Cecchi & Sab (2002). The

work of Massart, et al. (2007) takes the homogenisation approach a step further by

concurrently coupling both scales.

In these models a Representative Volume Element (RVE), which can generate an entire panel

by repetition, is introduced. Using the mechanical properties of the constituents and knowing

the geometry of the RVE, a boundary value problem can be solved on the RVE in order to

achieve average values for the homogenised material (Massart, 2003).

Three different scales can be identified for masonry, namely (1) Micro-scale or material scale

(2) Meso-scale or RVE scale (3) Macro-scale or structural scale (Figure 2.9). At micro-scale,

the mechanical behaviour of individual constituents is determined. Depending on the type of

considered phenomena the level of complexity varies at this scale. The meso-scale is the

characteristic size of masonry constituents such as unit and mortar. The variables determined

at this RVE scale are the failure patterns, local damage, stress and strain. Damage growth at

this scale results in phenomena such as damage-induced anisotropy which is observed in

macroscale. Finally, masonry structural scale is referred to as macroscale. The variables

defined at this scale are average stress-strain fields and external loads.

As stated before, formulating an accurate closed-form macroscopic constitutive law can pose

complexity to the multi-scale approach. This problem can be solved by fully coupling both

the meso and macro scales in the whole structure. In this multi-scale approach, by use of

computational homogenisation and introduction of a RVE the material behaviour can be

determined (Massart, 2003). It avoids complex closed-form formulations by simply using the

isotropic constitutive relations at micro levels in order to capture complex effects such as the

damage-induced anisotropy. The down side of this kind of multi-scale approach is that it is

computationally expensive, but it has become more acceptable due to growth in computing

power. Luciano & Sacco (1997) developed this approach with a brittle-elastic representation

of the constituents. Mercatoris & Massart (2009; 2011) applied a coupled two-scale

computational scheme for determination of the failure of masonry. Reccia, et al. (2014)

applied a full 3D homogenisation approach to investigate the behaviour of masonry arch

bridges.


The multi-scale models attempt to determine the constitutive properties of masonry from the

unit and mortar properties. This thesis is focussing on this type of models based on the

following theories:

Continuum Damage Mechanics (CDM)

Localisation and Mesh sensitivity

Transient-Gradient Non-local models

First-order Computational Homogenisation

The literature pertaining to these theories and their basic formulations are briefly listed in this

section.

Figure 2.9 Illustration of different scales in multi-scale methods

2.4 Continuum Damage Mechanics

Continuum Damage Mechanics provides a general framework for the derivation of consistent

material models affected by the nucleation of cracks and their subsequent growth (Kachanov,

1958; Rabotnov, 1969; Janson & Hult, 1977). In the 1980s it was established that damage

mechanics could model accurately the strain softening response of concrete (Krajcinovic,

1983; Kachanov, 1986) and more rigorous basis, based on thermodynamics of irreversible


processes which gave the framework to formulate the adapted constitutive laws (Mazars &

Pijaudier-Cabot, 1989). Since then, there have been numerous Continuum Damage

Mechanics models proposed for the constitutive modelling of materials in general and

particularly quasi-brittle material such as concrete, brick, masonry and mortar (Peerlings, et

al., 1996; Lourenço, et al., 1998; Peerlings, 1999; Zucchini & Lourenco, 2004; Pela, et al.,

2011; Pela, et al., 2014).

Continuum Damage Mechanics is used for materials so different as metals, ceramics, rock,

concrete and masonry. Such a large acceptance is due to several important factors, namely:

The consistency of the theory, which is formulated in a rigorous framework, i.e. the

Thermodynamics of irreversible processes.

The compatibility with other theories. For instance, the combination of damage

mechanics theory with plasticity is straightforward. Addition of plasticity may help in

better representation of some aspects such as shear. In addition, it is possible to

include thermal and rate dependent effects in such formulations.

The simplicity of the approach, compared to for example Fracture Mechanics and

leads to a much simpler formulation and interpretation. The damaged material is

assumed to remain as a continuum and the collective effect of cracks is modelled by

modifying the mechanical properties, i.e. stiffness and strength. One or more, scalar

or tensorial, field quantities are introduced into the constitutive equations as measures

of the degradation of the material.

The term Continuum Damage Mechanics was introduced by Janson and Hult (1977). The aim

of such a theory is to develop methods for the prediction of the load carrying capacity of

structures subjected to material damage evolution. It is a counterpart of Fracture Mechanics,

which deals with structures containing one or several cracks of finite size. In this latter

approach, the cracks are usually assumed to be embedded in a non-deteriorating material.

However, Fracture Mechanics and Continuum Damage Mechanics may be combined to

predict the damage growth and the resulting decrease of load carrying capacity (Krajcinovic,

1985; Janson & Hult, 1977).

The continuity ψ quantifies the absence of the material deterioration. The complementary

quantity 1 is therefore a measure of the state of deterioration or damage.


Although Kachanov assumed ψ to be a scalar field variable, later developments have led to

the study of tensorial quantities to describe damage, see Krajcinovic and Lemaitre (1987) and

the references therein. Scalar damage models have been used throughout the literature for

different purposes such as phenomenological damage models (Chaboche, 1981; Lemaitre,

1985; Ju, 1989), elasticity based damage models for quasi-brittle material (Peerlings, et al.,

2001; Massart, et al., 2005). Several anisotropic 4th

order and second order damage tensors

have been introduced in the literature for anisotropic damage models (Maire & Chaboche,

1997; Fichant, et al., 1998; Krajcinovic & Fanella, 1986; Chow & Wang, 1987; Lubarda &

Krajcinovic, 1993; Fichant, et al., 1998). Massart et al. (2004) considered a scalar damage

model to evaluate the need for incorporating damage induced anisotropy in brick masonry.

Pela et al. (2012) have developed a strain based continuum damage model similar to Massart

et al. (2004).

2.4.1 Damage Variable

From a physical point of view, damage is always related to plastic or irreversible strains and

more generally to a strain dissipation either on the mesoscale, the scale of the RVE, or on the

microscale, the scale of the discontinuities.

The damage variable is obtained from the amount of defects in a certain material point, see

Figure 2.10. Precise definitions of damage variable based on physical and measurable

quantities have been produced by some authors (Lemaitre & Chaboche, 1990; Lemaitre &

Desmorat, 2005).

In this thesis, since the main source of anisotropy in masonry is the geometric assemblage of

brick and mortar, we assume that damage development would not introduce anisotropy in the

material behaviour and thus a scalar damage parameter would be sufficient to model the local

damage condition.

To define the material damage at a point, let us consider a RVE identified by its normal n

and be the original cross sectional area. Scalar damage parameter 𝜔 which is the

mechanical measure of local damage relative to the direction n is defined such that

( ) 0n corresponds to undamaged or virgin state of the material;


( ) 1n represents a state that material has been completely damaged and has lost

its’ integrity;

0 ( ) 1n characterizes the damaged state;

The initial defects, microcracks and microvoids of the material have been accounted for in

the initial material parameters. Different regions of damage due to loading have been

illustrated in Figure 2.10. The material domain has been divided into three regions i.e.,

undamaged region Ω𝑖 , in which 𝜔 = 0, partially damaged region Ω𝑑 , in which 0 < 𝜔 < 1

and fully damaged region Ω𝑐 , in which 𝜔 = 1 and is a continuum damage mechanics

representation of crack.

Figure 2.10 Distribution of damage on an element and representation of microstructural

defects

In the classical continuum damage mechanics a scalar isotropic damage quantity 𝝎 has been

introduced in the stress-strain relationship in the following way

�̅�𝒊𝒋 = (𝟏 − 𝝎)𝝈𝒊𝒋 = (𝟏 − 𝝎)𝑪𝒊𝒋𝒌𝒍𝜺𝒊𝒋 (2.1)

Where 𝝈𝒊𝒋 is the Cauchy stress. Einstein’s summation convention applies (𝑖, 𝑗, 𝑘, 𝑙 = 1, 2, 3).

It has been assumed that the Poisson ratio 𝝂 is not affected by material damage. 𝜺𝒌𝒍

represents the linear strains


1

2

k lkl

l k

u u

x x

(2.2)

In which 𝒖𝒌 is the displacement with respect to Cartesian coordinate system. 𝑪𝒊𝒋𝒌𝒍, represents

the elasticity matrix components

ijkl ij kl ik jl il jkC (2.3)

This constitutes a non-rigorous hypothesis which assumes that all the different behaviours

accompanying damage (elasticity, plasticity, viscoplasticity) are affected in the same way by

the surface density of the damage defects. However, its simplicity allows the establishment of

a coherent and efficient formalism.

From Equation (2.1) it derives that damage variable 𝜔 introduces a reduction factor into

stress-strain equation. In the simple uniaxial form, the macroscopic (or apparent) tension σ is

related to the strain by means of a damaged Young’s modulus:

(1 )EE (2.4)

This expression is in fact in accordance with that observed in experiments (Nguyen, 2005).

The damage is irreversible, so

0, 0 E 0 (2.5)

2.4.2 Principle of Strain-Equivalence

A damage loading function in terms of the strain components which enables the model to

predict damage growth is essential to define the onset and progress of damage. The damage

onset is said to have occurred when the strain (or stress) exceeds a damage threshold :

,eq eqf (2.6)

where eq is called the equivalent strain and is based on the state of current strain

components. 𝜅 is a history parameter which depends on the maximum equivalent strain

experienced by the material so far. The shape and size of the loading surface depends on the

definition of the equivalent strain which in turn depends on the history parameter 𝜅.


When 𝑓 < 0, damage parameter will not grow. Change in the history parameter 𝜅 follows the

Kuhn-Tucker relations

0, 0, 0, 0 if f t (2.7)

These relations imply that in case of increasing damage (�̇� ≥ 0), the history parameter

satisfies, 𝜅 = 𝜀�̃�𝑞 which means damage growth will happen when 𝑓 = 0. 𝜅𝑖 is the initial

value equivalent strain from which damage onset occurs.

In order to properly model the behaviour of quasi-brittle material, an appropriate definition of

equivalent strain is required. The equivalent strain maps the strain tensor into a scalar by

properly weighting its components in order to imitate their effects on cracking. Damage

parameter depends on the energy release rate produced by crack propagation/ growth. This

energy release rate ( G ) for elasticity based damage is given in (Lemaitre & Chaboche, 1990)

1

2ij ijkl klC (2.8)

As can be seen in this equation, the energy release rate is only related to strain tensor and

weighted by the materials elasticity constants. Thus, this measure can be considered as the

equivalent strain eq (Peerlings, 1999). A modified and more natural dimensionless form can

be written as

1eq ij ijkl klC

E (2.9)

Since most engineering materials such as concrete, brick and mortar, have different tensile

and compressive strengths Mazars and Pijaudier-Cabot (1989), has introduced eq as a

function of principal strains i ( 1, 2, 3)i in the form of

32

1

eq i (2.10)

In which . is the McAuley brackets ( 1

2x x x ). i then indicates the positive part

of the principal values of the strain tensor.

Since the definition of equivalent strain based on Equation (2.10) depends on positive values

of principal strains, it is more sensitive to tensile strains than compressive strains. However, it

cannot predict compressive and tensile differences as high as ten to twenty times which have


experimentally been obtained for quasi-brittle materials such as concrete, brick and mortar.

To overcome this issue, de Vree et al. (1995) proposed a strain based modified von Mises

definition which was inspired from polymers plasticity models. This definition of equivalent

strain is in the form of

2

2

1 1 22 2

1 11 12

2 1 2 2 1 2 1eq

k k kI I J

k k

(2.11)

Where 𝐼1 = 𝜀𝑘𝑘 (𝑘 = 1, 2, 3), is the first invariant of strain tensor and 𝐽2 =1

6𝐼1

2 −1

2𝜀𝑖𝑗𝜀𝑖𝑗

(𝑖, 𝑗 = 1, 2, 3), is the second invariant of deviatoric strain tensor. Parameter 𝑘 controls the

sensitivity of the equivalent strain to tension and compression which is usually set to the ratio

of compressive strength of the material to its tensile strength. In this equation Poisson’s ratio

has been represented by 𝜐.

2.4.3 Damage Evolution Law

There have been several ways of representing the damage parameter , which can be a

single scalar for isotropic damage and a tensor for anisotropic damage. It can be represented

as a variable characterising the material deterioration with the concepts of effective stress or

effective strain (Simo & Ju, 1987; Mazars & Pijaudier‐Cabot, 1989; Lemaitre & Chaboche,

1990; Lee & Fenves, 1998; Peerlings, 1999; Jirásek, et al., 2004; Lemaitre & Desmorat,

2005) or as a function in terms of the position of the loading surface in stress space between

the initial and bounding surfaces (Li & Ansari, 1999); or even it can be a progressive

reducing function (Addessi, et al., 2002) representing the damage experienced by the material

and can hardly be directly related to the geometrical definition of damage. In fact, in

macroscopic constitutive modelling, physical interpretation of damage variables is not always

straightforward. However, the convincing physical interpretation of the damage variable

depends on the identification of the microscopic mechanism underlying the observed

macroscopic response (DeSimone, et al., 2001). The definition of damage variable

following the concepts of effective stress and effective strain, which has been presented

above, is probably the most well-known and widely used in literature.

The Continuum Damage Mechanics approach has been shown and proved by many authors

to be appropriate for constitutive models of quasi-brittle material like concrete and masonry


(Krajcinovic & Fonseka, 1981; Simo & Ju, 1987; Mazars & Pijaudier‐Cabot, 1989; Peerlings,

1999; Geers, et al., 2000; Jirásek, et al., 2004; Massart, et al., 2005; Lourenco, et al., 2007).

Due to the anisotropic nature of damage, even for initially isotropic materials, the damage

measure requires a tensorial representation. However damage models employing scalar

damage variables are still preferred because of their simplicity of the formulation, numerical

implementation and parameter identification (Burlion, et al., 2000). Only scalar damage is

considered in this study.

From the point of view of constitutive modelling, continuum damage mechanics alone can be

used exclusively in the case that the structures analysed are under monotonic loading, as it

can reproduce the softening response of the material without necessarily paying attention to

capturing the permanent deformation. In addition, the stiffness degradation, although

overestimated in pure damage models, can also be seen as an important feature to be reflected

in the constitute modelling of concrete materials. These features confirm the applicability of

pure damage models in the constitutive modelling of concrete materials, with promising

results obtained in the literature (Peerlings, 1999; Comi, 2001; Comi & Perego, 2001; Jirásek,

et al., 2004; Lourenco, et al., 2007).

A simple damage evolution law which is sometimes used for theoretical developments is

written as

0

1

cd

i

i

c

dc

if

if

if

(2.12)

In which, i is the threshold for damage initiation. d is the strain in which the material

completely loses its integrity ( 1 ). Figure 2.11 (a) shows damage evolution using this

equation. Figure 2.11 (b) shows the stress-strain curve assuming a uniaxial stress case and

also considering eq equals the axial strain.



corresponding to Equation (2.12)

Engineering materials soften nonlinearly with relatively steep stress drop when cracking

starts and a moderate decrease afterwards. Mazars & Pijaudier‐Cabot (1989) introduced an

exponential softening law for concrete in the form of

0

1 1 i

i

iie

(2.13)

This damage evolution law and its stress-strain relation are illustrated in Figure 2.12.


corresponding to Equation (2.13)

Due to crack bridging, the experimentally obtained load displacement data has a long tail

which is controlled by a numerical parameter (Hordijk, 1991). Using this expression,


when , stress approaches 1 iE which can represent this long tail. Parameter β

controls the damage growth rate which depends on the tensile fracture energy of the material.

When β is higher, the model will show a faster crack growth and a more brittle response.

In Jirásek & Patzák (2002) and Jirásek et al. (2004), an exponential curve was proposed and

can be calibrated based on the uniaxial behaviour of the material, with the area under the

uniaxial stress-strain curve representing the local (or specific) fracture energy Fg . The

damage evolution is of the form:

0

1 exp

i

i ii

f i

if

if

(2.14)

Where 𝜅𝑖 = 𝑓′𝑡/𝐸 is the strain at peak stress and 𝜅𝑓 a model parameter controlling the initial

slope of the softening curve, see Figure 2.13. This evolution law is in fact associated with the

damage criterion, ,eq eqf . The history variable κ here represents the maximum

equivalent strain eq obtained in the previous step.

The decomposition of the effective stress tensor used in this model properly captures the

tensile behaviour of the material. In addition, the damage loading function is only used to

define a failure criterion and does not play any role in the evolution law of damage. Although

using arbitrary assumptions in the formulation, the model here was shown to be adequate in

capturing the behaviour of the material in tensile-dominated stress states (Jirásek, et al.,

2004). However, it was also admitted (Jirásek, et al., 2004) that the model parameters cannot

be uniquely evaluated based solely on the input fracture energy Fg . This is because there can

be several stress-strain curves producing the same Fg (i.e., non-unique) and the problem of

evaluating model parameters becomes ill-conditioned unless additional constraints are

introduced. This non-uniqueness of the model parameters can also be observed in several

damage-based models (Comi, 2001; Comi & Perego, 2001; Borino, et al., 2003).

Several materials (rocks, concrete, ceramics) often show a unsymmetrical damage surface,

the yield stress in compression being several times that in tension. In order to overcome these

limitations, a damage model with two damage activation surfaces has been proposed by

Mazars and Pijaudier-Cabot (1989).


Figure 2.13 Exponential softening law from Jirásek, et al. (2004)

These two parameters, t and c , are evaluated from two evolution functions, tg and cg .

For a general loading path, different from uniaxial tension or compression, the value of

damage which enters in the constitutive law is proposed as the following combination

(Mazars & Pijaudier‐Cabot, 1989):

t t c c (2.15)

where t and c are taken in the form (Pijaudier-Cabot & Mazars, 2001):

3

21

( )ti ti cit

i

(2.16)

3

21

( )ci ti cic

i

(2.17)

in which ti is the positive strain due to positive stresses and ci the positive strain due to

negative stresses. The damage variables T and C describe the response in tension and

compression, respectively.

(1 )1

exp[ ( )]

i t tt

eq t eq i

A A

B

(2.18)

(1 )

1exp[ ( )]

i c cc

eq c eq i

A A

B

(2.19)

in which, tA , tB , cA and cB are material parameters. Using this expression, results in the

shear strength to be lower than the tensile strength and the energy dissipated in compression


and the energy dissipated in direct tension goes with the same ratio which is not realistic

(Pijaudier-Cabot, et al., 1991).

Alternatively, implicitly defined damage evolution laws have also been used (Luccioni, et al.,

1996; Comi, 2001; Comi & Perego, 2001; Nguyen, 2008; Salari, et al., 2004), in which the

damage growth and the plastic strain evolution are implicitly embedded in the coupling yield

and/or damage loading functions.

In the chapter 3 of this thesis, a damage evolution law is introduced which is capable of

capturing the behaviour of quasi-brittle constituents of masonry.

2.5 Localisation and Mesh Sensitivity in Quasi-Brittle Material

Quasi-brittle materials exhibit post-peak softening stress-strain characteristics. Finite element

modelling of damage of quasi-brittle materials are known to be significantly mesh sensitive

and the solution does not always improve by mesh refinement (de Vree, et al., 1995). Similar

phenomenon are also reported in softening plasticity and other material degradation models

that employ smear crack models (Bažant, et al., 1984; De Borst & Mühlhaus, 1992).

Engineering problems, which are usually inhomogeneous and have an anisotropic geometry,

exhibit damage localisation in a small area. Such strain localisation can be caused by

geometrical effects (e.g., necking of metallic bars) or by material instabilities (e.g.,

microcracking, frictional slip, or nonassociated plastic flow). This leads to ill-posedness of

the mathematical description and mesh sensitivity. Refined discretisation of the grid leads to

a faster crack initiation and growth which decreases the needed fracture energy for cracking

and causes a more brittle response.

Strain-softening continuum formulation leads to a solution that has several pathological

features including the total energy dissipated during failure process tending to zero, which is

partially due to loss of ellipticity of the governing equilibrium equations as soon as softening

appears. This ill-posedness is manifested by pathological sensitivity of the results to the size

of finite elements.

As a remedy, one of the following approaches can be used:

1) The cohesive crack model is a general model which can deal with the nonlinear zone ahead

of the crack tip (Elices, et al., 2002). This model admits the presence of a strong discontinuity


(jump in the displacement field) and describes softening by a traction-separation law. It

relates the traction transmitted by the crack to the crack opening (Planas, et al., 2003; Morel

& Dourado, 2011).

2) The crack band model represents the process zone and the macroscopic crack by a band of

highly localised strain, which is separated from the surrounding material by two weak

discontinuities (Bažant & Oh, 1983). Since the width of the numerically resolved band is

controlled by the size of finite elements, the softening part of the stress-strain law must be

adjusted according to the element size (Berton & Bolander, 2006).

3) Regularized models are based on generalized continuum theories (Bažant & Pijaudier-

Cabot, 1989; Wells, et al., 2002). These models incorporate a characteristic length and

prevent localisation of strain into an arbitrarily small volume. Since these models impose a

certain minimum width of the numerically resolved process zone, they are called localisation

limiters. Examples of such generalized continua include non-local integral formulations and

higher-order gradient theories (Peerlings, et al., 2002; Geers, 2004; Forest, 2009; Jirasek &

Rolshoven, 2009; Nguyen, et al., 2015). The nonlocal and gradient methods, which are

closely related, seem to be the most common approaches.

2.6 Non-Local Models

The standard “local” damage models fail to describe localised failure patterns in an objective

manner. A possible remedy consists in reformulating the constitutive model as non-local,

with the stress at a material point dependent on the strain history at that point including its

neighbourhood (Peerlings, et al., 2001; Le Bellego, et al., 2003; Engelen, et al., 2003).

The key idea of non-local damage models is to assume that the condition of growth of

damage is non local. This means that growth of damage variable depends on the average

deformation of the material in a certain neighbouring region. This neighbourhood is scaled by

an internal length parameter related to the size of the heterogeneities (Bažant & Pijaudier-

Cabot, 1989). Even though micro cracking is modelled in a crude fashion, i.e., by a scalar

defining the degradation of the Young’s modulus of the material which remains isotropic in

spite of the preferential orientation of the micro cracks, several numerical results show that


the paradox is avoided as far as the energy dissipation at failure is concerned (Forest, 2009;

Jirasek & Rolshoven, 2009).

2.6.1 Integral Non-local Model for Strain Softening Material

Bažant et al. (1984) introduced a non-local model for simulation of strain softening materials

in which deformation of a certain material point 𝒙 not only depends on the deformation of the

point 𝒙 itself but also on its surrounding area. This means that the damage field variable

should depend on a non-local equivalent strain eq , instead of local equivalent strain eq .The

non-local strain eq , is defined as the weighted average of eq over the volume of the element

Ω in the form of

1 1

with 1eq eqx x d d

(2.20)

Certain micromechanical justifications of the non-local damage model suggest that the non-

local weight function should depend on the orientation of principal strain axes at points x, but

the simplest and most practical formulation is obtained if the non-local length parameter is

considered as dependent only on the distance between the points.

is a homogenous and isotropic weight function which only depends on the relative

distance from point 𝒙 (Bažant, 1986).

2.6.2 Gradient Enhanced Non-local Model for Strain Softening

Material

By expanding the local equivalent strain eq with its Taylor series for sufficiently smooth eq

fields (Bažant, et al., 1984; Larsy & Belytschko, 1988; Peerlings, 1999; Jirásek & Patzák,

2002; Geers, 2004) we have

2 32 3

4 4

1 1. . .

2! 3!

1 .

4

!

eq eq eq eq eq

eq

x x x x x

x

(2.21)


In which 𝜵(𝒏) is the nth

order gradient and 𝒏

is the nth

order dyadic product of . By

substituting Equation (2.21) in Equation (2.20) and neglecting terms of order higher than two

we can derive an approximation for integral (2.20) as

2

eq eq eqc (2.22)

In which the coefficient 𝒄 is a function of length scale square which depends on the largest

heterogeneity of the material as discussed by Bažant and Planas (1997). 2

2

2i ix

is the

Laplacian operator. The importance of Equation (2.22) is that instead of integration �̅�𝒆𝒒 can

now be approximated by a differential Equation. The non-local equivalent strain 𝜀�̅�𝑞 can be

calculated explicitly in terms of local equivalent strain 𝜀𝑒𝑞 and its second-order derivatives.

Finite element solution of this explicit equation is not straight forward since, the substitution

of Equation (2.22) into the equilibrium equations will generate a set of fourth order partial

differential equations which requires higher deformation continuity and thus, additional

boundary conditions should be imposed on the domain.

Differentiating Equation (2.22) twice and neglecting terms of order higher than two an

alternative implicit approximation for non-local equivalent strain 𝜀�̅�𝑞 can be derived as

2

eq eq eqc (2.23)

The relevance of Equation (2.23) is that in contrast to Equation (2.22), �̅�𝒆𝒒 can now be

implicitly calculated in terms of 𝜀𝒆𝒒 using a 𝕮𝟎-continues finite element domain. To solve

this Helmholtz partial deferential equation (PDE) a boundary condition has to be specified.

Here we use the natural boundary condition as proposed by Peerlings et al. (1996; Jirásek &

Patzák, 2001; Jirásek, et al., 2004).

. 0eq n (2.24)

In which 𝒏 is the unit normal to the boundary 𝚪. This boundary condition is used on the

boundary surfaces of a RVE in homogenous materials. It may be also used on material

boundaries within a heterogeneous RVE.


2.6.3 Finite Element Implementation of Gradient Enhanced

Non-Local Model

Numerical solution of the gradient enhanced damage model requires a method consistent with

the continuum method to avoid a physically unrealistic behaviour. In order to implement the

gradient enhanced formulation, both the equilibrium equation and the implicit gradient

enhanced non-local partial differential Equation (2.23) are solved using finite element

method; for this their weak form is calculated using a Galerkin discritisation approach. By the

use of a test function 𝝋𝒊 , the divergence theorem equations (2.25) and (2.26) can be obtained

which illustrate the weak form of the equilibrium equation and Equation (2.23), respectively

(Peerlings, et al., 2002).

iij i j

i

d dx

(2.25)

Ω Γ

)( Ω Γeq eq

eq eq i

i i i

c d n dx x x

(2.26)

Using the natural boundary condition (Equation (2.24)) in Equation (2.26) yields

( 0)eq

eq eq

i i

c dx x

(2.27)

The equilibrium Equation (2.27) can be solved with a standard finite element procedure. Here

use of a finite element interpolation for calculation of non-local equivalent strain is explained.

Introducing a linear 𝕮𝟎-continues finite element interpolation for the non-local equivalent

strain �̅�𝒆𝒒, yields

eq N (2.28)

In which 𝜺 contains the nodal values of non-local equivalent strain �̅�𝒆𝒒. The derivatives of the

non-local equivalent strain can be obtained as

eq B (2.29)


The interpolation functions 𝑵 and 𝑩 are also used for 𝝋 and its derivatives. Discretising

Equation 2.34, and substituting Equations 2.28 and 2.29 whilst knowing that the resulting

equation holds for all admissible test functions we can conclude

T T T

eqcB B N N d N d

(2.30)

It is important to note that this discretisation can be defined independent of the discretisation

of the equilibrium equation and thus different interpolation polynomials can be used for the

finite element formulation (Massart, et al., 2007).

2.7 The Transient-Gradient Non-Local Model for Strain Softening

Material

Due to the heterogeneity of quasi-brittle material (e.g. concrete, rocks), diffuse micro-

cracking over a certain volume is usually observed at early stages of the fracturing process

(Haidar, et al., 2005; Otsuka & Date, 2000; Grassl & Jirásek, 2010).

The weight function of the averaging process is directly related to an internal length scale

which controls how a material point is affected by the behaviour of surrounding points. Initial

versions of these methods adopted a constant internal length scale which resulted in spurious

damage growth in inactive regions at high deformation levels (Pijaudier-Cabot, et al., 2004).

This is attributed (Geers, et al., 1998) to the fact that the nonlocal averaging process remains

active at all material points through the entire load history and, as a result, highly localised

strains within the process zone are mapped to surrounding points.

The increasing crack opening causes growing local equivalent crack strains. The gradient

damage averaging equation maps this increase onto the nonlocal equivalent strain, which

provokes an increase of damage. The weakening of the damage zone results again in larger

strains and the damage-strain localisation cycle is closed. The constant gradient damage

model combined with an exponential damage evolution law fails in the final stage where it

leads to an unacceptable growth of the damage zone. If the narrowing of the localisation zone

is insufficient, the averaging equation will map the local strain concentration onto a nonlocal

strain increase in an area outside the localisation zone at a growing distance of the actual

failure zone. The material next to the fracture zone remains sensitive to the opening of the


crack and even to the deformations at the other side of the fracture zone.

Although limited experimental knowledge is available regarding the evolution of the internal

length scale, micromechanical arguments have shown that the interaction between cracks and

voids is of a transient nature in the course of failure (Bažant, 1994). Pijaudier-Cabot et al.

(2004) made use of this argument to introduce a nonlocal model with evolving length scale,

whereby the internal length is made a function of local damage expressed in terms of the

local equivalent strain. Compared to the original nonlocal damage model, the model with

evolving length scale delivers physically acceptable damage profiles at high deformation

levels. Other nonlocal damage models with evolving length scale have been proposed (Giry,

et al., 2011; Nguyen, 2011). In particular, the stress-based model by Giry et al. (2011) is very

effective in solving the issues discussed by Simone et al. (2004) in terms of damage initiation

and propagation. Saroukhani, et al.(2013) introduced a simplified method in implementing

transient-gradient model.

2.7.1 Nonlocal Damage Model with a Damage-Dependant

Transient Length Scale

Inspired by Frémond & Nedjar (1996), and de Borst et al. (1995), the nonlocal variable can

also be derived from the damage. This suggestion was also made by Bažant & Pijaudier-

Cabot (1989). However, Jirásek (1998) already showed that such a nonlocal model cannot

provide meaningful results with a constant length parameter. A partial differential Equation

(2.30) with a variable length scale is therefore used instead

2( )eq eq eq (2.31)

while the boundary condition on Ω is given by

. 0n (2.32)

The nonlocal damage variable is then used in the constitutive relationship

(1 )Cij ijkl kl (2.33)

Equation 2.33 shows that it is no longer possible to keep the gradient parameter 2 / 2cc l

constant. If this would be the case, the nonlocal damage which enters Equation (2.33) will


never reach the ultimate point of failure ( 1) since the local damage is bounded by the

relations (Geers, et al., 2000). Accordingly, a nonlocal damage model in this sense can only

be realistic if approaches unity when its local counterpart does so. The gradient

parameter must therefore depend on the local damage variable . If the local damage equals

unity, the intrinsic length scale will vanish, and the nonlocal and local damage variables will

then coincide. Since the gradient term vanishes only after the crack is fully initiated, the well-

posedness remains preserved during the initiation phase.

2.7.2 Nonlocal Damage Model with a Strain-Dependant

Transient Length Scale

A strain-based transient gradient-damage model (or nonlocal equivalent strain model) has

been proposed in Geers, et al. (1998) and has been successfully applied to composite

materials (Geers, et al., 1999). The analysis of damage in discrete models with respect to the

consequences on continuum models has shown that the internal length scale of a continuum

increases with the damage of the system (Delaplace, et al., 1996). This observation is in

agreement with the transient nature adopted in (Geers, et al., 1998), where the gradient

parameter increases with the local equivalent strain eq . It has also been shown that the

transient character of the length scale alleviates spurious damage development with the use of

exponential damage evolution laws (Geers, et al., 1998; Nguyen, 2011; Saroukhani, et al.,

2013). The transient character of the intrinsic length scale', in this case is thus opposite to the

previous model, where the length scale decreased with increasing damage.

The partial differential Equation (2.23) with a variable length scale dependent on equivalent

strain is in the form of

2( )eq eq eq (2.34)

while the boundary condition on Ω is given by

. 0eq n (2.35)

in which , represents the transient gradient parameter and is defined in the following

equation.


n

eq

eq

eq

c

c

(2.36)

The above expression implies that nonlocal coupling starts as soon as deformation is detected

at a material point and reaches its maximum level, namely c, when the local equivalent strain

is . The consequences of this choice are twofold. On the one hand, nonlocal interaction is

progressively mobilized as the local equivalent strain increases. On the other hand, at

considerable deformation levels, vanishes in the unloaded material surrounding the

process zone. Hence, nonlocal interaction is appropriately confined and the unloaded material

behaves in a local manner.

2.8 First-Order Computational Homogenisation

To derive an enhanced constitutive material model for a complex composite like masonry

computational homogenisation can be used so that we can derive the global behaviour of the

masonry from its constituents such as concrete block and mortar. Fundamentals of

computational homogenisation have been introduced by Ghosh et al. (1995). This method

have been further developed by Smit (1998); Miehe et al. (1999); Feyel and Chaboche

(2000), Miehe and Koch (2002). The concepts of second-order computational

homogenisation were later introduced by Kouznetsova et al. (2001). Temizer & Wriggers

(2008), Coenen, et al. (2012) and (Holl, et al., 2013) and many more authors have developed

these models. Geers, et al. (2010) have reviewed and compared some of these models.

Homogenisation technique has been applied to upper and lower bound limit analysis of

masonry walls and vaults by Milani, et al. (2006), Cecchi, et al. (2007) and Milani, et al.

(2008). Cecchi & Di Marco (2000), Massart (2003), Massart, et al. (2007), Mercatoris, et al.

(2009), Milani & Cecchi (2013) and Reccia, et al. (2014) have all used homogenisation

methods to analyse the behaviour of different masonry systems. Using this technique,

through solution of the microstructural boundary value problem, the macroscopic response of

the structure can be obtained. In these techniques a Representative Volume Element (RVE) or

unit cell needs to be identified which is able to generate the entire structure by repetition.

Four steps for first-order computational homogenisation are as follows:


1) Consideration of a representative volume element (RVE) in which the constitutive

behaviour of all its components are known,

2) Calculation of the macroscopic deformation gradient on the RVE,

3) Application and analysis of the resulting microstructural boundary value problem and

calculation of macroscopic variables,

4) Derivation of numerical input and output macroscopic variables.

Homogenisation techniques have several advantages over the classical numerical techniques.

The macroscopic response of the structure is obtained from the underlying microscopic

boundary value problem, thus, no explicit assumption is needed for the macroscopic

constitutive response. Non-linear constitutive models can be used for the microstructural

constituents. Moreover, the boundary value problem on the RVE can be solved by any

suitable numerical method. Computational homogenisation is a powerful tool for the analysis

of systems with complex microstructural geometries and mechanical behaviour. This method

has been used in this report to simulate the biaxial behaviour of traditional masonry panels.

2.8.1 Choice of Homogenisation Boundary Condition

The scale transition consists in applying the strain tensor (macroscopic) to an RVE

(mesoscopic). Boundary conditions should be applied such that the imposed macroscopic

strain tensor equals the volume average of the mesoscopic strain on the RVE. Three types of

boundary conditions are classically used in homogenisation:

fully statically constrained boundaries also known as Taylor (or Voigt) assumption,

fully kinematically loaded boundaries or uniform displacement boundary condition,

Periodic boundary conditions.

Of the above choices fully statically constrained boundary condition is the most

computationally efficient choice; however, it usually significantly overestimates the stiffness.

The uniform displacement boundary condition also usually overestimates the material

properties. Generally, the periodic boundary conditions estimate stiffness more accurately

(van der Sluis, 2001; Massart, 2003). Furthermore, in case of a material such as masonry


which has a periodic mesostructure, use of periodic boundary conditions seems more

reasonable (Anthoine, 1995; Anthoine, 1997; Massart, et al., 2004).

2.8.2 Periodic Boundary Conditions for Computational

Homogenisation

For a complex composite such as masonry homogenisation technique can be used to develop

an enhanced constitutive material model from its constituents such as concrete block and

mortar. Uniform loading and periodic geometry for masonry has been assumed and thus,

homogenisation theory for periodic media which was adopted by Anthoine (1995; 1997) and

Massart (2003) seems suitable to use (also see Massart, et al. (2004) and Massart, et al.

(2005)). Computations have been performed on a single representative volume element

(RVE) which contains the information of the entire mesostructure. A boundary value problem

has been solved on the RVE using finite element method. Based on homogenisation theory

for periodic media, strains should be compatible and the stresses should be anti-periodic on

two opposite sides of the RVE. This will ensure that two neighbouring RVEs fit together. The

strain-periodic displacement field has the form (Massart, et al., 2004)

. u x ε x w x (2.37)

where 𝜺 is the macroscopic strain tensor, �⃗⃗� is the position vector and �⃗⃗⃗� (�⃗⃗� ) is a mesoscopic

displacement fluctuation field which distinguishes the real mesostructural displacement field

from the linear 𝜺. �⃗⃗� field. The fluctuation field is assumed to be periodic. The volume average

of the mesoscopic strain field resulting from relation (2.37) is given by

1 1 1

. M Ω Ω Γ

Ω Ω ΓΩ Ω Γ

RVE RVE RVE

sym sym

RVE RVE RVE

ε u d ε x w d ε wn d ε

(2.38)

Because of periodicity of �⃗⃗⃗� the boundary integral in Equation (2.38) equals to zero and thus,

the volume average of mesoscopic strain field is indeed equal to Mε . By the use of Hill-

Mandel work equivalence we have

1

: : Ω

ΩΩ

RVE

M m

RVE

σ ε σ δε d (2.39)


In which superscripts 𝑴 and 𝒎 refer to macro and meso, respectively. We can determine the

total or macro stress as the average of mesoscopic stresses as

1

Ω

ΩΩ

RVE

M m

RVE

σ σ d (2.40)

2.9 Concluding Remarks

Behaviour of different masonry systems under uniaxial and biaxial loading have been

investigated by a number of researchers. However, these different experiments have limited

applicability for other masonry systems. Based on the type of unit and mortar, their shape and

the type of assemblage, different strength envelopes are likely to be obtained for different

masonry systems. Obtaining failure envelopes through experimental work for each type of

masonry is time consuming and expensive. To overcome this issue, in this study,

development of a constitutive model for various types of ungrouted masonry systems based

on the behaviour of their individual constituents will be attempted. Utilizing this constitutive

law, will help in obtaining the behaviour of the complex masonry system using simple,

inexpensive tests on the constituents i.e. unit and mortar.

After comparing different types of modelling approaches developed in the literature,

computational homogenisation technique will be adopted to model the uniaxial and biaxial

behaviour of masonry. First, a damage evolution law capable of predicting the behaviour of

masonry constituents adopting Continuum Damage Mechanics (CDM) will be developed.

Then, the localisation issues of the model which is due to softening behaviour of the quasi-

brittle constituents will be resolved by adopting a transient gradient non-local model.

Different Representative Volume Elements (RVE) for masonry and the sensitivity of the

results to them will be studied similar to Massart (2004, 2005). To improve the models

performance in the fully bonded interface between the constituents, a transitional material

layer (Interphase) will be introduced to the RVE. Finally, the application of the model to

conventional and dry-stacked masonry will be examined against available experiments and

Australian Masonry Standard (AS 3700, 2011).

40 Development of a CDM Model Incorporating Variation of Poisson’s Ratio |

CHAPTER 3

Development of a CDM Model Incorporating

Variation of Poisson’s Ratio

3.1 Introduction

An accurate solution to problems involving the strength and deformation behaviour of

masonry significantly depends on the choice of the adopted constitutive laws of its

constituent materials. Constitutive laws are obtained considering a set of variables and a

thermodynamic potential. The choice of the potential and the damage variable is a crucial

part of the description of the constitutive law. Choice of damage evolution law or damage

variable assists in approximating and describing the underlying micromechanical growth of

micro-cracks (damage). With the growth of internal micro-cracks cause volume change,

which are treated in many different ways in numerical models, such as the phenomenon of

dilatancy (van Zijl, 2004; Burnett, et al., 2007) and variable Poisson’s ratio (Carol, et al.,

2002; Wang, et al., 2009).

Scalar damage models have been used throughout the literature for phenomenological

damage models (Chaboche, 1981; Lemaitre, 1985; Ju, 1989), and elasticity based damage

models for quasi-brittle material (Peerlings, et al., 2001; Massart, et al., 2005). The

formulation of these damage models (uncoupled with plasticity) to date have not accounted

for the independent variation in the softening response of the stress-strain behaviour of the

constituents under tension and compression. The novelty of this thesis is to specifically

introduce the damage evolution laws to explicitly differentiate the softening responses under

tension and compression independently and accounts for pre-peak cracking in compression.

This Chapter presents a Continuum Damage evolution law incorporating variation of

Poisson’s ratio. The proposed model is capable of describing the behaviour of masonry

constituents such as units (Clay brick, concrete block, stone block, etc.) and mortar. Basic


concepts are briefly described, together with the theoretical formulation. Then, an algorithm

for implementation of the formulations is presented. A simple numerical example is provided

to illustrate the capability of the model in dealing with masonry. The influence of key

parameters of the model on damage evolution and stress-strain behaviour of the material is

exhibited through a set of sensitivity analyses. The limitation of the model and the need for

improvement is described in the summary of this Chapter.

3.2 Uniaxial Behaviour of Quasi-brittle Material

A typical feature of quasi-brittle constituents (units and mortar) of masonry is the post-peak

softening behaviour that reflects growth of internal micro-cracks and gradual decrease of

stiffness under a monotonous increase of deformation. Figures 3.1 and 3.2 illustrate

characteristic stress-displacement diagrams for quasi-brittle materials under uniaxial tension

and uniaxial compression, respectively.

Figure 3.1 Typical behaviour of quasi brittle material under uniaxial tension

Figure 3.2 Typical behaviour of quasi brittle material under uniaxial compression


Proper prediction of these uniaxial performances of quasi-brittle material is essential for the

prediction under more complex loading cases (e.g., biaxial). As was reviewed in section 2.4

of this thesis, there still exists a need for developing a damage model capable of predicting

these behaviours without the need of coupling with plasticity theory.

3.3 Formulation

Different formulations for obtaining the equivalent strain eq (Equations (2.8), (2.9), (2.10)

and (2.11)) in order to map the components of the strain tensor to a scalar parameter have

been discussed in section 2.4.2 of the thesis. In this study, in order to deal with materials

exhibiting large difference between compressive and tensile strengths Equation (2.11)

proposed by de Vree et al. (1995), and extensively used by different authors (Mazars &

Pijaudier-Cabot, 1996; Peerlings, 1999; Massart, 2003; Massart, et al., 2007; Geers, et al.,

2010; Giry, et al., 2011) will be adopted.

In order to predict the uniaxial tensile and compressive behaviours (Figures (3.1) and (3.2),

respectively), a new damage evolution law is introduced here. The advantage of this model is

that it is able to predict both the pre- and the post- peak behaviours of quasi-brittle materials

in a more realistic manner than the pure Continuum Damage Mechanics (CDM) models

introduced in Equations (2.12), (2.13) and (2.14) used by many authors, previously (Mazars

& Pijaudier-Cabot, 1989; Jirásek & Patzák, 2002; Jirásek, et al., 2004; Massart, et al., 2005).

Moreover, this model is far simpler to implement, and calibrate than the coupled damage-

plasticity models used throughout the literature (Lourenco, 1996; Zucchini & Lourenço,

2002; Zucchini & Lourenco, 2004; Wei & Hao, 2009).

3.3.1 Refined Damage Evolution Law for Masonry Constituents

Damage evolution law is defined in the form of

1

1

1

1 1 0

0 ( )

( ) ( ) 0

ii

cc

c c c

c c cc

e if I

if I and

if I and

(3.1)


in which , and are material parameters. c and c are the threshold damage and

threshold strain beyond which damage grows rapidly, measured at a point of uniaxial

compression test, respectively. i is the threshold for damage initiation and is the

maximum equivalent strain eq , which the material has experienced so far. Parameter is

defined in the form of

2

2

2c c c c c

c c

(3.2)

The model considers independent energy dissipation in tension and compression and a more

gradual growth of microcraks in the beginning of the damaging process for compressive

loading. A qualitative description of damage growth in terms of equivalent strain has been

shown in Figure 3.3. When the equivalent strain is less than c , c , the material exhibits

increase in the rate of damage growth as the deformation increases. This gradual increase

continues until the strain reaches the threshold c . After reaching strain c the rate of

damage growth (slope of damage evolution) starts decreasing until the material is fully

damaged. The stress-strain behaviour of the quasi-brittle material in compression can be fully

simulated using this new damage evolution law.

Figure 3.3 Qualitative damage evolution in terms of equivalent strain under uniaxial

compression


This model uses an exponential softening behaviour for the tensile part of the equation. Using

this expression when 𝜀 → ∞, stress approaches (1 − 𝛼)𝐸𝜅𝑖 which can represent the long tail

which was experimentally obtained for quasi-brittle material. Parameter 𝛽 controls the

damage growth rate which depends on the tensile fracture energy of the material. When 𝛽 is

larger the model will show a faster crack growth and a more brittle response. The model is

sensitive to 𝛽 values up to 10000 and values larger than that seem to have minimum effect on

its behaviour.

A numerical example, considering the above explained model has been presented in section

3.5. Furthermore, all the influencing parameters have been discussed in detail in section 3.6.

3.3.2 Variable Poisson’s Ratio

In the scalar damage model (isotropic formulation) mentioned in previous sections, a single

scalar damage variable ω prescribes a proportional degradation to all components of the

secant stiffness and compliance according to Equation (3.1). It should be noted that this is

equivalent to reduction of Young’s modulus 0(1 )E E , Bulk modulus 0(1 )K K and

shear modulus 0(1 )G G . However, Poisson’s ratio remains constant 0 , which

implies a restricted form of isotropic degradation (see, Ju (1990)) – more specifically

volumetric changes due to micro-cracking remains unaccounted for in such formulations.

Various secant stiffness proposals were examined by Rizzi (1993,1995). In all cases, since all

the components of the secant stiffness are affected in the same manner which leads to

improper reproduction of volume changes of damaging material and a shear-dominated

failure. To resolve this issue (Mazars & Pijaudier-Cabot (1989) and Comi (2001) proposed

the tension and compression split of damage evolution law (see, Equations 2.15-19).

Moreover, Lubliner, et al. (1989), Comi & Perego (2001) and Wosatko (2011) amongst

others, used a volumetric-deviatoric decomposition approach. Pamin, et al. (2014) scaled the

intensity of damage growth for the volumetric part of the damage response based on the work

of Lemaitre & Desmorat (2005) and (Desmorat, et al. (2007).

According to experimental evidence (Launay & Gachon, 1970; Kotsovos & Newman, 1977;

Bongers, 1998; Ferretti, 2004) peak stress in concrete occurs under highly confined triaxial

compression which results in inelastic volume compaction. However, the damage evolution


law underestimates the material strength in the hydrostatic compression stress states in post-

peak stage.

Some Authors have attempted to resolve this issue in the Continuum Damage Mechanics

framework. Rizzi, et al. (1995) introduced a deviatoric degradation direction which resulted

in nonlinear increase of Poisson’s ratio with nonlinear degradation of stiffness. However, the

Bulk modulus remained constant in this case. Comi & Perego (2001) re-defined the bulk

modulus in compression in the form of 0(1 )K K in which 1 0 . Carol et al.

(2002) proposed an extension to this model based on independent evolution of deviatoric and

volumetric parts of stiffness. They introduced both a bi-dissipative and a single-dissipative

isotropic model for quasi-brittle material. The drawback of these models is that they violate

the classical definition of damage mechanics (Equation 2.1) as in Equation 3.3 for E for

example:

2

01 1

0 0

3(1 )

2(1 )(1 ) (1 2 )(1 )E E

(3.3)

In which η is a path parameter and 0 and 0E are the initial Poisson’s ratio and modulus of

Elasticity, respectively. A similar model based on Young’s modulus and Poisson’s ratio

decomposition was proposed by Carol, et al. (2005).

2(1 )

0E E (3.4)

and

4

0

(3.5)

In which η fixes the slope of constrained linear paths and is a function of damage

parameter.

In this thesis, a single dissipative model based on decomposition of Young’s modulus and

Poisson’s ratio proposed. Here, the influence of damage on the volumetric part of stress has

been modified in a way that the classical definition of isotropic damage mechanics (Equation

2.1) is not compromised.

Introducing a volumetric damage parameter K , the bulk modulus is expressed as


0(1 )KK K (3.6)

which reflects the Poisson’s ratio as

3

6

K E

K

(3.7)

Considering, 0(1 )E E and substituting Equation (3.6) into Equation (3.7) we can obtain

the Poisson’s ratio in the form of

0 0

0

3K (1 ) E (1 )

6 (1 )

K

KK

(3.8)

So far the model is bi-dissipative since the damage parameter and the volumetric damage

parameter K are independent. Different damage variables may be introduced at this stage to

describe the evolution of the volumetric damage parameter. In order to adopt a single loading

surface for the entire model we define the relation between volumetric damage parameter K

and damage parameter in the following form

1 (1 )K

(3.9)

which converts the model into a single-dissipative isotropic model. Here, parameter η

controls the effect of the volumetric damage parameter and is 0 . With this definition

K always holds and in case of 1.0 the two damage parameters become equal and it

results in a constant Poisson’s ratio and the classical damage model. For quasi-brittle

material, 0 1 which results in increase of Poisson’s ratio when damage progresses.

Placing Equation (3.9) into Equation (3.8) we can obtain the Poisson’s ratio in the form of

1

0 0 0

0 0

3K (1 ) E (1 ) E (1 )1

6 (1 ) 2 6K K

(3.10)

Now considering the value of the initial Bulk modulus in terms of Young’s modulus and

Poisson’s ratio

00

03(1 2 )

EK

(3.11)


and replacing it in Equation (3.10) we finally conclude the Poisson’s ratio in the form

1

01 (1 2 )(1 )

2

(3.12)

The shear modulus can be obtained in the same manner in the form of

1

0 0

(1 )

(3(1 ) (1 2 ))G

E

(3.13)

in terms of the initial Young’s modulus and initial Poisson’s ratio.

3.4 Implementation

The constitutive model described in Sections 3.1 to 3.3 was implemented through an

interactive code written by the author. The main program is a MATLAB code which

incorporates PYTHON scripting together with the commercial FE software ABAQUS/

Standard 6.11. Moreover a FORTRAN user-subroutine has been written for the use of the

ABAQUS software.

In the first step, after inputting all parameters in the MATLAB main program, it calls a

PYTHON code which constructs the geometry, discretises the domain and assigns

appropriate boundary conditions and loadings to the structure, see Figure 3.4.

In the second step, MATLAB will call ABAQUS to run the analysis for the first load

increment and write all the outputs in an ODB (output data base) file.

In the next step, after the analysis is completed, the program will call PYTHON to open the

ODB file and read all the nodal information.

In the fourth step, as shown in Figure 3.4, the local equivalent strains for all the material

integration points will be calculated from the outputs of the ABAQUS analysis. The modified

von Mises equivalent strain formulation (Equation 2.11) explained in section 2.3.2 has been

implemented for this purpose.


Figure 3.4 Implemented program’s flow chart

In the fifth step, Equation 3.1 is used to calculate the scalar damage variable for each material

point as shown in Figure 3.4. The proposed damage evolution law will be utilised for this

purpose.

In the sixth step, the program will check the convergence and check if it has reached the final

load increment. If Yes, the program will end the analysis.


If No, using the data from the previous steps, ABAQUS will be called to analyse the system

for the subsequent load increment. For this analysis, ABAQUS will require the material

properties of each integration points. These material properties are calculated from the stored

damage parameters which were calculated in the program previously.

Steps 2 – 6 will be repeated for all the load increments. These steps have been plotted into a

flow chart illustrated in Figure 3.4. PYTHON, MATLAB and FORTRAN codes used in this

procedure can be found in Appendices A, B and C, respectively.

The damage model proposed here permits to evaluate damage in an explicit way without the

need for any iterative procedure. Steps 4 and 5 of the above explained procedure for the nth

increment are described in more detail in Table 3.1.

Table 3.1 Algorithm for the implementation of the introduced damage model

1. Compute the strain increment and update the strain

nn n-1ε ε +=

2. Impose

1n nκ κ

1n nυ = υ

3. Compute the current equivalent strain

2

2

1 1 22 2

1 11 12

2 1 2 2 1 2 1n

k k kε I I J

k υ k υ υ

4. Check damage criterion

, n n

f ε κ ε κ 0

If Yes: No damage, Go to Step 9.

If No: Damage, proceed to 5.

5. Compute the damage variable n .


1

1

1

1 1 0

( 0 ( )

( ) ( ) 0

)

n ii

cn c

c c c

c c

n

c

n

e if I

if I and

if I and

ε

nn

n

n

n

ε

εε ε

ε

ε

ε cn

ε

6. Compute the volumetric damage variable ( )K n .

( ) 1 (1 )K n n

7. Compute current Poisson’s ratio

1

01 (1 2 )(1 )

2

nn

8. Update the damage threshold

n nκ ε

9. Update the stress

1 n ij ijkl klσ ω C ε

10. Return to Main Program

3.5 Numerical Example

In order to better understand the behaviour of the proposed model and its use for real datasets

from experiments we first consider a benchmark numerical example. This example consists

of a two-dimensional finite element. An eight noded element (see Figure 3.5) with size of

unity is subjected to uniaxial tension and compression.

Figure 3.5 The 8-noded element used for the numerical analysis


Table 3.2 Input properties of the tested element

Material Properties

E (MPa) 20000

0 0.2

k 10

α 1.0

β 1000

i 0.00005

c 0.00016

c 0.5

-0.25

This simple test illustrates the response of the improved model at the point level. A material

with 20000E MPa , 0 0.2 , strength ratio 10k have been considered. All mechanical

properties of the material can be found in Table 3.2. Calculations are performed using the

procedure explained in section 3.4.

3.5.1 Uniaxial tension

Uniaxial tensile strain has been imposed to the 8-noded element as shown in Figure 3.6. For

tension results of three different values for η (1.0, 1.1 and 1.2) have been compared in Figure

3.7. As can be seen, the model is able to capture the uniaxial behaviour of a quasi-brittle

material which was illustrated in Figure 3.1. As was mentioned in section 3.3.1.1 and can be

seen in Equation 3.1 the present model is using an exponential law for softening in tension. It

can also be observed that increasing the value for η (decrease of Poisson’s ratio) has minimal

effect on the tensile behaviour of the model. Further increase of this value results in decrease

of Poisson’s ratio to values less than negative one in small strains which is physically

impossible, see Figure 3.8. In this study, the value η is considered to be one for tensile state

of stresses.


Figure 3.6 The 8-noded element under uniaxial tension


tension test



uniaxial tension test

The damage evolution for in the uniaxial test using the proposed model for 𝜂 = 1.0 can be

seen in Figure 3.9.

Figure 3.9 Damage evolution of the material based on the proposed model in uniaxial tension

test for 𝜼 = 𝟏. 𝟎


3.5.2 Uniaxial compression

Uniaxial compressive strain has been imposed to the 8-noded element as shown in Figure

3.10. The material properties of the material used in this test can be found in Table 3.2.

Figure 3.10 The 8-noded element under uniaxial compression

For compression, again the results of three different values for η (0.1, 0.5 and 1.0) have been

compared in Figure 3.11. This figure illustrates that the model is able to capture the uniaxial

behaviour of a quasi-brittle material which was shown in Figure 3.2.



compression test


uniaxial compression test

Figure 3.12 shows the evolution of Poisson’s ratio for three values of η. The value for η

increases when loading progress which is what we expect from quasi-brittle materials.



compression test for 𝜼 = 𝟎. 𝟓

The damage growth with respect to uniaxial strain in a uniaxial compression test

corresponding to 𝜂 = 0.5 has been represented in Figure 3.13. Comparing the proposed

model and its damage evolution in Figure 3.13 with the damage evolution laws mentioned in

section 2.4.3, one can see that the model considers a more gradual growth of microcraks in

the beginning of the damaging process for compressive loading. This significantly improves

the prediction of the proposed model for the compressive behaviour of quasi-brittle materials

such as concrete, mortar and brick.

3.5.3 Pure Shear

Similarly to sections 3.5.1 and 3.5.2, in order to observe the behaviour of the model in pure

shear, an 8-noded element has been considered with boundary conditions as shown in Figure

3.14. The material properties of the material used in this test can be found in Table 3.2. For

shear, results of three different values for η (0.1, 0.5 and 1.0) have been compared in Figure

3.15.


Figure 3.14 The 8-noded element under pure shear

Figure 3.15 Stress-strain behaviour of the material based on the proposed model in pure

shear test

Figures 3.16 and 3.17 show the Poisson’s ratio and Damage’s evolution with respect to strain.

From Figure 3.15, it can be observed that the model shows the shear behaviour of quasi-

brittle material in a reasonable manner.



pure shear test

Figure 3.17 Damage evolution of the material based on the proposed model in pure shear test

3.6 Parametric Study

In order to verify the performance of the present damage model the effect of various material

parameters to the overall response of the system has been examined. This examination is

through a set of analyses using an eight noded two-dimensional finite element of a size unity


as shown in Figure 3.5. The influence of different parameters on the element which is

subjected to uniaxial tension and compression has been studied. This simple parametric study

illustrates the response of the improved model at the point level.

Effects of the following key parameters on the compressive and tensile behaviour of the

model have been discussed here.

Equivalent strain parameter k

Refined damage evolution law parameters , , and c

The volumetric damage parameter

The parameters employed in this parametric study are summarised in Table 3.2.

3.6.1 Uniaxial tension test

By examining Equation 3.1 it is clear that parameters α and β, have the most effects on the

softening branch of uniaxial tensile behaviour of the material. In the following sections, the

effects of these two parameters on stress-strain behaviour of the material have been

individually studied.

3.6.1.1 Effect of damage evolution law parameter α

This parameter was introduced to avoid numerical problems and does not represent any

physical phenomena. Results of a parametric study varying the parameter α ( 0 1 ) are

presented in Figures 3.18 and 3.19. As anticipated using this expression, when the strain

, stress approaches 1 iE , see Figure 3.18. For example if 0 ,then the stress

remains equal to iE after the equivalent strain is equal to initial threshold i and if 0.5

the stress gradually reduces to the stress remains equal to 0.5 iE . Increase of α results in a

steeper post-peak response in the model. Influence of the parameter α on the damage

evolution can be seen as not very sensitive in Figure 3.19. This can be related to the brittle

failure mode of masonry where the effect of α is not how it damages masonry under tension

but how it releases its post-peak energy.


Figure 3.18 Influence of parameter α on the stress-strain behaviour of the proposed model in


Figure 3.19 Influence of parameter α on the damage evolution of the proposed model in



3.6.1.2 Effect of damage evolution law parameters β

Parameter β controls the damage growth rate which depends on the tensile fracture energy of

the material. When β is higher, the model will show a faster crack growth and a more brittle

response. Results of a parametric study in uniaxial tension employing different values for

parameter β have been illustrated in Figures 3.20 and 3.21.

Figure 3.20 Influence of parameter β on the stress-strain behaviour of the proposed model in


Figure 3.21 Influence of parameter β on the damage evolution of the proposed model in



3.6.2 Uniaxial compression test

Uniaxial compression parametric study has been conducted to evaluate the performance of

the model in compressive stress states. The effects of parameters k, , and c are studied

in the following sections.

3.6.2.1 Effect of Equivalent strain parameter k

Results of a parametric study employing different values for parameter k in Equation (2.11)

on the stress-strain behaviour of the proposed model are shown in Figure 3.22.

The influence of this parameter to the damage evolution with respect to strain is presented in

Figure 3.23.

Parameter k controls the equivalent strain to tension and compression which is usually set to

the ratio of compressive strength of the material to its tensile strength. As mentioned in

section 3.3.1.1, this model is capable of modelling materials with high differences to their

tension and compression strengths.

Figure 3.22 Influence of parameter k on the stress-strain behaviour of the proposed model in



Figure 3.23 Influence of parameter k on damage evolution of the model in uniaxial

compression test

As the values of k increases, the damage growth decreases, see Figure 3.23. This results in

higher failure strength in compression. The difference between the failure strength in tension

and compression can be easily controlled with this parameter. When k = 1, the tensile and

compressive strengths are almost the same.

3.6.2.2 Effect of damage evolution law parameter ζ

Parameter ζ controls the steepness of the post-peak softening behaviour of the material. The

higher the value of ζ, the more brittle response the material will produce. Results of

parametric study, employing a range of ζ, to the stress-strain behaviour of the model under

uniaxial compression is shown in Figure 3.24. The influence of this parameter on the damage

evolution with respect to strain is presented in Figure 3.25. From this Figure, it can be

observed that this parameter controls the slope of the damage evolution curve at point c

which is the point from which all the curves are passing.


Figure 3.24 Influence of parameter ζ on the stress-strain behaviour of the proposed model in


Figure 3.25 Influence of parameter ζ on damage evolution of the model in uniaxial

compression test

3.6.2.3 Effect of damage evolution law parameter 𝝎𝒄

Parameter 𝜔𝑐 controls the dissipated energy in the present model due to the damage process.

Increase of this parameter leads to higher rate of damage dissipation in the model. The


influence of this parameter on the stress-strain behaviour of the proposed model under

uniaxial compression loading has been investigated and is shown in Figure 3.26.

Figure 3.26 Influence of parameter 𝛚𝐜 on the stress-strain behaviour of the proposed model

in uniaxial compression test

Figure 3.27 Influence of parameter 𝛚𝐜 on damage evolution of the proposed model in


Figure 3.27 shows the effect of parameter ωc on damage evolution in terms of strain. The

higher the value for 𝜔𝑐, the sooner the material loses its integrity. When 𝜔𝑐 = 1, the material

reaches full damage state at c which can be seen in Figures 3.26 and 3.27.


3.6.2.4 Effect of the volumetric damage parameter η

Sensitivity of parameter η to various properties of the material was performed. This

parameter controls the effect of the volumetric damage parameter and is 0 1 for quasi-

brittle material. With this definition K always holds and in case of 1.0 the two

damage parameters become equal and it results in a constant Poisson’s ratio and the classical

damage model.

Figure 3.28 shows the stress-strain behaviour of the proposed model for various values of η in

uniaxial compression test.

Figure 3.28 Influence of parameter η on the stress-strain behaviour of the proposed model in


Effect of change of parameter η to the relation between the damage variable and the

volumetric damage variable K is shown in Figure 3.29. This figure shows that when the

parameter η approaches unity, the difference between the two damage variables reduces.

In case of quasi-brittle materials such as brick, mortar and masonry, which are the focus of

this study, due to internal micro-cracking the Poisson’s ratio increases as loading progresses.

In the present model, as can be seen in Figure 3.30, as the value of η decreases the rate of

Poisson’s ratio growth increases.


Figure 3.29 Influence of parameter η on the volumetric damage variable K with respect to

the damage variable based on the proposed model in uniaxial compression test

Figure 3.30 Influence of parameter η on Poisson’s ratio with respect to strain based on the

proposed model in uniaxial compression test

Effects of parameter η on the growth of Poisson’s ratio with respect to damage variable

and volumetric damage variable K are shown in Figures 3.31 and 3.32, respectively.


Figure 3.31 Influence of parameter η on Poisson’s ratio with respect to the damage variable

based on the proposed model in uniaxial compression test

Figure 3.32 Influence of parameter η on Poisson’s ratio with respect to the volumetric

damage variable K based on the proposed model in uniaxial compression test

Only values of η, in the range of 0 1 have been considered in this study since, increase

of parameter η to values more than unity results in decrease of Poisson’s ratio which is not

the case for quasi-brittle material such as concrete, brick and mortar.


The influence of parameter η to the Bulk modulus is shown in Figure 3.33 with respect to

damage progression. Smaller values of parameter lead to slower decrease of Bulk modulus in

the model.

Figure 3.33 Influence of parameter η on Bulk’s modulus with respect to the damage variable

based on the proposed model in uniaxial compression test

3.7 Conclusion

In this chapter a new damage evolution law capable of differencing the variation in the

softening response of the stress-strain behaviour (unlike the existing damage models for

masonry applications) of quasi-brittle material such as units, mortar and masonry has been

introduced. The proposed damage evolution law considers a more gradual growth of

microcraks in the initial stages of the damaging process for compressive loading. This

significantly improves the prediction of the model for the compressive behaviour of quasi-

brittle material. Moreover, a single-dissipative formulation of the isotropic damage in quasi-

brittle material has been presented. This model is proposed based on decomposition of

Young’s modulus and Poisson’s ratio. This model extends the classical formulation of

damage mechanics to include the variation of Poisson’s ratio using a scalar damage

definition.


Subsequently, the upgraded model has been tested on a point level for uniaxial tension and

compression tests using a single finite element. Then, a parametric study was performed to

assess the effect of different material parameters to the pre-peak and post-peak behaviours of

the model under uniaxial loading states.

In order to implement this model to analyse the behaviour of different structures, a

Representative Volume Elements (RVE) enhanced with a nonlocal model is essential; the

following chapter delves in detail into formulation and enhancement of periodic RVEs.

71 Enhancement of the CDM Model through Non-Local Transient-Gradient Method |

CHAPTER 4

Enhancement of the CDM Model through a

Non-Local Transient-Gradient Method

4.1 Introduction

The continuum damage mechanics model embedded with variable Poisson’s ratio developed

in Chapter 3 is enhanced further through a nonlocal transient-gradient method in this Chapter.

As damage growth is highly dependent on the microstructure of the material, in particular

coarse aggregates in concrete units and mortar, micro cracks in these materials bridge

between the aggregates. Therefore, the fracture process should have direct relation to the

aggregate size. However, classical damage mechanics models do not explicitly consider the

scale of this microstructure and suffers from damage localisation (Bažant, et al., 1984), the

phenomenon of which is discussed in the section 2.5 of this thesis. To overcome this

localisation problem in the simulation of strain softening materials, we can introduce non-

locality to the constitutive relation so that the growth of damage variable depends on the

average deformation of the material in a certain regions. Addition of the non-local concept to

the damage model leads to a smooth damage growth depending on the length scale

(Peerlings, 1999). The nonlocal models, their ability to describe damage processes and their

principles of formulations are described in section 2.6 of this thesis. Among these models, the

implicit gradient enhanced damage models appear to be the most successful (Geers, et al.,

2000). There exist two types of intrinsic length parameter within the gradient enhanced

models (1) constant length scale (2) variable length scale. Jirásek (1998), illustrated that use

of constant nonlocal parameter can lead to residual stresses and unrealistic results. A partial

differential equation (Equation 2.31 in this thesis) has therefore been introduced with a

variable length scale to overcome this problem. Section 2.7 contains more explanation of the

transient-gradient nonlocal model. In the original formulation of the gradient-enhanced

damage models, the gradient activity parameter c is considered to be constant and equated to


2 / 2cl , where cl is the length scale parameter. However, the model involves an extra

continuity equation on either the gradient activity parameter or the local equivalent strain and

therefore adds a set of degrees of freedom to those of the standard model. A new approach in

implementing the transient-gradient model is introduced by Saroukhani, et al. (2013) to

eliminate the need the extra set of degrees of freedom with application to homogenised

materials. In this chapter, this transient gradient formulation is adapted to masonry for the

first time.

4.2 Formulation of a Transient-Gradient Model

An implicit gradient-enhanced damage model with an evolving length scale capable of

overcoming mesh pathological problem for concrete like materials was first introduced by

Geers et al. (1998). Saroukhani, et al. (2013) has further developed the transient gradient

enhanced CDM model by eliminating the need for any extra set of degrees of freedom.

These formulations have been adapted in this thesis for the first time in masonry applications;

the adapted formulation is computationally efficient as the extra degree of freedom required

in Geers et al. (1998) is eliminated.

As discussed in section 2.7, in the transient-gradient approach an extra partial differential

equation and a new set of continuity equation are introduced to the original gradient-

enhanced model. In order to avoid adding this extra set of governing equations, Saroukhani,

et al. (2013) suggested converting Equation (2.31) to a diffusion equation by dividing it by

the transient length scale 0 . This division leads to a new form of Equation (2.31) as

follows

2eq eq

eq

(4.1)

In which eq and eq are the local and nonlocal equivalent strains, respectively. Similarly to

the original gradient enhanced model explained in section 2.5.3 the Galerkin weighted

residual approach is used to discretise Equation (4.1). By multiplying a test function 𝝋𝒊 to the

equilibrium equation and Equation (4.1), and integrating over the domain Ω using the

divergence theorem Equations (4.2) and (4.3) are obtained.


iij i j

i

d dx

(4.2)

Ω Ω

( Ω) Ωeq eq eq

i i

d dx x

(4.3)

The equilibrium Equation (4.3) can be solved with a standard finite element procedure. Here

use of a finite element interpolation for the calculation of non-local equivalent strain is

explained. Introducing a linear 𝕮𝟎-continues finite element interpolation for the non-local

equivalent strain �̅�𝒆𝒒, yields

eq N (4.4)

In which 𝜺 contains the nodal nonlocal equivalent strains �̅�𝒆𝒒. The derivatives of the non-

local equivalent strain can be obtained as

eq B (4.5)

The interpolation functions 𝑵 and 𝑩 are also used for 𝝋 and its derivatives. Discretising

Equation 4.3, and substituting Equations 4.4 and 4.5 (whilst knowing that the resulting

equation holds for all admissible test functions) we obtain Equation (4.6).

TeqT T

eq

N NB B d N d

(4.6)

It is important to note that this discretisation can be defined independent of the discretisation

of the equilibrium equation and thus different interpolation polynomials can be used for this

finite element formulation. In other words, physical and transient finite elements can have

different shape functions.

The benefit of solving the transient-gradient model in this form is that it requires the same

number of continuity equations as in the original gradient enhanced model. The transient

gradient parameter was modified from Equation (2.36) by Geers, et al. (2000) to avoid

division by zero by Saroukhani, et al. (2013) as in Equation (4.7)

0 0( )

n

eq

eq

eq

c c c

c

(4.7)


In which and n are model parameters. 0c , is considered to be an arbitrary positive value

so that non-local interaction is prevented at the beginning of the analysis. Equation (4.7)

becomes identical to Equation (2.36) when 0 0c . In order to avoid division by zero in the

system of governing equations, 0c is set to an arbitrary positive value which is chosen such

that at the initial time step, nonlocal interaction between integration points does not occur

(Local and nonlocal strains are the same). It is sufficient to consider 0c to be less than the

square of the smallest distance between any two integration points.

As can be seen in Equation (4.6), the gradient of ( ) is no longer involved in the weak

form of the transient-gradient equation unlike the equation by Geers et al. (2000).

4.3 Homogenisation Technique for Modelling Masonry

Masonry is a composite material consisted of units and mortar and its global mechanical

behaviour can be derived from the properties of its constituents through the homogenisation

theory for periodic media. This technique is widely used by the researchers for different

mechanical models such as elasticity, plasticity, fracture, damage and limit state analysis of

masonry (De Buhan & De Felice, 1997; Anthoine, 1997; Massart, 2003; Milani, et al., 2006;

Mistler, et al., 2007; Salerno & de Felice, 2009; Cavalagli, et al., 2011; Addessi & Sacco,

2012; Stefanou, et al., 2015). In this thesis, a generalised homogenisation technique based on

the work of Kouznetsova, et al. (2001) and Massart, et al. (2007) has been adopted. Section

2.8 provides a brief review of these papers. The following sections explain the specific choice

of the Representative Volume Element (RVE) and its boundary conditions which is

considered in this thesis.

4.3.1 Choice of a Mesoscopic Representative Volume Element

(RVE)

In order to minimise the computational cost at the mesoscopic scale and also to capture all

possible failure mechanisms, representative volume elements (RVEs) are chosen carefully

through perusal of the periodicity in the real world structures. For stretcher bonded masonry,


different RVE shapes and sizes as shown in Figure (4.1) have been used by different

researchers, see, Anthoine (1995), Luciano and Sacco (1997), Massart (2003), Lourenco, et

al. (2007), Calderini, et al. (2010). It is expected that the average behaviour of any boundary

value problem can be appropriately predicted by any RVE provided the periodic boundary

conditions are appropriately enforced and the RVEs are formulated from thermodynamically

consistent theories and representative material scale parameters; in other words, infinitesimal

strain with no localisation can be determined using RVEs of different shapes and sizes.

Figure 4.1 Typical periodic RVEs used for masonry (Anthoine, 1995; Massart, 2003;

Lourenco, et al., 2007)

In this thesis, one of the objectives is to simulate the biaxial testing of masonry wallets (see

Figure 4.2(a)); a number of possibilities for RVE shapes and sizes are possible – two RVEs

were chosen (Figure 4.2(b). These two RVEs are further explored in this chapter.


Figure 4.2 Identified RVEs

4.3.2 Boundary Conditions for the Representative Volume

Element (RVE)

As explained in section 2.7.1 the fully kinematically constrained boundary conditions and the

uniform displacement boundary conditions usually significantly overestimate the stiffness of

the homogenised medium. Periodic boundary conditions are required to estimate stiffness

more appropriately (van der Sluis, 2001; Massart, 2003). In case of a material such as

masonry which has a periodic macrostructure, use of periodic boundary conditions has been

established by many researchers (Anthoine, 1995; Anthoine, 1997; Massart, et al., 2004;

Lourenco, et al., 2007; Calderini, et al., 2010).

In this thesis, the running (or, stretcher) bond masonry system is considered with the help of

two RVEs represented in Figure 4.2.

Periodic boundary conditions have been applied through periodically tying using three

controlling nodes A, B and C (see Figure 4.3) of the RVE. Similar approach is adopted by

Massart (2003) and justified by Anthoine (1995) and Smit (1998).


Figure 4.3 Controlling nodes and periodicity conditions on a typical masonry RVE (Massart,

2003)

The periodicity conditions for the two horizontal and two vertical edges can then be

formulated in terms of the controlling nodes as

, ,

, ,

, ,

eq 1 eq 4

eq 2 eq 5

eq 3 eq 6

ε ε

ε ε

ε ε

(4.8)

and

6 3

B A

C A

C B

4 1

5 2

u u u u

u u u u

u u u u

(4.9)

In which, ,eq 1

ε , ,eq 2

ε , ,eq 3

ε , ,eq 4

ε , ,5eq

ε and ,eq 6

ε are the equivalent strains on the boundary of

the RVE as illustrated in Figure 4.1. Displacements on the boundary of the RVE have been

represented by 1u , 2u , 3u , 4u , 5u and 6u . Displacements of the three controlling nodes A, B

and C are shown by Au , B

u and Cu , respectively. Equations (4.8) and (4.9) represent that

both the displacement and the damage will be periodic in the RVE. These periodicity

conditions lead to periodic predictions for mesoscopic stress and strain fields.

Loading is applied on the RVE by means of the three controlling points A, B and C as

illustrated in Figure 4.4. As can be seen in Figure 4.3, W, H and L are width, height and

length of the RVE, respectively. Moreover, 𝜎𝑥𝑥, 𝜎𝑦𝑦 𝑎𝑛𝑑 𝜎𝑥𝑦 are, stress parallel to bed joint,

stress perpendicular to bed joint and in plane shear stress, respectively.


Figure 4.4 Loading modes applied on the RVE

Using these RVEs it is shown in this chapter that the failure patterns achieved through

experimental work (see, Dhanasekar (1985)), are simulated. A reasonable illustration of the

behaviour of masonry is obtained, using this homogenisation method. Throughout this thesis,

some cases are analysed under load control and others under displacement control.

Obviously, the load control analyses could not trace post-peak curves and hence the final load

obtained was questioned onto whether it was physical or numerical. In this thesis, the final

load step was checked with experimental data and satisfied that. The final load was not overly

affected by numerical instabilities. Where experiments were conducted in displacement

control, analyses were also performed in displacement control and the peak was clearly

obtained from the complete response.

4.4 Implementation

The transient-gradient model described in sections 4.2 to 4.4, was implemented into the

constitutive model explained in sections 3.1 to 3.3 through an interactive code written by the

author. The main program is a MATLAB code which incorporates PYTHON scripting

together with the commercial FE software ABAQUS/ Standard 6.11. The procedure is as

explained in Chapter 3 and the main difference is introduction of the transient-gradient model

between steps 4 and 5. In this step, the transient-gradient boundary value problem is solved

on the domain, using Equations 4.6 and 4.7 and the nonlocal equivalent strains for all

material integration points are calculated.

Figure 4.5 shows an implementation flow chart for the program and Table 4.1 shows an

algorithm explaining steps 4 to 6 of the flow chart.


Figure 4.5 Flowchart for the analysis of RVE

Table 4.1 Algorithm for the implementation of the introduced transient gradient model on the

Representative Volume Element (RVE)

1. Compute the displacement increment for integration points from ABAQUS output

database

nu


nn n-1ε ε +=

3. Impose


1n nκ κ

1n nυ = υ

4. Compute the current equivalent strains

2

2

1 1 22 2

1 11 12

2 1 2 2 1 2 1n

k k kε I I J

k υ k υ υ

5. Solve the transient-gradient boundary value problem on the domain

TeqT T

eq

N NB B d N d

6. Loop over integration points;

Obtain the current nonlocal equivalent strains for integration points

Check damage criterion for integration points

, n n

f ε κ ε κ 0

Yes: No damage, Go to 10.

No: Damage, proceed to 6.

Compute the damage variable n for integration points

1

1

1

1 1 0

( 0 ( )

( ) ( ) 0

)

n ii

cn c

c c c

c c

n

c

n

e if I

if I and

if I and

ε

nn

n

n

n

ε

εε ε

ε

ε

ε cn

ε

Compute the volumetric damage variable ( )K n for integration points

( ) 1 (1 )K n n

Compute current Poisson’s ratio for integration points

1

01 (1 2 )(1 )

2

nn

Update the damage threshold for integration points

n nκ ε

Update the stress for integration points


7. End integration point loop


8. Compute the macro strain as the volume average mesoscopic strain field

1

n Ω

ΩΩ

RVERVE

ε ε u d

9. Compute the total or macro stress as the average of mesoscopic stress field

1

n Ω

ΩΩ

RVE

m

RVE

σ σ d

10. Return to Main Program

4.5 Numerical Example

In order to better understand the behaviour of the transient-gradient model and its use for real

datasets from experiments we first consider two RVEs illustrated in Figure 4.2. The

dimensions of both RVEs are considered to be 240 mm length (L), 60 mm height (H) and 110

mm width (W), see Figure 4.6. RVE-1 considers an entire block with dimensions of 230 mm

length and 50 mm height surrounded by 5 mm half mortar joint on all sides. RVE-2 considers

two half blocks with dimensions of 115 mm length and 50 mm height and two half mortar

bed joints of 5 mm and full head joint of 10 mm in the middle. Figure 4.6 illustrates how

these two RVEs were considered. Note that the overall dimensions of both RVEs are the

same.

This example is consisting of two-dimensional plane stress finite elements for both the mortar

and the unit. Reduced integration eight noded plane stress elements (CPS8R ABAQUS

documentation (2011)) were used for the entire RVEs. Choice of these elements will assist in

the solution of the nonlocal boundary value problem. Considering 8-noded elements, the

deformation domain will be approximated as a second order polynomial. Then, in order to

avoid stress oscillations explained by Peerlings (1999), a linear domain (4-noded elements)

should be considered for discretisation of the nonlocal boundary value problem. CPS8R

elements which were used for the solution of equilibrium equations, have four integration

points. These Integration points are used for the solution of the nonlocal equations.

4.5.1 Stress-Strain Behaviour of Individual Constituents

The geometry of the two RVEs and periodic boundary conditions from Equations (4.9) were

created through ABAQUS /6.11 CAE and PYTHON scripting. Brick with 16000E MPa ,


0 0.2 , strength ratio 10k and mortar with 4000E MPa , 0 0.18 , strength ratio

10k have been considered for this example. All mechanical properties for both the mortar

and the brick are summarised in Table 4.2. Calculations are performed using the procedure

explained in the previous section. Uniaxial tension and compression tests have been

performed on both RVE in directions parallel and perpendicular to bed joints. Loading was

applied as prescribed displacement.

Figure 4.6 Dimensions considered for the RVE

Table 4.2 Mechanical properties of the tested element

Material Properties

Brick Mortar

E (MPa) 16000 4000

0 0.2 0.18

k 10 10

α 1.0 1.0

β 1000 1000

i 0.0001 0.0001

c 0.00017 0.00018

c 0.5 0.5


-0.25 -0.25

The stress-strain behaviour of each constituent (Mortar and brick) under uniaxial tension and

compression loading, considering the mechanical properties in Table 4.2, are illustrated in

Figures 4.7 and 4.8.

Figure 4.7 Stress-strain behaviour of the RVE constituents (brick and mortar) under uniaxial

compression

Figure 4.8 Stress-strain behaviour of the RVE constituents (brick and mortar) under uniaxial

tension


From Figures 4.7 and 4.8 the strength of the tested brick in compression and tension are

considered to be 16.5 MPa and 1.7 MPa, respectively. The strength of mortar in tension and

compression have been considered 0.42 MPa and 4.4 MPa, respectively.

Next section delves into the numerical analysis of two RVEs with the two above mentioned

materials as their constituents.

4.5.2 Numerical Analysis of the RVEs

Typical discretisations used in the finite element computations of the RVEs are shown in

Figure 4.9; they consisted of 728 plane stress reduced integration eight noded elements.

Figure 4.9 Discretisation of individual RVEs

Each RVE was analysed under a displacement control uniaxial tension and compression,

parallel and perpendicular to bed joint and shear loading. Figures 4.10 to 4.12 illustrated the

loading configurations of RVE-1 under uniaxial tension perpendicular to bed joint, uniaxial

tension parallel to bed joint and pure shear. RVE-2 will be configured in the same manner.


Figure 4.10 Loading and boundary conditions of RVE-1 under uniaxial tension perpendicular

to bed joint

Figure 4.11 Loading and boundary conditions of RVE-1 under uniaxial tension parallel to

bed joint

Figure 4.12 Loading and boundary conditions of RVE-1 under pure shear


Both RVEs are loaded according to the configurations illustrated in Figures 4.10 to 4.12 and

their stress-strain behaviour is demonstrated in Figures 4.13 to 4.17. The transient-gradient

parameters used in this analysis have been summarised in Table 4.3.

Table 4.3 Transient-gradient properties of the constituents

Transient-gradient Parameters

Brick Mortar

cl (mm) 5 5

n 1 1

0.001 0.001

The stress-strain behaviour of both RVEs under uniaxial compression perpendicular to bed

joint has been demonstrated in Figure 4.13. It can be seen that as expected both RVEs predict

very close behaviour for the masonry system. Figure 4.14, shows the results from a test with

uniaxial tension loading perpendicular to bed joint. Similarly to compression, both RVEs

predict similar results for tensile loading perpendicular to bed joint.

Figure 4.13 Stress-strain behaviour of both RVEs under uniaxial compression perpendicular

to bed joint


Figure 4.14 Stress-strain behaviour of both RVEs under uniaxial tension perpendicular to

bed joint

Figures 4.15 and 4.16 show the stress-strain behaviour of the RVEs for compression and

tension parallel to bed joint, respectively. Once again, both RVEs produce similar results.

Figure 4.15 Stress-strain behaviour of both RVEs under uniaxial compression parallel to bed

joint


Figure 4.16 Stress-strain behaviour of both RVEs under uniaxial tension parallel to bed joint

Figure 4.17 shows the stress-strain behaviour of the RVEs under pure shear. From the results

demonstrated in Figures 4.13 to 4.17, it can be noted that the effect of different RVE types to

the predicted response of the masonry is negligible and therefore, either RVE could be used –

it was decided to use the RVE-1 in Figure 4.9 in the future analysis in this thesis.

Figure 4.17 Stress-strain behaviour of both RVEs under pure shear



In order to verify the performance of the present transient-gradient nonlocal model, the effect

of various nonlocal parameters to the overall response of the RVE (using RVE-1 based on

conclusion in subsection 4.5.2) was examined.

There are three nonlocal parameters (i.e. cl , n and ) that affect the behaviour of the

model. Influence of parameters n and , on the nonlocal model have been investigated by

Geers, et al.(1999) and Saroukhani, et al. (2013) previously and it was shown that these two

parameters have minimal effect on the performance of the RVE. Therefore, transient-gradient

evolution law (Equation 4.7) was used in its linear form for the purpose of sensitivity study in

this thesis. In this section, a parametric study on the influence of length scale parameter cl

subjected to compression both parallel and perpendicular to bed joint have been studied. The

parameters employed in this parametric study are summarised in Table 4.2 and 4.3.

4.6.1 Effect of the Non-Local Length Scale Parameter c

A parametric study is conducted in this section on the effect of nonlocal length scale

parameter c (𝑚𝑚2) on the stress-strain behaviour of the RVE under uniaxial compression

perpendicular to bed joints. 12 different length scales have been considered for Equation (4.7)

to show the influence of this parameter on the behaviour of the RVE.

Figure 4.18 Influence of different length scale parameters c (𝒎𝒎𝟐) on stress-strain

behaviour of the RVE under uniaxial compression perpendicular to bed joint


Figure 4.18 illustrates the effect of various length scales on the stress-strain behaviour of the

RVE under uniaxial compression perpendicular to bed joint. Comparison between the local

behaviour of the model and transient-gradient model with different length scales can be seen

in this Figure. A highly brittle response is obtained for the local model and the transient-

gradient model with length scale parameters less than 2.0 𝑚𝑚2 which is zoomed in and

illustrated in box 1 of Figure 4.18. By increasing the length scale parameter the response of

the model has improved gradually but the model is still localised at final stages as illustrated

in box 2 of Figure 4.18. Beyond the length scale parameter of 4.5 𝑚𝑚2 the influence of the

parameter is minimal and the localisation issue is resolved as shown in box 3 of Figure 4.18.

Figure 4.19 Strength of the RVE in terms of the nonlocal length scale parameter c

The length scale parameter controls the maximum width of the damage zone and localisation

zone in the RVE and its value must be in accordance with the physical nature of the material.

This parameter is commonly considered between 1.0 and 10.0 𝑚𝑚2 and should be selected

through calibration using experimental results. Figure 4.19 further supports the previous

argument by illustrating the dependence of the strength of the prism to the length scale

parameter for various length scales. The dashed line shows the tangential line in which the

predicted strengths are almost the same. In the rest of this study, the length scale parameter is

considered to be 5.0 𝑚𝑚2.

To further explain the effect of the length scale parameter on the behaviour of the RVE, stress

distribution on the interface of the two constituents for two different length scale parameters

is shown in Figures 4.20 and 4.21. Figure 4.20 shows the stress distribution of the RVE for


length scale parameter c = 1.0 and Figure 4.21 shows this distribution for length scale

parameter c = 5.0.

Figure 4.20 Stress distribution at interface between mortar and brick for c = 1.0

Figure 4.21 Stress distribution at interface between mortar and brick for c = 5.0


In these Figures stress distributions for three identical loading steps at the bed joint interface

have been plotted. The stress localisation issue is evident in Figure 4.20. As can be seen in

this Figure, in the third load step, when localisation occurs, the stress starts to jump at the

localised areas. By comparing Figure 4.20 to Figure 4.21, it can be observed that this issue

has been resolved and the stress has been continually distributed by the use of a larger length

scale parameter (c = 5.0).


A transient-gradient enhanced model published recently by Saroukhani, et al. (2013) for

concrete has been adapted for masonry in this thesis. The adapted model is computationally

more efficient than the original model by Geers et al. (1998) because the enhanced model

eliminates the need for an extra set of degrees of freedom. The nonlocal parameter in the

adapted model has been calibrated specifically for masonry application and a length scale of

5.0 𝑚𝑚2 was found to be sufficient.

Two types of RVEs have been introduced and tested to check their sensitivity to the predicted

results of masonry response under uniaxial compression and tension both perpendicular and

parallel to the bed joint as well as in-plane shear. The implementation of the transient

gradient nonlocal method for homogenisation has been discussed. It has been found that the

effect of the geometry of RVE has had minimal effect to the overall stress-strain behaviour of

the masonry (represented by the RVE). Therefore all further analyses have been carried out

with one of the RVEs (RVE-1). A parametric study has been conducted on the effect of

length scale parameter and the effect of stress localisation on the stress-strain behaviour and

the failure of the RVE.

In the next chapter, application of this transient-gradient model to the uniaxial and biaxial

behaviour prediction of masonry is presented and the predicted uniaxial and biaxial results

are compared to an experimental dataset reported in the literature.

93 Application of the Transient Enhanced RVE to Brick Masonry under Biaxial Loading |

CHAPTER 5

Application of the Transient Enhanced RVE to

Brick Masonry under Biaxial Loading

5.1 Introduction

In chapters 3 and 4 the formulation of a continuum damage constitutive model and its

implementation into ABAQUS/ 6.11 standard FE package through subroutine MATLAB

program and scripting have been presented. In this chapter the damage model enhanced with

transient-gradient nonlocal approach is devoted to the analysis of the orthotropic response of

burnt clay brick masonry under uniaxial and biaxial loadings.

The structural behaviour of masonry is affected by the properties of its principal constituents,

i.e. units and mortar. The bond between the unit and the mortar also significantly affects the

masonry behaviour. Strength of masonry under different loadings is affected by the principal

stresses and their orientation to bed joint.

The behaviour of masonry systems, their strength and their failure mode varies depending on

the orientation of loading to the bed joint direction. Due to its anisotropic nature, strength

envelope and behaviour of masonry is significantly affected by this orientation.

Consequently, the biaxial failure envelope is described in terms of both principal stresses 𝜎1

and 𝜎2 and the orientation of the loading axes with respect to masonry bed joints 𝜃.

Based on the range of available products of unit and mortar, their shape and the type of

assemblage, different masonry systems are likely to exhibit strength envelopes of varied

shapes and sizes. In section 2.1.4 of the thesis available failure surfaces in the literature have

been reviewed. Obtaining failure envelopes through experimental work for each type of

masonry is significantly time consuming and expensive. Therefore, a damage constitutive

model was developed in this thesis to describe the constitutive behaviour of different types of


masonry and is presented in Chapters 7, 8 and 9. In the following sections the capability of

the model in representing a set of experimental data is reported.

5.2 Problem Definition

Only a few studies were performed to obtain the full experimental stress-strain and failure

envelope for masonry as summarised in section 2.2 of this thesis. In this chapter, numerical

simulation of a series of tests by Dhanasekar (1985) has been performed. Dhanasekar (1985)

carried out a series of biaxial compression-compression and tension-compression tests to

investigate the failure envelope and strength characteristics of masonry panels subjected to

in-plane monotonic loading. A total of 186 wallet tests were conducted under five different

loading orientations, i.e. 0°, 22.5°, 45°, 67.5° and 90°. The biaxial load-control experiments

were conducted using testing rig shown in Figure 5.1. A set of brush platens were used for

load transfer to the panel, in order to lessen the restraining effect of the loading caps and to

ensure uniform distribution of the principal stresses applied in the vertical and horizontal

directions. The specimens were fabricated with five different bed joint angles (0°, 22.5°, 45°,

67.5° and 90°) so that a comprehensive failure surface could be developed.

Figure 5.1 Dhanasekar’s biaxial load-control test setup (Dhanasekar, 1985)


The failure surface was presented in terms of the two normal stresses and the inplane shear

stresses in the bed joints. Average compressive strength of brick and mortar was reported as

15.41 MPa and 5.08 MPa, respectively. Average modulus of elasticity of bricks was reported

as 15,650 MPa. No data were available on mortar modulus of elasticity.

5.2.1 Tensile and Compressive Behaviour of Individual

Constituents (Brick and Mortar)

In order to better simulate the behaviour of the tested masonry experiment, we first have to

calibrate the behaviour of its individual constituents. The best way to calibrate our model for

both material (mortar and brick) is through their individual compressive and tensile stress-

strain behaviour. However, due to lack of this information, the tested parameters such as

Young’s modulus, Poisson’s ratio and compressive strength were considered exactly as that

of the experiment. All other properties were chosen based on the parametric studies

conducted in chapters 3 and 4, in a manner that the material shows a realistic behaviour. All

mechanical properties, used in this model for both the brick and the mortar are listed in Table

5.1.

These individual material properties have been produced through a two-dimensional finite

element test on a single eight noded element with size of unity shown in Figure 5.3. The same

procedure explained in section 3.4 is adopted here.

Figure 5.2 The 8-noded element used for the numerical analysis


Table 5.1 Mechanical properties of the tested materials

Material Properties

Brick Mortar

E (MPa) 15650 4000

0 0.2 0.18

k 10 10

α 1.0 1.0

β 1000 1000

i 0.00009 0.00012

c 0.000162 0.00021

c 0.4 0.4

-0.25 -0.25

The stress-strain behaviour of mortar under uniaxial tension and compression loading, are

illustrated in Figures 5.3 and 5.4, respectively.

Figure 5.3 Stress-strain behaviour of mortar element under uniaxial tension


Figure 5.4 Stress-strain behaviour of mortar element under uniaxial compression

The stress-strain behaviour of brick under uniaxial tension and compression loading for one

element, are illustrated in Figures 5.5 and 5.6, respectively.

Figure 5.5 Stress-strain behaviour of brick element under uniaxial tension


Figure 5.6 Stress-strain behaviour of brick element under uniaxial compression

5.2.2 RVE Geometry, Discretisation and Loading Configuration

As was reported in section 4.5 of this thesis RVE-1 was selected for all further studies. The

RVE with its discretisation is shown in Figure 5.7. This RVE consists of 728 plane stress

reduced integration eight noded elements (CPS8R elements). Choice of these elements will

assist in the solution of the nonlocal boundary value problem as explained in section 4.5.

Figure 5.7 A typical discretisation of RVE

In order to simulate the experimental results on conventional masonry panels (half scale brick

dimensions of 55 mm width, 25 mm height and 115 mm length and mortar joint thickness of

5 mm) obtained by Dhanasekar (1985), a RVE with dimensions as shown in Figure 5.8 has


been considered. Note that a two-dimensional finite element analysis has been performed on

the RVE and the represented width only affected the thickness contributed to each element.

Figure 5.8 Typical dimensions of the modelled RVE

Table 5.1 illustrates parameters used in the analysis of the RVE. The panels were loaded

proportionally (that is, the ratio of the two principal stresses was kept unchanged throughout

the test) in the principal stress directions 1

and 2 along different orientations θ with

respect to the material axes. Figure 5.9 shows the loading configurations of the system.

Figure 5.9 Loading configuration of the panel


5.2.3 Stress-Strain Behaviour of the RVE under Different

Loading Configurations

In order to simulate the behaviour of the experiments conducted by Dhanasekar (1985), the

RVE is tested under a load control test. It was loaded incrementally based on the load ratio

considered for each test. Figure 5.10 illustrates the loading configuration and boundary

conditions used in this analysis. Stress-strain behaviour of the RVE under different biaxial

configurations has been produced using this procedure. The results are presented with the

experimental results from Dhanasekar (1985).

Table 5.2 Transient-gradient properties of RVE’s constituent


Brick Mortar

cl (mm) 5 5

n 1 1

0.0009 0.0012

Figure 5.10 Loading configuration and boundary conditions of the RVE

The RVE is loaded based on the configurations illustrated in Figure 5.9. When θ = 0°, 𝜎1 is

parallel to bed joint and when θ = 90°, 𝜎1 is perpendicular to bed joint. Load ratio is always


considered as 𝜎1/𝜎2. The transient-gradient parameters used in this analysis have been

summarised in Table 5.2.

Figure 5.11 Stress-Strain behaviour of the RVE under uniaxial compression parallel to bed

joint

Figure 5.11 shows the comparison of the stress-strain behaviour predicted by the model

against the experimental data for θ = 0° and 𝜎1/𝜎2 = ∞ which is the case of uniaxial

compression parallel to bed joint. Comparison between the model’s stress-strain prediction

and experimental data for θ = 0° under 𝜎1/𝜎2 = 1, 𝜎1/𝜎2 = 2 and 𝜎1/𝜎2 = 4 have been

illustrated in Figures 5.12, 5.13 and 5.14, respectively.



1 2/ 1


1 2/ 2


In all of these stress-strain behaviours, an over-estimation of the stiffness of the masonry can

be observed. For these types of loadings the presented RVE seems to be incapable of

simulating the beginning stages of the masonry behaviour in a realistic manner. In other

words, the model developed was not a true representation of the masonry.


1 2/ 4

Figure 5.15 illustrates the comparison of the stress-strain behaviour predicted by the model

against the experimental data for θ = 90° and 𝜎1/𝜎2 = ∞ which is the case of uniaxial

compression perpendicular to bed joint. Comparison between the model’s stress-strain

prediction and experimental data for θ = 0° under 𝜎1/𝜎2 = 2 and 𝜎1/𝜎2 = 4 are shown in

Figures 5.16 and 5.17, respectively.


Figure 5.15 Stress-Strain behaviour of the RVE under uniaxial compression perpendicular to

bed joint


1 2/ 2


Similar to the previous loading cases it can be observed that this RVE consistently over-

estimates the stiffness of masonry in the initial loading stages. The RVE seems to be

incapable of capturing the interfacial delamination between the two materials. The

assumption of perfect bond between mortar and brick that is inherent in the formulation of

the RVE could have contributed to this higher stiffness and poor stress-strain response

prediction. It is essential to allow for the damages at interfaces to realistically predict the

behaviour of masonry and to get better match with the experimental stress-strain curves.


1 2/ 4

5.3 Observations

The numerical analysis of the model illustrated that work is needed for further development

of the RVE. Although the model is able to predict fairly good failure strengths compared to

the experiments, the RVE failed to capture the interface failure of the two materials as

experimentally observed. It also did not predict the stress-strain responses of masonry

appropriately and its prediction is much stiffer than the experimental predictions and hence

far from satisfactory. This issue requires consideration and improvement to the assumption of

perfect bond between the unit and the mortar implied in the formulation of the RVE. This

assumption appears to be too restrictive for the RVE considered here.



The constitutive model developed in chapters 3 and 4 of the thesis and implemented in a RVE

has been used to simulate the experimental data from Dhanasekar (1985). The RVE was

analysed under different uniaxial and biaxial loading states. It has been observed that

assumption of a perfect bond between the two constituents of the RVE i.e. mortar and brick is

too restrictive and has made the current RVE over-stiff and failed to capture delamination

mode of failure of the interfaces between units and mortar.

In the ensuing chapter, an innovative method of allowing for interfacial damages will be

introduced to resolve the issue with the current RVE.

107 Enrichment of the RVE with Interfacial Transition Zone (ITZ) |

CHAPTER 6

Enrichment of the RVE with Interfacial

Transition Zone (ITZ)

6.1 Introduction

In chapter 5 application of the continuum damage constitutive model formulated in chapters 3

and 4 was demonstrated for a series of compression-compression tests. The damage

constitutive model represented both the mortar and the unit. The ability of the model to

predict the stress-strain behaviour and strength of traditional clay brick masonry tests was

examined through a two-phase RVE consisting of brick and mortar with a perfect bond along

their interfaces in Chapter 5. The result has shown that the assumption of a perfect bond

between the two constituents is too restrictive leading to stiffer formulation and

unconservative prediction of strength and stiffness of the masonry. The formulated RVE

therefore has required further development in order to capture delamination of the interfaces.

In this chapter, an innovative method is introduced to resolve the stiffer formulation issue of

the RVE without highly nonlinear options of contact stick-slip modelling between the units

and the mortar layers. For this purpose, concept of an interfacial transition zone (ITZ),

otherwise known as interphase modelling in concretes (Garboczi & Bentz, 1997; Li, et al.,

1999; Nadeau, 2003; Stroeven, et al., 2004; Hatami-Marbini & Shodja, 2008; Mihai &

Jefferson, 2011; Xu & Chen, 2013) has been introduced to implicitly represent damages at

the interfaces of the two constituents. This chapter presents the re-formulation of the RVE

incorporating the ITZ concept. A parametric study of the key parameters of the ITZ model is

also presented. Finally, application of the reformulated RVE to some real experimental

datasets is shown to predict the results appropriately and conservatively. It is concluded the

reformulated RVE enriched with ITZ is capable of predicting the behaviour of masonry under

complex states of stress.


6.2 Interfacial Transition Zone (ITZ) Concept

The inspiration for the concept of Interfacial Transition Zone (ITZ) in this work has been

derived from the successful implementation of layered interphase model usage to simulate

interface damages in composites including concretes in the literature (Ramesh, et al., 1996;

Garboczi & Bentz, 1997; Lutz, et al., 1997; Yang, 1998; Garboczi & Berryman, 2001;

Stroeven, et al., 2004; Duan, et al., 2007; Hirschberger, et al., 2009; Nguyen, et al., 2010;

Klusemann & Svendsen, 2012). This method has been extensively used throughout the

literature for concrete (Li, et al., 1999; Nadeau, 2003; Zheng, et al., 2012; Xiao, et al., 2013;

Zhou, et al., 2014), cementitious materials (Mihai & Jefferson, 2011; Duplan, et al., 2014),

fiber-reinforced composites (Sevostianov, et al., 2012) and other quasi-brittle materials

(Garboczi & Berryman, 2001; Hatami-Marbini & Shodja, 2008; Xu & Chen, 2013). In all the

above literatures, the interphase was considered as a transitional layer between the aggregate

inclusions and the cement matrix as shown in Figure 6.1 or nano-particles within the RVE in

a micro- or nano-scale. In this thesis, motivated by this idea of the transition zone, we

introduce an Interfacial Transition Zone (ITZ) between the mortar and the unit in meso-scale

(RVE). It should be noted that the origin of appearance of the ITZ is different than the

classical concept in concrete (difference in water content close to aggregates) and only the

mechanical consequences of a weak transition zone are modelled here. Utilising this concept

eliminates the need for introduction of an interface element/ nonlinear contact concept

between the two constituents which would add to the complexity of the model. In the

following sections, the RVE formulated in Chapter 4 is reformulated enriched with ITZ and

applied to the masonry under complex stress states.

Figure 6.1 Representation of the Interfacial Transition Zone (ITZ) for concrete


6.2.1 Enrichment of the RVE with an Interfacial Transition

Zone (ITZ)

The concept of ITZ enriched in an RVE is shown in Figure 6.2. It can be seen in this figure

that a transitional layer has been introduced to the RVE between the two constituents i.e.

mortar and unit. These set of new layers will have different mechanical properties than that of

the mortar and the unit.

Figure 6.2 Representation of the masonry RVE utilised with a 5-layered Interfacial



Here it is assumed that the ITZ commences from the edge of the unit and progresses through

the thickness of mortar across the whole width (not shown in the figure) of the mortar. Where

weak units are used (e.g., unburnt mud bricks), the ITZ layers could also be assumed to

progress into the unit. The mechanical properties and stiffness of this layer gradually

changes, from the weakest (highly damaged state) at the interface between the mortar and

unit towards the strongest (undamaged state) at the point of connection to the mortar layer.

The maximum strength of this layer will be equal to the strength of mortar. In Figure 6.2 a 5-

layer Interfacial Transition Zone (ITZ) is shown for simple illustration.

The RVE is utilised with the Interfacial Transition Zone (ITZ) model in order for it to be able

to capture the inelastic sliding at the interface of the two materials. The capability of the

model to accurately predict the stress-strain behaviour of some real experiments have been

examined and reported in the following sections of this chapter.

6.2.2 Elastic Properties of the Layers of the Interfacial


The stiffness of each layer of the Interfacial Transition Zone (ITZ) is different to each other

and that of the undamaged mortar (the ITZ layers are enriched within mortar layer only in

this formulation). The elastic properties of each layer of the ITZ vary, based on the type and

strength of the two materials that create the bond and the thickness of the layer. Numerous

interface effects, such as friction and delamination further complicate the prediction of these

properties. Depending on the materials considered and the thickness and the number of layers

in the ITZ, different authors have considered different formulations for the elastic properties

of this zone in concretes (Garboczi & Berryman, 2001; Duan, et al., 2007; Mihai & Jefferson,

2011; Klusemann & Svendsen, 2012; Grondin & Matallah, 2014). Here we introduce a power

law for the initial Young’s modulus of each layer of the ITZ as in Equation (6.1).

0

(2 1)(t ) 2( ) ( )

2Ej ITZ ITZ

M

j

n T NTE n E

Nt

(6.1)

In which n is the number of the specific layer from the interface between the two materials

(edge of the unit in this case) and N is the total number of layers within the ITZ. 0( )E n is

the initial Young’s modulus of that specific layer. ME is the Young’s modulus of the mortar


layer. The thickness of the mortar joint and the thickness of the total ITZ have been

represented by jt and

ITZT , respectively. E is a parameters controlling the stiffness

degradation. The Poisson’s ratio of each layer of the Interfacial Transition Zone (ITZ) has

been defined in the same manner as follows

0

(2 1)(t ) 2( ) ( )

2

j ITZ ITZ

M

j

n T NTn

Nt

(6.2)

In which 0( )n is the initial Poisson’s ratio of the n

th layer of the ITZ.

M is the Poisson’s

ratio of the mortar layer and is a parameters controlling the decrease of Poisson’s ratio.

For a better description of some of these parameters the reader can refer to Figure 6.3.

Figure 6.3 Representation of the masonry RVE thicknesses with an n-layered Interfacial



With a view to verifying the performance of the present RVE, enhanced with ITZ, the effect

of the transition stiffness parameters on the overall response of the system has been

examined. This examination is through an analysis of the RVE with the same geometry as the

one studied in section 4.6 of this thesis. The influences of thickness of the ITZ and the

number of layers have been considered in this parametric study.


The mechanical properties used in this analysis for brick and mortar elements are summarised

in Table 6.1. Transient-gradient properties of the mortar and the brick are also shown in

Table 6.2. This simple parametric study illustrates the response of the ITZ enhanced RVE for

the uniaxial compression tests. The mechanical properties of each layer within the ITZ are

shown separately for each test.

Table 6.1 Properties of the constituent materials

Material Properties

Unit Mortar

E (MPa) 15650 4000

0 0.2 0.18

k 10 10

α 1.0 1.0

β 1000 1000

i 0.000009 0.000012

c 0.000162 0.00021

c 0.4 0.4

-0.25 -0.25



Unit Mortar

cl (mm) 5 5

n 1 1

0.00009 0.00012

The dimensions of the RVE are 240 mm length (L), 60 mm height (H) and 110 mm width

(W). Two-dimensional finite element analysis was carried out using reduced integration eight


noded plane stress elements (CPS8R ABAQUS documentation (2011)) to discretise the entire

RVE. The finite element discretisation of the RVE is illustrated in Figure 6.4.

Figure 6.4 Finite Element discretisation of the RVE

6.3.1 Effect of Thickness of the Interfacial Transition Zone

(ITZ)

In this section, the influence of the thickness of the Interfacial Transition Zone (ITZ) to the

stress-strain behaviour of the masonry is examined. For this purpose, five ITZ thicknesses

were considered. The thickness of each layer within the ITZ was kept constant (equal to 0.25

mm). The parameters of the ITZ for each test are presented in Table 6.3. A linear initial

Young’s modulus transition and a constant initial Poisson’s ratio have been considered for

this part of the study.


Table 6.3 The Interfacial Transition Zone parameters for each test

Interfacial Transition Zone (ITZ) parameters

Test # ITZT (mm)

jt (mm) N E

1 1.25 2.5 5 1.0 0.0

2 1.0 2.5 4 1.0 0.0

3 0.75 2.5 3 1.0 0.0

4 0.5 2.5 2 1.0 0.0

5 0.25 2.5 1 1.0 0.0

Table 6.4 Young’s modulus and Poisson’s ratio of each individual layer

Mechanical Properties

Test # ME (MPa)

0(1)E

(MPa) 0(2)E

(MPa) 0(3)E

(MPa) 0(4)E

(MPa) 0(5)E

(MPa) M

0( )n

1 4000 2200 2600 3000 3400 3800 0.18 0.18

2 4000 1900 2500 3100 3700 0.18 0.18

3 4000 1667 2600 3533 0.18 0.18

4 4000 1600 3200 0.18 0.18

5 4000 1760 0.18 0.18

The change in material properties of the ITZ layers for each test is presented in Table 6.4.

The material properties of the undamaged mortar and brick are kept the same as per original

analysis (see Table 5.1). Figure 6.5 illustrates the influence of the ITZ thickness to the stress-

strain behaviour of the masonry under uniaxial compression perpendicular to bed joint. It can

be observed that with the increase in the thickness of the ITZ, the response of masonry has

softened, which is consistent to the logical expectation.

Figure 6.6 shows the effect of ITZ thickness on the stress-strain behaviour of the masonry

under uniaxial compression parallel to bed joint. Five thicknesses of ITZ have been tested

and similar observations can be seen on the effect of ITZ thickness for the case of

compression perpendicular to bed joint shown in Figure 6.5. Again, lower the thickness of the

ITZ, higher the stiffness and larger the strength of masonry. Thickness of ITZ larger than


1.5mm has no significant effect on the deformation and strength characteristics of masonry

for both cases of uniaxial compression parallel and perpendicular to bed joint.




compression parallel to bed joint


Influence of thickness of the ITZ layer on the stress-strain behaviour of the RVE under

tensile loading perpendicular and parallel to bed joints is illustrated in Figures 6.7 and 6.8,

respectively.


tension perpendicular to bed joint


tension parallel to bed joint


Figure 6.9 shows the stress strain behaviour of the RVE under pure shear loading for different

ITZ thicknesses.

Figure 6.9 Influence of ITZ thickness on stress-strain behaviour of the RVE under pure shear

loading

Similarly to uniaxial compression cases, in uniaxial tension and shear case, lower the

thickness of the ITZ, higher the stiffness and failure strength of masonry. Quite similar

response can also be observed in these loading cases for ITZ thicknesses between 0.75 to

1.25 mm, which shows a damage layer thickness of more than 0.75mm (15% thickness of the

mortar layer) does not affect the behaviour of masonry. From the parametric study conducted

on the ITZ thickness, it can be concluded that a thickness between 0.75 to 1.25 mm is

desirable for the ITZ.

6.3.2 Effect of the Interfacial Transition Zone (ITZ) Parameter

𝝀𝑬

Influence of the ITZ parameter 𝜆𝐸 (varied from 0.5 to 4.0) to the behaviour of masonry is

investigated by assuming all other parameters constant as shown in Table 6.5 and keeping the


thickness of the RVE as 1.25 mm (5 layers in the ITZ). A constant initial Poisson’s ratio is

considered for this part of the study.

The change in material properties of the ITZ layers for each test is as shown in Table 6.6.

Figure 6.10 illustrates the influence of the ITZ parameter E on the stress-strain behaviour of

the RVE under uniaxial compression perpendicular to bed joint for five different values.

Table 6.5 The Interfacial Transition Zone parameters for each test

Interfacial Transition Zone (ITZ) parameters

Test # ITZT (mm)

jt (mm) N E

1 1.25 2.5 5 0.5 0.0

2 1.25 2.5 5 1.0 0.0

3 1.25 2.5 5 2.0 0.0

4 1.25 2.5 5 3.0 0.0

5 1.25 2.5 5 4.0 0.0


uniaxial compression perpendicular to bed joint


Figure 6.11 shows the effect of ITZ parameter E on stress-strain behaviour of the model

under uniaxial compression parallel to bed joint for five different values. It can be observed

in these figures that the higher the value of E , the weaker the RVE will become. As can be

seen in Tables 6.5 and 6.6, the higher the ITZ parameter E , the bigger the range of stiffness

change within the ITZ.

Table 6.6 Young’s modulus and Poisson’s ratio of each individual layer


Test # ME (MPa)

0(1)E

(MPa) 0(2)E

(MPa) 0(3)E

(MPa) 0(4)E

(MPa) 0(5)E

(MPa) M

0( )n

1 4000 2966 3225 3464 3688 3899 0.18 0.18

2 4000 2200 2600 3000 3400 3800 0.18 0.18

3 4000 1210 1690 2250 2890 3610 0.18 0.18

4 4000 665 1098 1687 2456 3429 0.18 0.18

5 4000 366 714 1266 2088 3258 0.18 0.18


uniaxial compression parallel to bed joint


Influence of the ITZ parameter E on the stress-strain behaviour of the RVE under tensile

loading perpendicular and parallel to bed joints is illustrated in Figures 6.12 and 6.13,

respectively.


uniaxial tension perpendicular to bed joint


uniaxial tension parallel to bed joint


Figure 6.14 shows the stress strain behaviour of the RVE under pure shear loading for

different ITZ parameters E .

Figure 6.14 Influence of ITZ parameter E on stress-strain behaviour of masonry under pure

shear loading

Similar observations can be made from Figures 6.10 to 6.14. As the parameter E increase,

the range of stiffness change within the ITZ increases and the overall stiffness of the RVE

decreases.

In the next section, the proposed Interfacial Transition Zone (ITZ) enhanced RVE is validated

for a number of uniaxial experimental datasets available in the literature for conventional

masonry. In the first part of this section, the prediction of the proposed ITZ enhanced RVE is

compared with the results of the RVE studied in chapter 5 of the thesis.


6.4 Validation of the Interfacial Transition Zone (ITZ) Enriched

CDM Model

The predictions of the ITZ enriched CDM model has been validated using several

experimental datasets reported in the literature and presented in this section. The superiority

of the ITZ enhanced RVE over the basic RVE formulated in Chapter 4 is also shown in this

section. Only the uniaxial test datasets were considered for simplicity; biaxial test cases are

reported in Chapter 7 for conventional masonry.

6.4.1 Validation of Uniaxial Tests Conducted by Dhanasekar

(1985)

In order to simulate the compressive test results on solid brick conventional masonry (half

scale brick dimensions of 55 mm width, 25 mm height and 115 mm length and mortar joint

thickness of 5 mm) obtained by Dhanasekar (1985), a RVE with dimensions as shown in

Figure 6.15 has been considered. The finite element discretisation of the RVE has been

shown in Figure 6.4. The material properties used for mortar and brick can be found in Table

6.1. The transient-gradient properties of brick and mortar can be found in Table 6.2. The

initial Poisson’s ratio for all layers has been considered to be 0.18. Table 6.7 illustrates the

mechanical properties of each individual ITZ layer.

Table 6.7 Young’s modulus and Poisson’s ratio of each ITZ layer


0(1)E

(MPa) 0(2)E

(MPa) 0(3)E

(MPa) 0(4)E

(MPa) 0(5)E

(MPa) 0( )n

2200 2600 3000 3400 3800 0.18

Within the RVE a 1.25 mm ITZ has been considered as shown in Figure 6.15. Each ITZ layer

has a thickness of 0.25 mm and the ITZ parameter is considered to be 0.5E .



Dhanasekar (1985)

The stress-strain behaviour of the RVE presented in Chapter 4 and the new ITZ enriched

RVE are compared with the uniaxial experimental results from Dhanasekar (1985) in this

section. Figure 6.16 shows the predicted stress-strain behaviour by the two RVEs and the

experimental datasets for θ = 0° and 𝜎1/𝜎2 = ∞ which is the case of uniaxial compression

parallel to bed joint. Figure 6.17 presents the stress-strain behaviour predicted by the two

RVEs and the experimental datasets for θ = 90° and 𝜎1/𝜎2 = ∞ (uniaxial compression

perpendicular to bed joint). Globally good agreement is found for the model and experimental

data. The model slightly under-estimates the strength obtained by the experiment, however,

comparing to the previous RVE, it favourably predicts the overall stress-strain behaviour of

the masonry panels under uniaxial compression.


Figure 6.16 Comparison of the stress-strain behaviour of the RVE under uniaxial

compression parallel to bed joint

Figure 6.17 Comparison of the stress-strain behaviour of the RVEs under uniaxial



The stress-strain behaviour of both RVEs compared with experimental results under uniaxial

tension parallel and perpendicular to bed joints is presented in Figures 6.18 and 6.19,

respectively.

Figure 6.18 Comparison of the stress-strain behaviour of the RVE under uniaxial tension

parallel to bed joint

Figure 6.19 Comparison of the stress-strain behaviour of the RVEs under uniaxial tension

perpendicular to bed joint


It can be observed from the above mentioned tests that enriching the RVE with an ITZ

enhances its stress-strain performance. The over-estimation of stiffness for the initial stages

of the loading process has been resolved with this method. However, it should be noted that a

slight under-estimation of the final strength of the RVE can be seen in all cases, which is

desirable for computational models as the predictions are conservative and can be used in

design.

6.4.2 Validation of Uniaxial Compression Tests Conducted by

Barbosa & Hanai (2009)

Barbosa & Hanai (2009) conducted a series of uniaxial compression tests on hollow block

prisms with four different unit strengths. The dimensions of the tested blocks are shown in

Figure 6.20.

Figure 6.20 Hollow concrete block with dimensions from Barbosa & Hanai (2009) and

Barbosa, et al. (2010)


Figure 6.21 Idealised dimensions of the modelled RVE

Table 6.8 Material properties of the tested constituents


Prism Material E(MPa) ν 𝑓𝑐(MPa) 𝑓𝑡(𝑀𝑃𝑎)

P1 Mortar 9745 0.127 9.4 1.1

Concrete 20595 0.203 22.8 2.2

P2 Mortar 8121 0.134 7.7 0.9

Concrete 17449 0.195 18.6 1.7

P3 Mortar 13195 0.151 15.5 1.8

Concrete 22175 0.204 24.9 2.4

P4 Mortar 16672 0.153 22.2 2.6

Concrete 27104 0.207 36.2 3.1

Barbosa, et al. (2010) has presented the experimental results for four different block

strengths. The behaviour of the prisms has also been numerically simulated using a

micromodelling strategy with plane-strain (PE), plane-stress (PS) and three-dimensional (3D)

elements. In order to simulate the compressive test results (block dimensions of 140 mm


width, 190 mm height and 390 mm length and mortar joint thickness of 10 mm) a RVE with

idealised dimensions as shown in Figure 6.21 has been considered. Table 6.8 summarises the

tested elastic properties of mortar and concrete, considered for each test. The mechanical

properties used in the constitutive model for each test has been summarised in Tables 6.9-12.

Table 6.9 Mechanical properties of the tested prism 1 (P1)

Material Properties

Block Mortar

E (MPa) 20595 9745

0 0.203 0.127

k 10 9

α 1.0 1.0

β 1000 1000

i 0.00011 0.00011

c 0.00017 0.000165

c 0.4 0.4

-0.25 -0.25


Material Properties

Block Mortar

E (MPa) 17449 8121

0 0.195 0.134

k 11 9

α 1.0 1.0

β 1000 1000

i 0.00011 0.00011

c 0.000155 0.000165

c 0.4 0.4

-0.25 -0.25


An Interfacial Transition Zone (ITZ) with 5 layers is considered with 2.5 mm thickness as

shown in Figure 6.22. Each ITZ layer has a thickness of 0.5 mm and the ITZ parameter is

considered to be 0.5E .


Material Properties

Block Mortar

E (MPa) 22175 13195

0 0.204 0.151

k 10 9

α 1.0 1.0

β 1000 1000

i 0.00011 0.00011

c 0.00017 0.000165

c 0.4 0.4

-0.25 -0.25


Material Properties

Block Mortar

E (MPa) 27104 16672

0 0.207 0.153

k 12 10

α 1.0 1.0

β 1000 1000

i 0.00011 0.00011

c 0.00017 0.00025

c 0.4 0.4

-0.25 -0.25



Barbosa & Hanai (2009)

The finite element discretisation of the RVE has been shown in Figure 6.23. The stress-strain

behaviour of masonry under uniaxial compression perpendicular to bed joints is compared

with the experimental data an numerical analysis produced by Barbosa, et al. (2010) shown in

Figures 6.24 to 6.27. Globally good agreement is found for the model and experimental data.

The model slightly under-estimates the strength obtained by the experiment in tests P1 and P2

and over-estimates the strength for test P4. This might be due to the slight difference between

the actual block dimensions and the RVE’s dimensions. The model favourably predicts the

overall stress-strain behaviour of the masonry prisms under uniaxial compression.




against the experimental data and numerical analysis by Barbosa, et al. (2010)






It should also be observed that the trials of Barbosa, et al. (2010) using Plane Strain (PE in

Figures 6.24 to 6.27) and Plane Stress (PS in Figures 6.24 to 6.27) have predicted the

experimental data poorly. Barbosa, et al. (2010) has subsequently concluded that a 3D FE

model is essential for the predicted stress-strain responses. In contrast, in this thesis a 2D

RVE is shown to effectively predict the behaviour of masonry close to the 3D FE model


predictions. The computational superiority of the RVE introduced in the thesis is thus

evident.




In this chapter, an innovative method was introduced to resolve the issue with the simple

RVE used in chapter 5 while keeping the assumption of the perfect bond between the

constituents. An Interfacial Transition Zone (ITZ) was introduced between the two

constituents i.e. mortar and unit. A RVE was developed considering this concept and a

parametric study has been conducted on the key parameters influencing the response of the

RVE. Finally, the ITZ enhanced RVE was validated by simulating the stress-strain behaviour

of some real experimental datasets. It has been found that the transient-gradient constitutive

model, utilized with an ITZ enhanced RVE can predict the uniaxial behaviour of

conventional masonry in a sound manner.

A comparative analysis of experimental and finite element analysis presented by Barbosa, et

al. (2010) has established the computational efficiency of the 2D RVE developed in this

thesis as predictions are shown comparable to the 3D FE results by Barbosa, et al. (2010).

In the next chapter, the application of the model will be investigated through simulation of

biaxial behaviour of experimental data.

134 Application of the ITZ Enriched CDM Model-1: Constitutive Properties of Conventional

Masonry under Biaxial Stresses |

CHAPTER 7

Application of the ITZ Enriched CDM Model

– 1: Constitutive Properties of Conventional

Masonry under Biaxial Stresses

7.1 Introduction

In chapter 6 an interface transition zone (ITZ) enriched continuum damage mechanics (CDM)

model based transient-gradient nonlocal constitutive characteristics was presented and its

capability to predict the behaviour of masonry systems under uniaxial stress state was

demonstrated through comparison with several experimental datasets presented in the

literature. The comparison was quite good and hence the ITZ enriched CDM model was

regarded as validated. Through a sensitivity analysis, an ideal thickness of the ITZ was

determined as 1.25mm. In this section, the ability of the present model to predict the biaxial

failure envelope of conventional masonry is presented; failure envelope of dry stack masonry

is attended to in the ensuing chapter.

7.2 Problem Definition

From a phenomenological perspective, masonry is an anisotropic composite material in spite

of its constituents can be regarded isotropic. This anisotropy is mainly due to the geometrical

arrangements of units and mortar. Considering the anisotropy of masonry, the failure

envelope cannot be shown in terms of principal stresses only. For the in-plane stress state

which is the case of this study, the failure envelope should be represented in terms of all

stress components x , y and xy (plane stress). Here, we consider x to be the bed joints



direction and y to be the head joints direction. The failure envelope can also be represented in

terms of the principal stresses and an angle θ which is the angle between the material axes

and the principal axes which is shown in Figure 7.1.

Figure 7.1 Representation of material and principal axes in masonry

Only a few studies were performed to obtain the full experimental stress-strain and failure

envelope for masonry. These studies have been reviewed in section 2.2 of this thesis. This

Chapter presents an investigation of the numerical application of the proposed model to some

experimental biaxial tests on masonry. The ability of the present constitutive model to predict

the failure envelope of different conventional masonry under in-plane loading is

demonstrated through a comparison with available experimental data. Here, the peak of the

average stress-strain response of the RVE is considered to be the point of failure.

7.3 Masonry RVEs, their Dimensions and Properties

In this chapter, the performance of the transient-gradient constitutive model using an RVE

enhanced with an Interfacial Transition Zone (ITZ), for prediction of biaxial failure envelop

of masonry is examined for two sets of experimental date. The biaxial tension-compression

and compression-compression tests done by Dhanasekar (1985) on solid brick conventional

masonry, and uniaxial compression tests done by Barbosa & Hanai (2009) on hollow block



conventional masonry. To this end, two RVEs with the following dimensions and properties

were considered. Both RVEs are enhanced with a 5-layer ITZ.

7.3.1 RVE Considered to Simulate Biaxial Tests by Dhanasekar

(1985)

In order to simulate the biaxial tests on solid brick conventional masonry (brick dimensions

of 55 mm width, 25 mm height and 115 mm length and mortar joint thickness of 5 mm)

carried out by Dhanasekar (1985), a RVE of dimensions 120 mm long (L), 30 mm high (H)

and 55 mm wide (W), shown in Figure 7.2 was considered. The finite element discretisation

of the RVE is shown in Figure 7.3.

The material properties used for mortar and brick are reported in Table 6.1. The transient-

gradient properties of brick and mortar are contained in Table 6.2. The initial Poisson’s ratio

for all layers was taken as 0.18.

Within the RVE a 1.25 mm Interfacial Transition Zone (ITZ) was considered as shown in

Figure 7.2. Each ITZ layer had a thickness of 0.25 mm and the ITZ parameter considered was

0.5E . The mechanical properties of each ITZ layer are shown in Table 7.1.

Table 7.1 Young’s modulus and Poisson’s ratio of each ITZ layer


0(1)E

(MPa) 0(2)E

(MPa) 0(3)E

(MPa) 0(4)E

(MPa) 0(5)E

(MPa) 0( )n

2200 2600 3000 3400 3800 0.18



Figure 7.2 Typical dimensions of the modelled RVE and its ITZ




7.3.2 RVE for the Simulation of Biaxial Tests on Hollow

Concrete Masonry

Barbosa & Hanai (2009) conducted a series of uniaxial compression tests on hollow block

conventional masonry prisms of gross dimensions of 140 mm wide, 190 mm high and 390

mm long as shown in Figure 7.4. The mortar joint thickness was 10 mm.

Barbosa, et al. (2010) has presented the experimental results for four different block

strengths. In order to simulate these test results on hollow block conventional masonry the

dimensions of the RVE were set as 400 mm long (L), 200 mm high (H) and 140 mm wide

(W) as shown in Figure 7.5.

Figure 7.4 Typical dimensions considered for the tested blocks

Table 7.2 shows their transient-gradient properties of mortar and concrete for each prism test

and Table 6.8 summarises their elastic properties. The mechanical properties used in the

constitutive model for each test has been summarised in Tables 6.9, 6.10, 6.11 and 6.12

respectively for the four types of concrete blocks and mortar.



Figure 7.5 Typical dimensions of the modelled RVE and its ITZ



Block Mortar

cl (mm) 5 5

n 1 1

0.0009 0.0012

7.4 Numerical Modelling of Biaxial Testings

In this section, the procedure of analysis using the RVEs (for clay brick and concrete

masonry) are provided under different loading configurations i.e. uniaxial tension, uniaxial

compression, biaxial tension- compression and biaxial compression-compression.




Totally 22 combinations of stress states were simulated using the proposed model for bed

joint angles of 0°, 22.5° , 45°, 67.5° and 90° to principal stresses for each RVE. Table 7.3

illustrates the loading combinations which are applied to the RVE according to Figure 7.1 for

each bed joint orientation angle. Values shown in this table are the ratio between normal and

parallel loads for each load case.

Figure 7.7 Loading configuration and boundary conditions of the RVE for 6 load cases

considering θ = 90°



Mohr circle is used to obtain 𝜎𝑥𝑥, 𝜎𝑦𝑦 and 𝜎𝑥𝑦 for all loading combinations, according to

Table 7.3. Using Equations 7.1, 7.2 and 7.3, the loading configuration for each test can be

calculated. The RVE is loaded for each case according to Figure 7.6 and using its obtained

𝜎𝑛, 𝜎𝑝 and 𝜏.

1 2 1 2 cos 22 2

xx

(7.1)

1 2 1 2 cos 22 2

yy

(7.2)

1 2 sin 22

xy

(7.3)

Table 7.3 Load factors for each load case

Load Case 1 2 3 4 5 6 7 8 9 10 11

1 2/ ∞ 0 ∞ 0 +1 +0.5 +0.25 +0.125 +0.083 +2 +4

Load Case 12 13 14 15 16 17 18 19 20 21 22

1 2/ +8 +12 -4 -2 -1 -0.25 -0.5 -1 +1 +0.5 +2

7.4.1 Simulation of clay brick masonry Experimental tests

The RVE illustrated in Figures 7.2 and 7.3 was loaded incrementally based on the load ratio

considered for each test. Stress-strain behaviour of the RVE under different biaxial

configurations has been produced using this procedure. The results have been compared with

the experimental results in Dhanasekar (1985). The RVE is loaded based on the

configurations illustrated in Figure 7.6. Ratio between 1 and 2 principle stresses are

illustrated in Table 7.5 and will be applied to the RVE based on their orientation to the bed

joint according to Figure 7.1.



7.4.1.1 Bed Joint Angles of 𝜽 = 𝟎° and 𝜽 = 𝟗𝟎°(Zero-shear

state)

According to Equations 7.1 to 7.3 and Figures 7.1 and 7.7, when θ = 0°, 1xx , 2yy

and 0xy . Similarly, when θ = 90°, 2xx , 1yy and 0xy . In order to better

understand how the loading procedure is done, the RVE, its prescribed boundary conditions

and its loading configurations for six different loading cases considering θ = 90° are shown

in Figure 7.7. Loading cases 1 to 6 from Table 7.3 are shown in this Figure.

Figures D-1 to D-22 show all 22 loading combinations. In these Figure F is the load

magnitude on a specific load increment. Width, height and length of the RVE are equal to 55,

30 and 120 mm, respectively. Since the load ratio is important in setting up the RVEs, all load

magnitudes have been divided by the larger load.

Here, the uniaxial compression, biaxial compression-compression, uniaxial tension and

biaxial tension-compression cases were analysed to get the masonry failure envelope of bed

joint angles of 0° and 90°. When θ = 0°, 𝜎1 is parallel to bed joint and when θ = 90°, 𝜎1 is

perpendicular to bed joint. Load ratio is always considered as 𝜎1/𝜎2. The RVE was loaded

according to Table 7.5 and Table F-1 of Appendix F and as demonstrated in Appendix D. The

ultimate failure capacity for all stress states are summarised in Table E-1 of Appendix E.

Figures 7.8 and 7.9 illustrate the RVE’s predicted failure envelopes against the available

experimental results in terms of principle stresses for bed joint orientations of 0° and 90°,

respectively. These values are taken from Table F-1 of Appendix F.




The failure envelop of bed joint angle 90° were taken from bed joint angle 0° and plotted to

reverse principle stress axis due to symmetry, see Figure 7.9. It can be seen that in these

loading configurations shear stresses will not be produced (zero shear state).

Generally, good agreement is found between the model and experiments. The model has

slightly under-estimated the strength of masonry in biaxial compression-compression. Figures

7.8 and 7.9 show that the strength of masonry increases in biaxial compression-compression

states by nearly 50% from the original strength in uniaxial compression.




7.4.1.2 Bed Joint Angles of 𝜽 = 𝟐𝟐. 𝟓° and 𝜽 = 𝟔𝟕. 𝟓°

The 22 combinations are obtained considering load cases in Table 7.3 and using Equations

7.1 to 7.3. These load combinations are shown in Table 7.3 and were simulated using the

proposed model for bed joint angles of 22.5° and 67.5°. The ultimate failure capacity for all

cases analysed is summarised in Table E-2 of Appendix E. Figures 7.10 and 7.11 illustrate

the RVE’s predicted failure envelopes against the available experimental results in terms of

principle stresses for bed joint orientations of 22.5° and 67.5°, respectively. These values are

taken from Table F-2 of Appendix F.




22.5°


67.5°

Figure 7.12 shows failure envelope of masonry in the normal-shear stress plane and Figure

7.13 shows the failure envelope in parallel-shear stress plane. The model has reproduced the



failure envelope for solid brick masonry for loading with bed joint angles of 22.5° and 67.5°,

with good accuracy.


joint n for bed joint angles of 22.5° and 67.5°


joint p for bed joint angles of 22.5° and 67.5°



7.4.1.3 Bed Joint Angle of 𝜽 = 𝟒𝟓

The 22 combinations of stress states were simulated using the proposed model for bed joint

angle of 45°. The RVE was loaded according to Table F-3 of Appendix F. Figure 7.14

illustrates the RVE’s predicted failure envelope against the available experimental results in

terms of principle stresses for bed joint orientation of 45°.



joint n for bed joint angle of 45°




joint p for bed joint angle of 45°

The failure envelopes of masonry with bed joint angle of 45° in normal-shear stress plane and

parallel-shear stress plane are illustrated in Figures 7.15 and 7.16, respectively. The ultimate

failure capacity for all analysed stress states is shown in Table E-3 of Appendix E. It can be

seen that the model has predicted the failure envelope for solid brick masonry for loading

with bed joint angle of 45° with fairly good accuracy.

7.4.1.4 Failure Envelope

In order to develop the full failure envelope of masonry, projection of failure envelopes of

Figures 7.9 to 7.12 and 7.15 in the normal-parallel stress plane for bed joint orientations of

0°, 22.5°, 45°, 67.5° and 90° is plotted in Figure 7.17. The Failure stress of all previously

mentioned cases has been plotted against the experimental results by Dhanasekar (1985) for

normal-shear stress plane and parallel-shear stress plane in Figures 7.18 and 7.19,

respectively. Good agreement can be observed between the model’s numerical predictions

and the experimental datasets with slight under-estimation of strength which is desirable for

design purposes.



Figure 7.17 Failure envelope of the RVE in terms of principle stresses for bed joint angles

0°, 22.5°, 45°, 67.5° and 90°


joint n for bed joint angles of 0°, 22.5°, 45°, 67.5° and 90°




joint p for bed joint angle of 0°, 22.5°, 45°, 67.5° and 90°

Figure 7.20 3-D failure envelope of the RVE in terms of shear stress τ, stress normal to bed

joint n and stress parallel to bed joint p



In order to get a better understanding of the behaviour of the model and develop the full

three-dimensional failure envelope of masonry 330 load cases were analysed. These cases

were constructed by adding 10% of the normal load to shear load in each of the 22 load

combinations shown in Table 7.3. Load factors for all load sets have been shown in Tables F-

4 to F-18 of Appendix F.

Figure 7.21 Failure envelope of the RVE in the normal-parallel stress plane

The 3D failure envelop of masonry was obtained using these 330 cases as shown in Figure

7.20. Projection of the 3-D failure envelope on the normal-parallel stress plane is shown in

Figure 7.21. Two ellipsoids were fitted to these 330 data using the minimum volume

enclosing ellipsoid based on Khachiyan Algorithm (Todd and Yildirim, 2005). One was fitted

to cases with the summation of the normal and parallel stresses less than -1 and one was fitted

to the rest of the cases. The fitted ellipsoid is in the form of

( ) ' ( ) 1x c A x c (7.4)



Figure 7.22 shows the fitted ellipsoid along with the failure points for cases with the

summation of the normal and parallel stresses less than -1. The centre of this ellipsoid is

4.02

3.63

0

c

and matrix

0.0315 -0.0031 0.0000

-0.0031 0.0336 0.0001

0.0000 0.0001 0.2357

A

.

Figure 7.22 Fitted ellipsoid and its corresponding failure cases for cases with the summation

of the normal and parallel stresses less than -1

Similarly, Figure 7.23 shows the fitted ellipsoid along with numerical failure points for the

rest of the loading cases. For this ellipsoid, the centre is

4.02

3.63

0

c

and matrix

0.0315 -0.0031 0.0000

-0.0031 0.0336 0.0001

0.0000 0.0001 0.2357

A

.



Figure 7.23 Fitted ellipsoid and its corresponding failure cases for cases with the summation

of the normal and parallel stresses more than -1

7.4.1.5 Prediction of Modes of Failure

Failure patterns are an important feature of numerical models which needs to be investigated.

Here, the damage progression in the ITZ and the failure mode of five cases (Uniaxial

compression normal and parallel to bed joint, uniaxial tension normal and parallel to bed joint

and biaxial compression-compression) have been investigated and compared to failure modes

presented in Dhanasekar (1985). In all these figures light grey shade represents higher

damage.



Figure 7.24 Failure mode and damage progression in the ITZ layers for uniaxial compression

parallel to bed joint in the initial loading stages

First masonry under uniaxial loading parallel to bed joint is considered. Figures 7.24 and 7.25

show damage progression at six different elevations of the RVE for two loading steps (one in

the beginning loading stages and one in later stages). It can be observed in both Figures that

damage levels are highest at the first layer of the ITZ from the interface (Y=25.5 mm).

Damage parameter decrease as the layer gets further away from the interface and is the

smallest at the outer edge of the RVE (Y=30 mm).




parallel to bed joint in the later loading stages

As can be seen in the above mentioned Figures, failure in case of uniaxial compression

parallel to bed joints is occurring in the bed joints. This observation is in agreement with the

failure pattern presented in Dhanasekar (1985) and shown in Figure 2.2.

Similarly, for the case of compression normal to bed joint, damage progression for two load

steps at six layers from the interface are shown in Figures 7.26 and 7.27. It can also be seen



here that damage parameter is the largest at the first ITZ layer from the interface (X=115.5

mm) and decrease as we approach the outer edge of the RVE (X=120 mm).


normal to bed joint in the initial loading stages

It can be observed from these two figures that in case of uniaxial compression normal to bed

joints, failure is occurring in the head joints. This observation is partially in line with the

failure mode presented by Dhanasekar (1985), see Figure 2.2. In these experiments failure

occurs with vertical cracking of the units and head joints.




normal to bed joint in the later loading stages

Figures 7.28 and 7.29 demonstrate damage progression and failure mode for uniaxial tension

parallel and normal to bed joints at the later loading stages, respectively. From Figure 7.28, it

can be seen that failure occurs in the head joints by starting from the closest ITZ layer to the

interface. This failure mode complies with the failure mode reported by Dhanasekar (1985)

and shown in Figure 2.3.




parallel to bed joint in the later loading stages

Figure 7.30 shows failure mode and damage progression for the uniaxial tension loading

normal to bed joints in later loading stages. It can be observed here that similar to Dhanasekar

(1985) failure is happening in the bed joints, see Figure 2.3. The damage parameter is the

biggest at the layer closest to the interface (Y=25.5 mm) and decreases as we go closer to the

external edge of the RVE (Y=30 mm).

The case of biaxial compression-compression loading is considered in Figure 7.30. As can be

observed in this Figure, in case of biaxial compression loading, damage progresses in both

bed and head joints and also within the masonry brick, this process continues until final

failure happens. It was reported in Dhanasekar (1985) that failure in biaxial compression

cases occurs by splitting of the specimen in a plane parallel to the free surface as shown in

Figure 2.6. Since the model considered here is a 2D model it cannot predict the splitting of



the specimen, however, damage progression in the manner explained previously implies the

same failure mode. Basically, damage progresses in both brick and mortar until the whole

RVE loses its integrity.


normal to bed joint in the later loading stages

Finally, the case of tension-compression loading with bed joint orientation of 45º is shown in

Figure 7.31. As can be seen in this Figure in this case which is close to the case of pure shear,

failure is occurring in a stair way manner, which is in line with Dhanasekar (1985)

predictions, see Figure 2.5.



Figure 7.30 Failure mode and damage progression in the ITZ layers for biaxial compression-

compression (bed joint angle of 0°) in the later loading stages



Figure 7.31 Failure mode and damage progression in the ITZ layers for biaxial compression-

tension (bed joint angle of 45°) in the later loading stages

The capability of the ITZ enriched RVE for the predictions of the modes of failure of

masonry panels subjected to biaxial loading provides further confidence in the failure

surfaces presented in sections 7.4.1.4. The model is therefore further extended for the

analysis of other types of masonry with confidence in the subsequent sections of this chapter.



7.4.2 Simulation of Concrete Block Masonry Biaxial

Experiments by Barbosa, et al. (2010)

The RVE shown in Figures 7.5 was loaded incrementally based on the load ratio considered

for each test. The boundary condition and loading scenario are as shown in section 7.4.

Stress-strain behaviour of the RVE under different biaxial configurations has been produced

using this procedure. The uniaxial compression tests have been calibrated in section 6.5.2 for

experimental results by Barbosa, et al. (2010). The RVE is loaded based on the

configurations illustrated in Figure 7.1. The 1 and 2 principle stresses are also illustrated

in Figure 7.1.

7.4.2.1 Analysing Case 1 (P1)

The material properties used in the RVE for the first case have been shown in Table 6.9. The

RVE was loaded for 22 load combinations (zero-shear) according to Table 7.3. The predicted

failure envelope is illustrated in Figure 7.32. The ultimate failure capacity for all analysed

stress states is shown in Table D-4 of Appendix D.

Figure 7.32 Failure envelope of the RVE in terms of principle stresses in zero-shear (P1)




For the second case, the material properties of the constituents i.e. concrete and mortar are

shown in Table 6.10 and the RVE is loaded according to Table 7.3 for cases of zero-shear

stress. The ultimate failure capacity for all analysed stress states can be seen in Table D-5 of

Appendix D and the predicted failure envelope is illustrated in Figure 7.33.



Similarly, 22 combinations of stress states shown in Table 7.3 were simulated using the

proposed model for the third case (P3). The ultimate failure capacity for all analysed stress

states is summarised in Table D-6 of Appendix D. Figure 7.34 illustrates the RVE’s predicted

failure envelope.





Finally, the fourth test was analysed for the same load combinations shown in Table 7.3 and

the failure envelope of this set of analysis can be seen in Figure 7.35. The ultimate failure

capacity for all analysed stress states can be found in Table D-7 of Appendix D.



Figure 7.35 Failure envelope of the RVE in terms of principle stresses in zero-shear state

(P4)

7.4.3 Comparison of Failure Envelopes for Conventional

Masonry

In this section, using the results reported in sections 7.4.1 and 7.4.2, a comparative study of

the biaxial failure of various masonry systems was carried out by dimensionlising the biaxial

failure envelopes. Figure 7.36 shows the projection of strength of the previously investigated

biaxial tests in the n

-p

principal stress space. Stresses in both orientations for all loading

cases have been divided by the unit characteristic strength to non-dimensionalise the ultimate

strength. The mean strength of the units from each test is reported in Table 7.4.



Table 7.4 Characteristic strength of units in different experiments


Experiment Characteristic Strength (MPa)

Dhanasekar, et al. (1985) 15.41

Barbosa, et al. (2010) P1 22.8




Figure 7.36 Comparison of the failure envelopes of conventional masonry with different

strength and geometry


In this chapter, the application of the transient-gradient CDM model with enriched ITZ to

predicting the biaxial failure surfaces and modes of failures of the conventional masonry has

been presented. Two experimental tests reported in the literature (one for clay brick masonry



and the other for the concrete block masonry) have been chosen for this purpose. The

dimensions of the RVEs were idealised to represent the dimension of the units and mortar

layer as close as possible. All forms of biaxial loading conditions such as biaxial

compression-compression, tension-compression, uniaxial tension and uniaxial compression

cases were analysed and reported. The variation of damage in the layers of the ITZ was used

as an illustration of the mode of failure and was compared to the experimental mode of

failure. The modes of failure and the biaxial failure envelope predicted by the model were

found to be in very good agreement with the experimental results. . In a non-dimensional

stress space, the failure envelope of the concrete block masonry has exhibited higher

compressive strength than that of the clay brick masonry in the direction of parallel to the bed

joint.

In the next chapter, the model will be further developed to enable simulation of the behaviour

of dry-stacked (or, mortarless) masonry systems and used for the prediction of biaxial failure

envelopes of dry-stacked masonry.

168 Application of the ITZ Enriched CDM Model – II: Constitutive Properties of Dry-stack


CHAPTER 8

Application of the ITZ Enriched CDM Model

– II: Constitutive Properties of Dry-stack

Masonry under Biaxial Stresses

8.1 Introduction

Masonry is labour intensive; this has two consequences – variability in the properties of the bonded

assemblage and higher labour cost. With a view to finding novel solutions to the spiralling costs of

masonry buildings, dry-stack systems have been introduced by the industry with specially shaped

interlockable blocks for ease of construction.

In the past decades much research about the behaviour of solid and hollow conventional masonry

using different homogenisation techniques has been carried out but there is only limited information

available for dry-stack masonry; further, the available information is limited to only the uniaxial

behaviour. Thus, it is important to develop biaxial failure surfaces for dry-stack masonry to enable

computational modelling. In this chapter examination of the dry-stack hollow concrete masonry

behaviour using a modified ITZ enriched CDM model introduced in chapter 6 to implicitly account

for the interacting dry interfaces (in the absence of mortar layers) is presented. The model has been

validated with the limited available uniaxial experimental datasets in the literature and then used to

predict biaxial failure of the dry-stack masonry.

8.2 Dry-Stack Masonry

Mortared block masonry is labour intensive and workmanship is a determining factor in various

operation involved. Mortar joints have relatively weak bond strength. As labour is expensive,

masonry is losing its market share to other new generation products, e.g., tilt-up concrete walls/ glass

panel systems. With a view to reversing this trend, the masonry industry is developing several



innovative products in recent times; one such is interlockable dry-stack systems that could be

constructed with minimal skilled mason time and hence can be cheaper.

The behaviour of the dry-stack systems is significantly affected by the behaviour of the mortarless dry

joint. Hence, modelling the behaviour of these joints is crucial in predicting the behaviour of these

systems (Gasser, et al., 2004; Thanoon, et al., 2008; Andreev, et al., 2012). Imperfect initial contact

and its closure require careful attention in the modelling dry-stack masonry. Unfortunately, there is

little information on this phenomena and its simulation. Thus, it is important to introduce a

computational method for modelling dry-stack masonry joints and their effect on overall behaviour of

the system.

In this chapter, in order to simulate joint closure and its effect on overall performance of dry-stack

masonry, the previously introduced ITZ enriched CDM model is further developed to implicitly

account for the damage of the interacting dry interfaces.

8.3 Constitutive Modelling of Dry-Stack Masonry

In dry-stack masonry, due to absence of mortar, blocks are not fully connected in the initial

loading stages and some gaps exist in the joints, see Figure 8.1. When loading progresses,

these gaps commence closing up, a phenomenon defined as ‘joint closure’ and is observed by

Oh (1994), Gasser, et al. (2004), Thanoon, et al. (2008), Nguyen, et al. (2009) and Andreev,

et al. (2012). Qualitative stress-strain behaviour of dry-stack masonry until joint closure is in

effect is shown in Figure 8.2 (a).

Figure 8.1 Representation of a dry joint in dry-stack masonry



Figure 8.2 Qualitative stress-strain and damage evolution of dry-stack masonry due to joint

closure

In order to simulate this part of the stress-strain behaviour, the damage evolution law requires

modification in a way that the stiffness of the material first increases (due to crack closure)

and a better contact between the two neighbouring block is eventually achieved. A qualitative

damage evolution curve, representing this increase in stiffness is shown in Figure 8.2 (b).

The dashed line represents damage evolution considering no joint closure (for mortared

masonry, for example) and the solid line shows damage evolution considering the joint

closure phenomenon. In case of dry-stack masonry, when loading initiates, the gap between

the two blocks acts like the material has been cracked and lost some of its area. This results in

the damage evolution starting from a non-zero state as can be seen in Figure 8.2 (b). After

loading initiates, due to the joint closure effect, the gap between the two blocks reduces and

the material acts as if it is gaining contact area and the damage parameter decreases, see

Figure 8.2 (b). After the joint is fully closed (where the solid line and the dashed line

intersect), the material continues its behaviour in the same path that a material without joint

closure was acting. In this section joint closure due to initial joint imperfections is introduced

as a form of an enhanced damage evolution law, see Figure 8.2(c).

With the increase in loading, it is conceptualised that the contact area increases due to

smoothening of the rough surfaces – or, the ‘damage’ decreases - as narrated in Figure 8.2

(b). In spite of the reduction in damage from the initial damage, the body continues to gain

energy until a conforming contact is established between the two bodies, where the damage is

conceptualised to have fully ‘healed’. With further ongoing increase in loading, damage re-

initiates due to exceedance of energy stored in usual manner. Therefore, although damage

curve shows non-monotonic appearance in the damage-strain plane (Figure 8.2 (b)), the

energy only increases monotonically and gets released in the usual way. Therefore the

assumption does not violate the fundamentals of the second law of thermodynamics.



8.3.1 Formulation of Damage Evolution Law for Dry-Stack

Joint

This model incorporates all parameters used in Chapter 3 which have been explained in

sections 3.3 and 3.6. Here there are three new parameters introduced to the model i.e. jc , 0I

and jS , in which

jc is the strain at which the joint is fully closed by establishing full contact

between the two block surfaces . Beyond this strain, the joint will act same as the block.

The initial imperfection parameter 0I is always 00 1I . The higher the

0I , the lower the

contact between the two neighbouring blocks. When 0I is equal to zero, the two

neighbouring blocks are in full contact. On the other hand, when 0I is equal to unity the

surfaces of the two blocks are at an infinitesimal gap analogues to full damage state. Damage

slope parameter jS controls the rate of damage evolution in the joint closure part of

damaging process.

Damage evolution law is defined in the form of

1

2

0 1

1

1 1 0

0

0 ( )

(

ii

j jc

cjc

c c c

e if I

S I if I and

if I and

1

) ( ) 0 c c c

cif I and

(8.1)

If jc c and in the form of

1

1

2

0 1

1 1 0

0 ( )

0

( )

ii

cc

c c c

j c

c

e if I

if I and

S I if I and

1

( ) 0 c c

jcif I and

(8.2)

If jc c . Parameter μ is determined as follows



0

2

0

2 2

( ( ) )

( ) ( )

( )

j jccjc c

c jc c c jc jc

jc c c c j jc

jc jc

jc jc jc

I Sif

I Sif

(8.3)

In these equations , and are material parameters. c and c are the level of damage

and the magnitude of strain, measured at a point of a uniaxial compression test, respectively.

Parameter is defined in the form of

2

2

2c c c c c

c c

(8.4)

Effects of each of these parameters on the damage evolution and stress-strain behaviour of

the material are shown in a parametric study done on Section 8.4 of this chapter. The non-

local effects are considered in the same manner as the previous chapters.

8.3.2 Implementation

The constitutive model described in Sections 3.1-3.3 and 8.3.1 was implemented into

ABAQUS/ Standard 6.11 through a MATLAB program. Implementation of the constitutive

law presented in section 8.3.1 is explained through an algorithm presented in Table 8.1.

Similar procedure as the one explained in section 4.4 has been used here. PYTHON,

MATLAB and FORTRAN codes used in this procedure can be found in Appendices A, B

and C, respectively.

Table 8.1 Algorithm for the implementation of the dry-surface damage model

1. Retrieve the displacement increment of an integration point n from the ABAQUS

output database

nu


nn n-1ε ε +=



3. Impose

1n nκ κ

1n nυ = υ

4. Compute the current equivalent strain

2

2

1 1 22 2

1 11 12

2 1 2 2 1 2 1n

k k kε I I J

k υ k υ υ

5. Check damage criterion

, n n

f ε κ ε κ 0

If Yes: No damage, Go to Step10.

If No: Damage, proceed to 6.

6. Compute the damage variable n .

If jc c

1

2

0 1

1

1 1 0

0

0 ( )

(

ii

j jc

n cjc

c c c

e if I

S I if I and

if I and

1

) ( ) 0 c c c

cif I and

If jc c

1

1

2

0 1

1 1 0

0 ( )

0

( )

ii

cc

c c c

j c

c

e if I

if I and

S I if I and

1

( ) 0 c c

jcif I and

7. Compute the volumetric damage variable ( )K n .

( ) 1 (1 )K n n

8. Compute the current Poisson’s ratio

1

01 (1 2 )(1 )

2

nn



9. Update the damage threshold

n nκ ε

10. Update the stress


11. Return to ABAQUS for further load/ displacement increment


In order to verify the performance of the dry-surface damage model the effect of different

material parameters to the overall response of the system has been examined. This

examination is carried out through a set of analyses using an eight noded two-dimensional

isoparametric finite element, see Figure 8.3. The influence of different parameters to the

uniaxial compression behaviour of dry-stack masonry has been studied. This simple

parametric study illustrates the response of the improved model at a point level.

Effects of the following key parameters to the compressive behaviour of the model

are discussed here.

Full contact strain jc

Initial imperfection parameter 0I

The damage slope parameter jS

The parameters employed in this parametric study are summarised in Table 8.2.

Figure 8.3 Eight-noded plane stress element used in the analysis (CPS8R)



Table 8.2 Mechanical properties of the tested element

Material Properties

E (MPa) 20000

0 0.2

k 10

α 1.0

β 1000

i 0.00005

c 0.00016

c 0.5

-0.25

jc 0.00015

0I 0.8

jS -1000

8.4.1 Effect of the Full Contact Strain 𝒌𝒋𝒄

Parameter jc controls the strain at which the joint is fully closed. From this strain onwards,

the damage evolution law and behaviour of the material will follow the same path as that of

the material without initial damage explained in Chapters 3 and 4. Influence of employing

different constants for parameter jc , on the evolution of damage with respect to strain is

illustrated in Figure 8.4. The stress-strain behaviour of the material, considering five

constants for jc , are also shown in Figure 8.5. For all cases, 0I is kept as 0.8.



Figure 8.4 Influence of different values of the full contact strain parameter jc on damage

evolution in uniaxial compression

Figure 8.5 Influence of different values of the full contact strain parameter on stress-strain

behaviour under uniaxial compression



8.4.2 Effect of Initial Imperfection Parameter 𝑰𝟎

The initial imperfection parameter 0I should fall with the limit of 00 1I . The higher the

value of 0I , the lower the contact between the two contacting dry surfaces. When 0I is equal

to zero, the two neighbouring blocks are in full contact from the beginning of loading

procedure (no initial damage) and there exist a perfect contact between them. On the other

hand, when 0I is equal to unity the two blocks surfaces have no initial contact.

Figure 8.6 shows the effect of the initial imperfection parameter 0I to the damage evolution

of the material against uniaxial compression strain. Influence of this parameter to the stress-

strain behaviour of the material using the present model is also illustrated in Figure 8.7.

Figure 8.6 Influence of different values of the initial imperfection parameter 0I on damage




Figure 8.7 Influence of different values of the initial imperfection parameter 0I on stress-

strain behaviour under uniaxial compression

8.4.3 Effect of Damage Slope Parameter 𝑺𝒋

Damage slope parameter jS controls the rate of damage evolution in the joint closure part of

damaging process. This parameter is always 0jS and as can be seen in Figures 8.8 and 8.9

the smaller its value, the faster the rate of reaching full contact of neighbouring blocks.



Figure 8.8 Influence of different values of the damage slope parameter jS on damage


Figure 8.9 Influence of different values of the damage slope parameter jS on stress-strain

behaviour under uniaxial compression



Figure 8.8 shows the effect of damage slope parameter jS , on the damage evolution of the

material under uniaxial compression. Stress-strain behaviour of the material under the same

stress state for different values of jS is also illustrated in Figure 8.9.

8.5 Numerical Validation of the Model

There are only few studies in the literature on the behaviour of dry-stack masonry. An

investigation of the failure mechanism of interlocking dry-stack masonry systems was

conducted by Oh (1994). Ferozkhan (2005) did experiments on a dry-stack system to

investigate its behaviour under out-of-plane bending. There is also some research available

on the compressive behaviour of these systems (Thanoon, et al., 2008; Andreev, et al., 2012).

Gasser, et al. (2004) did an investigation on the modelling of the effects of joints to the

behaviour of refractory linings and Nguyen, et al. (2009) did biaxial tests on refractory

linings and simulated the behaviour using a homogenisation method.

8.5.1 Simulation of Experimental Tests Conducted by Oh (1994)

In this section, the model presented in Section 8.4 is validated with the experimental work

reported by Oh (1994). Two types of blocks were used in this set of experiments. Figure 8.10

shows the dimensions of a H-block used by Oh (1994) and complete details of all

measurements are contained in Oh (1994).

Two experiments were conducted on blocks with the dimensions (in mm) shown in Figure

8.10. First experiment was performed using the H-blocks. The second experiment was

conducted by using improved prisms in which all the visible gaps on the surface of the blocks

have been eliminated (using an advanced moulding procedure). We refer to these prisms as

the modified H-block. The dimensions of the conventional hollow block measured (in mm)

by Oh (1994) can also be seen in Figure 8.11. Average compressive strength of the H-block

and the conventional hollow block were reported as 32.88 MPa and 13.09 MPa, respectively.



Figure 8.10 Measured dimensions of the H-block presented by Oh (1994)

Figure 8.11 Measured dimensions of the conventional hollow block presented by Oh (1994)



8.5.2 Masonry Unit Properties

In order to better simulate the behaviour of the masonry, calibration of the compressive and

tensile stress-strain behaviour of its individual constituents is essential. However, due to lack

of these information, only the compressive strength was considered exactly as that of the

experiment. All other properties were chosen based on the parametric studies conducted in

section 8.4 and are listed in Table 8.3. The properties used for the fictitious joint of the

conventional hollow block, the H-block and the modified H-block are listed in Table 8.4.

Table 8.3 Properties of the H-block and conventional block

Material Properties

H-Block Conventional

E (MPa) 21000 10000

0 0.2 0.2

k 10 10

α 1.0 1.0

β 1000 1000

i 0.00005 0.00005

c 0.00023 0.0002

c 0.4 0.4

-0.25 -0.25

Table 8.4 Properties of interface layers

Interface Properties

H-Block Modified H-Block Conventional Hollow Block

jc 0.0002 0.0002 0.00018

0I 0.9 0.5 0.9

jS -1000 -1000 -1000



The original geometry of the H and the conventional blocks were idealised for computational

efficiency. The idealised geometry of the H and the conventional blocks are shown in

Figures 8.12 and 8.13 respectively. The Figures also include their respective RVEs.

Figure 8.12 Idealised dimensions of the H-block and modified H-block and their RVE



The dimensions of the idealised H-block used in the analysis are shown in Figure 8.12. As

can be observed in this Figure a fictitious joint of 2 mm thickness is considered in the RVE. It

should be noted that the dimensions of the modified H-block was kept unchanged to the

original H block. Figure 8.13 illustrates the idealised dimensions of the conventional hollow

block

Figure 8.13 Idealised dimensions of the conventional hollow block and its RVE



The RVE was constructed based on these measurements and considering a 2 mm fictitious

joint around the block. It should be noted that the fictitious joint in both RVEs have the same

material properties as the block with the exception of joint closure properties.

Figure 8.14 Stress-strain behaviour of H-block under uniaxial compression

Figure 8.15 Stress-strain behaviour of conventional hollow block under uniaxial compression

The stress-strain behaviour of the H-block and conventional hollow block under uniaxial

compression loading considering mechanical parameters shown in Table 8.3 are graphically

illustrated in Figures 8.14 and 8.15, respectively.



The stress-strain behaviour for the fictitious joint of the H-block and the modified H-block

are graphically illustrated in Figure 8.16. The stress-strain behaviour of the conventional

hollow block under uniaxial compression loading is also shown in Figure 8.17.

Figure 8.16 Stress-strain behaviour of the fictitious joint of H-block under uniaxial

compression

Figure 8.17 Stress-strain behaviour of the fictitious joint of conventional hollow block under

uniaxial compression



8.5.3 Analysis of the RVEs for Uniaxial Compression Test

The stress-strain behaviour of the prisms using the present model has been illustrated in

Figures 8.18 to 8.20.

Figure 8.18 Stress-strain behaviour of conventional hollow block under uniaxial compression

Figure 8.18 illustrates the stress-strain behaviour predicted by the model for the conventional

hollow block under compressive loading along with its experimental data (C-1 to C-4).

Figure 8.19 Stress-strain behaviour of the H-block under uniaxial compression



Figure 8.20 Stress-strain behaviour of the modified H-block under uniaxial compression

Figures 8.19 and 8.20 compare the results obtained in the experiment (H-1 to H-3) with the

present model’s prediction under axial compressive loading for the H-block and modified H-

block, respectively. The only difference in the simulation of these two tests is the joint

closure parameters which were previously presented.

As can be seen in the previous figures, the model is showing good agreement with the

experimental data for uniaxial compression.

8.6 Prediction of Biaxial Failure Envelope for Dry-Stack Masonry

Using the properties presented in Tables 8.2 to 8.4,the biaxial failure envelopes of dry stack

masonry made from these blocks (H and conventional) were evaluated. These predicted

failure envelopes, although cannot be independently validated due to lack of experimental

evidences, based on the validation of the predicted failure surfaces in Chapter 7 and the

uniaxial behaviour in this chapter, the biaxial failure surface of the dry stack masonry

predicted can be regarded as rational.

Twenty two combinations of stress states were simulated using the proposed model for the

zero-shear stress states, see Table 8.5. The uniaxial compression, biaxial compression-



compression, uniaxial tension and biaxial tension-compression cases were analysed to get the

masonry failure envelopes. Table 8.5 illustrates the loading combinations which are applied

to the RVE according to Figure 8.21 for each bed joint orientation angle. Values shown in

this table are the ratio between normal and parallel loads for each load case.

Mohr circle is used to obtain 𝜎𝑥𝑥, 𝜎𝑦𝑦 and 𝜎𝑥𝑦 for all loading combination, according to

Table 8.5. Using Equations 7.1, 7.2 and 7.3 (Mohr equations), the loading configuration for

each test can be calculated. The RVE is loaded for each case according to Figure 8.22 and

using its obtained 𝜎𝑛, 𝜎𝑝 and 𝜏. All twenty two load cases with their boundary conditions are

shown in Figures E-1 to E-22 of Appendix E.

Figure 8.21 Representation of material and principal axes in masonry




Table 8.5 Load factors for each load case

Load Case 1 2 3 4 5 6 7 8 9 10 11

1 -1 0 +1 0 -1 -1 -1 -1 -1 -2 -4

2 0 -1 0 +1 -1 -2 -4 -8 -12 -1 -1

1 2/ ∞ 0 ∞ 0 +1 +0.5 +0.25 +0.125 +0.083 +2 +4

Load Case 12 13 14 15 16 17 18 19 20 21 22

1 -8 -12 -4 -2 -1 +1 +1 +1 +1 +1 +2

2 -1 -1 +1 +1 +1 -4 -2 -1 +1 +2 +1

1 2/ +8 +12 -4 -2 -1 -0.25 -0.5 -1 +1 +0.5 +2

8.6.1 Prediction of Biaxial Failure Envelope for the H-Block

The RVE shown in Figure 8.12 was simulated under the 22 load combinations of zero shear

stress state (Table 8.5) using the proposed model. The ultimate failure capacity for all

analysed stress states is summarised in Table D-9 of Appendix D. Figure 8.23 illustrates the

RVE’s predicted failure envelope for the H-block.

Figure 8.23 Failure envelope of the H-block in terms of principal stresses



8.6.2 Prediction of Biaxial Failure Envelope for Modified H-

Block

The same loading procedure explained in Section 8.6 has been followed for the simulation of

biaxial failure envelope of the modified H-block. Same dimensions have been considered for

the modified H-block and the H-block, see Figure 8.12. Similarly, 22 combinations of zero

shear stress states were simulated using the proposed model. The ultimate failure capacity for

all analysed stress states is summarised in Table D-10 of Appendix D and the predicted

failure envelope has been plotted in Figure 8.24. H -blocks typically show lower strengths

due to absence of end webs. 13 MPa masonry strength for a 32 MPa unit strength is

considerably low and is attributed to the low stiffness of the units (Figure 8.14). The biaxial

case generally exhibits higher strengths than the uniaxial case.

Figure 8.24 Failure envelope of the modified H-block in terms of principal stresses

8.6.3 Prediction of Biaxial Failure Envelope for the

Conventional Hollow Block

The RVE shown in Figure 8.13 was loaded incrementally based on the load ratio considered

for each test. Stress-strain behaviour of the masonry under different biaxial configurations

has been produced using this procedure. The RVE is loaded based on the configurations

illustrated in Table 8.5. Figure 8.25 shows the RVEs failure envelope in terms of principal



stresses. The ultimate failure capacity for all stress states are summarised in Table D-8 of

Appendix D.

Figure 8.25 Failure envelope of the conventional hollow block in terms of principal stresses

8.6.4 Comparison of Biaxial Failure Envelopes of Different

Masonry Systems

In this section, based on the analyses done in sections 8.6.1 to 8.6.3 and also 7.4.1 and 7.4.2,

a dimensionless study is illustrated on the biaxial failure envelope for conventional masonry.

Table 8.6 Characteristic strength of units in different experiments


Experiment Characteristic Strength (MPa)

Dhanasekar, et al. (1985) 15.41





Oh (1994) Conventional Block 12.89

Oh (1994) H-Block 30.7

Oh (1994) Modified H-Block 30.7



Figure 8.26 shows the projection of strength of the previously investigated biaxial tests in the

1/ f -

2/ f (in which f is the strength of each specimen) principal stress space. Stresses in

both orientations for all loading cases have been divided by the unit characteristic strengths to

obtain the dimensionless values. The characteristic strength of the units from each test is

demonstrated in Table 8.6.

It can be observed from Figure 8.26 that the dry-stack H-block is the least efficient masonry

system. However, by improving the prisms used in this masonry system (modified H-block),

its performance significantly improves.

Figure 8.26 Comparison of the failure envelopes of different masonry systems with various

strengths

The range of failure strengths for different masonry systems can be obtained from Figure

8.26. The shaded envelope of different masonry systems from Figure 8.26 has been shown in

Figure 8.27. The H-block failure envelop is the minimum failure strength in this range. The

maximum failure strength is obtained from a combination of the hollow block prisms and the

modified H-block.



Figure 8.27 Failure envelop zone of masonry systems with various strengths and geometries


This Chapter presented a continuum damage evolution law capable of describing the

behaviour of dry-stack masonry. The behaviour of the dry-stack masonry is significantly

affected by the behaviour of the mortarless dry joint. The proposed damage evolution law

considers the joint closure phenomena in the initial stages of the damaging process. By

implicitly accounting for the interacting dry interfaces, the prediction of the model for dry-

stack masonry significantly improves.

Basic concepts were briefly described, together with the theoretical formulation. Then, an

algorithm for implementation of the formulations has been presented. The influences of key

parameters of the model are investigated through a set of sensitivity analyses. Then, the

model was validated using real experimental data on compressive behaviour of three different

blocks.

Finally, considering the validated parameters the predicted failure envelope for each block

has been illustrated and failure envelopes of different masonry systems have been compared.



In the next chapter, the clauses relating to compressive strength of masonry in Australian

Masonry Standard AS 3700 (2011) is reviewed and the strength of masonry obtained from

the model are compared with the strengths obtained by this standard.

196 Application of the Model for the Masonry Compressive Strength Prediction |

CHAPTER 9

Application of the Model for the Masonry

Compressive Strength Prediction

9.1 Introduction

The ITZ enriched transient-gradient CDM model formulated in Chapter 6 have been

successfully applied in Chapters 7 and 8 for the development of biaxial failure surfaces for

conventional and dry-stacked masonry respectively. The use of this model for the prediction

of compressive strength of a range of masonry utilising unburnt clay (or mud) bricks to

dressed stones and two types of mortar (low and high strength) is presented in this chapter. In

addition to these strength properties, the height of the unit and the thickness of the mortar

layer were also modified, which provided seven ratios (5 – 19) of height of unit to thickness

of mortar joint. The predicted strengths have been compared with the strengths calculated

using the provisions of the Australian Masonry Standard (AS 3700).

9.2 Current AS 3700 (2011) Provisions for Compressive Strength of

Masonry

Clause 3.3.2 in AS 3700 (2011) refers to the determination of the characteristic compressive

strength of masonry mf ' from the characteristic compressive strength of units uc

f ' ,

mortar grade 2 3 4M M M, , , height of unit uh and thickness of mortar joints j

t . The

clause is provided in Eq. (9.1).

m m h ucf k k f' ' (9.1)

Where, m

K compressive strength factor shown in Table 9.1.


hK joint thickness factor

0 291 3u j

h t .. ( / ) but to not exceed 1.3

ucf ' characteristic unconfined compressive strength of units in MPa

uh masonry unit height (mm)

jt mortar bed joint thickness (mm)

Table 9.1 Compressive strength factor m

K for masonry

Type of Masonry Unit Bedding Type Mortar Class Compressive strength factor

Clay Full M3 1.4

Clay Full M4 2.0

Concrete Full M3 1.4

Concrete Face shell M3 1.6

Calcium silicate Full M3 1.4

Calcium silicate Full M4 2.0

In order to investigate the effect of the key parameters influencing the compressive strength

of masonry, units with ten characteristic unconfined compressive strengths uc

f ' , which

typically stand for unburnt clay (or mud) bricks (3MPa – 6MPa), concrete blocks (10MPa –

25MPa), burnt clay bricks (30MPa – 50MPa) and dressed stone (100MPa) have been

considered as shown in Table 9.2. All units have been analysed using two different mortar

grades i.e. M3 (compressive strength 5MPa) and M4 (compressive strength 12MPa). For each

set of unit and mortar strengths, seven masonry unit height to mortar bed joint thicknesses,

ranging from 5 to 19, have been considered as shown in Table 9.3. In total of

10 2 7 140 cases have been analysed using the ITZ enriched CDM model formulated in

Chapter 6.

9.3 Effect of Different Parameters on Compressive Strength of

Masonry

In order to investigate the effect of different parameters influencing the compressive strength

of masonry, units with ten different characteristic unconfined compressive strengths uc

f '


have been considered, ranging from mud or unburnt clay to stone units, see Table 9.2. All

units have been analysed using two different mortar grades i.e. M3 and M4. For each set of

unit and mortar strengths, seven different masonry unit height to mortar bed joint thicknesses,

ranging from 5 to 19, have been tested as shown in Table 9.3. In total 140 different cases

have been analysed.

Table 9.2 Compressive strengths considered for units

Unit number 1 2 3 4 5 6 7 8 9 10

Unit Strength (MPa) 3.0 6.0 10 14 16 18 25 30 50 100

Table 9.3 Masonry unit height to mortar bed joint thicknesses considered for units

𝒉𝒖/𝒕𝒋

5.0 5.7 6.5 7.6 9.0 11 19

Figure 9.1 Numbering system for each case

Each case has been given a name based on their unit strength, mortar type and their height to

thickness ratio. The numbering system is explained in Figure 9.1. As can be seen in this

figure, the first number represents type of mortar (1.0 for type M3 and 2.0 for type M4). The

second number represents the height to thickness ratio of the prism. And the last number

represents the unit numbers shown in Table 9.2. All case names and their corresponding unit

strength, mortar strength and height to thickness ratios can be found in Appendix G. Figure

9.1 shows the example of Mortar type M4, height to thickness ratio 5.7 and unit number 6.


Figure 9.2 Dimensions of the RVEs used for different height to thickness ratios

Dimensions of the RVEs used for this set of analysis are shown in Figure 9.2. Total

dimensions of the RVE are kept as constant. However, the height and length of the unit (and

mortar) is changing within the RVE in order to obtain different height to thickness ratios. The

RVEs are loaded under uniaxial compression normal to bed joint as shown in Figure E-1 of

Appendix E.


9.3.1 Effect of Characteristic Unconfined Compressive Strength

of Masonry Unit 𝒇′𝒖𝒄

The influence of characteristic unconfined compressive strength of masonry unit 𝑓′𝑢𝑐,

utilizing the proposed model has been investigated in this section. Table 9.2 shows the

different unit compressive strengths analysed here.

Figures 9.3 and 9.4 illustrate the predicted masonry compressive strengths of the proposed

model and AS3700 for height to thickness ratios of 7.6 and 19, respectively. Unit strengths

corresponding to 3, 6 10, 16, 18, 25, 30, 50 and 100 MPa are plotted for AS3700. These

Figures illustrate the prism strength for two mortar strengths i.e. 5.0 MPa and 12 MPa

corresponding to type M3 and M4 mortar, respectively.


AS3700 for compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 7.6 for two mortar strengths

From Figure 9.3, it can be observed that for the case of height to thickness ratio equal to 7.6,

the AS3700 standard is giving fairly similar results as the proposed model for masonry with

unit compressive strengths less than 20 MPa for both mortar types. However, for unit

strengths more than 20 MPa, the standard seems to be over-estimating the prism’s

compressive strength. This over-estimation is more severe in case of the M3 mortar (5.0 MPa

strength).


Figure 9.4 Comparison of the predicted results using the proposed model and AS3700 for

compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 19 for two mortar strengths

Figure 9.4, illustrates the comparison between the predicted compressive strength of prism

using the model and AS3700 for height to thickness ratio of 19 for the two mortar types. It

can be seen in this Figure the both the model and standard are predicting fairly identical

prism compressive strengths except for case M1.0H/T19U10, in which the standard is over-

estimating the prism strength.

Predicted compressive strength of masonry prism using the proposed model has been

compared with AS3700 for all height to thickness ratios and can be seen in Figures F-1 to F-

14 of Appendix F. From these Figures, it can be observed that for the M3 mortar the standard

over-estimates the prism strength for unit strengths over 20 MPa and height to thickness

ratios under 11. As the height to thickness ratio increases, the over-estimation of the standard

becomes smaller.

In case of the M4 mortar, the predictions of the AS3700 standard and the model are fairly

close in most cases except M2.0H/T5.0U50, M2.0H/T5.0U100, M2.0H/T5.7U100 and

M2.0H/T6.5U100, which are the cases of low height to thickness ratio and high unit strength

(stone masonry).


9.3.2 Effect of Masonry Unit Height to Mortar Bed Joint

Thickness 𝒉𝒖/𝒕𝒋

The influence of masonry unit height to mortar bed joint thickness utilizing the proposed

model has been investigated in this section. The seven different ratios considered here are

mentioned in Table 9.3. Effect of this parameter for two different mortar strengths i.e. 5 MPa

and 12 MPa have been illustrated here.

Figures 9.5 to 9.14 compare the masonry compressive strength obtained by the proposed

model with AS3700 standard’s predictions. Here, we considered ten different unit strengths

ranging from a 3 MPa unit which can represent mud or unburnt clay to 100 MPa units

representing stone masonry. For these set of analysis two mortar types M3 and M4 have been

considered.



for two mortar strengths

Figure 9.5 illustrates a comparison between the proposed model and the AS3700 standard for

the masonry compressive strength. It can be seen that for both mortar strengths, the model is

showing almost similar compressive strengths for all height to thickness ratios considered in

the analysis. The AS3700 standard is over-estimating the masonry compressive strength in

most cases except M1.0H/T5.0U01, M1.0H/T5.7U01 and M1.0H/T6.5U01, in which it is

under-estimating the strength. The standard predicts masonry strength higher than the unit


strength for cases where unit strength is 3MPa with M4 mortar for all height to thickness

ratios, which is not rational.

Figure 9.6 Comparison of the predicted compressive strength of masonry using the proposed

model and AS3700 with unit characteristic compressive strength 𝑓′𝑢𝑐 = 6.0 𝑀𝑃𝑎 for two

mortar strengths


model and AS3700 with unit characteristic compressive strength 𝑓′𝑢𝑐 = 10 𝑀𝑃𝑎 for two

mortar strengths

It can be observed from Figures 9.6 to 9.10 that the standard and the model are predicting

quite identical masonry strengths for M3 mortar for unit strengths between 6.0 and 18 MPa.

However, the standard over-estimates the compressive strength of masonry for type M4


mortar (12 MPa strength) for unit strengths up to 10 MPa, see Figures 9.5 to 9.7. This over-

estimation seems to improve when the unit strength is higher than the mortar strength, see

Figures 9.8 to 9.10.



mortar strengths



mortar strengths


Figure 9.10 Comparison of the predicted compressive strength of masonry using the






For the mortar with the M3 type mortar (5 MPa strength), by increasing the unit strength to

values more than 25 MPa the standard over-estimates the compressive strength for smaller


height to thickness ratios (Figures 9.11 to 9.14). By increasing the unit strength the standard

over-estimates the strength for higher height to thickness ratios. For example as can be seen

in Figure 9.14, for the prism with unit strength of 100 MPa the standard over-estimates the

strength for all height to thickness ratios considered here.








On the other hand, for the masonry with type M4 mortar, both the model and the standard

illustrate similar predictions for almost all unit strengths, see Figure 9.9 to 9.13. Masonry

prisms with unit strength of 100 MPa are the exceptions of this observation. As can be seen

in Figure 9.14, in cases M2.0H/T5.0U10, M2.0H/T6.5U10 and M2.0H/T6.5U10 (low height

to thickness ratios) the standard over-estimates the masonry compressive strength.





In this section, the ability of the proposed model to produce compressive design information

of the masonry with different strengths ranging from mud or unburnt clay to stone masonry

has been investigated. Units with ten different characteristic unconfined compressive

strengths uc

f ' and mortars with two different strengths have been considered. For each set of

unit and mortar strengths, seven different masonry unit height to mortar bed joint thicknesses,

ranging from 5 to 19, have been considered.

Several observations can be made from the comparative study done in this section. The

standard seems to be over-estimating the compressive strength of masonry prisms with unit

strengths higher than 18 MPa and type M3 mortar. It is also over-estimating the masonry


compressive strength for unit strengths higher than 50 MPa and type M4 mortar with low

height to thickness ratios. The standard also over-estimates the prism compressive strength

when unit strength is lower than mortar strength (for both mortar types M3 and M4).

The AS3700 standard considers the type of mortar used in the masonry (M2, M3 and M4).

However, the mortar strength seems to be having an important effect on the masonry

compressive strength and the standards inability to incorporate this strength seems to be its

shortcoming.

The proposed model in this study can be incorporated to produce improved design clause for

the compressive strength of masonry in AS3700 standard, incorporating height to thickness

ratio, unit strength and mortar strength.

209 Conclusions and Recommendations |

CHAPTER 10

Conclusions and Recommendations

10.1 Summary

Behaviour of masonry systems under uniaxial and biaxial loading vary due to type and

dimensions of units, type and thickness of mortar layers, and workmanship reflected as the

tensile and shear bond characteristics at the unit, mortar interfaces. Under biaxial loading,

masonry systems constructed from varied units and mortar using varied skills of masons

exhibit varied shapes and sizes of failure envelopes. Obtaining failure envelopes through

experimental work for each type of masonry, although the most desirable, is time consuming

and expensive. To overcome this issue, in this thesis, a Continuum Damage Mechanics

(CDM) constitutive model for various types of ungrouted masonry systems is developed.

Utilizing this constitutive law helps in obtaining the behaviour of the complex masonry

system using simple, inexpensive tests on the constituents.

To this end, a transient-gradient continuum damage model enriched with an Interfacial

Transition Zone (ITZ) on conventional masonry has been introduced. The study focused on

the development of the biaxial failure envelop for conventional and dry-stack masonry with

different strengths and geometries and its comparison with available data in the literature. All

forms of biaxial loading conditions such as biaxial compression-compression, tension-

compression, uniaxial tension and uniaxial compression cases were analysed and reported.

Good agreement was found between the model prediction and available experiments. Finally,

the proposed model has been used to produce compressive design information of the masonry

with different strengths ranging from mud or unburnt clay to dressed stone masonry. The

predictions have been compared with the Australian masonry standard AS3700.


10.2 Conclusion

The general conclusions obtained from this research are:

1. The continuum damage mechanics (CDM) model incorporating variable Poisson’s

ratio enhanced with transient-gradient nonlocal formulation is capable of predicting

the behaviour of masonry under uniaxial and biaxial loading conditions.

2. A representative volume element containing a whole unit surrounded by half

thickness of binder will be sufficient to represent the behaviour of the masonry.

3. RVE consisting of fully bonded unit and mortar layers predicts the behaviour of

masonry unconservatively.

4. Enriching the RVE with Interfacial Transition Zone significantly improves the

predictions of the uniaxial and biaxial behaviour of all forms of masonry.

The specific conclusions with reference to the modelling method presented in this thesis are:

1. The geometry of the representative volume element has minimal effect on the overall

stress-strain behaviour of the masonry.

2. The length scale parameter for masonry in the transient-gradient nonlocal model

should be considered 5.0 𝑚𝑚2.

3. From the parametric study conducted on the ITZ thickness, it can be concluded that a

thickness between 15 to 25 percent of the joint thickness is desirable for the ITZ.

4. The continuum damage mechanics (CDM) model incorporating joint closure effects is

capable of predicting the behaviour of dry-stack masonry under uniaxial and biaxial

loading conditions.

The generic conclusions determined on the behaviour of masonry are:

1. Failure envelopes in the non-dimensional zero-shear stress plane for all masonry

systems fall in a zone with the hollow block dry-stack masonry as its lower bound and

a combination of the conventional hollow block prisms and the modified hollow block

dry-stack masonry as its upper bound envelopes.

2. The lower the imperfections on the surfaces of the masonry units, the higher the

uniaxial and biaxial strengths of dry-stack masonry.


The specific conclusions with reference to the Australian masonry standard AS3700 are:

1. The standard predicts masonry strength higher than the unit strength for cases where

unit strength is lower than 5MPa with M4 mortar for all height to thickness ratios,

which is not rational.

2. The standard over-estimates the compressive strength of masonry prisms for unit

strengths higher than 18 MPa constructed with type M3 mortar.

3. The standard also over-estimates the masonry compressive strength for unit strengths

higher than 50 MPa and constructed with type M4 mortar with low height to thickness

ratios.

4. The standard over-estimates the prism compressive strength when the unit strength is

lower than mortar strength (for both mortar types M3 and M4).

10.3 Recommendations for Future Work

In retrospect, the present thesis has successfully achieved the objectives defined in Chapter 1.

In particular, a constitutive model has been developed which is capable of simulating the

stress-strain behaviour and failure envelope of different masonry systems. The constitutive

model can be used as an effective tool to deal with analysis of masonry systems. Moreover, it

is crucial to implement the proposed model to a variety of elements and structures to gain

more experience on its usage. Some key suggestions for the future work are as follows:

A well-structured experimental program to identify the models parameters for its

constituents i.e. mortar and units.

The extension of the damage model to the three-dimensional case. This should be

fairly straightforward; however, some additional parameters might have to be

introduced to the model.

Further development of the RVE to incorporate grouting and rendering as shown in

Figures 10.1 and 10.2. For this purpose homogenisation of the RVE in two directions

is required.

Multi-scale modelling of structural elements such as masonry walls and beams by

coupling the proposed model with a macro model, considering the scale transition

concepts.

Extension of the proposed damage model to incorporate out-of-plane loads.


Incorporating the proposed model to produce improved design clause for the

compressive strength of masonry in AS3700 standard, considering height to thickness

ratio, unit strength and mortar strength.

Figure 10.1 An RVE incorporating grouting and rendering 3D view

Figure 10.2 An RVE incorporating grouting and rendering 2D view

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APPENDIX-A PYTHON codes

A-1 Appendix A |

Code A-1 Creating an ITZ embedded RVE from abaqus import * from abaqusConstants import * session.Viewport(name='Viewport: 1', origin=(0.0, 0.0), width=362.829681396484, height=267.758331298828) session.viewports['Viewport: 1'].makeCurrent() session.viewports['Viewport: 1'].maximize() from caeModules import * from driverUtils import executeOnCaeStartup executeOnCaeStartup() session.viewports['Viewport: 1'].partDisplay.geometryOptions.setValues( referenceRepresentation=ON) Mdb() #: A new model database has been created. #: The model "Model-1" has been created. session.viewports['Viewport: 1'].setValues(displayedObject=None) s = mdb.models['Model-1'].ConstrainedSketch(name='__profile__', sheetSize=400.0) g, v, d, c = s.geometry, s.vertices, s.dimensions, s.constraints s.setPrimaryObject(option=STANDALONE) s.rectangle(point1=(-120.0, -30.0), point2=(120.0, 30.0)) p = mdb.models['Model-1'].Part(name='Part-1', dimensionality=TWO_D_PLANAR, type=DEFORMABLE_BODY) p = mdb.models['Model-1'].parts['Part-1'] p.BaseShell(sketch=s) s.unsetPrimaryObject() p = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p) del mdb.models['Model-1'].sketches['__profile__'] p = mdb.models['Model-1'].parts['Part-1'] f, e, d1 = p.faces, p.edges, p.datums t = p.MakeSketchTransform(sketchPlane=f[0], sketchPlaneSide=SIDE1, origin=(0.0, 0.0, 0.0)) s1 = mdb.models['Model-1'].ConstrainedSketch(name='__profile__', sheetSize=494.77, gridSpacing=12.36, transform=t) g, v, d, c = s1.geometry, s1.vertices, s1.dimensions, s1.constraints s1.setPrimaryObject(option=SUPERIMPOSE) p = mdb.models['Model-1'].parts['Part-1'] p.projectReferencesOntoSketch(sketch=s1, filter=COPLANAR_EDGES) s1.rectangle(point1=(-120.0, -30.0), point2=(-117.5, 0.0)) s1.unsetPrimaryObject() del mdb.models['Model-1'].sketches['__profile__'] p = mdb.models['Model-1'].parts['Part-1'] f1, e1, d2 = p.faces, p.edges, p.datums t = p.MakeSketchTransform(sketchPlane=f1[0], sketchPlaneSide=SIDE1, origin=( 0.0, 0.0, 0.0)) s = mdb.models['Model-1'].ConstrainedSketch(name='__profile__', sheetSize=494.77, gridSpacing=12.36, transform=t) g, v, d, c = s.geometry, s.vertices, s.dimensions, s.constraints s.setPrimaryObject(option=SUPERIMPOSE) p = mdb.models['Model-1'].parts['Part-1'] p.projectReferencesOntoSketch(sketch=s,

filter=COPLANAR_EDGES) s.rectangle(point1=(-120.0, -30.0), point2=(-117.5, -27.5)) s.rectangle(point1=(-117.5, -30.0), point2=(-117.0, -27.5)) s.rectangle(point1=(-117.0, -30.0), point2=(-116.5, -27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.125, farPlane=496.42, width=19.9362, height=12.0737, cameraPosition=(-110.59, -26.259, 494.773), cameraTarget=(-110.59, -26.259, 0)) s.rectangle(point1=(-116.5, -30.0), point2=(-116.0, -27.5)) s.rectangle(point1=(-116.0, -30.0), point2=(-115.5, -27.5)) s.rectangle(point1=(-115.5, -30.0), point2=(-115.0, -27.5)) s.rectangle(point1=(-115.0, -30.0), point2=(-5.0, -27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.365, farPlane=497.18, width=29.1355, height=17.645, cameraPosition=(-1.9192, -23.9051, 494.773), cameraTarget=(-1.9192, -23.9051, 0)) s.rectangle(point1=(-5.0, -30.0), point2=(-2.5, -27.5)) s.rectangle(point1=(-2.5, -30.0), point2=(-2.0, -27.5)) s.rectangle(point1=(-2.0, -30.0), point2=(-1.5, -27.5)) s.rectangle(point1=(-1.5, -30.0), point2=(-1.0, -27.5)) s.rectangle(point1=(-1.0, -30.0), point2=(-0.5, -27.5)) s.rectangle(point1=(-0.5, -30.0), point2=(0.0, -27.5)) s.rectangle(point1=(0.0, -30.0), point2=(0.5, -27.5)) s.rectangle(point1=(0.5, -30.0), point2=(1.0, -27.5)) s.rectangle(point1=(1.0, -30.0), point2=(1.5, -27.5)) s.rectangle(point1=(1.5, -30.0), point2=(2.0, -27.5)) s.rectangle(point1=(2.0, -30.0), point2=(2.5, -27.5)) s.rectangle(point1=(2.5, -30.0), point2=(5.0, -27.5)) s.rectangle(point1=(5.0, -30.0), point2=(115.0, -27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.732, farPlane=496.814, width=24.6985, height=14.9579, cameraPosition=(117.05, -24.6164, 494.773), cameraTarget=(117.05, -24.6164, 0)) s.rectangle(point1=(115.0, -30.0), point2=(115.5, -27.5)) s.rectangle(point1=(115.5, -30.0), point2=(116.0, -27.5)) s.rectangle(point1=(116.0, -30.0), point2=(116.5, -27.5)) s.rectangle(point1=(116.5, -30.0), point2=(117.0, -27.5)) s.rectangle(point1=(117.0, -30.0), point2=(117.5, -27.5)) s.rectangle(point1=(117.5, -30.0), point2=(120.0, -27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=489.588, farPlane=499.957, width=62.7374, height=37.9949, cameraPosition=(-93.4505, -17.3211, 494.773), cameraTarget=(-93.4505, -17.3211, 0)) s.rectangle(point1=(-120.0, -27.0), point2=(-117.5, -26.5)) s.rectangle(point1=(-117.5, -27.0), point2=(-117.0, -26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.846, farPlane=496.699, width=26.3825, height=15.9777, cameraPosition=(-107.497, -22.9933, 494.773), cameraTarget=(-107.497, -22.9933, 0))

A-2 Appendix A |

s.rectangle(point1=(-117.0, -27.0), point2=(-116.5, -26.5)) s.rectangle(point1=(-116.5, -27.0), point2=(-116.0, -26.5)) s.rectangle(point1=(-116.0, -27.0), point2=(-115.5, -26.5)) s.rectangle(point1=(-115.5, -27.0), point2=(-115.0, -26.5)) s.rectangle(point1=(-115.0, -27.0), point2=(-5.0, -26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.715, farPlane=496.83, width=24.9011, height=15.0805, cameraPosition=(-1.21643, -23.8212, 494.773), cameraTarget=(-1.21643, -23.8212, 0)) s.rectangle(point1=(-5.0, -27.0), point2=(-2.5, -26.5)) s.rectangle(point1=(-2.5, -27.0), point2=(-2.0, -26.5)) s.rectangle(point1=(-2.0, -27.0), point2=(-1.5, -26.5)) s.rectangle(point1=(-1.5, -27.0), point2=(-1.0, -26.5)) s.rectangle(point1=(-1.0, -27.0), point2=(-0.5, -26.5)) s.rectangle(point1=(-0.5, -27.0), point2=(0.0, -26.5)) s.rectangle(point1=(0.0, -27.0), point2=(0.5, -26.5)) s.rectangle(point1=(0.5, -27.0), point2=(1.0, -26.5)) s.rectangle(point1=(1.0, -27.0), point2=(1.5, -26.5)) s.rectangle(point1=(1.5, -27.0), point2=(2.0, -26.5)) s.rectangle(point1=(2.0, -27.0), point2=(2.5, -26.5)) s.rectangle(point1=(2.5, -27.0), point2=(5.0, -26.5)) s.rectangle(point1=(5.0, -27.0), point2=(115.0, -26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.042, farPlane=496.503, width=20.9373, height=12.68, cameraPosition=(111.303, -25.0139, 494.773), cameraTarget=(111.303, -25.0139, 0)) s.rectangle(point1=(115.0, -27.0), point2=(115.5, -26.5)) s.rectangle(point1=(115.5, -27.0), point2=(116.0, -26.5)) s.rectangle(point1=(116.0, -27.0), point2=(116.5, -26.5)) s.rectangle(point1=(116.5, -27.0), point2=(117.0, -26.5)) s.rectangle(point1=(117.0, -27.0), point2=(117.5, -26.5)) s.rectangle(point1=(117.5, -27.0), point2=(120.0, -26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.907, farPlane=495.638, width=11.8502, height=7.17671, cameraPosition=(-114.234, -25.8016, 494.773), cameraTarget=(-114.234, -25.8016, 0)) s.rectangle(point1=(-120.0, -27.5), point2=(-117.5, -27.0)) s.rectangle(point1=(-117.5, -27.5), point2=(-117.0, -27.0)) s.rectangle(point1=(-117.0, -27.5), point2=(-116.5, -27.0)) s.rectangle(point1=(-116.5, -27.5), point2=(-116.0, -27.0)) s.rectangle(point1=(-116.0, -27.5), point2=(-115.5, -27.0)) s.rectangle(point1=(-115.5, -27.5), point2=(-115.0, -27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.359, farPlane=496.186, width=17.1065, height=10.36, cameraPosition=(-4.50565, -26.0558, 494.773), cameraTarget=(-4.50565, -26.0558, 0)) s.rectangle(point1=(-115.0, -27.5), point2=(-5.0, -27.0)) s.rectangle(point1=(-5.0, -27.5), point2=(-2.5, -27.0)) s.rectangle(point1=(-2.5, -27.5), point2=(-2.0, -27.0))

s.rectangle(point1=(-2.0, -27.5), point2=(-1.5, -27.0)) s.rectangle(point1=(-1.5, -27.5), point2=(-1.0, -27.0)) s.rectangle(point1=(-1.0, -27.5), point2=(-0.5, -27.0)) s.rectangle(point1=(-0.5, -27.5), point2=(0.0, -27.0)) s.rectangle(point1=(0.0, -27.5), point2=(0.5, -27.0)) s.rectangle(point1=(0.5, -27.5), point2=(1.0, -27.0)) s.rectangle(point1=(1.0, -27.5), point2=(1.5, -27.0)) s.rectangle(point1=(1.5, -27.5), point2=(2.0, -27.0)) s.rectangle(point1=(2.0, -27.5), point2=(2.5, -27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.594, farPlane=495.952, width=14.2665, height=8.64006, cameraPosition=(4.05735, -26.504, 494.773), cameraTarget=(4.05735, -26.504, 0)) s.rectangle(point1=(2.5, -27.5), point2=(5.0, -27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.893, farPlane=495.652, width=10.6427, height=6.44539, cameraPosition=(117.294, -26.4963, 494.773), cameraTarget=(117.294, -26.4963, 0)) s.rectangle(point1=(5.0, -27.5), point2=(115.0, -27.0)) s.rectangle(point1=(115.0, -27.5), point2=(115.5, -27.0)) s.rectangle(point1=(115.5, -27.5), point2=(116.0, -27.0)) s.rectangle(point1=(116.0, -27.5), point2=(116.5, -27.0)) s.rectangle(point1=(116.5, -27.5), point2=(117.0, -27.0)) s.rectangle(point1=(117.0, -27.5), point2=(117.5, -27.0)) s.rectangle(point1=(117.5, -27.5), point2=(120.0, -27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.113, farPlane=496.432, width=20.0846, height=12.1636, cameraPosition=(-114.278, -23.2024, 494.773), cameraTarget=(-114.278, -23.2024, 0)) s.rectangle(point1=(-120.0, -26.0), point2=(-117.5, -25.5)) s.rectangle(point1=(-117.5, -26.0), point2=(-117.0, -25.5)) s.rectangle(point1=(-117.0, -26.0), point2=(-116.5, -25.5)) s.rectangle(point1=(-116.5, -26.0), point2=(-116.0, -25.5)) s.rectangle(point1=(-116.0, -26.0), point2=(-115.5, -25.5)) s.rectangle(point1=(-115.5, -26.0), point2=(-115.0, -25.5)) s.rectangle(point1=(-115.0, -26.0), point2=(-5.0, -25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.371, farPlane=496.174, width=16.9566, height=10.2693, cameraPosition=(-0.982128, -23.9971, 494.773), cameraTarget=(-0.982128, -23.9971, 0)) s.rectangle(point1=(-5.0, -26.0), point2=(-2.5, -25.5)) s.rectangle(point1=(-2.5, -26.0), point2=(-2.0, -25.5)) s.rectangle(point1=(-2.0, -26.0), point2=(-1.5, -25.5)) s.rectangle(point1=(-1.5, -26.0), point2=(-1.0, -25.5)) s.rectangle(point1=(-1.0, -26.0), point2=(-0.5, -25.5)) s.rectangle(point1=(-0.5, -26.0), point2=(0.0, -25.5)) s.rectangle(point1=(0.0, -26.0), point2=(0.5, -25.5)) s.rectangle(point1=(0.5, -26.0), point2=(1.0, -25.5)) s.rectangle(point1=(1.0, -26.0), point2=(1.5, -25.5)) s.rectangle(point1=(1.5, -26.0), point2=(2.0, -25.5)) s.rectangle(point1=(2.0, -26.0), point2=(2.5, -25.5)) s.rectangle(point1=(2.5, -26.0), point2=(5.0, -25.5)) session.viewports['Viewport:

A-3 Appendix A |

1'].view.setValues(nearPlane=493.044, farPlane=496.502, width=20.9216, height=12.6705, cameraPosition=(110.593, -22.9149, 494.773), cameraTarget=(110.593, -22.9149, 0)) s.rectangle(point1=(5.0, -26.0), point2=(115.0, -25.5)) s.rectangle(point1=(115.0, -26.0), point2=(115.5, -25.5)) s.rectangle(point1=(115.5, -26.0), point2=(116.0, -25.5)) s.rectangle(point1=(116.0, -26.0), point2=(116.5, -25.5)) s.rectangle(point1=(116.5, -26.0), point2=(117.0, -25.5)) s.rectangle(point1=(117.0, -26.0), point2=(117.5, -25.5)) s.rectangle(point1=(117.5, -26.0), point2=(120.0, -25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.427, farPlane=497.119, width=28.3895, height=17.1932, cameraPosition=(-106.263, -22.8564, 494.773), cameraTarget=(-106.263, -22.8564, 0)) s.rectangle(point1=(-120.0, -26.5), point2=(-117.5, -26.0)) s.rectangle(point1=(-117.5, -26.5), point2=(-117.0, -26.0)) s.rectangle(point1=(-117.0, -26.5), point2=(-116.5, -26.0)) s.rectangle(point1=(-116.5, -26.5), point2=(-116.0, -26.0)) s.rectangle(point1=(-116.0, -26.5), point2=(-115.5, -26.0)) s.rectangle(point1=(-115.5, -26.5), point2=(-115.0, -26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.54, farPlane=497.005, width=27.015, height=16.3608, cameraPosition=(4.93193, -26.5892, 494.773), cameraTarget=(4.93193, -26.5892, 0)) s.rectangle(point1=(-115.0, -26.5), point2=(-5.0, -26.0)) s.rectangle(point1=(-5.0, -26.5), point2=(-2.5, -26.0)) s.rectangle(point1=(-2.5, -26.5), point2=(-2.0, -26.0)) s.rectangle(point1=(-2.0, -26.5), point2=(-1.5, -26.0)) s.rectangle(point1=(-1.5, -26.5), point2=(-1.0, -26.0)) s.rectangle(point1=(-1.0, -26.5), point2=(-0.5, -26.0)) s.rectangle(point1=(-0.5, -26.5), point2=(0.0, -26.0)) s.rectangle(point1=(0.0, -26.5), point2=(0.5, -26.0)) s.rectangle(point1=(0.5, -26.5), point2=(1.0, -26.0)) s.rectangle(point1=(1.0, -26.5), point2=(1.5, -26.0)) s.rectangle(point1=(1.5, -26.5), point2=(2.0, -26.0)) s.rectangle(point1=(2.0, -26.5), point2=(2.5, -26.0)) s.rectangle(point1=(2.5, -26.5), point2=(5.0, -26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.741, farPlane=495.805, width=12.4868, height=7.56223, cameraPosition=(114.378, -26.1904, 494.773), cameraTarget=(114.378, -26.1904, 0)) s.rectangle(point1=(5.0, -26.5), point2=(115.0, -26.0)) s.rectangle(point1=(115.0, -26.5), point2=(115.5, -26.0)) s.rectangle(point1=(115.5, -26.5), point2=(116.0, -26.0)) s.rectangle(point1=(116.0, -26.5), point2=(116.5, -26.0)) s.rectangle(point1=(116.5, -26.5), point2=(117.0, -26.0)) s.rectangle(point1=(117.0, -26.5), point2=(117.5, -26.0)) s.rectangle(point1=(117.5, -26.5), point2=(120.0, -26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.277, farPlane=497.268, width=30.1998, height=18.2896, cameraPosition=(-106.458, -24.8071, 494.773), cameraTarget=(-106.458, -

24.8071, 0)) s.rectangle(point1=(-120.0, -25.0), point2=(-117.5, 25.0)) s.rectangle(point1=(-117.5, -25.0), point2=(-117.0, 25.0)) s.rectangle(point1=(-117.0, -25.0), point2=(-116.5, 25.0)) s.rectangle(point1=(-116.5, -25.0), point2=(-116.0, 25.0)) s.rectangle(point1=(-116.0, -25.0), point2=(-115.5, 25.0)) s.rectangle(point1=(-115.5, -25.0), point2=(-115.0, 25.0)) s.rectangle(point1=(-115.0, -25.0), point2=(-5.0, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.044, farPlane=496.501, width=20.9192, height=12.669, cameraPosition=(-1.23187, -22.8889, 494.773), cameraTarget=(-1.23187, -22.8889, 0)) s.rectangle(point1=(-5.0, -25.0), point2=(-25.5, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.381, farPlane=499.164, width=53.1374, height=32.181, cameraPosition=(-18.8275, -22.2424, 494.773), cameraTarget=(-18.8275, -22.2424, 0)) s.delete(objectList=(g[556], )) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.381, farPlane=499.164, width=60.1374, height=36.4203, cameraPosition=(-18.3292, -21.9957, 494.773), cameraTarget=(-18.3292, -21.9957, 0)) s.rectangle(point1=(-5.0, -25.0), point2=(-2.5, 25.0)) s.rectangle(point1=(-2.5, -25.0), point2=(-2.0, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.926, farPlane=496.619, width=22.3455, height=13.5328, cameraPosition=(-1.48716, -22.0288, 494.773), cameraTarget=(-1.48716, -22.0288, 0)) s.rectangle(point1=(-2.0, -25.0), point2=(-1.5, 25.0)) s.rectangle(point1=(-1.5, -25.0), point2=(-1.0, 25.0)) s.rectangle(point1=(-1.0, -25.0), point2=(-0.5, 25.0)) s.rectangle(point1=(-0.5, -25.0), point2=(0.0, 25.0)) s.rectangle(point1=(0.0, -25.0), point2=(0.5, 25.0)) s.rectangle(point1=(0.5, -25.0), point2=(1.0, 25.0)) s.rectangle(point1=(1.0, -25.0), point2=(1.5, 25.0)) s.rectangle(point1=(1.5, -25.0), point2=(2.0, 25.0)) s.rectangle(point1=(2.0, -25.0), point2=(2.5, 25.0)) s.rectangle(point1=(2.5, -25.0), point2=(5.0, 25.0)) s.rectangle(point1=(5.0, -25.0), point2=(115.0, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.818, farPlane=495.727, width=11.5468, height=6.99296, cameraPosition=(114.386, -24.8328, 494.773), cameraTarget=(114.386, -24.8328, 0)) s.rectangle(point1=(115.0, -25.0), point2=(115.5, 25.0)) s.rectangle(point1=(115.5, -25.0), point2=(116.0, 25.0)) s.rectangle(point1=(116.0, -25.0), point2=(116.5, 25.0)) s.rectangle(point1=(116.5, -25.0), point2=(117.0, 25.0)) s.rectangle(point1=(117.0, -25.0), point2=(117.5, 25.0)) s.rectangle(point1=(117.5, -25.0), point2=(120.0, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.652, farPlane=495.894, width=13.5649, height=8.21512, cameraPosition=(-114.093, -25.0033, 494.773), cameraTarget=(-114.093, -25.0033, 0)) s.rectangle(point1=(-120.0, -25.5), point2=(-117.5, -25.0))

A-4 Appendix A |

s.rectangle(point1=(-117.5, -25.5), point2=(-117.0, -25.0)) s.rectangle(point1=(-117.0, -25.5), point2=(-116.5, -25.0)) s.rectangle(point1=(-116.5, -25.5), point2=(-116.0, -25.0)) s.rectangle(point1=(-116.0, -25.5), point2=(-115.5, -25.0)) s.rectangle(point1=(-115.5, -25.5), point2=(-115.0, -25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.843, farPlane=497.703, width=35.4554, height=21.4724, cameraPosition=(-21.2624, -23.5603, 494.773), cameraTarget=(-21.2624, -23.5603, 0)) s.rectangle(point1=(-115.0, -25.5), point2=(-5.0, -25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.174, farPlane=496.371, width=19.3433, height=11.7146, cameraPosition=(-0.73084, -24.4841, 494.773), cameraTarget=(-0.73084, -24.4841, 0)) s.rectangle(point1=(-5.0, -25.5), point2=(-2.5, -25.0)) s.rectangle(point1=(-2.5, -25.5), point2=(-2.0, -25.0)) s.rectangle(point1=(-2.0, -25.5), point2=(-1.5, -25.0)) s.rectangle(point1=(-1.5, -25.5), point2=(-1.0, -25.0)) s.rectangle(point1=(-1.0, -25.5), point2=(-0.5, -25.0)) s.rectangle(point1=(-0.5, -25.5), point2=(0.0, -25.0)) s.rectangle(point1=(0.0, -25.5), point2=(0.5, -25.0)) s.rectangle(point1=(0.5, -25.5), point2=(1.0, -25.0)) s.rectangle(point1=(1.0, -25.5), point2=(1.5, -25.0)) s.rectangle(point1=(1.5, -25.5), point2=(2.0, -25.0)) s.rectangle(point1=(2.0, -25.5), point2=(2.5, -25.0)) s.rectangle(point1=(2.5, -25.5), point2=(5.0, -25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.506, farPlane=496.04, width=15.3337, height=9.28634, cameraPosition=(113.931, -24.8814, 494.773), cameraTarget=(113.931, -24.8814, 0)) s.rectangle(point1=(5.0, -25.5), point2=(115.0, -25.0)) s.rectangle(point1=(115.0, -25.5), point2=(115.5, -25.0)) s.rectangle(point1=(115.5, -25.5), point2=(116.0, -25.0)) s.rectangle(point1=(116.0, -25.5), point2=(116.5, -25.0)) s.rectangle(point1=(116.5, -25.5), point2=(117.0, -25.0)) s.rectangle(point1=(117.0, -25.5), point2=(117.5, -25.0)) s.rectangle(point1=(117.5, -25.5), point2=(120.0, -25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.27, farPlane=497.275, width=30.2859, height=18.3417, cameraPosition=(-108.227, 28.9904, 494.773), cameraTarget=(-108.227, 28.9904, 0)) s.rectangle(point1=(-120.0, 25.5), point2=(-117.5, 26.0)) s.rectangle(point1=(-117.5, 25.5), point2=(-117.0, 26.0)) s.rectangle(point1=(-117.0, 25.5), point2=(-116.5, 26.0)) s.rectangle(point1=(-116.5, 25.5), point2=(-116.0, 26.0)) s.rectangle(point1=(-116.0, 25.5), point2=(-115.5, 26.0)) s.rectangle(point1=(-115.5, 25.5), point2=(-115.0, 26.0)) s.rectangle(point1=(-115.0, 25.5), point2=(-5.0, 26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.208, farPlane=496.337, width=18.93, height=11.4644, cameraPosition=(-0.0648434, 28.0401, 494.773), cameraTarget=(-0.0648434, 28.0401, 0))

s.rectangle(point1=(-5.0, 25.5), point2=(-25.5, 26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.61, farPlane=495.935, width=14.0641, height=8.51744, cameraPosition=(-23.4347, 28.1219, 494.773), cameraTarget=(-23.4347, 28.1219, 0)) s.undo() session.viewports['Viewport: 1'].view.setValues(nearPlane=493.916, farPlane=495.629, width=10.3639, height=6.27656, cameraPosition=(-2.2029, 26.8322, 494.773), cameraTarget=(-2.2029, 26.8322, 0)) s.rectangle(point1=(-5.0, 25.5), point2=(-2.5, 26.0)) s.rectangle(point1=(-2.5, 25.5), point2=(-2.0, 26.0)) s.rectangle(point1=(-2.0, 25.5), point2=(-1.5, 26.0)) s.rectangle(point1=(-1.5, 25.5), point2=(-1.0, 26.0)) s.rectangle(point1=(-1.0, 25.5), point2=(-0.5, 26.0)) s.rectangle(point1=(-0.5, 25.5), point2=(0.0, 26.0)) s.rectangle(point1=(0.0, 25.5), point2=(0.5, 26.0)) s.rectangle(point1=(0.5, 25.5), point2=(1.0, 26.0)) s.rectangle(point1=(1.0, 25.5), point2=(1.5, 26.0)) s.rectangle(point1=(1.5, 25.5), point2=(2.0, 26.0)) s.rectangle(point1=(2.0, 25.5), point2=(2.5, 26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.669, farPlane=497.877, width=37.5616, height=22.748, cameraPosition=(8.38624, 28.3411, 494.773), cameraTarget=(8.38624, 28.3411, 0)) s.rectangle(point1=(2.5, 25.5), point2=(5.0, 26.0)) s.rectangle(point1=(5.0, 25.5), point2=(115.0, 26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.525, farPlane=496.02, width=15.0922, height=9.14013, cameraPosition=(115.253, 27.2694, 494.773), cameraTarget=(115.253, 27.2694, 0)) s.rectangle(point1=(115.0, 25.5), point2=(115.5, 26.0)) s.rectangle(point1=(115.5, 25.5), point2=(116.0, 26.0)) s.rectangle(point1=(116.0, 25.5), point2=(116.5, 26.0)) s.rectangle(point1=(116.5, 25.5), point2=(117.0, 26.0)) s.rectangle(point1=(117.0, 25.5), point2=(117.5, 26.0)) s.rectangle(point1=(117.5, 25.5), point2=(120.0, 26.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.789, farPlane=495.756, width=11.9013, height=7.20764, cameraPosition=(-114.357, 26.1304, 494.773), cameraTarget=(-114.357, 26.1304, 0)) s.rectangle(point1=(-120.0, 25.0), point2=(-117.5, 25.5)) s.rectangle(point1=(-117.5, 25.0), point2=(-117.0, 25.5)) s.rectangle(point1=(-117.0, 25.0), point2=(-116.5, 25.5)) s.rectangle(point1=(-116.5, 25.0), point2=(-116.0, 25.5)) s.rectangle(point1=(-116.0, 25.0), point2=(-115.5, 25.5)) s.rectangle(point1=(-115.5, 25.0), point2=(-115.0, 25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.821, farPlane=495.724, width=11.5116, height=6.97161, cameraPosition=(-7.19121, 25.3347, 494.773), cameraTarget=(-7.19121, 25.3347, 0)) s.rectangle(point1=(-115.0, 25.0), point2=(-5.0, 25.5)) s.rectangle(point1=(-5.0, 25.0), point2=(-2.5, 25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.425,

A-5 Appendix A |

farPlane=496.121, width=16.3122, height=9.87895, cameraPosition=(1.61507, 24.3844, 494.773), cameraTarget=(1.61507, 24.3844, 0)) s.rectangle(point1=(-2.5, 25.0), point2=(-2.0, 25.5)) s.rectangle(point1=(-2.0, 25.0), point2=(-1.5, 25.5)) s.rectangle(point1=(-1.5, 25.0), point2=(-1.0, 25.5)) s.rectangle(point1=(-1.0, 25.0), point2=(-0.5, 25.5)) s.rectangle(point1=(-0.5, 25.0), point2=(0.0, 25.5)) s.rectangle(point1=(0.0, 25.0), point2=(0.5, 25.5)) s.rectangle(point1=(0.5, 25.0), point2=(1.0, 25.5)) s.rectangle(point1=(1.0, 25.0), point2=(1.5, 25.5)) s.rectangle(point1=(1.5, 25.0), point2=(2.0, 25.5)) s.rectangle(point1=(2.0, 25.0), point2=(2.5, 25.5)) s.rectangle(point1=(2.5, 25.0), point2=(5.0, 25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.931, farPlane=495.615, width=10.19, height=6.17124, cameraPosition=(114.961, 24.8012, 494.773), cameraTarget=(114.961, 24.8012, 0)) s.rectangle(point1=(5.0, 25.0), point2=(115.0, 25.5)) s.rectangle(point1=(115.0, 25.0), point2=(115.5, 25.5)) s.rectangle(point1=(115.5, 25.0), point2=(116.0, 25.5)) s.rectangle(point1=(116.0, 25.0), point2=(116.5, 25.5)) s.rectangle(point1=(116.5, 25.0), point2=(117.0, 25.5)) s.rectangle(point1=(117.0, 25.0), point2=(117.5, 25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.759, farPlane=495.787, width=13.8846, height=8.40878, cameraPosition=(114.895, 24.4394, 494.773), cameraTarget=(114.895, 24.4394, 0)) s.rectangle(point1=(117.5, 25.0), point2=(120.0, 25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.463, farPlane=496.082, width=15.8425, height=9.59453, cameraPosition=(-0.996821, -26.1855, 494.773), cameraTarget=(-0.996821, -26.1855, 0)) s.delete(objectList=(c[292], )) s.delete(objectList=(c[293], )) s.delete(objectList=(c[271], )) s.delete(objectList=(g[52], )) s.delete(objectList=(g[54], )) s.delete(objectList=(g[48], )) s.delete(objectList=(g[50], )) s.delete(objectList=(g[44], )) s.delete(objectList=(g[46], )) s.delete(objectList=(g[40], )) s.delete(objectList=(g[42], )) s.delete(objectList=(c[291], )) s.delete(objectList=(c[206], )) s.delete(objectList=(c[209], )) s.delete(objectList=(g[36], )) s.delete(objectList=(g[38], )) s.delete(objectList=(c[185], )) s.delete(objectList=(c[188], )) s.delete(objectList=(g[60], )) s.delete(objectList=(g[64], )) s.delete(objectList=(g[62], )) s.delete(objectList=(g[56], )) s.delete(objectList=(g[58], )) s.delete(objectList=(g[66], )) s.delete(objectList=(g[68], )) s.delete(objectList=(g[70], )) s.delete(objectList=(g[72], ))

s.delete(objectList=(g[74], )) s.delete(objectList=(g[76], )) s.delete(objectList=(g[78], )) s.delete(objectList=(c[418], )) s.delete(objectList=(c[417], )) s.rectangle(point1=(-5.0, -30.0), point2=(-4.5, -27.5)) s.rectangle(point1=(-4.5, -30.0), point2=(-4.0, -27.5)) s.rectangle(point1=(-4.0, -30.0), point2=(-3.5, -27.5)) s.rectangle(point1=(-3.5, -30.0), point2=(-3.0, -27.5)) s.rectangle(point1=(-3.0, -30.0), point2=(-2.5, -27.5)) s.rectangle(point1=(-2.5, -30.0), point2=(0.0, -27.5)) s.rectangle(point1=(0.0, -30.0), point2=(2.5, -27.5)) s.rectangle(point1=(2.5, -30.0), point2=(3.0, -27.5)) s.rectangle(point1=(3.0, -30.0), point2=(3.5, -27.5)) s.rectangle(point1=(3.5, -30.0), point2=(4.0, -27.5)) s.rectangle(point1=(4.0, -30.0), point2=(4.5, -27.5)) s.rectangle(point1=(4.5, -30.0), point2=(5.0, -27.5)) s.delete(objectList=(c[1278], )) s.delete(objectList=(g[244], )) s.delete(objectList=(g[246], )) s.delete(objectList=(g[248], )) s.delete(objectList=(g[250], )) s.delete(objectList=(g[252], )) s.delete(objectList=(g[254], )) s.delete(objectList=(g[256], )) s.delete(objectList=(g[258], )) s.delete(objectList=(g[260], )) s.delete(objectList=(g[262], )) s.delete(objectList=(g[264], )) s.delete(objectList=(g[266], )) s.delete(objectList=(g[268], )) s.delete(objectList=(g[270], )) s.delete(objectList=(g[272], )) s.delete(objectList=(g[274], )) s.delete(objectList=(g[276], )) s.delete(objectList=(g[278], )) s.delete(objectList=(g[280], )) s.delete(objectList=(g[282], )) s.delete(objectList=(c[1488], )) s.delete(objectList=(c[1489], )) s.delete(objectList=(c[1280], )) s.delete(objectList=(c[1277], )) s.delete(objectList=(c[734], )) s.delete(objectList=(c[731], )) s.delete(objectList=(c[2372], )) s.delete(objectList=(c[2369], )) s.delete(objectList=(c[1826], )) s.delete(objectList=(c[1823], )) s.delete(objectList=(c[3485], )) s.delete(objectList=(c[3482], )) s.delete(objectList=(g[140], )) s.delete(objectList=(g[142], )) s.delete(objectList=(g[452], )) s.delete(objectList=(g[454], )) s.delete(objectList=(g[348], )) s.delete(objectList=(g[350], )) s.delete(objectList=(c[3506], )) s.delete(objectList=(g[664], )) s.delete(objectList=(g[666], )) s.delete(objectList=(g[668], )) s.delete(objectList=(g[670], )) s.delete(objectList=(g[352], )) s.delete(objectList=(g[354], )) s.delete(objectList=(g[456], )) s.delete(objectList=(g[458], )) s.delete(objectList=(g[144], )) s.delete(objectList=(g[146], ))

A-6 Appendix A |

s.delete(objectList=(g[148], )) s.delete(objectList=(g[150], )) s.delete(objectList=(g[152], )) s.delete(objectList=(g[154], )) s.delete(objectList=(g[156], )) s.delete(objectList=(g[158], )) s.delete(objectList=(g[160], )) s.delete(objectList=(g[162], )) s.delete(objectList=(g[164], )) s.delete(objectList=(g[166], )) s.delete(objectList=(g[168], )) s.delete(objectList=(g[170], )) s.delete(objectList=(g[172], )) s.delete(objectList=(g[174], )) s.delete(objectList=(g[176], )) s.delete(objectList=(g[178], )) s.delete(objectList=(c[1323], )) s.delete(objectList=(g[460], )) s.delete(objectList=(g[462], )) s.delete(objectList=(c[756], c[777], c[798])) s.delete(objectList=(g[464], g[466], g[468], g[470], g[472], g[474], g[476], g[478], g[480], g[482], g[484], g[486], g[488], g[490], c[819], c[840], c[861], c[882], c[903], c[924], c[945], c[2434], c[2455], c[2456], c[2476], c[2477], c[2497], c[2498], c[2518], c[2519], c[2539], c[2540], c[2560], c[2561], c[2582])) s.delete(objectList=(c[1281], c[1302], c[1344], c[1365], c[1386], c[1407], c[1428], c[1449], c[1470], c[1491])) s.delete(objectList=(c[210], c[231], c[252], c[273], c[294], c[315], c[336], c[357], c[378], c[399], c[5082], c[5103])) s.delete(objectList=(g[356], g[358], g[360], g[362], g[364], g[366], g[368], g[370], g[372], g[374], g[376], g[378], g[380], g[382], g[384], g[386], c[1867], c[1888], c[1889], c[1909], c[1910], c[1930], c[1931], c[1951], c[1952], c[1972], c[1973], c[1993], c[1994], c[2014], c[2015], c[2036], c[2373], c[2394], c[2415], c[2436], c[2457], c[2478], c[2499], c[2520], c[2541], c[2562], c[2583])) s.delete(objectList=(c[735], )) s.delete(objectList=(g[672], g[674], g[676], g[678], g[680], g[682], g[684], g[686], g[688], g[690], g[692], g[694], g[696], g[698], g[700], g[702], c[1827], c[1848], c[1869], c[1890], c[1911], c[1932], c[1953], c[1974], c[1995], c[2016], c[2037], c[3526], c[3547], c[3548], c[3568], c[3569], c[3589], c[3590], c[3610], c[3611], c[3631], c[3632], c[3652], c[3653], c[3673], c[3674], c[3695])) s.delete(objectList=(g[560], g[562], g[564], g[566], g[568], g[570], g[572], g[574], g[576], g[578], g[580], g[582], g[584], g[586], g[588], g[590], g[592], g[594], g[596], g[598], c[2938], c[2959], c[2960], c[2980], c[2981], c[3001], c[3002], c[3022], c[3023], c[3043], c[3044], c[3064], c[3065], c[3085], c[3086], c[3106], c[3107], c[3127],

c[3128], c[3149], c[3486], c[3507], c[3528], c[3549], c[3570], c[3591], c[3612], c[3633], c[3654], c[3675], c[3696])) s.delete(objectList=(g[39], g[43], g[47], g[51], g[55], g[249], g[253], g[257], g[261], g[265], g[967])) s.delete(objectList=(g[2], g[41], g[45], g[49], g[53], g[57], g[969])) s.delete(objectList=(g[61], g[65], g[69], g[73], g[77], g[973], c[5102])) s.rectangle(point1=(-2.5, -30.0), point2=(0.0, -27.5)) s.delete(objectList=(g[59], g[63], g[67], g[71], g[75], g[269], g[273], g[277], g[281], g[285], g[971])) s.rectangle(point1=(0.0, -30.0), point2=(2.5, -27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.653, farPlane=495.892, width=13.5479, height=8.20484, cameraPosition=(-1.50289, -25.4435, 494.773), cameraTarget=(-1.50289, -25.4435, 0)) s.rectangle(point1=(-5.0, -27.5), point2=(-4.5, -27.0)) s.rectangle(point1=(-4.5, -27.5), point2=(-4.0, -27.0)) s.rectangle(point1=(-4.0, -27.5), point2=(-3.5, -27.0)) s.delete(objectList=(g[141], )) s.delete(objectList=(g[243], )) s.delete(objectList=(g[145], g[149], g[153], g[157], g[161], g[165], g[169], g[173], g[177], g[181], g[247], g[251], g[255], g[259], g[263], g[267], g[271], g[275], g[279], g[283])) s.delete(objectList=(g[139], g[143], g[147], g[151], g[155], g[159], g[163], g[167], g[171], g[175], g[179], g[347], g[349], g[351], g[353], g[355], g[357], g[359], g[361], g[363], g[365], g[367], g[369], g[371], g[373], g[375], g[377], g[379], g[381], g[383], g[385], g[387], g[389], g[451], g[453], g[455], g[457], g[459], g[461], g[463], g[465], g[467], g[469], g[471], g[473], g[475], g[477], g[479], g[481], g[483], g[485], g[487], g[489], g[491], g[493], g[561], g[565], g[569], g[573], g[577], g[581], g[585], g[589], g[593], g[597], g[601], g[663], g[665], g[667], g[669], g[671], g[673], g[675], g[677], g[679], g[681], g[683], g[685], g[687], g[689], g[691], g[693], g[695], g[697], g[699], g[701], g[703], g[705])) s.rectangle(point1=(-3.5, -27.5), point2=(-3.0, -27.0)) s.rectangle(point1=(-3.0, -27.5), point2=(-2.5, -27.0)) s.rectangle(point1=(-2.5, -27.5), point2=(2.5, -27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.339, farPlane=496.207, width=19.6383, height=11.8933, cameraPosition=(-1.19556, -24.7085, 494.773), cameraTarget=(-1.19556, -24.7085, 0)) s.undo() s.rectangle(point1=(-2.5, -27.5), point2=(0.0, -27.0)) s.delete(objectList=(g[180], g[182], g[183], g[185], g[284], g[286], g[287], g[388], g[390], g[391], g[393], g[492], g[494], g[495],

A-7 Appendix A |

g[497], g[600], g[602], g[605], g[704], g[706], g[707], g[709], c[420], c[962], c[963], c[964], c[965], c[966], c[1508], c[1509], c[1510], c[1511], c[1512], c[2054], c[2055], c[2056], c[2057], c[2058], c[2600], c[2601], c[2602], c[2603], c[2604], c[3169], c[3170], c[3713], c[3714], c[3715], c[3716], c[3717], c[5124], c[5145], c[5166], c[5187], c[5208])) s.rectangle(point1=(0.0, -27.5), point2=(2.5, -27.0)) s.rectangle(point1=(2.5, -27.5), point2=(3.0, -27.0)) s.rectangle(point1=(3.0, -27.5), point2=(3.5, -27.0)) s.rectangle(point1=(3.5, -27.5), point2=(4.0, -27.0)) s.rectangle(point1=(4.0, -27.5), point2=(4.5, -27.0)) s.rectangle(point1=(4.5, -27.5), point2=(5.0, -27.0)) s.rectangle(point1=(-5.0, -26.5), point2=(-4.5, -26.0)) s.rectangle(point1=(-4.5, -26.5), point2=(-4.0, -26.0)) s.rectangle(point1=(-4.0, -26.5), point2=(-3.0, -26.0)) s.undo() s.rectangle(point1=(-4.0, -26.5), point2=(-3.5, -26.0)) s.rectangle(point1=(-3.5, -26.5), point2=(-3.0, -26.0)) s.rectangle(point1=(-3.0, -26.5), point2=(-2.5, -26.0)) s.rectangle(point1=(-2.5, -26.5), point2=(0.0, -26.0)) s.rectangle(point1=(0.0, -26.5), point2=(2.5, -26.0)) s.rectangle(point1=(2.5, -26.5), point2=(3.0, -26.0)) s.rectangle(point1=(3.0, -26.5), point2=(3.5, -26.0)) s.rectangle(point1=(3.5, -26.5), point2=(4.0, -26.0)) s.rectangle(point1=(4.0, -26.5), point2=(4.5, -26.0)) s.rectangle(point1=(4.5, -26.5), point2=(5.0, -26.0)) s.rectangle(point1=(-5.0, -27.0), point2=(-4.5, -26.5)) s.rectangle(point1=(-4.5, -27.0), point2=(-4.0, -26.5)) s.rectangle(point1=(-4.0, -27.0), point2=(-3.5, -26.5)) s.rectangle(point1=(-3.5, -27.0), point2=(-3.0, -26.5)) s.rectangle(point1=(-3.0, -27.0), point2=(-2.5, -26.5)) s.rectangle(point1=(-2.5, -27.0), point2=(0.0, -26.5)) s.rectangle(point1=(0.0, -27.0), point2=(2.5, -26.5)) s.rectangle(point1=(2.5, -27.0), point2=(3.0, -26.5)) s.rectangle(point1=(3.0, -27.0), point2=(3.5, -26.5)) s.rectangle(point1=(3.5, -27.0), point2=(4.0, -26.5)) s.rectangle(point1=(4.0, -27.0), point2=(4.5, -26.5)) s.rectangle(point1=(4.5, -27.0), point2=(5.0, -26.5)) s.rectangle(point1=(-5.0, -25.5), point2=(-4.5, -25.0)) s.rectangle(point1=(-4.5, -25.5), point2=(-4.0, -25.0)) s.rectangle(point1=(-4.0, -25.5), point2=(-3.5, -25.0)) s.rectangle(point1=(-3.5, -25.5), point2=(-3.0, -25.0)) s.rectangle(point1=(-3.0, -25.5), point2=(-2.5, -25.0)) s.rectangle(point1=(-2.5, -25.5), point2=(0.0, -25.0)) s.rectangle(point1=(0.0, -25.5), point2=(2.5, -25.0)) s.rectangle(point1=(2.5, -25.5), point2=(3.0, -25.0)) s.rectangle(point1=(3.0, -25.5), point2=(3.5, -25.0)) s.rectangle(point1=(3.5, -25.5), point2=(4.0, -25.0)) s.rectangle(point1=(4.0, -25.5), point2=(4.5, -25.0)) s.rectangle(point1=(4.5, -25.5), point2=(5.0, -25.0)) s.rectangle(point1=(-5.0, -26.0), point2=(-4.5, -25.5)) s.rectangle(point1=(-4.5, -26.0), point2=(-4.0, -25.5)) s.rectangle(point1=(-4.0, -26.0), point2=(-3.5, -25.5)) s.rectangle(point1=(-3.5, -26.0), point2=(-3.0, -25.5)) s.rectangle(point1=(-3.0, -26.0), point2=(-2.5, -25.5)) s.rectangle(point1=(-2.5, -26.0), point2=(0.0, -25.5)) s.rectangle(point1=(0.0, -26.0), point2=(2.5, -25.5)) s.rectangle(point1=(2.5, -26.0), point2=(3.0, -25.5)) s.rectangle(point1=(3.0, -26.0), point2=(3.5, -25.5)) s.rectangle(point1=(3.5, -26.0), point2=(4.0, -25.5)) s.rectangle(point1=(4.0, -26.0), point2=(4.5, -25.5)) s.rectangle(point1=(4.5, -26.0), point2=(5.0, -25.5)) session.viewports['Viewport:

1'].view.setValues(nearPlane=493.143, farPlane=496.402, width=19.7187, height=11.942, cameraPosition=(1.20528, 24.058, 494.773), cameraTarget=(1.20528, 24.058, 0)) s.delete(objectList=(g[559], g[563], g[567], g[571], g[575], g[579], g[583], g[587], g[591], g[595], g[599], g[603], g[767], g[768], g[769], g[770], g[771], g[772], g[773], g[774], g[775], g[776], g[777], g[778], g[779], g[780], g[781], g[782], g[783], g[784], g[785], g[786], g[787], g[788], g[789], g[790], g[791], g[792], g[793], g[794], g[795], g[796], g[797], g[798], g[799], g[800], g[801], g[802], g[803], g[804], g[805], g[806], g[807], g[808], g[809], g[810], g[811], g[813], g[871], g[872], g[873], g[874], g[875], g[876], g[877], g[878], g[879], g[880], g[881], g[882], g[883], g[884], g[885], g[886], g[887], g[888], g[889], g[890], g[891], g[892], g[893], g[894], g[895], g[896], g[897], g[898], g[899], g[900], g[901], g[902], g[903], g[904], g[905], g[906], g[907], g[908], g[909], g[910], g[911], g[912], g[913], g[914], g[915], g[917], c[2940], c[2961], c[2982], c[3003], c[3024], c[3045], c[3066], c[3087], c[3108], c[3129], c[3150], c[3171], c[4029], c[4030], c[4032], c[4049], c[4050], c[4051], c[4052], c[4053], c[4070], c[4071], c[4072], c[4073], c[4074], c[4091], c[4092], c[4093], c[4094], c[4095], c[4112], c[4113], c[4114], c[4115], c[4116], c[4133], c[4134], c[4135], c[4136], c[4137], c[4154], c[4155], c[4156], c[4157], c[4158], c[4175], c[4176], c[4177], c[4178], c[4179], c[4196], c[4197], c[4198], c[4199], c[4200], c[4217], c[4218], c[4219], c[4220], c[4221], c[4238], c[4239], c[4240], c[4241], c[4242], c[4259], c[4262], c[4263], c[4575], c[4576], c[4578], c[4595], c[4596], c[4597], c[4598], c[4599], c[4616], c[4617], c[4618], c[4619], c[4620], c[4637], c[4638], c[4639], c[4640], c[4641], c[4658], c[4659], c[4660], c[4661], c[4662], c[4679], c[4680], c[4681], c[4682], c[4683], c[4700], c[4701], c[4702], c[4703], c[4704], c[4721], c[4722], c[4723], c[4724], c[4725], c[4742], c[4743], c[4744], c[4745], c[4746], c[4763], c[4764], c[4765], c[4766], c[4767], c[4784], c[4785], c[4786], c[4787], c[4788], c[4805], c[4808], c[4809])) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.966, farPlane=495.58, width=9.76518, height=5.91397, cameraPosition=(-3.03115, -25.1376, 494.773), cameraTarget=(-3.03115, -25.1376, 0)) s.rectangle(point1=(-5.0, -25.0), point2=(-4.5, 25.0)) session.viewports['Viewport:

A-8 Appendix A |

1'].view.setValues(nearPlane=493.935, farPlane=495.61, width=10.1307, height=6.13531, cameraPosition=(-2.29185, -23.8043, 494.773), cameraTarget=(-2.29185, -23.8043, 0)) s.rectangle(point1=(-4.5, -25.0), point2=(-4.0, 25.0)) s.rectangle(point1=(-4.0, -25.0), point2=(-3.5, 25.0)) s.rectangle(point1=(-3.5, -25.0), point2=(-3.0, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.017, farPlane=496.529, width=21.2503, height=12.8696, cameraPosition=(0.357925, -21.3761, 494.773), cameraTarget=(0.357925, -21.3761, 0)) s.rectangle(point1=(-3.0, -25.0), point2=(-2.5, 25.0)) s.rectangle(point1=(-2.5, -25.0), point2=(0.0, 25.0)) s.rectangle(point1=(0.0, -25.0), point2=(2.5, 25.0)) s.rectangle(point1=(2.5, -25.0), point2=(3.0, 25.0)) s.rectangle(point1=(3.0, -25.0), point2=(3.5, 25.0)) s.rectangle(point1=(3.5, -25.0), point2=(4.0, 25.0)) s.rectangle(point1=(4.0, -25.0), point2=(4.5, 25.0)) s.rectangle(point1=(4.5, -25.0), point2=(5.0, 25.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.762, farPlane=495.783, width=12.2261, height=7.40436, cameraPosition=(-0.101947, 25.2013, 494.773), cameraTarget=(-0.101947, 25.2013, 0)) s.rectangle(point1=(-5.0, 25.5), point2=(-4.5, 26.0)) s.rectangle(point1=(-4.5, 25.5), point2=(-4.0, 26.0)) s.rectangle(point1=(-4.0, 25.5), point2=(-3.5, 26.0)) s.rectangle(point1=(-3.5, 25.5), point2=(-3.0, 26.0)) s.undo() s.rectangle(point1=(-3.5, 25.5), point2=(-3.0, 26.0)) s.rectangle(point1=(-3.0, 25.5), point2=(-2.5, 26.0)) s.rectangle(point1=(-2.5, 25.5), point2=(0.0, 26.0)) s.rectangle(point1=(0.0, 25.5), point2=(2.5, 26.0)) s.rectangle(point1=(2.5, 25.5), point2=(3.0, 26.0)) s.rectangle(point1=(3.0, 25.5), point2=(3.5, 26.0)) s.rectangle(point1=(3.5, 25.5), point2=(4.0, 26.0)) s.rectangle(point1=(4.0, 25.5), point2=(4.5, 26.0)) s.rectangle(point1=(4.5, 25.5), point2=(5.0, 26.0)) s.rectangle(point1=(-5.0, 25.0), point2=(-4.5, 25.5)) s.rectangle(point1=(-4.5, 25.0), point2=(-4.0, 25.5)) s.rectangle(point1=(-4.0, 25.0), point2=(-3.5, 25.5)) s.rectangle(point1=(-3.5, 25.0), point2=(-3.0, 25.5)) s.rectangle(point1=(-3.0, 25.0), point2=(-2.5, 25.5)) s.rectangle(point1=(-2.5, 25.0), point2=(0.0, 25.5)) s.rectangle(point1=(0.0, 25.0), point2=(2.5, 25.5)) s.rectangle(point1=(2.5, 25.0), point2=(3.0, 25.5)) s.rectangle(point1=(3.0, 25.0), point2=(3.5, 25.5)) s.rectangle(point1=(3.5, 25.0), point2=(4.0, 25.5)) s.rectangle(point1=(4.0, 25.0), point2=(4.5, 25.5)) s.rectangle(point1=(4.5, 25.0), point2=(5.0, 25.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.22, farPlane=496.325, width=18.7865, height=11.3774, cameraPosition=(-112.031, 24.8639, 494.773), cameraTarget=(-112.031, 24.8639, 0)) s.rectangle(point1=(-120.0, 26.5), point2=(-117.5, 27.0)) s.rectangle(point1=(-117.5, 26.5), point2=(-117.0, 27.0)) s.rectangle(point1=(-117.0, 26.5), point2=(-116.5, 27.0)) s.rectangle(point1=(-116.5, 26.5), point2=(-116.0, 27.0)) s.rectangle(point1=(-116.0, 26.5), point2=(-115.5, 27.0)) s.rectangle(point1=(-115.5, 26.5), point2=(-115.0, 27.0)) s.rectangle(point1=(-115.0, 26.5), point2=(-5.0, 27.0))

session.viewports['Viewport: 1'].view.setValues(nearPlane=493.546, farPlane=496, width=14.8483, height=8.9924, cameraPosition=(-0.141583, 25.9309, 494.773), cameraTarget=(-0.141583, 25.9309, 0)) s.rectangle(point1=(-5.0, 26.5), point2=(-4.5, 27.0)) s.rectangle(point1=(-4.5, 26.5), point2=(-4.0, 27.0)) s.rectangle(point1=(-4.0, 26.5), point2=(-3.5, 27.0)) s.rectangle(point1=(-3.5, 26.5), point2=(-3.0, 27.0)) s.rectangle(point1=(-3.0, 26.5), point2=(-2.5, 27.0)) s.rectangle(point1=(-2.5, 26.5), point2=(0.0, 27.0)) s.rectangle(point1=(0.0, 26.5), point2=(2.5, 27.0)) s.rectangle(point1=(2.5, 26.5), point2=(3.0, 27.0)) s.rectangle(point1=(3.0, 26.5), point2=(3.5, 27.0)) s.rectangle(point1=(3.5, 26.5), point2=(4.0, 27.0)) s.rectangle(point1=(4.0, 26.5), point2=(4.5, 27.0)) s.rectangle(point1=(4.5, 26.5), point2=(5.0, 27.0)) s.rectangle(point1=(5.0, 26.5), point2=(115.0, 27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.905, farPlane=495.64, width=10.4974, height=6.35743, cameraPosition=(115.088, 26.2927, 494.773), cameraTarget=(115.088, 26.2927, 0)) s.rectangle(point1=(115.0, 26.5), point2=(115.5, 27.0)) s.rectangle(point1=(115.5, 26.5), point2=(116.0, 27.0)) s.rectangle(point1=(116.0, 26.5), point2=(116.5, 27.0)) s.rectangle(point1=(116.5, 26.5), point2=(117.0, 27.0)) s.rectangle(point1=(117.0, 26.5), point2=(117.5, 27.0)) s.rectangle(point1=(117.5, 26.5), point2=(120.0, 27.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.768, farPlane=496.777, width=24.258, height=14.6911, cameraPosition=(-109.456, 26.1291, 494.773), cameraTarget=(-109.456, 26.1291, 0)) s.rectangle(point1=(-120.0, 26.0), point2=(-117.5, 26.5)) s.rectangle(point1=(-117.5, 26.0), point2=(-117.0, 26.5)) s.rectangle(point1=(-117.0, 26.0), point2=(-116.5, 26.5)) s.rectangle(point1=(-116.5, 26.0), point2=(-116.0, 26.5)) s.rectangle(point1=(-116.0, 26.0), point2=(-115.5, 26.5)) s.rectangle(point1=(-115.5, 26.0), point2=(-115.0, 26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.045, farPlane=496.5, width=20.9024, height=12.6589, cameraPosition=(2.06147, 24.9273, 494.773), cameraTarget=(2.06147, 24.9273, 0)) s.rectangle(point1=(-115.0, 26.0), point2=(-5.0, 26.5)) s.rectangle(point1=(-5.0, 26.0), point2=(-4.5, 26.5)) s.rectangle(point1=(-4.5, 26.0), point2=(-4.0, 26.5)) s.rectangle(point1=(-4.0, 26.0), point2=(-3.5, 26.5)) s.rectangle(point1=(-3.5, 26.0), point2=(-3.0, 26.5)) s.rectangle(point1=(-3.0, 26.0), point2=(-2.5, 26.5)) s.rectangle(point1=(-2.5, 26.0), point2=(0.0, 26.5)) s.rectangle(point1=(0.0, 26.0), point2=(2.5, 26.5)) s.rectangle(point1=(2.5, 26.0), point2=(3.0, 26.5)) s.rectangle(point1=(3.0, 26.0), point2=(3.5, 26.5)) s.rectangle(point1=(3.5, 26.0), point2=(4.0, 26.5)) s.rectangle(point1=(4.0, 26.0), point2=(4.5, 26.5)) s.rectangle(point1=(4.5, 26.0), point2=(5.0, 26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.5, farPlane=496.046, width=15.4022, height=9.32786, cameraPosition=(115.876, 26.3474, 494.773), cameraTarget=(115.876, 26.3474,

A-9 Appendix A |

0)) s.rectangle(point1=(5.0, 26.0), point2=(115.0, 26.5)) s.rectangle(point1=(115.0, 26.0), point2=(115.5, 26.5)) s.rectangle(point1=(115.5, 26.0), point2=(116.0, 26.5)) s.rectangle(point1=(116.0, 26.0), point2=(116.5, 26.5)) s.rectangle(point1=(116.5, 26.0), point2=(117.0, 26.5)) s.rectangle(point1=(117.0, 26.0), point2=(117.5, 26.5)) s.rectangle(point1=(117.5, 26.0), point2=(120.0, 26.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.965, farPlane=496.58, width=21.8739, height=13.2472, cameraPosition=(-111.757, 30.2881, 494.773), cameraTarget=(-111.757, 30.2881, 0)) s.rectangle(point1=(-120.0, 27.5), point2=(-117.5, 30.0)) s.rectangle(point1=(-117.5, 27.5), point2=(-117.0, 30.0)) s.rectangle(point1=(-117.0, 27.5), point2=(-116.5, 30.0)) s.rectangle(point1=(-116.5, 27.5), point2=(-116.0, 30.0)) s.rectangle(point1=(-116.0, 27.5), point2=(-115.5, 30.0)) s.rectangle(point1=(-115.5, 27.5), point2=(-115.0, 30.0)) s.rectangle(point1=(-115.0, 27.5), point2=(-5.0, 30.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.038, farPlane=496.507, width=20.9855, height=12.7092, cameraPosition=(1.74713, 29.7805, 494.773), cameraTarget=(1.74713, 29.7805, 0)) s.rectangle(point1=(-5.0, 27.5), point2=(-4.5, 30.0)) s.rectangle(point1=(-4.5, 27.5), point2=(-4.0, 30.0)) s.rectangle(point1=(-4.0, 27.5), point2=(-3.5, 30.0)) s.rectangle(point1=(-3.5, 27.5), point2=(-3.0, 30.0)) s.rectangle(point1=(-3.0, 27.5), point2=(-2.5, 30.0)) s.rectangle(point1=(-2.5, 27.5), point2=(0.0, 30.0)) s.rectangle(point1=(0.0, 27.5), point2=(2.5, 30.0)) s.rectangle(point1=(2.5, 27.5), point2=(3.0, 30.0)) s.rectangle(point1=(3.0, 27.5), point2=(3.5, 30.0)) s.rectangle(point1=(3.5, 27.5), point2=(4.0, 30.0)) s.rectangle(point1=(4.0, 27.5), point2=(4.5, 30.0)) s.rectangle(point1=(4.5, 27.5), point2=(5.0, 30.0)) s.rectangle(point1=(5.0, 27.5), point2=(115.0, 30.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.396, farPlane=496.149, width=16.6568, height=10.0877, cameraPosition=(115.908, 29.1335, 494.773), cameraTarget=(115.908, 29.1335, 0)) s.rectangle(point1=(115.0, 27.5), point2=(115.5, 30.0)) s.rectangle(point1=(115.5, 27.5), point2=(116.0, 30.0)) s.rectangle(point1=(116.0, 27.5), point2=(116.5, 30.0)) s.rectangle(point1=(116.5, 27.5), point2=(117.0, 30.0)) s.rectangle(point1=(117.0, 27.5), point2=(117.5, 30.0)) s.rectangle(point1=(117.5, 27.5), point2=(120.0, 30.0)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.366, farPlane=496.179, width=17.019, height=10.307, cameraPosition=(-113.52, 27.3936, 494.773), cameraTarget=(-113.52, 27.3936, 0)) s.rectangle(point1=(-120.0, 27.0), point2=(-117.5, 27.5)) s.rectangle(point1=(-117.5, 27.0), point2=(-117.0, 27.5)) s.rectangle(point1=(-117.0, 27.0), point2=(-116.5, 27.5)) s.rectangle(point1=(-116.5, 27.0), point2=(-116.0, 27.5)) s.rectangle(point1=(-116.0, 27.0), point2=(-115.5, 27.5)) s.rectangle(point1=(-115.5, 27.0), point2=(-115.0, 27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.728, farPlane=495.818, width=12.6442, height=7.65756,

cameraPosition=(-4.28608, 27.0401, 494.773), cameraTarget=(-4.28608, 27.0401, 0)) s.rectangle(point1=(-115.0, 27.0), point2=(-5.0, 27.5)) s.rectangle(point1=(-5.0, 27.0), point2=(-4.5, 27.5)) s.rectangle(point1=(-4.5, 27.0), point2=(-4.0, 27.5)) s.rectangle(point1=(-4.0, 27.0), point2=(-3.5, 27.5)) s.rectangle(point1=(-3.5, 27.0), point2=(-3.0, 27.5)) s.rectangle(point1=(-3.0, 27.0), point2=(-2.5, 27.5)) s.rectangle(point1=(-2.5, 27.0), point2=(0.0, 27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.58, farPlane=495.965, width=14.4272, height=8.73737, cameraPosition=(3.55404, 27.0009, 494.773), cameraTarget=(3.55404, 27.0009, 0)) s.rectangle(point1=(0.0, 27.0), point2=(2.5, 27.5)) s.rectangle(point1=(2.5, 27.0), point2=(3.0, 27.5)) s.rectangle(point1=(3.0, 27.0), point2=(3.5, 27.5)) s.rectangle(point1=(3.5, 27.0), point2=(4.0, 27.5)) s.rectangle(point1=(4.0, 27.0), point2=(4.5, 27.5)) s.rectangle(point1=(4.5, 27.0), point2=(5.0, 27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.81, farPlane=495.735, width=11.6447, height=7.05225, cameraPosition=(112.813, 27.82, 494.773), cameraTarget=(112.813, 27.82, 0)) s.rectangle(point1=(5.0, 27.0), point2=(115.0, 27.5)) s.rectangle(point1=(115.0, 27.0), point2=(115.5, 27.5)) s.rectangle(point1=(115.5, 27.0), point2=(116.0, 27.5)) s.rectangle(point1=(116.0, 27.0), point2=(116.5, 27.5)) s.rectangle(point1=(116.5, 27.0), point2=(117.0, 27.5)) s.rectangle(point1=(117.0, 27.0), point2=(117.5, 27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.97, farPlane=495.575, width=9.71148, height=5.88145, cameraPosition=(117.338, 27.665, 494.773), cameraTarget=(117.338, 27.665, 0)) s.rectangle(point1=(117.5, 27.0), point2=(120.0, 27.5)) session.viewports['Viewport: 1'].view.setValues(nearPlane=475.222, farPlane=514.323, width=267.743, height=162.15, cameraPosition=(3.52915, 16.5952, 494.773), cameraTarget=(3.52915, 16.5952, 0)) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces pickedFaces = f.getSequenceFromMask(mask=('[#1 ]', ), ) e, d1 = p.edges, p.datums p.PartitionFaceBySketch(faces=pickedFaces, sketch=s) s.unsetPrimaryObject() del mdb.models['Model-1'].sketches['__profile__'] session.viewports['Viewport: 1'].view.setValues(nearPlane=474.843, farPlane=514.703, width=241.849, height=146.468, viewOffsetX=3.90336, viewOffsetY=7.31571) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.meshOptions.setValues( meshTechnique=ON) session.viewports['Viewport: 1'].partDisplay.geometryOptions.setValues( referenceRepresentation=OFF) elemType1 = mesh.ElemType(elemCode=CPS8R,

A-10 Appendix A |

elemLibrary=STANDARD) elemType2 = mesh.ElemType(elemCode=CPS6M, elemLibrary=STANDARD) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#ffffffff:10 #3ffff ]', ), ) pickedRegions =(faces, ) p.setElementType(regions=pickedRegions, elemTypes=(elemType1, elemType2)) elemType1 = mesh.ElemType(elemCode=CPS8R, elemLibrary=STANDARD) elemType2 = mesh.ElemType(elemCode=CPS6M, elemLibrary=STANDARD) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#ffffffff:10 #3ffff ]', ), ) pickedRegions =(faces, ) p.setElementType(regions=pickedRegions, elemTypes=(elemType1, elemType2)) session.viewports['Viewport: 1'].view.setValues(nearPlane=483.105, farPlane=506.44, width=140.99, height=85.6323, viewOffsetX=-59.5001, viewOffsetY=-9.50505) p = mdb.models['Model-1'].parts['Part-1'] p.seedPart(size=1.25, deviationFactor=0.1, minSizeFactor=0.1) p = mdb.models['Model-1'].parts['Part-1'] p.seedPart(size=0.5, deviationFactor=0.1, minSizeFactor=0.1) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#0:22 #f ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#0:19 #20000000 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=473.457, farPlane=516.089, width=256.536, height=155.811, viewOffsetX=-3.00951, viewOffsetY=4.24907) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:2 #100028 #4000040 #10000 #1000 #100 #40000008', ' #2000000 #100000 #8000 #400 #20 #8000001 #400000', ' #200000 #80000 #4000 #10000040 #80010000 #4 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:11 #2800000 #200000 #40000 #8000 #10000

#8000', ' #1000 #80000080 #200000 #5200 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] s1 = p.features['Partition face-1'].sketch mdb.models['Model-1'].ConstrainedSketch(name='__edit__', objectToCopy=s1) s2 = mdb.models['Model-1'].sketches['__edit__'] g, v, d, c = s2.geometry, s2.vertices, s2.dimensions, s2.constraints s2.setPrimaryObject(option=SUPERIMPOSE) p.projectReferencesOntoSketch(sketch=s2, upToFeature=p.features['Partition face-1'], filter=COPLANAR_EDGES) session.viewports['Viewport: 1'].view.setValues(nearPlane=484.229, farPlane=505.316, width=144.395, height=87.448, cameraPosition=(-14.7667, 11.0758, 494.773), cameraTarget=(-14.7667, 11.0758, 0)) s2.delete(objectList=(g[551], )) s2.delete(objectList=(g[555], )) s2.delete(objectList=(g[553], )) s2.delete(objectList=(g[557], )) session.viewports['Viewport: 1'].view.setValues(nearPlane=483.418, farPlane=506.128, width=155.505, height=94.1764, cameraPosition=(-56.9664, 9.46199, 494.773), cameraTarget=(-56.9664, 9.46199, 0)) s2.rectangle(point1=(-115.0, -25.0), point2=(-5.0, 25.0)) s2.unsetPrimaryObject() p = mdb.models['Model-1'].parts['Part-1'] p.features['Partition face-1'].setValues(sketch=s2) del mdb.models['Model-1'].sketches['__edit__'] p = mdb.models['Model-1'].parts['Part-1'] p.regenerate() session.viewports['Viewport: 1'].view.setValues(nearPlane=480.797, farPlane=508.748, width=168.937, height=102.606, viewOffsetX=-21.1305, viewOffsetY=3.19824) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#0:16 #8000 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#0:17 #800 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=475.934, farPlane=513.611, width=227.861, height=138.395, viewOffsetX=39.109, viewOffsetY=2.22115) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=(

A-11 Appendix A |

'[#0 #1400000 #20000200 #80000 #800 #200 #80', ' #10 #40000002 #48000000 #1 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0 #1400000 #20000200 #80000 #800 #200 #80', ' #10 #40000002 #48000000 #1 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0 #1400000 #20000200 #80000 #800 #200 #80', ' #10 #40000002 #48000000 #1 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=2.5, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=477.98, farPlane=511.565, width=228.841, height=138.99, viewOffsetX=12.1797, viewOffsetY=20.4505) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:11 #2800000 #200000 #40000 #8000 #10000 #8000', ' #800 #20000020 #80000 #1480 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=2.55, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=489.774, farPlane=499.771, width=60.3538, height=36.6568, viewOffsetX=-99.793, viewOffsetY=-19.1051) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:19 #10000000 #40004000 #81001000 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=484.366, farPlane=505.18, width=141.936, height=86.2069, viewOffsetX=-71.3827, viewOffsetY=1.97981) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:15 #50 #8 #4000001 #40000 #100 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:11 #1000000 #80000 #4000 #200 #128 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport:

1'].view.setValues(nearPlane=487.109, farPlane=502.436, width=92.5637, height=56.2199, viewOffsetX=4.13205, viewOffsetY=-17.6692) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#0:11 #200 #40 #8 #20000001 #5 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:12 #10 #40000002 #8000000 #90000000 #2 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:9 #80000000 #4000000 #200000 #10000 #800 #40', ' #2 #20000001 #800000 #8000 #2000020 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=483.025, farPlane=506.52, width=160.189, height=97.2933, viewOffsetX=8.87101, viewOffsetY=-4.48457) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0 #800000 #2000080 #2000 #80000008 #8000000 #400000', ' #20000 #1000 #80 #20000004 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=487.562, farPlane=501.984, width=87.0907, height=52.8958, viewOffsetX=93.3664, viewOffsetY=-15.8382) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:2 #400010 #80000400 #15000000 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:4 #28000000 #1000000 #80000 #4000 #200 #10 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=484.393, farPlane=505.153, width=125.422, height=76.1768, viewOffsetX=85.7214, viewOffsetY=1.35996)

A-12 Appendix A |

p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#2008082a #200 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=('[#4010105 #100040 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=485.576, farPlane=503.969, width=111.094, height=67.4743, viewOffsetX=32.2409, viewOffsetY=10.7152) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:6 #a00000 #80000 #10000 #2000 #400 #80 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:5 #14000000 #1000000 #200000 #40000 #8000 #1000 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=1.25, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport: 1'].view.setValues(nearPlane=475.774, farPlane=513.772, width=258.846, height=157.214, viewOffsetX=5.17472, viewOffsetY=20.6233) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0 #1400000 #20000200 #80000 #800 #200 #80', ' #10 #40000002 #48000000 #1 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=4.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0 #1400000 #20000200 #80000 #800 #200 #80', ' #10 #40000002 #48000000 #1 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) p = mdb.models['Model-1'].parts['Part-1'] e = p.edges pickedEdges = e.getSequenceFromMask(mask=( '[#0:11 #2800000 #200000 #40000 #8000 #10000 #8000', ' #800 #20000020 #80000 #1480 ]', ), ) p.seedEdgeBySize(edges=pickedEdges, size=5.0, deviationFactor=0.1, minSizeFactor=0.1, constraint=FINER) session.viewports['Viewport:

1'].view.setValues(nearPlane=473.373, farPlane=516.172, width=291.467, height=177.027, viewOffsetX=-4.42112, viewOffsetY=9.68117) p = mdb.models['Model-1'].parts['Part-1'] p.generateMesh() session.viewports['Viewport: 1'].view.setValues(nearPlane=482.331, farPlane=507.214, width=150.362, height=91.3243, viewOffsetX=-7.26895, viewOffsetY=23.2027) a = mdb.models['Model-1'].rootAssembly session.viewports['Viewport: 1'].setValues(displayedObject=a) session.viewports['Viewport: 1'].assemblyDisplay.setValues(loads=ON, bcs=ON, predefinedFields=ON, connectors=ON, optimizationTasks=OFF, geometricRestrictions=OFF, stopConditions=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(sectionAssignments=ON, engineeringFeatures=ON, mesh=OFF) session.viewports['Viewport: 1'].partDisplay.meshOptions.setValues( meshTechnique=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].view.setValues(nearPlane=486.949, farPlane=502.596, width=94.7688, height=57.3937, viewOffsetX=-101.078, viewOffsetY=-7.83979) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:41 #40000000 ]', ), ) p.Set(nodes=nodes, name='L-1') #: The set 'L-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:41 #20000000 ]', ), ) p.Set(nodes=nodes, name='L-2') #: The set 'L-2' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:11 #400000 ]', ), ) p.Set(nodes=nodes, name='L-3') #: The set 'L-3' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.33,

A-13 Appendix A |

farPlane=497.216, width=29.5711, height=17.9088, viewOffsetX=-114.105, viewOffsetY=-21.5734) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:11 #20000 ]', ), ) p.Set(nodes=nodes, name='L-4') #: The set 'L-4' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:11 #800 ]', ), ) p.Set(nodes=nodes, name='L-5') #: The set 'L-5' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:11 #10 ]', ), ) p.Set(nodes=nodes, name='L-6') #: The set 'L-6' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:10 #10000000 ]', ), ) p.Set(nodes=nodes, name='L-7') #: The set 'L-7' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:10 #80000 ]', ), ) p.Set(nodes=nodes, name='L-8') #: The set 'L-8' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #8000000 ]', ), ) p.Set(nodes=nodes, name='L-9') #: The set 'L-9' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #4000000 ]', ), )

p.Set(nodes=nodes, name='L-10') #: The set 'L-10' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=485.385, farPlane=504.16, width=113.733, height=68.8786, viewOffsetX=-89.6446, viewOffsetY=3.4605) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #2000000 ]', ), ) p.Set(nodes=nodes, name='L-11') #: The set 'L-11' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #1000000 ]', ), ) p.Set(nodes=nodes, name='L-12') #: The set 'L-12' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #800000 ]', ), ) p.Set(nodes=nodes, name='L-13') #: The set 'L-13' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=485.363, farPlane=504.182, width=128.709, height=77.9487, viewOffsetX=-86.3953, viewOffsetY=-0.319077) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #400000 ]', ), ) p.Set(nodes=nodes, name='L-14') #: The set 'L-14' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #200000 ]', ), )

A-14 Appendix A |

p.Set(nodes=nodes, name='L-15') #: The set 'L-15' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #100000 ]', ), ) p.Set(nodes=nodes, name='L-16') #: The set 'L-16' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:38 #80000 ]', ), ) p.Set(nodes=nodes, name='L-17') #: The set 'L-17' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:10 #200 ]', ), ) p.Set(nodes=nodes, name='L-18') #: The set 'L-18' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.186, farPlane=496.36, width=19.2096, height=11.6337, viewOffsetX=-114.326, viewOffsetY=24.3795) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:9 #40000000 ]', ), ) p.Set(nodes=nodes, name='L-19') #: The set 'L-19' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:9 #40000 ]', ), ) p.Set(nodes=nodes, name='L-20') #: The set 'L-20' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport:

1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:9 #20 ]', ), ) p.Set(nodes=nodes, name='L-21') #: The set 'L-21' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:8 #800000 ]', ), ) p.Set(nodes=nodes, name='L-22') #: The set 'L-22' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:8 #100 ]', ), ) p.Set(nodes=nodes, name='L-23') #: The set 'L-23' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:32 #100 ]', ), ) p.Set(nodes=nodes, name='L-24') #: The set 'L-24' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.636, farPlane=497.91, width=37.9768, height=22.9994, viewOffsetX=110.642, viewOffsetY=-20.9581) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:18 #400 ]', ), ) p.Set(nodes=nodes, name='R-1') #: The set 'R-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:18 #800 ]', ), )

A-15 Appendix A |

p.Set(nodes=nodes, name='R-2') #: The set 'R-2' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:2 #2000000 ]', ), ) p.Set(nodes=nodes, name='R-3') #: The set 'R-3' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:2 #1000000 ]', ), ) p.Set(nodes=nodes, name='R-4') #: The set 'R-4' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:2 #800 ]', ), ) p.Set(nodes=nodes, name='R-5') #: The set 'R-5' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0 #80000000 ]', ), ) p.Set(nodes=nodes, name='R-6') #: The set 'R-6' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0 #100000 ]', ), ) p.Set(nodes=nodes, name='R-7') #: The set 'R-7' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0 #400 ]', ), ) p.Set(nodes=nodes, name='R-8') #: The set 'R-8' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1']

n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:13 #20000000 ]', ), ) p.Set(nodes=nodes, name='R-9') #: The set 'R-9' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:13 #40000000 ]', ), ) p.Set(nodes=nodes, name='R-10') #: The set 'R-10' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:13 #80000000 ]', ), ) p.Set(nodes=nodes, name='R-11') #: The set 'R-11' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=485.902, farPlane=503.643, width=107.461, height=65.0802, viewOffsetX=93.5091, viewOffsetY=0.19087) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:14 #1 ]', ), ) p.Set(nodes=nodes, name='R-12') #: The set 'R-12' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:14 #2 ]', ), ) p.Set(nodes=nodes, name='R-13') #: The set 'R-13' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:14 #4 ]', ), ) p.Set(nodes=nodes, name='R-14') #: The set 'R-14' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:14 #8 ]', ),

A-16 Appendix A |

) p.Set(nodes=nodes, name='R-15') #: The set 'R-15' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:14 #10 ]', ), ) p.Set(nodes=nodes, name='R-16') #: The set 'R-16' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:14 #20 ]', ), ) p.Set(nodes=nodes, name='R-17') #: The set 'R-17' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.274, farPlane=496.271, width=18.1342, height=10.9824, viewOffsetX=115.683, viewOffsetY=24.6752) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0 #2 ]', ), ) p.Set(nodes=nodes, name='R-18') #: The set 'R-18' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#2000000 ]', ), ) p.Set(nodes=nodes, name='R-19') #: The set 'R-19' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#40000 ]', ), ) p.Set(nodes=nodes, name='R-20') #: The set 'R-20' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON)

p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#80 ]', ), ) p.Set(nodes=nodes, name='R-21') #: The set 'R-21' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#1000 ]', ), ) p.Set(nodes=nodes, name='R-21') #: The set 'R-21' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#80 ]', ), ) p.Set(nodes=nodes, name='R-22') #: The set 'R-22' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#2 ]', ), ) p.Set(nodes=nodes, name='R-23') #: The set 'R-23' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:11 #8000000 ]', ), ) p.Set(nodes=nodes, name='R-24') #: The set 'R-24' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport:

A-17 Appendix A |

1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.427, farPlane=496.119, width=16.2922, height=9.86686, viewOffsetX=-118.446, viewOffsetY=-26.2846) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:174 #20 ]', ), ) p.Set(nodes=nodes, name='C-L-1') #: The set 'C-L-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE LEFT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.388, farPlane=496.158, width=16.7632, height=10.1521, viewOffsetX=116.991, viewOffsetY=-26.0183) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:78 #100 ]', ), ) p.Set(nodes=nodes, name='C-R-1') #: The set 'C-R-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE RIGHT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.459, farPlane=499.086, width=52.2256, height=31.6287, viewOffsetX=115.494, viewOffsetY=-15.1682) session.viewports['Viewport:

1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:81 #4 ]', ), ) p.Set(nodes=nodes, name='C-R-4') #: The set 'C-R-4' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) del mdb.models['Model-1'].parts['Part-1'].sets['C-R-'] p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=488.128, farPlane=501.418, width=80.4807, height=48.7405, viewOffsetX=114.128, viewOffsetY=12.6566) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.57, farPlane=496.976, width=26.6641, height=16.1483, viewOffsetX=117.917, viewOffsetY=22.3209) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:59 #20000000 ]', ), ) p.Set(nodes=nodes, name='C-R-23') #: The set 'C-R-23' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:59 #8000 ]', ), ) p.Set(nodes=nodes, name='C-R-24') #: The set 'C-R-24' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:59

A-18 Appendix A |

#100000 ]', ), ) p.Set(nodes=nodes, name='C-R-25') #: The set 'C-R-25' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=486.959, farPlane=502.587, width=94.6511, height=57.3224, viewOffsetX=-79.5292, viewOffsetY=-14.2896) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:162 #200000 ]', ), ) p.Set(nodes=nodes, name='C-L-15') #: The set 'C-L-15' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.441, farPlane=498.104, width=40.3337, height=24.4268, viewOffsetX=-107.608, viewOffsetY=23.1471) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:162 #10000 ]', ), ) p.Set(nodes=nodes, name='C-L-16') #: The set 'C-L-16' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:162 #800 ]', ), ) p.Set(nodes=nodes, name='C-L-17') #: The set 'C-L-17' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1']

n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:162 #40 ]', ), ) p.Set(nodes=nodes, name='C-L-18') #: The set 'C-L-18' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:159 #10000 ]', ), ) p.Set(nodes=nodes, name='C-L-19') #: The set 'C-L-19' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:156 #1000000 ]', ), ) p.Set(nodes=nodes, name='C-L-20') #: The set 'C-L-20' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:136 #80 ]', ), ) p.Set(nodes=nodes, name='C-L-21') #: The set 'C-L-21' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:132 #2000000 ]', ), ) p.Set(nodes=nodes, name='C-L-22') #: The set 'C-L-22' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:136 #80 ]',

A-19 Appendix A |

), ) p.Set(nodes=nodes, name='C-L-23') #: The set 'C-L-23' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:132 #2000000 ]', ), ) p.Set(nodes=nodes, name='C-L-24') #: The set 'C-L-24' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:132 #10000000 ]', ), ) p.Set(nodes=nodes, name='C-L-25') #: The set 'C-L-25' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.965, farPlane=496.58, width=21.8809, height=13.2514, viewOffsetX=-118.61, viewOffsetY=26.9694) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:153 #40000000 ]', ), ) p.Set(nodes=nodes, name='C-L-21') #: The set 'C-L-21' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes

nodes = n.getSequenceFromMask(mask=('[#0:139 #100 ]', ), ) p.Set(nodes=nodes, name='C-L-22') #: The set 'C-L-22' has been edited (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.139, farPlane=498.406, width=49.7344, height=30.1201, viewOffsetX=-96.9559, viewOffsetY=27.4579) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:8 #80 ]', ), ) p.Set(nodes=nodes, name='TL-1') #: The set 'TL-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP LEFT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.489, farPlane=496.057, width=15.5392, height=9.41083, viewOffsetX=115.293, viewOffsetY=30.3636) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#4 ]', ), ) p.Set(nodes=nodes, name='TR-1') #: The set 'TR-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP RIGHT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport:

A-20 Appendix A |

1'].view.setValues(nearPlane=491.432, farPlane=498.114, width=40.4468, height=24.4953, viewOffsetX=-16.0783, viewOffsetY=-25.0366) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:8 #10 ]', ), ) p.Set(nodes=nodes, name='BL-1') #: The set 'BL-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE BOTTOM LEFT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.402, farPlane=498.144, width=46.1477, height=27.9479, viewOffsetX=23.0406, viewOffsetY=-25.1749) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:32 #20 ]', ), ) p.Set(nodes=nodes, name='BR-1') #: The set 'BR-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE BOTTOM RIGHT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=483.918, farPlane=505.627, width=148.458, height=89.9087, viewOffsetX=58.5602, viewOffsetY=-16.7173) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport:

1'].view.setValues(nearPlane=494.046, farPlane=495.5, width=8.79994, height=5.3294, viewOffsetX=-115.407, viewOffsetY=29.2167) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:132 #8000000 ]', ), ) p.Set(nodes=nodes, name='C-TL-1') #: The set 'C-TL-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP LEFT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=484.433, farPlane=505.113, width=141.421, height=85.647, viewOffsetX=-50.3832, viewOffsetY=36.19) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.538, farPlane=497.007, width=27.0503, height=16.3821, viewOffsetX=109.542, viewOffsetY=29.9303) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:59 #200000 ]', ), ) p.Set(nodes=nodes, name='C-TR-1') #: The set 'C-TR-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP RIGHT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.321, farPlane=496.224, width=17.5694, height=10.6404, viewOffsetX=-8.82433, viewOffsetY=-27.2811) session.viewports['Viewport:

A-21 Appendix A |

1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:135 #1000000 ]', ), ) p.Set(nodes=nodes, name='C-BL-1') #: The set 'C-BL-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE BOTTOM LEFT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=483.787, farPlane=505.758, width=132.759, height=80.4013, viewOffsetX=-64.3148, viewOffsetY=-15.5757) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.914, farPlane=497.631, width=34.6029, height=20.9561, viewOffsetX=15.9304, viewOffsetY=-25.5518) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) p = mdb.models['Model-1'].parts['Part-1'] n = p.nodes nodes = n.getSequenceFromMask(mask=('[#0:132 #1000 ]', ), ) p.Set(nodes=nodes, name='C-BR-1') #: The set 'C-BR-1' has been created (1 node). session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE BOTTOM RIGHT HAND SIDE OF THE RVE #----------------------------------------------------------- session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) session.viewports['Viewport: 1'].view.setValues(nearPlane=473.509, farPlane=516.036, width=256.645, height=155.429, viewOffsetX=11.1351, viewOffsetY=-2.84002) p1 = mdb.models['Model-1'].parts['Part-1'] session.viewports['Viewport: 1'].setValues(displayedObject=p1) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=ON) session.viewports['Viewport: 1'].view.setValues(nearPlane=473.509, farPlane=516.036, width=256.645, height=155.429,

viewOffsetX=-3.6264, viewOffsetY=-21.7407) mdb.models['Model-1'].Material(name='Brick') mdb.models['Model-1'].Material(name='Mortar') mdb.models['Model-1'].Material(name='Interphase1') mdb.models['Model-1'].Material(name='Interphase2') mdb.models['Model-1'].Material(name='Interphase3') mdb.models['Model-1'].Material(name='Interphase4') mdb.models['Model-1'].Material(name='Interphase5') mdb.models['Model-1'].HomogeneousSolidSection(name='Brick', material='Brick', thickness=110.0) mdb.models['Model-1'].HomogeneousSolidSection(name='Mortar', material='Mortar', thickness=110.0) mdb.models['Model-1'].HomogeneousSolidSection(name='Interphase1', material='Interphase1', thickness=110.0) mdb.models['Model-1'].HomogeneousSolidSection(name='Interphase2', material='Interphase2', thickness=110.0) mdb.models['Model-1'].HomogeneousSolidSection(name='Interphase3', material='Interphase3', thickness=110.0) mdb.models['Model-1'].HomogeneousSolidSection(name='Interphase4', material='Interphase4', thickness=110.0) mdb.models['Model-1'].HomogeneousSolidSection(name='Interphase5', material='Interphase5', thickness=110.0) session.viewports['Viewport: 1'].partDisplay.setValues(mesh=OFF) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:2 #400000 #20010008 #800400 #40020010 #1000800 #100040', ' #2 ]', ), ) region = p.Set(faces=faces, name='Set-417') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Brick', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#1020844b #802010 #20010008 #800400 #40020010 #1000800 #80040020', ' #4001 ]', ), ) region = p.Set(faces=faces, name='Set-418') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#1020844b #802010 #20010008 #800400 #40020010 #1000800 #80040020', ' #4001 ]', ), ) region = p.Set(faces=faces, name='Set-418') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0,

A-22 Appendix A |

offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:2 #10008004 #400200 #20010008 #800400 #40020010 #4001000', ' #40080080 #2020100 #11841 ]', ), ) region = p.Set(faces=faces, name='Set-419') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#8104225 #401008 #4006 ]', ), ) region = p.Set(faces=faces, name='Set-420') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#1 ]', ), ) region = p.Set(faces=faces, name='Set-421') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=487.417, farPlane=502.129, width=113.74, height=63.2443, viewOffsetX=-77.4966, viewOffsetY=3.24942) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:7 #8002000 #80100100 #4040200 #2082 ]', ), ) region = p.Set(faces=faces, name='Set-422') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=483.525, farPlane=506.02, width=153.743, height=85.4873, viewOffsetX=53.0181, viewOffsetY=7.85238) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#1 ]', ), ) region = p.Set(faces=faces, name='Set-423') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) del mdb.models['Model-1'].parts['Part-1'].sectionAssignments[6] del mdb.models['Model-1'].parts['Part-1'].sectionAssignments[4] del mdb.models['Model-1'].parts['Part-

1'].sectionAssignments[3] p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#8104224 #401008 #4002 ]', ), ) region = p.Set(faces=faces, name='Set-424') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Mortar', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.Viewport(name='Viewport: 1', origin=(0.0, 0.0), width=362.829681396484, height=267.758331298828) session.viewports['Viewport: 1'].makeCurrent() session.viewports['Viewport: 1'].maximize() from caeModules import * from driverUtils import executeOnCaeStartup executeOnCaeStartup() session.viewports['Viewport: 1'].partDisplay.geometryOptions.setValues( referenceRepresentation=ON) execfile( 'H:/Work Directory/Creating Sketch/RVE #3/Ready to input interphase section .py', __main__.__dict__) #: A new model database has been created. #: The model "Model-1" has been created. session.viewports['Viewport: 1'].setValues(displayedObject=None) session.viewports['Viewport: 1'].view.setValues(nearPlane=489.369, farPlane=500.176, width=65.4339, height=39.628, viewOffsetX=85.4842, viewOffsetY=17.8852) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #10000000 #100 ]', ), ) region = p.Set(faces=faces, name='Set-425') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase1', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.429, farPlane=499.117, width=52.5959, height=31.853, viewOffsetX=-18.281, viewOffsetY=22.5693) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:2 #200000 #10008004 #400200 #20010008 #800400 #80020 ]', ), ) region = p.Set(faces=faces, name='Set-426') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase1', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.826, farPlane=498.719, width=47.7813, height=28.9372, viewOffsetX=-109.966, viewOffsetY=24.4911)

A-23 Appendix A |

p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:8 #2001 ]', ), ) region = p.Set(faces=faces, name='Set-427') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase1', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.846, farPlane=497.699, width=35.4278, height=21.4557, viewOffsetX=-106.679, viewOffsetY=-24.1404) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:8 #2004000 ]', ), ) region = p.Set(faces=faces, name='Set-428') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase1', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.608, farPlane=497.938, width=38.3133, height=23.2032, viewOffsetX=4.87539, viewOffsetY=-26.6616) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:3 #40020010 #1000800 #80040020 #2001000 #200080 #4 ]', ), ) region = p.Set(faces=faces, name='Set-429') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase1', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.595, farPlane=497.95, width=38.469, height=23.2975, viewOffsetX=109.27, viewOffsetY=-24.4743) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:2 #800200 ]', ), ) region = p.Set(faces=faces, name='Set-430') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase1', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=489.037, farPlane=500.508, width=69.4563, height=42.064, viewOffsetX=103.468, viewOffsetY=-19.3217) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:2 #1000000 #40020 ]', ), ) region = p.Set(faces=faces, name='Set-431') p = mdb.models['Model-1'].parts['Part-1']

p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #20000000 #400 ]', ), ) region = p.Set(faces=faces, name='Set-432') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.625, farPlane=496.92, width=25.9975, height=15.7446, viewOffsetX=115.932, viewOffsetY=19.5188) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #40100 ]', ), ) region = p.Set(faces=faces, name='Set-433') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #8020000 ]', ), ) region = p.Set(faces=faces, name='Set-434') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.292, farPlane=497.253, width=30.0261, height=18.1843, viewOffsetX=3.41771, viewOffsetY=22.3745) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:2 #100080 #8004002 #200100 #10008004 #400200 #10 ]', ), ) region = p.Set(faces=faces, name='Set-435') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.934, farPlane=498.611, width=46.4718, height=28.1441, viewOffsetX=-103.7, viewOffsetY=21.5535) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:7 #80040000 ]', ), ) region = p.Set(faces=faces, name='Set-436') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region,

A-24 Appendix A |

sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:8 #1001000 ]', ), ) region = p.Set(faces=faces, name='Set-437') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.085, farPlane=497.46, width=32.5307, height=19.7012, viewOffsetX=-105.832, viewOffsetY=-24.2005) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #2008 ]', ), ) region = p.Set(faces=faces, name='Set-438') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:8 #4000000 #10 ]', ), ) region = p.Set(faces=faces, name='Set-439') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.059, farPlane=498.486, width=44.9614, height=27.2294, viewOffsetX=0.161634, viewOffsetY=-23.3242) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:3 #80000000 #2001000 #80040 #4002001 #400100 #8008 ]', ), ) region = p.Set(faces=faces, name='Set-440') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase2', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:4 #4002001 #100080 #8004002 #800200 #8010010 #20 ]', ), ) region = p.Set(faces=faces, name='Set-441') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.328,

farPlane=497.217, width=29.592, height=17.9215, viewOffsetX=105.699, viewOffsetY=-22.2611) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:2 #2000000 #80040 ]', ), ) region = p.Set(faces=faces, name='Set-442') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #40080000 #800 ]', ), ) region = p.Set(faces=faces, name='Set-443') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.594, farPlane=496.952, width=26.3761, height=15.9739, viewOffsetX=116.341, viewOffsetY=21.1993) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#1000000 #201 ]', ), ) region = p.Set(faces=faces, name='Set-444') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.983, farPlane=497.562, width=38.1861, height=23.1262, viewOffsetX=113.497, viewOffsetY=20.288) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#80000000 #10080 ]', ), ) region = p.Set(faces=faces, name='Set-445') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.882, farPlane=496.664, width=22.8896, height=13.8624, viewOffsetX=-0.387309, viewOffsetY=22.964) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0 #4000000 #80040 #4002001 #100080 #8004002 #200100 ]', ), ) region = p.Set(faces=faces, name='Set-446') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0,

A-25 Appendix A |

offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.442, farPlane=497.104, width=28.2167, height=17.0885, viewOffsetX=-103.229, viewOffsetY=21.3697) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:7 #40020008 ]', ), ) region = p.Set(faces=faces, name='Set-447') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.664, farPlane=497.881, width=37.6305, height=22.7897, viewOffsetX=-102.15, viewOffsetY=-21.9989) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:8 #800800 #4 ]', ), ) region = p.Set(faces=faces, name='Set-448') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #20201000 ]', ), ) region = p.Set(faces=faces, name='Set-449') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #404000 ]', ), ) region = p.Set(faces=faces, name='Set-450') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase3', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #40808000 #210 ]', ), ) region = p.Set(faces=faces, name='Set-451') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #10100800 #8 ]', ), ) region = p.Set(faces=faces, name='Set-452')

p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.974, farPlane=498.571, width=45.9874, height=27.8508, viewOffsetX=5.66844, viewOffsetY=-23.3537) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:4 #8004000 #200100 #10008004 #1000400 #10020020 #40 ]', ), ) region = p.Set(faces=faces, name='Set-453') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.575, farPlane=497.971, width=38.7144, height=23.4461, viewOffsetX=111.603, viewOffsetY=-24.3076) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:2 #4000000 #100080 #2 ]', ), ) region = p.Set(faces=faces, name='Set-454') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #80100400 #1000 ]', ), ) region = p.Set(faces=faces, name='Set-455') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.91, farPlane=497.635, width=34.6533, height=20.9866, viewOffsetX=111.23, viewOffsetY=28.1377) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#2041000 #2 ]', ), ) region = p.Set(faces=faces, name='Set-456') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#40820000 #40 ]', ), ) region = p.Set(faces=faces, name='Set-457') p = mdb.models['Model-1'].parts['Part-1']

A-26 Appendix A |

p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.057, farPlane=499.488, width=57.099, height=34.5802, viewOffsetX=-5.67037, viewOffsetY=26.6189) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0 #2008000 #80040020 #2001000 #80040 #4002001 #80 ]', ), ) region = p.Set(faces=faces, name='Set-458') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=490.207, farPlane=499.338, width=55.2779, height=33.4773, viewOffsetX=-106.223, viewOffsetY=21.9879) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:6 #100000 #20010004 ]', ), ) region = p.Set(faces=faces, name='Set-459') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:8 #400400 #2 ]', ), ) region = p.Set(faces=faces, name='Set-460') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase4', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:7 #10000000 #200200 #80401 ]', ), ) region = p.Set(faces=faces, name='Set-461') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:5 #2000000 #80040 #8002 ]', ), ) region = p.Set(faces=faces, name='Set-462') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.911,

farPlane=496.634, width=22.5293, height=13.6441, viewOffsetX=-110.478, viewOffsetY=-25.4618) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #8000000 #24104 ]', ), ) region = p.Set(faces=faces, name='Set-463') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:9 #81000000 #8420 ]', ), ) region = p.Set(faces=faces, name='Set-464') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=491.8, farPlane=497.745, width=35.9816, height=21.7911, viewOffsetX=0.722275, viewOffsetY=-26.0431) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0:4 #10000000 #400200 #20010008 #2000800 #20040040 #10080 ]', ), ) region = p.Set(faces=faces, name='Set-465') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.401, farPlane=497.144, width=28.707, height=17.3855, viewOffsetX=112.433, viewOffsetY=-24.9473) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0:2 #8000000 #200100 #8004 ]', ), ) region = p.Set(faces=faces, name='Set-466') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#0 #200804 #2001 ]', ), ) region = p.Set(faces=faces, name='Set-467') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=493.717, farPlane=495.828, width=12.7714, height=7.73457,

A-27 Appendix A |

viewOffsetX=114.781, viewOffsetY=25.6363) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#4082110 ]', ), ) region = p.Set(faces=faces, name='Set-468') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=('[#20410880 ]', ), ) region = p.Set(faces=faces, name='Set-469') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=492.947, farPlane=496.598, width=22.0973, height=13.3825, viewOffsetX=-1.95299, viewOffsetY=26.7499) p = mdb.models['Model-1'].parts['Part-1'] f = p.faces faces = f.getSequenceFromMask(mask=( '[#0 #1004020 #40020010 #1000800 #80040020 #1000 ]', ), ) region = p.Set(faces=faces, name='Set-470') p = mdb.models['Model-1'].parts['Part-1'] p.SectionAssignment(region=region, sectionName='Interphase5', offset=0.0, offsetType=MIDDLE_SURFACE, offsetField='', thicknessAssignment=FROM_SECTION) session.viewports['Viewport: 1'].view.setValues(nearPlane=476.605, farPlane=512.94, width=248.606, height=138.235, viewOffsetX=-4.14577, viewOffsetY=4.6687) a = mdb.models['Model-1'].rootAssembly session.viewports['Viewport: 1'].setValues(displayedObject=a) session.viewports['Viewport: 1'].assemblyDisplay.setValues(loads=OFF, bcs=OFF, predefinedFields=OFF, connectors=OFF) a = mdb.models['Model-1'].rootAssembly a.DatumCsysByDefault(CARTESIAN) p = mdb.models['Model-1'].parts['Part-1'] a.Instance(name='Part-1-1', part=p, dependent=ON) session.viewports['Viewport: 1'].assemblyDisplay.setValues( adaptiveMeshConstraints=ON) mdb.models['Model-1'].StaticStep(name='Step-1', previous='Initial') session.viewports['Viewport: 1'].assemblyDisplay.setValues(step='Step-1') session.viewports['Viewport: 1'].assemblyDisplay.setValues(loads=ON, bcs=ON, predefinedFields=ON, connectors=ON, adaptiveMeshConstraints=OFF) a = mdb.models['Model-1'].rootAssembly v1 = a.instances['Part-1-1'].vertices verts1 = v1.getSequenceFromMask(mask=('[#0:11

#800000 ]', ), ) region = a.Set(vertices=verts1, name='Set-1') mdb.models['Model-1'].DisplacementBC(name='BC-1', createStepName='Step-1', region=region, u1=0.0, u2=0.0, ur3=UNSET, amplitude=UNSET, fixed=OFF, distributionType=UNIFORM, fieldName='', localCsys=None) a = mdb.models['Model-1'].rootAssembly v1 = a.instances['Part-1-1'].vertices verts1 = v1.getSequenceFromMask(mask=('[#0:2 #10000000 ]', ), ) region = a.Set(vertices=verts1, name='Set-2') mdb.models['Model-1'].DisplacementBC(name='BC-2', createStepName='Step-1', region=region, u1=UNSET, u2=0.0, ur3=UNSET, amplitude=UNSET, fixed=OFF, distributionType=UNIFORM, fieldName='', localCsys=None) a = mdb.models['Model-1'].rootAssembly v1 = a.instances['Part-1-1'].vertices verts1 = v1.getSequenceFromMask(mask=('[#0:3 #800 ]', ), ) region = a.Set(vertices=verts1, name='Set-3') mdb.models['Model-1'].ConcentratedForce(name='Load-1', createStepName='Step-1', region=region, cf2=1.0, distributionType=UNIFORM, field='', localCsys=None) a = mdb.models['Model-1'].rootAssembly v1 = a.instances['Part-1-1'].vertices verts1 = v1.getSequenceFromMask(mask=('[#0:3 #800 ]', ), ) region = a.Set(vertices=verts1, name='Set-4') mdb.models['Model-1'].ConcentratedForce(name='Load-2', createStepName='Step-1', region=region, cf1=2.0, distributionType=UNIFORM, field='', localCsys=None) a = mdb.models['Model-1'].rootAssembly v1 = a.instances['Part-1-1'].vertices verts1 = v1.getSequenceFromMask(mask=('[#0:2 #10000000 ]', ), ) region = a.Set(vertices=verts1, name='Set-5') mdb.models['Model-1'].ConcentratedForce(name='Load-3', createStepName='Step-1', region=region, cf1=3.0, distributionType=UNIFORM, field='', localCsys=None) #--------------------------------------------------------------------- mdb.models['Model-1'].Equation(name='Constraint-1', terms=((1.0, 'Part-1-1.R-1', 1), (-1.0, 'Part-1-1.L-1', 1), (-1.0, 'Part-1-1.BR-36', 1), (1.0, 'Part-1-1.BL-37', 1))) mdb.models['Model-1'].Equation(name='Constraint-2', terms=((1.0, 'Part-1-1.R-1', 2), (-1.0, 'Part-1-1.L-1', 2), (-1.0, 'Part-1-1.BR-36', 2), (1.0, 'Part-1-1.BL-37', 2))) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE VERTICAL SIDES OF THE RVE #----------------------------------------------------------- #---------------------------------------------------------

A-28 Appendix A |

mdb.models['Model-1'].Equation(name='Constraint-49', terms=((1.0, 'Part-1-1.BL-1', 1), (-1.0, 'Part-1-1.TR-1', 1), (1.0, 'Part-1-1.TR-37', 1), (-1.0, 'Part-1-1.BL-37', 1))) mdb.models['Model-1'].Equation(name='Constraint-50', terms=((1.0, 'Part-1-1.BL-1', 2), (-1.0, 'Part-1-1.TR-1', 2), (1.0, 'Part-1-1.TR-37', 2), (-1.0, 'Part-1-1.BL-37', 2))) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP LEFT AND BOTTOM RIGHT OF THE RVE #----------------------------------------------------------- #------------------------------ #mdb.models['Model-1'].Equation(name='Constraint-48', terms=((1.0, # 'Part-1-1.BR-1', 1), (-1.0, 'Part-1-1.TL-1', 1), (1.0, 'Part-1-1.TR-16', # 1), (-1.0, 'Part-1-1.BR-15', 1))) #mdb.models['Model-1'].Equation(name='Constraint-49', terms=((1.0, # 'Part-1-1.BR-1', 2), (-1.0, 'Part-1-1.TL-1', 2), (1.0, 'Part-1-1.TR-16', # 2), (-1.0, 'Part-1-1.BR-15', 2))) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP RIGHT AND BOTTOM LEFTOF THE RVE #----------------------------------------------------------- mdb.models['Model-1'].Equation(name='Constraint-191', terms=((1.0, 'Part-1-1.TR-1', 1), (-1.0, 'Part-1-1.TL-1', 1), (-1.0, 'Part-1-1.BR-36', 1), (1.0, 'Part-1-1.BL-37', 1))) mdb.models['Model-1'].Equation(name='Constraint-192', terms=((1.0, 'Part-1-1.TR-1', 2), (-1.0, 'Part-1-1.TL-1', 2), (-1.0, 'Part-1-1.BR-36', 2), (1.0, 'Part-1-1.BL-37', 2))) #------------------------------------------------------------- mdb.models['Model-1'].Equation(name='Constraint-193', terms=((1.0, 'Part-1-1.C-BR-1', 1), (-1.0, 'Part-1-1.C-TL-1', 1), (1.0, 'Part-1-1.TR-37', 1), (-1.0, 'Part-1-1.BR-36', 1))) mdb.models['Model-1'].Equation(name='Constraint-194', terms=((1.0, 'Part-1-1.C-BR-1', 2), (-1.0, 'Part-1-1.C-TL-1', 2), (1.0, 'Part-1-1.TR-37', 2), (-1.0, 'Part-1-1.BR-36', 2))) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP LEFT AND BOTTOM RIGHT OF THE RVE #----------------------------------------------------------- #------------------------------------------------------------------------- mdb.models['Model-1'].Equation(name='Constraint-265', terms=((1.0, 'Part-1-1.C-BL-1', 1), (-1.0, 'Part-1-1.C-TR-1', 1), (1.0,

'Part-1-1.TR-37', 1), (-1.0, 'Part-1-1.BL-37', 1))) mdb.models['Model-1'].Equation(name='Constraint-266', terms=((1.0, 'Part-1-1.C-BL-1', 2), (-1.0, 'Part-1-1.C-TR-1', 2), (1.0, 'Part-1-1.TR-37', 2), (-1.0, 'Part-1-1.BL-37', 2))) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE TOP RIGHT AND BOTTOM LEFT OF THE RVE #----------------------------------------------------------- #----------------------------------------------------------------------------- mdb.models['Model-1'].Equation(name='Constraint-337', terms=((1.0, 'Part-1-1.C-R-1', 1), (-1.0, 'Part-1-1.C-L-1', 1), (-1.0, 'Part-1-1.BR-36', 1), (1.0, 'Part-1-1.BL-37', 1))) mdb.models['Model-1'].Equation(name='Constraint-338', terms=((1.0, 'Part-1-1.C-R-1', 2), (-1.0, 'Part-1-1.C-L-1', 2), (-1.0, 'Part-1-1.BR-36', 2), (1.0, 'Part-1-1.BL-37', 2))) #----------------------------------------------------------- #REPEAT FOR ALL NODES ON THE SIDES OF THE RVE #-----------------------------------------------------------

A-29 Appendix A |

Code A-2 Reading an ABAQUS ODB file and writing a full txt report

from abaqus import * from abaqusConstants import * session.Viewport(name='Viewport: 1', origin=(0.0, 0.0), width=333.5625, height=207.199981689453) session.viewports['Viewport: 1'].makeCurrent() session.viewports['Viewport: 1'].maximize() from caeModules import * from driverUtils import executeOnCaeStartup executeOnCaeStartup() session.viewports['Viewport: 1'].partDisplay.geometryOptions.setValues( referenceRepresentation=ON) #: Writing Field Report********************************* o1 = session.openOdb(name='initialStep.odb') session.viewports['Viewport: 1'].setValues(displayedObject=o1) #: Model: H:/Work Directory/Main Program/restartCheck1.odb #: Number of Assemblies: 1 #: Number of Assembly instances: 0 #: Number of Part instances: 1 #: Number of Meshes: 1 #: Number of Element Sets: 1 #: Number of Node Sets: 1 #: Number of Steps: 1 odb = session.odbs['initialStep.odb'] session.fieldReportOptions.setValues(printTotal=OFF, printMinMax=OFF) session.writeFieldReport(fileName='abaqus.rpt', append=OFF, sortItem='Node Label', odb=odb, step=0, frame=1, outputPosition=ELEMENT_NODAL, variable=(('E', INTEGRATION_POINT, (( COMPONENT, 'E11'), (COMPONENT, 'E22'), (COMPONENT, 'E33'), (COMPONENT, 'E12'), )), ('FV1', INTEGRATION_POINT), ('S', INTEGRATION_POINT, (( COMPONENT, 'S11'), (COMPONENT, 'S22'), (COMPONENT, 'S33'), (COMPONENT, 'S12'), )), )) #: Closing Odb ***************************************** #: session.odbs['H:/Work Directory/Main Program/restartCheck1.odb'].close() #: Opening Odb ***************************************** #: odb = openOdb(path='restartCheck1.odb') #: Open txt ******************************************** h = open('Node and Connectivity.txt','w') #: Catching Last Frame

********************************* #: lastFrame = odb.steps['Step-1'].frames[-1] h.write('************Nodes***********\n') assembly = odb.rootAssembly elements = odb.rootAssembly.instances['PART-1-1'].elements nodes = odb.rootAssembly.instances['PART-1-1'].nodes sections = odb.sections d = 0 for s in nodes: d = d + 1 h.write('Number of Nodes\n') d = str(d) h.write(d) h.write('\n') for s in nodes: sLabel = s.label Label = str(sLabel) sCoordx = s.coordinates[0] Coordx = str(sCoordx) sCoordy = s.coordinates[1] Coordy = str(sCoordy) h.write(Label) h.write(' ') h.write(Coordx) h.write(' ') h.write(Coordy) h.write('\n') h.write('************Element Connectivity***********\n') d = 0 for v in elements: d = d + 1 print v print print v.sectionCategory h.write('Number of Elements\n') d = str(d) h.write(d) h.write('\n') for v in elements: matName = v.sectionCategory.name print v.sectionCategory.sectionPoints h.write(matName) h.write('\n') vLabel = v.label Label = str(vLabel) nod1 = v.connectivity[0] node1 = str(nod1) nod2 = v.connectivity[1] node2 = str(nod2) nod3 = v.connectivity[2] node3 = str(nod3) nod4 = v.connectivity[3] node4 = str(nod4) nod5 = v.connectivity[4]

A-30 Appendix A |

node5 = str(nod5) nod6 = v.connectivity[5] node6 = str(nod6) nod7 = v.connectivity[6] node7 = str(nod7) nod8 = v.connectivity[-1] node8 = str(nod8) h.write(Label) h.write(' ') h.write(node1) h.write(' ') h.write(node2) h.write(' ') h.write(node3) h.write(' ') h.write(node4) h.write(' ') h.write(node5) h.write(' ') h.write(node6) h.write(' ') h.write(node7) h.write(' ') h.write(node8) h.write(' ') h.write('\n') #: Closing Odb ***************************************** session.odbs['initialStep.odb'].close() h.close()

Code A-3 Reading an ABAQUS ODB file and writing stress-strain report

from abaqus import * from abaqusConstants import * session.Viewport(name='Viewport: 1', origin=(0.0, 0.0), width=333.5625, height=207.199981689453) session.viewports['Viewport: 1'].makeCurrent() session.viewports['Viewport: 1'].maximize() from caeModules import * from driverUtils import executeOnCaeStartup executeOnCaeStartup() session.viewports['Viewport: 1'].partDisplay.geometryOptions.setValues( referenceRepresentation=ON) #: Writing Field Report********************************* o1 = session.openOdb(name='Report.odb') session.viewports['Viewport: 1'].setValues(displayedObject=o1) #: Model: H:/Work Directory/Main Program/restartCheck1.odb #: Number of Assemblies: 1 #: Number of Assembly instances: 0 #: Number of Part instances: 1 #: Number of Meshes: 1 #: Number of Element Sets: 1 #: Number of Node Sets: 1 #: Number of Steps: 1 odb = session.odbs['Report.odb'] session.fieldReportOptions.setValues(printTotal=OFF, printMinMax=OFF) session.writeFieldReport(fileName='abaqus.rpt', append=OFF, sortItem='Node Label', odb=odb, step=0, frame=1, outputPosition=ELEMENT_NODAL, variable=(('E', INTEGRATION_POINT, (( COMPONENT, 'E11'), (COMPONENT, 'E22'), (COMPONENT, 'E33'), (COMPONENT, 'E12'), )), ('FV1', INTEGRATION_POINT), ('S', INTEGRATION_POINT, (( COMPONENT, 'S11'), (COMPONENT, 'S22'), (COMPONENT, 'S33'), (COMPONENT, 'S12'), )), )) session.odbs['Report.odb'].close()

APPENDIX-B

MATLAB codes

B-1 Appendix B |

Code B-1 Main Program

clc clear !abaqus job=initialStep interactive d=0; stat = 0; pause(8); variables; loadVariables; ll=1; while stat == 0 stat = exist('initialStep.msg','file'); pause(1); d=d+1; end numSLoad = 0; numBLoad = 0; newBLoadStep = loadStep * 15; newSLoadStep = loadStep; pause(8); fidd = fopen('caseNumber.txt'); line = fgetl(fidd); st = str2num(line); RVEnumber = st(1); LengthScaleNumber = st(2); caseNumber = st(3); copyFile1 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB']; copyfile('initialStep.odb',copyFile1) moveFile1 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB\initialStep.odb']; moveFile2 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB\Report.odb']; movefile(moveFile1,moveFile2) !abaqus cae noGUI=initialReport.py pause(8); copyFile1 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport']; copyfile('abaqus.rpt',copyFile1) moveFile1 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport\abaqus.rpt']; moveFile2 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport\initial.rpt']; movefile(moveFile1,moveFile2) lambdaBrick = -(ecBrick*dcBrick ^ 2 - 2 * zetaBrick * ecBrick * dcBrick + zetaBrick * ecBrick)/(-dcBrick ^ 2 + zetaBrick*dcBrick); lambdaMortar = -(ecMortar*dcMortar ^ 2 - 2 * zetaMortar * ecMortar * dcMortar + zetaMortar * ecMortar)/(-dcMortar ^ 2 + zetaMortar*dcMortar); lambdaInterphase1 = -(ecInterphase1*dcInterphase1 ^ 2 - 2 * zetaInterphase1 * ecInterphase1 * dcInterphase1 + zetaInterphase1 * ecInterphase1)/(-dcInterphase1 ^ 2 + zetaInterphase1*dcInterphase1);

B-2 Appendix B |

lambdaInterphase2 = -(ecInterphase2*dcInterphase2 ^ 2 - 2 * zetaInterphase2 * ecInterphase2 * dcInterphase2 + zetaInterphase2 * ecInterphase2)/(-dcInterphase2 ^ 2 + zetaInterphase2*dcInterphase2); lambdaInterphase3 = -(ecInterphase3*dcInterphase3 ^ 2 - 2 * zetaInterphase3 * ecInterphase3 * dcInterphase3 + zetaInterphase3 * ecInterphase3)/(-dcInterphase3 ^ 2 + zetaInterphase3*dcInterphase3); lambdaInterphase4 = -(ecInterphase4*dcInterphase4 ^ 2 - 2 * zetaInterphase4 * ecInterphase4 * dcInterphase4 + zetaInterphase4 * ecInterphase4)/(-dcInterphase4 ^ 2 + zetaInterphase4*dcInterphase4); lambdaInterphase5 = -(ecInterphase5*dcInterphase5 ^ 2 - 2 * zetaInterphase5 * ecInterphase5 * dcInterphase5 + zetaInterphase5 * ecInterphase5)/(-dcInterphase5 ^ 2 + zetaInterphase5*dcInterphase5); readGeo; localEqStrainCalc; nonLocalEqStrains; damageParCalc; %% fid31 = fopen('Results11.txt','w'); fprintf(fid31,'%d ',macroStrain(ll,1)); fprintf(fid31,'%d',macroStress(ll,1)); fprintf(fid31,'\n'); fid32 = fopen('Results22.txt','w'); fprintf(fid32,'%d ',macroStrain(ll,2)); fprintf(fid32,'%d',macroStress(ll,2)); fprintf(fid32,'\n'); fid33 = fopen('Results33.txt','w'); fprintf(fid33,'%d ',macroStrain(ll,3)); fprintf(fid33,'%d',macroStress(ll,3)); fprintf(fid33,'\n'); fid34 = fopen('Results12.txt','w'); fprintf(fid34,'%d ',macroStrain(ll,4)); fprintf(fid34,'%d',macroStress(ll,4)); fprintf(fid34,'\n'); maxStrain = max([abs(macroStrain(ll,1)) abs(macroStrain(ll,2)) abs(macroStrain(ll,4))]); if abs(maxStrain) < 0.005 numBLoad = numBLoad + 1; else numSLoad = numSLoad + 1; end %% ll = 2; writeInput; !abaqus job=step-2 oldjob=initialStep interactive stat = 0; pause(8); while stat == 0 stat = exist('step-2.msg','file'); pause(1); d=d+1;

B-3 Appendix B |

end pause(8); stepCopy = ['step-' num2str(ll) '.odb']; copyFile1 = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB']; copyfile(stepCopy,copyFile1) stepMove = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB\step-' num2str(ll) '.odb']; copyfile(stepMove,['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\Report.odb']) !abaqus cae noGUI=Report.py pause(5); copyfile('abaqus.rpt',['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport']) movefile(['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport\abaqus.rpt'],['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport\step-2.rpt']) readGeo; localEqStrainCalc; nonLocalEqStrains; damageParCalc; %% fid31 = fopen('Results11.txt','a'); fprintf(fid31,'%d ',macroStrain(ll,1)); fprintf(fid31,'%d',macroStress(ll,1)); fprintf(fid31,'\n'); fid32 = fopen('Results22.txt','a'); fprintf(fid32,'%d ',macroStrain(ll,2)); fprintf(fid32,'%d',macroStress(ll,2)); fprintf(fid32,'\n'); fid33 = fopen('Results33.txt','a'); fprintf(fid33,'%d ',macroStrain(ll,3)); fprintf(fid33,'%d',macroStress(ll,3)); fprintf(fid33,'\n'); fid34 = fopen('Results12.txt','a'); fprintf(fid34,'%d ',macroStrain(ll,4)); fprintf(fid34,'%d',macroStress(ll,4)); fprintf(fid34,'\n'); maxStrain = max([abs(macroStrain(ll,1)) abs(macroStrain(ll,2)) abs(macroStrain(ll,4))]); if abs(maxStrain) < 0.005 numBLoad = numBLoad + 1; else numSLoad = numSLoad + 1; end %% for ll = 3:numStep

B-4 Appendix B |

writeInput; runStepNumber = ['!abaqus job=step-' num2str((ll)) ' oldjob=step-' num2str((ll-1)) ' interactive']; eval(runStepNumber) stat = 0; pause(8); stepPause = ['step-' num2str(ll) '.msg']; while stat == 0 stat = exist(stepPause,'file'); pause(1); d=d+1; end pause(8); stepCopy = ['step-' num2str(ll) '.odb']; copyfile(stepCopy,['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB']) stepMove = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempODB\step-' num2str(ll) '.odb']; movefile(stepMove,['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\Report.odb']) !abaqus cae noGUI=Report.py pause(5); copyfile('abaqus.rpt',['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport']) moveFileNumber = ['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport\step-' num2str(ll) '.rpt']; movefile(['\\home\n8412031\Main\RVE' num2str(RVEnumber) '\LengthScale' num2str(LengthScaleNumber) '\Case' num2str(caseNumber) '\TempReport\abaqus.rpt'],moveFileNumber) readGeo; localEqStrainCalc; nonLocalEqStrains; damageParCalc; %% fid31 = fopen('Results11.txt','a'); fprintf(fid31,'%d ',macroStrain(ll,1)); fprintf(fid31,'%d',macroStress(ll,1)); fprintf(fid31,'\n'); fid32 = fopen('Results22.txt','a'); fprintf(fid32,'%d ',macroStrain(ll,2)); fprintf(fid32,'%d',macroStress(ll,2)); fprintf(fid32,'\n'); fid33 = fopen('Results33.txt','a'); fprintf(fid33,'%d ',-

B-5 Appendix B |

(macroStress(ll,1)+macroStress(ll,2))*nooB/brickE); fprintf(fid33,'%d',macroStress(ll,3)); fprintf(fid33,'\n'); fid34 = fopen('Results12.txt','a'); fprintf(fid34,'%d ',macroStrain(ll,4)); fprintf(fid34,'%d',macroStress(ll,4)); fprintf(fid34,'\n'); maxStrain = max([abs(macroStrain(ll,1)) abs(macroStrain(ll,2)) abs(macroStrain(ll,4))]); if abs(maxStrain) < 0.005 numBLoad = numBLoad + 1; else numSLoad = numSLoad + 1; end %% if abs(macroStrain(ll,1)) > 0.005 break elseif abs(macroStrain(ll,2)) > 0.005 break elseif abs(macroStrain(ll,3)) > 0.005 break elseif abs(macroStrain(ll,4)) > 0.005 break end end sketch; fclose(all);

B-6 Appendix B |

Code B-2 Reading Geometry

fid = fopen('materialSet.txt','r'); line = fgetl(fid); line = fgetl(fid); mortarElement = str2num(line); material = struct('elemNumber',{}); for ii = 1:length(mortarElement) material(mortarElement(ii)).elemNumber = 'MORTAR'; end numMortarElem = length(mortarElement); line = fgetl(fid); line = fgetl(fid); brickElement = str2num(line); for ii = 1:length(brickElement) material(brickElement(ii)).elemNumber = 'BRICK'; end numBrickElem = length(brickElement); numberOfInterphases = 0; for ii = 1:5 line = fgetl(fid); interphaseName = ['Interphase' num2str(ii) '']; if strcmp(line, interphaseName) == 1 else break end if ii == 1 line = fgetl(fid); interphase1Element = str2num(line); for jj = 1:length(interphase1Element) material(interphase1Element(jj)).elemNumber = 'INTERPHASE1'; end numberOfInterphases = 1; numInterphase1Elem = length(interphase1Element); elseif ii == 2 line = fgetl(fid); interphase2Element = str2num(line); for jj = 1:length(interphase2Element) material(interphase2Element(jj)).elemNumber = 'INTERPHASE2'; end numberOfInterphases = 2; numInterphase2Elem = length(interphase2Element);

B-7 Appendix B |

elseif ii == 3 line = fgetl(fid); interphase3Element = str2num(line); for jj = 1:length(interphase3Element) material(interphase3Element(jj)).elemNumber = 'INTERPHASE3'; end numberOfInterphases = 3; numInterphase3Elem = length(interphase3Element); elseif ii == 4 line = fgetl(fid); interphase4Element = str2num(line); for jj = 1:length(interphase4Element) material(interphase4Element(jj)).elemNumber = 'INTERPHASE4'; end numberOfInterphases = 4; numInterphase4Elem = length(interphase4Element); elseif ii == 5 line = fgetl(fid); interphase5Element = str2num(line); for jj = 1:length(interphase5Element) material(interphase5Element(jj)).elemNumber = 'INTERPHASE5'; end numberOfInterphases = 5; numInterphase5Elem = length(interphase5Element); end end fid2 = fopen('Node and Connectivity.txt','r'); line = fgetl(fid2); line = fgetl(fid2); line = fgetl(fid2); numNode = str2double(line); nodeRep = zeros(numNode,1); for ii=1:numNode line=fgetl(fid2); st=str2num(line); node(st(1),1) = st(2); node(st(1),2) = st(3); end line=fgetl(fid2); line=fgetl(fid2); line=fgetl(fid2); numElem=str2double(line); for ii=1:numElem line=fgetl(fid2); line = fgetl(fid2); st=str2num(line); eleCon(st(1),1) = st(2); eleCon(st(1),2) = st(3);

B-8 Appendix B |

eleCon(st(1),3) = st(4); eleCon(st(1),4) = st(5); extraCon(st(6),1) = 1; extraCon(st(7),1) = 1; extraCon(st(8),1) = 1; extraCon(st(9),1) = 1; end stat = fclose(fid); stat = fclose(fid2); fid=fopen('abaqus.rpt','r'); for ii = 1:22 line = fgetl(fid); end nodeLabel = 0; line = fgetl(fid); st = str2num(line); for ii = 1:numNode if isempty(st) == 1 break end if st(2) == nodeLabel line = fgetl(fid); st = str2num(line); if isempty(st) == 1 break end end if st(2) == nodeLabel line = fgetl(fid); st = str2num(line); if isempty(st) == 1 break end end if st(2) == nodeLabel line = fgetl(fid); st = str2num(line); if isempty(st) == 1 break end end if nodeRep(st(2),1) == 0 strain(st(2),1) = st(3); strain(st(2),2) = st(4); strain(st(2),3) = st(5); strain(st(2),4) = st(6); FV1(st(2),ll)= st(7); stress(st(2),1) = st(8); stress(st(2),2) = st(9); stress(st(2),3) = st(10); stress(st(2),4) = st(11); nodeRep(st(2),1) = 1; nodeLabel = st(2); line = fgetl(fid); st = str2num(line);

B-9 Appendix B |

elseif nodeRep(st(2),1) == 1; strain(st(2),1) = (strain(st(2),1) + st(3))/2; strain(st(2),2) = (strain(st(2),2) + st(4))/2; strain(st(2),3) = (strain(st(2),3) + st(5))/2; strain(st(2),4) = (strain(st(2),4) + st(6))/2; FV1(st(2),ll) = (FV1(st(2),ll) + st(7))/2; stress(st(2),1) = (stress(st(2),1) + st(8))/2; stress(st(2),2) = (stress(st(2),2) + st(9))/2; stress(st(2),3) = (stress(st(2),3) + st(10))/2; stress(st(2),4) = (stress(st(2),4) + st(11))/2; nodeRep(st(2),1) = 2; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); elseif nodeRep(st(2),1) == 2; strain(st(2),1) = (strain(st(2),1)*2 + st(3))/3; strain(st(2),2) = (strain(st(2),2)*2 + st(4))/3; strain(st(2),3) = (strain(st(2),3)*2 + st(5))/3; strain(st(2),4) = (strain(st(2),4)*2 + st(6))/3; FV1(st(2),ll) = (FV1(st(2),ll)*2 + st(7))/3; stress(st(2),1) = (stress(st(2),1)*2 + st(8))/3; stress(st(2),2) = (stress(st(2),2)*2 + st(9))/3; stress(st(2),3) = (stress(st(2),3)*2 + st(10))/3; stress(st(2),4) = (stress(st(2),4)*2 + st(11))/3; nodeRep(st(2),1) = 3; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); elseif nodeRep(st(2),1) == 3; strain(st(2),1) = (strain(st(2),1)*3 + st(3))/4; strain(st(2),2) = (strain(st(2),2)*3 + st(4))/4; strain(st(2),3) = (strain(st(2),3)*3 + st(5))/4; strain(st(2),4) = (strain(st(2),4)*3 + st(6))/4; FV1(st(2),ll) = (FV1(st(2),ll)*3 + st(7))/4; stress(st(2),1) = (stress(st(2),1)*3 + st(8))/4; stress(st(2),2) = (stress(st(2),2)*3 + st(9))/4; stress(st(2),3) = (stress(st(2),3)*3 + st(10))/4; stress(st(2),4) = (stress(st(2),4)*3 + st(11))/4; nodeRep(st(2),1) = 4; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); end end for jj =1:(numRegion-1) for ii = 1:9 line = fgetl(fid); end nodeLabel = 0; line = fgetl(fid); st = str2num(line); for ii = 1:numNode*4 if isempty(st) == 1

B-10 Appendix B |

break end if st(2) == nodeLabel line = fgetl(fid); st = str2num(line); if isempty(st) == 1 break end end if st(2) == nodeLabel line = fgetl(fid); st = str2num(line); if isempty(st) == 1 break end end if st(2) == nodeLabel line = fgetl(fid); st = str2num(line); if isempty(st) == 1 break end end if nodeRep(st(2),1) == 0 strain(st(2),1) = st(3); strain(st(2),2) = st(4); strain(st(2),3) = st(5); strain(st(2),4) = st(6); FV1(st(2),ll)= st(7); stress(st(2),1) = st(8); stress(st(2),2) = st(9); stress(st(2),3) = st(10); stress(st(2),4) = st(11); nodeRep(st(2),1) = 1; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); elseif nodeRep(st(2),1) == 1; strain(st(2),1) = (strain(st(2),1) + st(3))/2; strain(st(2),2) = (strain(st(2),2) + st(4))/2; strain(st(2),3) = (strain(st(2),3) + st(5))/2; strain(st(2),4) = (strain(st(2),4) + st(6))/2; FV1(st(2),ll) = (FV1(st(2),ll) + st(7))/2; stress(st(2),1) = (stress(st(2),1) + st(8))/2; stress(st(2),2) = (stress(st(2),2) + st(9))/2; stress(st(2),3) = (stress(st(2),3) + st(10))/2; stress(st(2),4) = (stress(st(2),4) + st(11))/2; nodeRep(st(2),1) = 2; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); elseif nodeRep(st(2),1) == 2; strain(st(2),1) = (strain(st(2),1)*2 + st(3))/3; strain(st(2),2) = (strain(st(2),2)*2 + st(4))/3; strain(st(2),3) = (strain(st(2),3)*2 + st(5))/3; strain(st(2),4) = (strain(st(2),4)*2 + st(6))/3; FV1(st(2),ll) = (FV1(st(2),ll)*2 + st(7))/3;

B-11 Appendix B |

stress(st(2),1) = (stress(st(2),1)*2 + st(8))/3; stress(st(2),2) = (stress(st(2),2)*2 + st(9))/3; stress(st(2),3) = (stress(st(2),3)*2 + st(10))/3; stress(st(2),4) = (stress(st(2),4)*2 + st(11))/3; nodeRep(st(2),1) = 3; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); elseif nodeRep(st(2),1) == 3; strain(st(2),1) = (strain(st(2),1)*3 + st(3))/4; strain(st(2),2) = (strain(st(2),2)*3 + st(4))/4; strain(st(2),3) = (strain(st(2),3)*3 + st(5))/4; strain(st(2),4) = (strain(st(2),4)*3 + st(6))/4; FV1(st(2),ll) = (FV1(st(2),ll)*3 + st(7))/4; stress(st(2),1) = (stress(st(2),1)*3 + st(8))/4; stress(st(2),2) = (stress(st(2),2)*3 + st(9))/4; stress(st(2),3) = (stress(st(2),3)*3 + st(10))/4; stress(st(2),4) = (stress(st(2),4)*3 + st(11))/4; nodeRep(st(2),1) = 4; nodeLabel = st(2); line = fgetl(fid); st = str2num(line); end end end stat = fclose(fid); for ii = 1:numElem Connectivity = eleCon(ii,:); if node(Connectivity(1),1) > node(Connectivity(2),1) X(1) = 1; X(2) = 0; if node(Connectivity(1),1) == node(Connectivity(3),1) X(3) = 1; X(4) = 0; else X(4) = 1; X(3) = 0; end elseif node(Connectivity(1),1) == node(Connectivity(2),1) if node(Connectivity(1),1) > node(Connectivity(3),1) X(1) = 1; X(2) = 1; X(3) = 0; X(4) = 0; else X(1) = 0; X(2) = 0; X(3) = 1; X(4) = 1; end else X(1) = 0; X(2) = 1; if node(Connectivity(1),1) == node(Connectivity(3),1) X(3) = 0; X(4) = 1;

B-12 Appendix B |

else X(4) = 0; X(3) = 1; end end if node(Connectivity(1),2) > node(Connectivity(2),2) Y(1) = 1; Y(2) = 0; if node(Connectivity(1),2) == node(Connectivity(3),2) Y(3) = 1; Y(4) = 0; else Y(4) = 1; Y(3) = 0; end elseif node(Connectivity(1),2) == node(Connectivity(2),2) if node(Connectivity(1),2) > node(Connectivity(3),2) Y(1) = 1; Y(2) = 1; Y(3) = 0; Y(4) = 0; else Y(1) = 0; Y(2) = 0; Y(3) = 1; Y(4) = 1; end else Y(1) = 0; Y(2) = 1; if node(Connectivity(1),2) == node(Connectivity(3),2) Y(3) = 0; Y(4) = 1; else Y(4) = 0; Y(3) = 1; end end for jj=1:4 if (X(jj)==1 && Y(jj)==1) node3 = Connectivity(jj); elseif (X(jj)==0 && Y(jj)==0) node1 = Connectivity(jj); elseif (X(jj)==0 && Y(jj)==1) node4 = Connectivity(jj); else node2 = Connectivity(jj); end end eleConNew(ii,:) = [node1 node2 node3 node4]; end fid = fopen('IntPoint.txt','r'); for ii = 1:numElem line = fgetl(fid); st1 = str2num(line); line = fgetl(fid); st = str2num(line);

B-13 Appendix B |

st(1) = round(st(1)*10000) / 10000.0; st(2) = round(st(2)*10000) / 10000.0; intPoint(1,:) = [st(1) st(2)]; line = fgetl(fid); st2 = str2num(line); line = fgetl(fid); st = str2num(line); st(1) = round(st(1)*10000) / 10000.0; st(2) = round(st(2)*10000) / 10000.0; intPoint(2,:) = [st(1) st(2)]; line = fgetl(fid); st3 = str2num(line); line = fgetl(fid); st = str2num(line); st(1) = round(st(1)*10000) / 10000.0; st(2) = round(st(2)*10000) / 10000.0; intPoint(3,:) = [st(1) st(2)]; line = fgetl(fid); st4 = str2num(line); line = fgetl(fid); st = str2num(line); st(1) = round(st(1)*10000) / 10000.0; st(2) = round(st(2)*10000) / 10000.0; intPoint(4,:) = [st(1) st(2)]; if ii ==723 end Connectivity = [st1(2) st2(2) st3(2) st4(2)]; if intPoint(Connectivity(1),1) > intPoint(Connectivity(2),1) X(1) = 1; X(2) = 0; if intPoint(Connectivity(1),1) == intPoint(Connectivity(3),1) X(3) = 1; X(4) = 0; else X(4) = 1; X(3) = 0; end elseif intPoint(Connectivity(1),1) == intPoint(Connectivity(2),1) if intPoint(Connectivity(1),1) > intPoint(Connectivity(3),1) X(1) = 1; X(2) = 1; X(3) = 0; X(4) = 0; else X(1) = 0; X(2) = 0; X(3) = 1; X(4) = 1; end else X(1) = 0; X(2) = 1; if intPoint(Connectivity(1),1) == intPoint(Connectivity(3),1) X(3) = 0; X(4) = 1; else X(4) = 0;

B-14 Appendix B |

X(3) = 1; end end if intPoint(Connectivity(1),2) > intPoint(Connectivity(2),2) Y(1) = 1; Y(2) = 0; if intPoint(Connectivity(1),2) == intPoint(Connectivity(3),2) Y(3) = 1; Y(4) = 0; else Y(4) = 1; Y(3) = 0; end elseif intPoint(Connectivity(1),2) == intPoint(Connectivity(2),2) if intPoint(Connectivity(1),2) > intPoint(Connectivity(3),2) Y(1) = 1; Y(2) = 1; Y(3) = 0; Y(4) = 0; else Y(1) = 0; Y(2) = 0; Y(3) = 1; Y(4) = 1; end else Y(1) = 0; Y(2) = 1; if intPoint(Connectivity(1),2) == intPoint(Connectivity(3),2) Y(3) = 0; Y(4) = 1; else Y(4) = 0; Y(3) = 1; end end for jj=1:4 if (X(jj)==1 && Y(jj)==1) node3 = Connectivity(jj); elseif (X(jj)==0 && Y(jj)==0) node1 = Connectivity(jj); elseif (X(jj)==0 && Y(jj)==1) node4 = Connectivity(jj); else node2 = Connectivity(jj); end end eleIntNew(ii,:) = [node1 node2 node3 node4]; end stat = fclose(fid); macroStress(ll,1) = mean(stress(:,1)); macroStress(ll,2) = mean(stress(:,2)); macroStress(ll,3) = mean(stress(:,3)); macroStress(ll,4) = mean(stress(:,4)); stressMax(ll,1) = max(stress(:,1)); stressMax(ll,2) = max(stress(:,2)); stressMax(ll,3) = max(stress(:,3)); stressMax(ll,4) = max(stress(:,4)); stressMin(ll,1) = min(stress(:,1)); stressMin(ll,2) = min(stress(:,2));

B-15 Appendix B |

stressMin(ll,3) = min(stress(:,3)); stressMin(ll,4) = min(stress(:,4)); macroStrain(ll,1) = mean(strain(:,1)); macroStrain(ll,2) = mean(strain(:,2)); macroStrain(ll,3) = mean(strain(:,3)); macroStrain(ll,4) = mean(strain(:,4)); strainMax(ll,1) = max(strain(:,1)); strainMax(ll,2) = max(strain(:,2)); strainMax(ll,3) = max(strain(:,3)); strainMax(ll,4) = max(strain(:,4)); strainMin(ll,1) = min(strain(:,1)); strainMin(ll,2) = min(strain(:,2)); strainMin(ll,3) = min(strain(:,3)); strainMin(ll,4) = min(strain(:,4)); fprintf('**********************************************************************\n'); fprintf('**********************************************************************\n'); fprintf('Macro Strain(11) = %d Macro Stress(11) = %d \n',macroStrain(ll,1),macroStress(ll,1)); fprintf('Macro Strain(22) = %d Macro Stress(22) = %d \n',macroStrain(ll,2),macroStress(ll,2)); fprintf('Macro Strain(33) = %d Macro Stress(33) = %d \n',macroStrain(ll,3),macroStress(ll,3)); fprintf('Macro Strain(12) = %d Macro Stress(12) = %d \n',macroStrain(ll,4),macroStress(ll,4)); fprintf('**********************************************************************\n'); fprintf('**********************************************************************\n'); fclose all;

B-16 Appendix B |

Code B-3 Calculation of local equivalent strain

fid = fopen('elementDamage.txt','r'); for jj = 1:numElem if ll == 1 elementDamage(jj) = 0; else line = fgetl(fid); elementDamage(jj) = str2num(line); end end fclose(fid); nodalDamage = zeros(numNode); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% for ii = 1:numNode [refElement,position] = find(eleConNew == ii); bb = 0; mm = 0; II1 = 0; II2 = 0; II3 = 0; II4 = 0; II5 = 0; for nn=1:size(refElement) if strcmp(material(refElement(nn)).elemNumber,'BRICK') == 1 bb = bb+1; end if strcmp(material(refElement(nn)).elemNumber,'MORTAR') == 1 mm = mm+1; end if strcmp(material(refElement(nn)).elemNumber,'INTERPHASE1') == 1 II1 = II1+1; end if strcmp(material(refElement(nn)).elemNumber,'INTERPHASE2') == 1 II2 = II2+1; end if strcmp(material(refElement(nn)).elemNumber,'INTERPHASE3') == 1 II3 = II3+1; end if strcmp(material(refElement(nn)).elemNumber,'INTERPHASE4') == 1 II4 = II5+1; end if strcmp(material(refElement(nn)).elemNumber,'INTERPHASE5') == 1 II5 = II5+1; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nodalDamage(ii) = nodalDamage(ii) + elementDamage(refElement(nn)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end if mm == 4 kapa = kapaM; noo = nooM;

B-17 Appendix B |

elseif bb == 4 kapa = kapaB; noo = nooB; end if II1 == 3 kapa = (kapaI1 * 3 + kapaB * 1)/4; noo = (nooI1 * 3 + nooB * 1) / 4; elseif II1 == 2 if bb == 2 kapa = (kapaI1 * 2 + kapaB * 2)/4; noo = (nooI1 * 2 + nooB * 2) / 4; elseif II2 == 2 kapa = (kapaI1 * 2 + kapaI2 * 2)/4; noo = (nooI1 * 2 + nooI2 * 2) / 4; end elseif II1 == 1 kapa = (kapaI1 * 1 + kapaI2 * 3)/4; noo = (nooI1 * 1 + nooI2 * 3) / 4; end if II2 == 1 kapa = (kapaI2 * 1 + kapaI3 * 3)/4; noo = (nooI2 * 1 + nooI3 * 3) / 4; elseif II2 == 2 if II3 == 2 kapa = (kapaI2 * 2 + kapaI3 * 2)/4; noo = (nooI2 * 2 + nooI3 * 2) / 4; end end if II3 == 1 kapa = (kapaI3 * 1 + kapaI4 * 3)/4; noo = (nooI3 * 1 + nooI4 * 3) / 4; elseif II3 == 2 if II4 == 2 kapa = (kapaI3 * 2 + kapaI4 * 2)/4; noo = (nooI3 * 2 + nooI4 * 2) / 4; end end if II4 == 1 kapa = (kapaI4 * 1 + kapaI5 * 3)/4; noo = (nooI4 * 1 + nooI5 * 3) / 4; elseif II4 == 2 if II5 == 2 kapa = (kapaI4 * 2 + kapaI5 * 2)/4; noo = (nooI4 * 2 + nooI5 * 2) / 4; end end if II5 == 1 kapa = (kapaI5 * 1 + kapaM * 3)/4; noo = (nooI5 * 1 + nooM * 3) / 4; elseif II5 == 2 if mm == 2 kapa = (kapaI5 * 2 + kapaM * 2)/4; noo = (nooI5 * 2 + nooM * 2) / 4;

B-18 Appendix B |

end end if mm == 4 kapa = kapaM; noo = nooM; elseif mm == 2 if II5 == 0 kapa = kapaM; noo = nooM; end elseif mm == 1 if II5 == 0 kapa = kapaM; noo = nooM; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dv = 1 - (1 - nodalDamage(ii)) ^ betaV; v = 0.5 - 0.5 * (1-2*noo)*(1 - nodalDamage(ii))/(1-Dv); if v >= 0.5 v = 0.48; elseif v <= -1 v = -0.98; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% localStrain = [strain(ii,1) strain(ii,4) 0; strain(ii,4) strain(ii,2) 0;0 0 (-noo/(1 - noo) * (strain(ii,1) + strain(ii,2)))]; eigLocal = eig(localStrain); I1(ii) = eigLocal(1) + eigLocal(2) + eigLocal(3); volStrain = (localStrain(1,1) + localStrain(2,2) + localStrain(3,3)) / 3; divStrain = localStrain - [volStrain 0 0;0 volStrain 0; 0 0 volStrain]; eigDivLocal = eig(divStrain); J2 = eigDivLocal(1) ^ 2 + eigDivLocal(2) ^ 2 + eigDivLocal(3) ^ 2; localEqStrain(ii,1) = ((kapa - 1) * I1(ii)) / (2 * kapa * (1 - 2 * noo)) + 1 / (2 * kapa) * sqrt((kapa-1) ^2 * I1(ii) ^2 / (1 - 2 * noo) ^ 2 + 12 * kapa * J2 / (1 + noo) ^ 2); end fprintf(' Local Equivalent Strains have been calculated \n'); fprintf('**********************************************************************\n'); fprintf('**********************************************************************\n');

B-19 Appendix B |

Code B-4 Calculation of nonlocal equivalent strain

numNode = numNode - sum(extraCon); Kt = zeros(numNode,numNode); strainF = zeros(numNode,1); for ii= 1:numElem NtNs = zeros(4,1); Ns = 0.0; K = zeros(4,4); xNode1 = node(eleConNew(ii,1),1); xNode2 = node(eleConNew(ii,2),1); Lx = abs(xNode1 - xNode2); yNode1 = node(eleConNew(ii,1),2); yNode3 = node(eleConNew(ii,3),2); Ly = abs(yNode1 - yNode3); gX(1) = 0.211325 * Lx; gX(2) = 0.788675 * Lx; gY(1) = 0.211325 * Ly; gY(2) = 0.788675 * Ly; if strcmp(material(ii).elemNumber,'BRICK') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleB; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1); lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); elseif strcmp(material(ii).elemNumber,'MORTAR') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleM; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1); lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); elseif strcmp(material(ii).elemNumber,'INTERPHASE1') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleI1; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1);

B-20 Appendix B |

lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); elseif strcmp(material(ii).elemNumber,'INTERPHASE2') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleI2; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1); lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); elseif strcmp(material(ii).elemNumber,'INTERPHASE3') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleI3; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1); lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); elseif strcmp(material(ii).elemNumber,'INTERPHASE4') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleI4; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1); lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); elseif strcmp(material(ii).elemNumber,'INTERPHASE5') == 1 for mm = 1:4 curStrain = localEqStrain(eleConNew(ii,mm),1); transLengthScaleI5; lengthScale(mm,1) = tNLenScale; end lengthS(1,1) = lengthScale(1,1); lengthS(2,1) = lengthScale(4,1); lengthS(3,1) = lengthScale(2,1); lengthS(4,1) = lengthScale(3,1); end t = 0; for kk = 1:2 for jj= 1:2 Gx = gX(kk); Gy = gY(jj); t = t + 1; N(1,1) = 1 - Gx/Lx -Gy/Ly + Gx * Gy / Lx / Ly; N(1,2) = Gx/Lx - Gx * Gy / Lx / Ly; N(1,3) = Gx * Gy / Lx / Ly; N(1,4) = Gy/Ly - Gx * Gy / Lx / Ly;

B-21 Appendix B |

Ns = N(1,1) * localEqStrain(eleConNew(ii,1),1); Ns = Ns + N(1,2) * localEqStrain(eleConNew(ii,2),1); Ns = Ns + N(1,3) * localEqStrain(eleConNew(ii,3),1); Ns = Ns + N(1,4) * localEqStrain(eleConNew(ii,4),1); Nt = N'; NtNs(1,1) = NtNs(1,1) + Nt(1,1) * Ns / lengthS(t); NtNs(2,1) = NtNs(2,1) + Nt(2,1) * Ns / lengthS(t); NtNs(3,1) = NtNs(3,1) + Nt(3,1) * Ns / lengthS(t); NtNs(4,1) = NtNs(4,1) + Nt(4,1) * Ns / lengthS(t); NtN = N' * N; B(1,1) = - 1 / Lx + Gy / Lx / Ly; B(2,1) = - 1 / Ly + Gx / Lx / Ly; B(1,2) = + 1 / Lx - Gy / Lx / Ly; B(2,2) = - Gx / Lx / Ly; B(1,3) = + Gy / Lx / Ly; B(2,3) = + Gx / Lx / Ly; B(1,4) = - Gy / Lx / Ly; B(2,4) = + 1 / Ly - Gx / Lx / Ly; BtB = B'*B; K = K + NtN / lengthS(t) + BtB; end end for jj = 1:4 for kk = 1:4 Kt(eleConNew(ii,jj),eleConNew(ii,kk)) = Kt(eleConNew(ii,jj),eleConNew(ii,kk)) + K(jj,kk); end strainF(eleConNew(ii,jj),1) = strainF(eleConNew(ii,jj),1) + NtNs(jj,1); end end nLocalEqStrain = Kt \ strainF;

B-22 Appendix B |

Code B-5 Calculation of damage parameter

fid = fopen('damagePar.txt', 'w'); for ii = 1:numElem if strcmp(material(ii).elemNumber,'BRICK') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >= damageStrainBrick if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainBrick / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaBrick * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainBrick))); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecBrick damagePar(ii,jj) = dcBrick * zetaBrick * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcBrick * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcBrick - zetaBrick) * ecBrick); else damagePar(ii,jj) = (abs(nLocalEqStrain(nodeNumber,1)-0) + (dcBrick*ecBrick - dcBrick*lambdaBrick - ecBrick)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaBrick); end end if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elseif strcmp(material(ii).elemNumber,'MORTAR') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >= damageStrainMortar if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainMortar / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaMortar * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainMortar) )); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecMortar

B-23 Appendix B |

damagePar(ii,jj) = dcMortar * zetaMortar * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcMortar * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcMortar - zetaMortar) * ecMortar); else damagePar(ii,jj) = (abs(nLocalEqStrain(nodeNumber,1)-0) + (dcMortar*ecMortar - dcMortar*lambdaMortar - ecMortar)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaMortar); end end if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elseif strcmp(material(ii).elemNumber,'INTERPHASE1') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >= damageStrainInterphase1 if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainInterphase1 / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaInterphase1 * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainInterphase1) )); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecInterphase1 damagePar(ii,jj) = dcInterphase1 * zetaInterphase1 * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcInterphase1 * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcInterphase1 - zetaInterphase1) * ecInterphase1); else damagePar(ii,jj) = (abs(nLocalEqStrain(nodeNumber,1)-0) + (dcInterphase1*ecInterphase1 - dcInterphase1*lambdaInterphase1 - ecInterphase1)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaInterphase1); end end

B-24 Appendix B |

if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elseif strcmp(material(ii).elemNumber,'INTERPHASE2') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >= damageStrainInterphase2 if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainInterphase2 / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaInterphase2 * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainInterphase2) )); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecInterphase2 damagePar(ii,jj) = dcInterphase2 * zetaInterphase2 * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcInterphase2 * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcInterphase2 - zetaInterphase2) * ecInterphase2); else damagePar(ii,jj) = (abs(nLocalEqStrain(nodeNumber,1)-0) + (dcInterphase2*ecInterphase2 - dcInterphase2*lambdaInterphase2 - ecInterphase2)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaInterphase2); end end if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end

B-25 Appendix B |

end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elseif strcmp(material(ii).elemNumber,'INTERPHASE3') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >= damageStrainInterphase3 if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainInterphase3 / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaInterphase3 * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainInterphase3) )); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecInterphase3 damagePar(ii,jj) = dcInterphase3 * zetaInterphase3 * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcInterphase3 * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcInterphase3 - zetaInterphase3) * ecInterphase3); else damagePar(ii,jj) = (abs(nLocalEqStrain(nodeNumber,1)-0) + (dcInterphase3*ecInterphase3 - dcInterphase3*lambdaInterphase3 - ecInterphase3)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaInterphase3); end end if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elseif strcmp(material(ii).elemNumber,'INTERPHASE4') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >=

B-26 Appendix B |

damageStrainInterphase4 if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainInterphase4 / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaInterphase4 * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainInterphase4) )); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecInterphase4 damagePar(ii,jj) = dcInterphase4 * zetaInterphase4 * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcInterphase4 * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcInterphase4 - zetaInterphase4) * ecInterphase4); else damagePar(ii,jj) = (abs(nLocalEqStrain(nodeNumber,1)-0) + (dcInterphase4*ecInterphase4 - dcInterphase4*lambdaInterphase4 - ecInterphase4)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaInterphase4); end end if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end % end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elseif strcmp(material(ii).elemNumber,'INTERPHASE5') == 1 for jj = 1:4 nodeNumber = eleConNew(ii,jj); % if abs(nLocalEqStrain(nodeNumber,1)) >= damageStrainInterphase5 if I1(nodeNumber) >= 0 damagePar(ii,jj) = 1 - damageStrainInterphase5 / abs(nLocalEqStrain(nodeNumber,1)) * (1 - alpha + alpha * exp(- betaInterphase5 * (abs(nLocalEqStrain(nodeNumber,1)) - damageStrainInterphase5) )); else if abs(nLocalEqStrain(nodeNumber,1)) <= ecInterphase5 damagePar(ii,jj) = dcInterphase5 * zetaInterphase5 * (abs(nLocalEqStrain(nodeNumber,1))-0) / (dcInterphase5 * (abs(nLocalEqStrain(nodeNumber,1))-0) - (dcInterphase5 - zetaInterphase5) * ecInterphase5); else damagePar(ii,jj) =

B-27 Appendix B |

(abs(nLocalEqStrain(nodeNumber,1)-0) + (dcInterphase5*ecInterphase5 - dcInterphase5*lambdaInterphase5 - ecInterphase5)) / ((abs(nLocalEqStrain(nodeNumber,1))-0) - lambdaInterphase5); end end if damagePar(ii,jj) >= 1.0 damagePar(ii,jj) = 1.0; end if damagePar(ii,jj) <= 0.0 damagePar(ii,jj) = 0.0; end % else % % damagePar(ii,jj) = 0; % % end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% elementDamage(ii) = (damagePar(ii,1)+damagePar(ii,2)+damagePar(ii,3)+damagePar(ii,4))/4; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end fprintf(fid,'%d \n', ii); if eleIntNew(ii,1) == 1 fprintf(fid,'%f \n', damagePar(ii,1)); if eleIntNew(ii,2) == 2 fprintf(fid,'%f \n', damagePar(ii,2)); if eleIntNew(ii,3) == 3 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,4)); else fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,3)); end elseif eleIntNew(ii,2) == 3 if eleIntNew(ii,3) == 2 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,4)); elseif eleIntNew(ii,3) == 4 fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,3)); end elseif eleIntNew(ii,2) == 4 if eleIntNew(ii,3) == 2 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,2)); elseif eleIntNew(ii,3) == 3 fprintf(fid,'%f \n', damagePar(ii,4));

B-28 Appendix B |

fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,2)); end end elseif eleIntNew(ii,1) == 2 if eleIntNew(ii,2) == 1 fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,1)); if eleIntNew(ii,3) == 3 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,4)); else fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,3)); end elseif eleIntNew(ii,2) == 3 if eleIntNew(ii,3) == 1 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,4)); elseif eleIntNew(ii,3) == 4 fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,3)); end elseif eleIntNew(ii,2) == 4 if eleIntNew(ii,3) == 1 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,2)); elseif eleIntNew(ii,3) == 3 fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,2)); end end elseif eleIntNew(ii,1) == 3 if eleIntNew(ii,2) == 1 fprintf(fid,'%f \n', damagePar(ii,2)); if eleIntNew(ii,3) == 2 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,4)); else fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,3)); end elseif eleIntNew(ii,2) == 2 if eleIntNew(ii,3) == 1 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,4)); elseif eleIntNew(ii,3) == 4 fprintf(fid,'%f \n', damagePar(ii,4));

B-29 Appendix B |

fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,3)); end elseif eleIntNew(ii,2) == 4 if eleIntNew(ii,3) == 1 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,2)); elseif eleIntNew(ii,3) == 2 fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,1)); fprintf(fid,'%f \n', damagePar(ii,2)); end end elseif eleIntNew(ii,1) == 4 if eleIntNew(ii,2) == 1 fprintf(fid,'%f \n', damagePar(ii,2)); if eleIntNew(ii,3) == 2 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,1)); else fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,1)); end elseif eleIntNew(ii,2) == 2 if eleIntNew(ii,3) == 1 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,1)); elseif eleIntNew(ii,3) == 3 fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,1)); end elseif eleIntNew(ii,2) == 3 if eleIntNew(ii,3) == 1 fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,1)); elseif eleIntNew(ii,3) == 2 fprintf(fid,'%f \n', damagePar(ii,4)); fprintf(fid,'%f \n', damagePar(ii,3)); fprintf(fid,'%f \n', damagePar(ii,2)); fprintf(fid,'%f \n', damagePar(ii,1)); end end end end

B-30 Appendix B |

fclose all; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fid = fopen('elementDamage.txt','w'); for ii = 1:numElem fprintf(fid,'%f \n', elementDamage(ii)); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% traceElemX; traceElemY;

B-31 Appendix B |

Code B-6 writing ABAQUS input files

fprintf('**********************************************************************\n'); fprintf('**********************************************************************\n'); fprintf('******* Case-%d.inp *******\n',caseNumber); fprintf('******* step-%d.inp *******\n',(ll)); fprintf('**********************************************************************\n'); fprintf('**********************************************************************\n'); stepNumber = ['step-' num2str((ll)) '.inp']; fid = fopen(stepNumber,'w'); fprintf(fid,'*Restart, read, step=%d \n', (ll-1)); fprintf(fid,'** STEP: Step-%d \n', (ll)); fprintf(fid,'** \n'); fprintf(fid,'*Step, INC=1 \n'); fprintf(fid,'*Static \n'); fprintf(fid,'1., 1., 1e-05, 1. \n'); fprintf(fid,'** \n'); fprintf(fid,'** LOADS \n'); fprintf(fid,'** \n'); fprintf(fid,'** Name: Load-1 Type: Concentrated force \n'); fprintf(fid,'*Cload \n'); fprintf(fid,'Set-3, 2, %d \n',(firstLoad * loadRatio1 + (numBLoad) * newBLoadStep* loadRatio1 + (numSLoad) * newSLoadStep* loadRatio1 )); fprintf(fid,'** Name: Load-2 Type: Concentrated force \n'); fprintf(fid,'*Cload \n'); fprintf(fid,'Set-4, 1, %d \n',(firstLoad * loadRatio2 + (numBLoad) * newBLoadStep* loadRatio2 + (numSLoad) * newSLoadStep* loadRatio2 )); fprintf(fid,'** \n'); fprintf(fid,'** Name: Load-3 Type: Concentrated force \n'); fprintf(fid,'*Cload \n'); fprintf(fid,'Set-5, 1, %d \n',(firstLoad * loadRatio3 + (numBLoad) * newBLoadStep* loadRatio3 + (numSLoad) * newSLoadStep* loadRatio3 )); fprintf(fid,'** \n'); fprintf(fid,'** OUTPUT REQUESTS \n'); fprintf(fid,'** \n'); fprintf(fid,'*Restart, write, frequency=1 \n'); fprintf(fid,'** \n'); fprintf(fid,'** FIELD OUTPUT: F-Output-1 \n'); fprintf(fid,'*OUTPUT, FIELD, FREQUENCY=1 \n'); fprintf(fid,'*ELEMENT OUTPUT \n'); fprintf(fid,'S, EE, FV \n'); fprintf(fid,'** \n'); fprintf(fid,'*Output, field, variable=PRESELECT \n'); fprintf(fid,'** \n'); fprintf(fid,'** HISTORY OUTPUT: H-Output-1 \n'); fprintf(fid,'** \n'); fprintf(fid,'*Output, history, variable=PRESELECT \n'); fprintf(fid,'*End Step \n'); fclose all;

B-32 Appendix B |

Code B-6 writing output data

nodeX = node(:,1); geoX = sort((nodeX),'descend'); for ii =length(nodeX):-1:2 if geoX(ii) == geoX(ii-1) geoX(ii) = []; end end for ii = ((length(geoX)-1)/2):-1:1 geoX(2*ii) = []; end for jj = 1:length(targetX) nn = 0; for ii=1:numElem if node(eleConNew(ii,3),1) == geoX(targetX(jj)) if node(eleConNew(ii,1),1) == geoX(targetX(jj)+1) fid = fopen(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.out'],'a+'); nn = nn + 1; targetElemX(nn,1) = ii; fprintf(fid,'Element number \n',ii,eleConNew(ii,3)); fprintf(fid,'%d \n',ii); fprintf(fid,'Node number \n'); fprintf(fid,'%d \n',eleConNew(ii,3)); fprintf(fid,'Coordinates \n'); fprintf(fid,'%f %f \n',node(eleConNew(ii,3),1),node(eleConNew(ii,3),2)); fprintf(fid,'Damage Parameter \n'); fprintf(fid,'%f \n',damagePar(ii,3)); fprintf(fid,'Stress \n'); fprintf(fid,'%f %f %f %f \n',stress(eleConNew(ii,3),1),stress(eleConNew(ii,3),2),stress(eleConNew(ii,3),3),stress(eleConNew(ii,3),4)); fprintf(fid,'Strain \n'); fprintf(fid,'%f %f %f %f \n',strain(eleConNew(ii,3),1),strain(eleConNew(ii,3),2),strain(eleConNew(ii,3),3),strain(eleConNew(ii,3),4)); fprintf(fid,'************************************************************ \n'); nodeNumbers(nn) = eleConNew(ii,3); Y(nn) = node(eleConNew(ii,3),2); S11(nn) = stress(eleConNew(ii,3),1); S22(nn) = stress(eleConNew(ii,3),2); S33(nn) = stress(eleConNew(ii,3),3); S12(nn) = stress(eleConNew(ii,3),4);

B-33 Appendix B |

E11(nn) = strain(eleConNew(ii,3),1); E22(nn) = strain(eleConNew(ii,3),2); E33(nn) = strain(eleConNew(ii,3),3); E12(nn) = strain(eleConNew(ii,3),4); DP(nn) = damagePar(ii,3); end end end fclose all; end fid = fopen(['Step-' num2str(ll-1) 'X = ' num2str(node(eleConNew(ii,3),1)) '.out']); for ii=1:100 line = fgetl(fid); if line == -1 break end line = fgetl(fid); st = str2num(line); elementNumber = st(1); line = fgetl(fid); line = fgetl(fid); st = str2num(line); nodeNumbers(ii) = st(1); line = fgetl(fid); line = fgetl(fid); st = str2num(line); X(ii) = st(1); Y(ii) = st(2); line = fgetl(fid); line = fgetl(fid); st = str2num(line); DP(ii) = st(1); line = fgetl(fid); line = fgetl(fid); st = str2num(line); S11(ii) = st(1);

B-34 Appendix B |

S22(ii) = st(2); S33(ii) = st(3); S12(ii) = st(4); line = fgetl(fid); line = fgetl(fid); st = str2num(line); E11(ii) = st(1); E22(ii) = st(2); E33(ii) = st(3); E12(ii) = st(4); line = fgetl(fid); end xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],nodeNumbers(:),'A2:A80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],Y(:),'C2:C80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],S11(:),'E2:E80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],S22(:),'F2:F80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],'G2:G80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],S12(:),'I2:I80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],E11(:),'K2:K80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],E22(:),'L2:L80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],E33(:),'M2:M80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],E12(:),'N2:N80'); xlswrite(['Step-' num2str(ll-1) ' X = ' num2str(node(eleConNew(ii,3),1)) '.xls'],DP(:),'P2:P80'); fclose all; nodeY = node(:,2); geoY = sort((nodeY),'descend'); for ii =length(nodeY):-1:2 if geoY(ii) == geoY(ii-1) geoY(ii) = []; end end for ii = ((length(geoY)-1)/2):-1:1 geoY(2*ii) = []; end

B-35 Appendix B |

for jj = 1:length(targetY) mm = 0; for ii=1:numElem if node(eleConNew(ii,3),2) == geoY(targetY(jj)) if node(eleConNew(ii,1),2) == geoY(targetY(jj)+1) fid = fopen(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.out'],'a+'); mm = mm + 1; targetElemY(mm,1) = ii; fprintf(fid,'Element number \n',ii,eleConNew(ii,3)); fprintf(fid,'%d \n',ii); fprintf(fid,'Node number \n'); fprintf(fid,'%d \n',eleConNew(ii,3)); fprintf(fid,'Coordinates \n'); fprintf(fid,'%f %f \n',node(eleConNew(ii,3),1),node(eleConNew(ii,3),2)); fprintf(fid,'Damage Parameter \n'); fprintf(fid,'%f \n',damagePar(ii,3)); fprintf(fid,'Stress \n'); fprintf(fid,'%f %f %f %f \n',stress(eleConNew(ii,3),1),stress(eleConNew(ii,3),2),stress(eleConNew(ii,3),3),stress(eleConNew(ii,3),4)); fprintf(fid,'Strain \n'); fprintf(fid,'%f %f %f %f \n',strain(eleConNew(ii,3),1),strain(eleConNew(ii,3),2),strain(eleConNew(ii,3),3),strain(eleConNew(ii,3),4)); fprintf(fid,'************************************************************ \n'); X(mm) = node(eleConNew(ii,3),1); S11(mm) = stress(eleConNew(ii,3),1); S22(mm) = stress(eleConNew(ii,3),2); S33(mm) = stress(eleConNew(ii,3),3); S12(mm) = stress(eleConNew(ii,3),4); E11(mm) = strain(eleConNew(ii,3),1); E22(mm) = strain(eleConNew(ii,3),2); E33(mm) = strain(eleConNew(ii,3),3); E12(mm) = strain(eleConNew(ii,3),4); DP(mm) = damagePar(ii,3); end end end fclose all;

B-36 Appendix B |

end fid = fopen(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.out']); for ii=1:100 line = fgetl(fid); if line == -1 break end line = fgetl(fid); st = str2num(line); elementNumber = st(1); line = fgetl(fid); line = fgetl(fid); st = str2num(line); nodeNumbers(ii) = st(1); line = fgetl(fid); line = fgetl(fid); st = str2num(line); X(ii) = st(1); Y(ii) = st(2); line = fgetl(fid); line = fgetl(fid); st = str2num(line); DP(ii) = st(1); line = fgetl(fid); line = fgetl(fid); st = str2num(line); S11(ii) = st(1); S22(ii) = st(2); S33(ii) = st(3); S12(ii) = st(4); line = fgetl(fid); line = fgetl(fid); st = str2num(line); E11(ii) = st(1); E22(ii) = st(2); E33(ii) = st(3); E12(ii) = st(4); line = fgetl(fid); end xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],nodeNumbers(:),'A2:A80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],X(:),'C2:C80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],S11(:),'E2:E80'); xlswrite(['Step-' num2str(ll-1) ' Y = '

B-37 Appendix B |

num2str(node(eleConNew(ii,3),2)) '.xls'],S22(:),'F2:F80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],S33(:),'G2:G80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],S12(:),'I2:I80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],E11(:),'K2:K80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],E22(:),'L2:L80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],E33(:),'M2:M80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],E12(:),'N2:N80'); xlswrite(['Step-' num2str(ll-1) ' Y = ' num2str(node(eleConNew(ii,3),2)) '.xls'],DP(:),'P2:P80'); fclose all;

APPENDIX-C

FORTRAN codes

C-1 Appendix C |

Code C-1 ABAQUS usdfld subroutine

subroutine usdfld(field,statev,pnewdt,direct,t,celent, 1 time,dtime,cmname,orname,nfield,nstatv,noel,npt,layer, 2 kspt,kstep,kinc,ndi,nshr,coord,jmac,jmatyp,matlayo, 3 laccfla) c include 'aba_param.inc' c CHARACTER*80 CMNAME,ORNAME CHARACTER*3 FLGRAY(15) DIMENSION FIELD(NFIELD),STATEV(NSTATV),DIRECT(3,3), 1 T(3,3),TIME(2) DIMENSION ARRAY(15),JARRAY(15),JMAC(*),JMATYP(*),COORD(*) c -------------------------------------------------------------------------- c USER CODE START write(*,*) 'Material = ', cmname 1 write(*,*) '1', field 2 write(*,*) '2', statev 3 write(*,*) '3', pnewdt 4 write(*,*) '4', direct 5 write(*,*) '5', t 6 write(*,*) '6', celent 7 write(*,*) '7', time 8 write(*,*) '8', dtime 9 write(*,*) '9', orname 10 write(*,*) '10', nfield 11 write(*,*) '11', nstatv 12 write(*,*) '12', noel 13 write(*,*) '13', npt 14 write(*,*) '14', layer 15 write(*,*) '15', kspt 16 write(*,*) '16', kstep 17 write(*,*) '17', kinc 18 write(*,*) '18', ndi 19 write(*,*) '19', nshr 20 write(*,*) '20x', coord(1) 21 write(*,*) '20y', coord(2) Open(1,File= 1 '//home/n8412031/Main/RVE/LengthScale/Case/damagePar.txt')

C-2 Appendix C |

if (TIME(2) .eq. 0.0) then field(1) = 0.0 open(2,File= 1 '//home/n8412031/Main/RVE/LengthScale/Case/IntPoint.txt') Write(2,*), noel, npt Write(2,*), coord(1), coord(2) else if (noel .eq. 1) then do 100 km = 1,(npt+1) Read(1,'(F10.5)') A 100 enddo else do 200 km = 1,((noel-1) * 5) Read(1,'(F10.5)') A 200 enddo do 300 km = 1,(npt+1) Read(1,'(F10.5)') A 300 enddo endif field(1) = A endif write(*,*) 'FV1 = ', field(1) close(1) c close(2) c USER CODE END c -------------------------------------------------------------------------- return end

APPENDIX-D

Loading configurations and boundary conditions of RVEs

D-1 Appendix D |

Figure D-1 Loading configuration and boundary conditions of the RVE for load case-1






D-2 Appendix D |







D-3 Appendix D |







D-4 Appendix D |







D-5 Appendix D |







D-6 Appendix D |







D-7 Appendix D |







D-8 Appendix D |



APPENDIX-E

Failure stresses of RVEs

E-1 Appendix E |

Table E-1 Summary of failure stresses of the RVE for bed joint angles of 0° and 90°

Failure Summary

Load Case Parallel Stress

(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State

1 -0.347626 -6.0753412 0.000 Uniaxial Compression


3 0.002817 0.178062 0.000 Uniaxial Tension


5 -7.5334404 -8.1844068 0.000 Compression-Compression









14 0.390742 -1.886089 0.000 Tension-Compression



17 -0.12 0.36 0.000 Tension-Compression



20 0.09832 0.104307 0.000 Tension-Tension

21 0.110298 0.059733 0.000 Tension-Tension

22 0.071023 0.149401 0.000 Tension-Tension

E-2 Appendix E |

Table E-2 Summary of failure stresses of the RVE for bed joint angles of 22.5° and 67.5°

Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State

1 -0.660078 -3.0463802 1.0166938 C-C with Shear

2 -2.738155 -0.394886 -1.164851 C-C with Shear

3 0.04735 0.287836 -0.117752 T-T with Shear

4 0.103615 0.019929 0.034831 T-T with Shear

5 -7.44372272 -8.08338888 0 Compression-Compression

6 -7.011774 -4.7937582 -1.4405076 C-C with Shear

7 -6.0476374 -2.815533 -2.0663244 C-C with Shear

8 -4.9664461 -1.7215187 -2.2188507 C-C with Shear

9 -4.50558 -1.3721576 -1.9726936 C-C with Shear

10 -4.0800955 -6.9062526 1.3773656 C-C with Shear

11 -1.97454105 -4.89175605 1.4513526 C-C with Shear

12 -1.22493625 -3.98707125 1.30095 C-C with Shear

13 -0.831383 -3.3536635 1.1209616 C-C with Shear

14 0.062713 -1.344967 0.592482 T-C with Shear

15 0.105851 -0.341359 0.157645 T-C with Shear

16 0.080655 -0.088681 0.10941 T-C with Shear

17 -1.247429 0.102804 -0.989026 T-C with Shear

18 -0.385164 0.155657 -0.420975 T-C with Shear

19 -0.103132 0.114531 -0.100793 T-C with Shear

20 0.069079 0.074251 0 Tension-Tension

21 0.097443 0.064924 0.020366 T-T with Shear

22 0.057227 0.098029 -0.019077 T-T with Shear

E-3 Appendix E |

Table E-3 Summary of failure stresses of the RVE for bed joint angle of 45°

Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State

1 -1.7913574 -1.9426064 1.9870704 C-C with Shear

2 -1.3950524 -1.5125292 -1.5487892 C-C with Shear

3 0.166429 0.166656 -0.166603 T-T with Shear

4 0.061999 0.066133 0.067448 T-T with Shear


6 -4.936204 -5.3378738 -1.8222666 C-C with Shear

7 -3.6301636 -3.9339146 -2.4106866 C-C with Shear

8 -3.9703958 -4.2929936 -3.4044206 C-C with Shear

9 -2.7074888 -2.9166288 -2.5243912 C-C with Shear

10 -4.5295081 -4.8990591 1.6723551 C-C with Shear

11 -2.4769584 -2.685319 1.650124 C-C with Shear

12 -1.8682076 -2.0254668 1.6135266 C-C with Shear

13 -1.7084914 -1.8518052 1.6049656 C-C with Shear

14 -0.6504736 -0.7069944 1.203216 C-C with Shear

15 -0.14079 -0.157281 0.439898 C-C with Shear

16 -0.000006 0.000001 0.04547 T-C with Shear

17 0.000276 -0.000143 -0.118156 T-C with Shear

18 -0.283756 -0.316512 -0.916635 C-C with Shear

19 0.000276 -0.000143 -0.118156 T-C with Shear

20 0.069079 0.074251 0 Tension-Tension

21 0.070518 0.075734 0.025764 T-T with Shear

22 0.074265 0.079391 -0.026983 T-T with Shear

E-4 Appendix E |

Table E-4 Summary of failure stresses of the RVE for Case 1 (P1)

Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State

1 -0.433208 -8.916751 0 Uniaxial Compression


3 0.0070217 0.8422183 0 Uniaxial Tension











14 1.010030945 -2.92296725 0 Tension-Compression



17 -4.22096661 0.63517734 0 Tension-Compression



20 0.5429837 0.3958213 0 Tension-Tension

21 0.7136269 0.2701286 0 Tension-Tension

22 0.372897 0.5308352 0 Tension-Tension

E-5 Appendix E |


Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State




















20 0.023449134 0.017057118 0 Tension-Tension

21 0.030478828 0.011491253 0 Tension-Tension

22 0.016102935 0.022885785 0 Tension-Tension

E-6 Appendix E |


Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State




















20 0.026709104 0.018933116 0 Tension-Tension

21 0.033783855 0.012464442 0 Tension-Tension

22 0.019318341 0.026754337 0 Tension-Tension

E-7 Appendix E |


Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State




















20 0.90469841 0.63978376 0 Tension-Tension

21 1.12258608 0.3744979 0 Tension-Tension

22 0.65380546 0.90251238 0 Tension-Tension

E-8 Appendix E |

Table E-8 Summary of failure stresses of the conventional hollow block

Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State




















20 0.30530773 0.22208368 0.000 Tension-Tension

21 0.39683434 0.14961611 0.000 Tension-Tension

22 0.20966022 0.29797292 0.000 Tension-Tension

E-9 Appendix E |

Table E-9 Summary of failure stresses of the H-block

Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State

















17 -8.1304686 -0.19424259 0.000 Tension-Compression



20 0.6650417 0.34898195 0.000 Tension-Tension

21 0.7144488 0.2408509 0.000 Tension-Tension

22 0.5867958 0.5027818 0.000 Tension-Tension

E-10 Appendix E |

Table E-10 Summary of failure stresses of the modified H-block

Failure Summary


(MPa)

Normal Stress

(MPa)

Shear Stress

(MPa)

Stress State




















20 0.568610654 0.298379567 0.000 Tension-Tension

21 0.610853724 0.20592752 0.000 Tension-Tension

22 0.501710409 0.429878439 0.000 Tension-Tension

APPENDIX-F

Load factors applied on RVEs

F-1 Appendix F |

Table F-1 Load factors for 𝜽 = 𝟎° and 𝜽 = 𝟗𝟎°

Load Case nσ pσ τ

1 -1 0 0

2 0 -1 0

3 +1 0 0

4 0 +1 0

5 -1 -0.25 0

6 -1 -0.125 0

7 -1 -0.0625 0

8 -1 -0.03125 0

9 -1 -0.020833 0

10 -1 -0.5 0

11 -1 -1 0

12 -0.5 -1 0

13 -0.33 -1 0

14 -1 +1 0

15 -1 +0.5 0

16 -1 +0.25 0

17 +1 -0.0625 0

18 +1 -0.125 0

19 +1 -0.25 0

20 +1 +0.25 0

21 +1 +0.125 0

22 +1 +0.5 0

F-2 Appendix F |

Table F-2 Load factors for 𝜽 = 𝟐𝟐.𝟓° and 𝜽 = 𝟔𝟔.𝟓°


1 -0.853553391 -0.146446609 0.353553391

2 -0.146446609 -0.853553391 -0.353553391

3 0.853553391 0.146446609 -0.353553391

4 0.146446609 0.853553391 0.353553391

5 -1 -1 0

6 -1.146446609 -1.853553391 -0.353553391

7 -1.439339828 -3.560660172 -1.060660172

8 -2.025126266 -6.974873734 -2.474873734

9 -2.610912703 -10.3890873 -3.889087297

10 -1.853553391 -1.146446609 0.353553391

11 -3.560660172 -1.439339828 1.060660172

12 -6.974873734 -2.025126266 2.474873734

13 -10.3890873 -2.610912703 3.889087297

14 -3.267766953 0.267766953 1.767766953

15 -1.560660172 0.560660172 1.060660172

16 -0.707106781 0.707106781 0.707106781

17 0.267766953 -3.267766953 -1.767766953

18 0.560660172 -1.560660172 -1.060660172

19 0.707106781 -0.707106781 -0.707106781

20 1 1 0

21 1.146446609 1.853553391 0.353553391

22 1.853553391 1.146446609 -0.353553391

F-3 Appendix F |

Table F-3 Load factors for 𝜽 = 𝟗𝟎


1 -0.5 -0.5 0.5

2 -0.5 -0.5 -0.5

3 0.5 0.5 -0.5

4 0.5 0.5 0.5

5 -1 -1 0

6 -1.5 -1.5 -0.5

7 -2.5 -2.5 -1.5

8 -4.5 -4.5 -3.5

9 -6.5 -6.5 -5.5

10 -1.5 -1.5 0.5

11 -2.5 -2.5 1.5

12 -4.5 -4.5 3.5

13 -6.5 -6.5 5.5

14 -1.5 -1.5 2.5

15 -0.5 -0.5 1.5

16 -6.12323E-17 6.12323E-17 1

17 -1.5 -1.5 -2.5

18 -0.5 -0.5 -1.5

19 6.12323E-17 -6.12323E-17 -1

20 1 1 0

21 1.5 1.5 0.5

22 1.5 1.5 -0.5

F-4 Appendix F |

Table F-4 Load factors for 𝝉 = 𝟎.𝟏𝝈𝒏


1 -1 0 -0.1

2 0 -1 -0.1

3 1 0 0.1

4 0 1 0.1

5 -1 -1 -0.1

6 -1 -2 -0.1

7 -1 -4 -0.1

8 -1 -8 -0.1

9 -1 -12 -0.1

10 -2 -1 -0.2

11 -4 -1 -0.4

12 -8 -1 -0.8

13 -12 -1 -1.2

14 -4 1 -0.4

15 -2 1 -0.2

16 -1 1 -0.1

17 1 -4 0.1

18 1 -2 0.1

19 1 -1 0.1

20 1 1 0.1

21 1 2 0.1

22 2 1 0.2

F-5 Appendix F |

Table F-5 Load factors for 𝝉 = 𝟎.𝟐𝝈𝒏


1 -1 0 -0.2

2 0 -1 -0.2

3 1 0 0.2

4 0 1 0.2

5 -1 -1 -0.2

6 -1 -2 -0.2

7 -1 -4 -0.2

8 -1 -8 -0.2

9 -1 -12 -0.2

10 -2 -1 -0.4

11 -4 -1 -0.8

12 -8 -1 -1.6

13 -12 -1 -2.4

14 -4 1 -0.8

15 -2 1 -0.4

16 -1 1 -0.2

17 1 -4 0.2

18 1 -2 0.2

19 1 -1 0.2

20 1 1 0.2

21 1 2 0.2

22 2 1 0.4

F-6 Appendix F |

Table F-6 Load factors for 𝝉 = 𝟎.𝟑𝝈𝒏


1 -1 0 -0.3

2 0 -1 -0.3

3 1 0 0.3

4 0 1 0.3

5 -1 -1 -0.3

6 -1 -2 -0.3

7 -1 -4 -0.3

8 -1 -8 -0.3

9 -1 -12 -0.3

10 -2 -1 -0.6

11 -4 -1 -1.2

12 -8 -1 -2.4

13 -12 -1 -3.6

14 -4 1 -1.2

15 -2 1 -0.6

16 -1 1 -0.3

17 1 -4 0.3

18 1 -2 0.3

19 1 -1 0.3

20 1 1 0.3

21 1 2 0.3

22 2 1 0.6

F-7 Appendix F |

Table F-7 Load factors for 𝝉 = 𝟎.𝟒𝝈𝒏


1 -1 0 -0.4

2 0 -1 -0.4

3 1 0 0.4

4 0 1 0.4

5 -1 -1 -0.4

6 -1 -2 -0.4

7 -1 -4 -0.4

8 -1 -8 -0.4

9 -1 -12 -0.4

10 -2 -1 -0.8

11 -4 -1 -1.6

12 -8 -1 -3.2

13 -12 -1 -4.8

14 -4 1 -1.6

15 -2 1 -0.8

16 -1 1 -0.4

17 1 -4 0.4

18 1 -2 0.4

19 1 -1 0.4

20 1 1 0.4

21 1 2 0.4

22 2 1 0.8

F-8 Appendix F |

Table F-8 Load factors for 𝝉 = 𝟎.𝟓𝝈𝒏


1 -1 0 -0.5

2 0 -1 -0.5

3 1 0 0.5

4 0 1 0.5

5 -1 -1 -0.5

6 -1 -2 -0.5

7 -1 -4 -0.5

8 -1 -8 -0.5

9 -1 -12 -0.5

10 -2 -1 -1

11 -4 -1 -2

12 -8 -1 -4

13 -12 -1 -6

14 -4 1 -2

15 -2 1 -1

16 -1 1 -0.5

17 1 -4 0.5

18 1 -2 0.5

19 1 -1 0.5

20 1 1 0.5

21 1 2 0.5

22 2 1 1

F-9 Appendix F |

Table F-9 Load factors for 𝝉 = 𝟎.𝟔𝝈𝒏


1 -1 0 -0.6

2 0 -1 -0.6

3 1 0 0.6

4 0 1 0.6

5 -1 -1 -0.6

6 -1 -2 -0.6

7 -1 -4 -0.6

8 -1 -8 -0.6

9 -1 -12 -0.6

10 -2 -1 -1.2

11 -4 -1 -2.4

12 -8 -1 -4.8

13 -12 -1 -7.2

14 -4 1 -2.4

15 -2 1 -1.2

16 -1 1 -0.6

17 1 -4 0.6

18 1 -2 0.6

19 1 -1 0.6

20 1 1 0.6

21 1 2 0.6

22 2 1 1.2

F-10 Appendix F |

Table F-10 Load factors for 𝝉 = 𝟎.𝟔𝝈𝒏


1 -1 0 -0.7

2 0 -1 -0.7

3 1 0 0.7

4 0 1 0.7

5 -1 -1 -0.7

6 -1 -2 -0.7

7 -1 -4 -0.7

8 -1 -8 -0.7

9 -1 -12 -0.7

10 -2 -1 -1.4

11 -4 -1 -2.8

12 -8 -1 -5.6

13 -12 -1 -8.4

14 -4 1 -2.8

15 -2 1 -1.4

16 -1 1 -0.7

17 1 -4 0.7

18 1 -2 0.7

19 1 -1 0.7

20 1 1 0.7

21 1 2 0.7

22 2 1 1.4

F-11 Appendix F |

Table F-11 Load factors for 𝝉 = 𝟎.𝟖𝝈𝒏


1 -1 0 -0.8

2 0 -1 -0.8

3 1 0 0.8

4 0 1 0.8

5 -1 -1 -0.8

6 -1 -2 -0.8

7 -1 -4 -0.8

8 -1 -8 -0.8

9 -1 -12 -0.8

10 -2 -1 -1.6

11 -4 -1 -3.2

12 -8 -1 -6.4

13 -12 -1 -9.6

14 -4 1 -3.2

15 -2 1 -1.6

16 -1 1 -0.8

17 1 -4 0.8

18 1 -2 0.8

19 1 -1 0.8

20 1 1 0.8

21 1 2 0.8

22 2 1 1.6

F-12 Appendix F |

Table F-12 Load factors for 𝝉 = 𝟎.𝟗𝝈𝒏


1 -1 0 -0.9

2 0 -1 -0.9

3 1 0 0.9

4 0 1 0.9

5 -1 -1 -0.9

6 -1 -2 -0.9

7 -1 -4 -0.9

8 -1 -8 -0.9

9 -1 -12 -0.9

10 -2 -1 -1.8

11 -4 -1 -3.6

12 -8 -1 -7.2

13 -12 -1 -10.8

14 -4 1 -3.6

15 -2 1 -1.8

16 -1 1 -0.9

17 1 -4 0.9

18 1 -2 0.9

19 1 -1 0.9

20 1 1 0.9

21 1 2 0.9

22 2 1 1.8

F-13 Appendix F |

Table F-13 Load factors for 𝝉 = 𝟏.𝟎𝝈𝒏


1 -1 0 -1

2 0 -1 -1

3 1 0 1

4 0 1 1

5 -1 -1 -1

6 -1 -2 -1

7 -1 -4 -1

8 -1 -8 -1

9 -1 -12 -1

10 -2 -1 -2

11 -4 -1 -4

12 -8 -1 -8

13 -12 -1 -12

14 -4 1 -4

15 -2 1 -2

16 -1 1 -1

17 1 -4 1

18 1 -2 1

19 1 -1 1

20 1 1 1

21 1 2 1

22 2 1 2

F-14 Appendix F |

Table F-14 Load factors for 𝝉 = 𝟏.𝟏𝝈𝒏


1 -1 0 -1.1

2 0 -1 -1.1

3 1 0 1.1

4 0 1 1.1

5 -1 -1 -1.1

6 -1 -2 -1.1

7 -1 -4 -1.1

8 -1 -8 -1.1

9 -1 -12 -1.1

10 -2 -1 -2.2

11 -4 -1 -4.4

12 -8 -1 -8.8

13 -12 -1 -13.2

14 -4 1 -4.4

15 -2 1 -2.2

16 -1 1 -1.1

17 1 -4 1.1

18 1 -2 1.1

19 1 -1 1.1

20 1 1 1.1

21 1 2 1.1

22 2 1 2.2

F-15 Appendix F |

Table F-15 Load factors for 𝝉 = 𝟏.𝟐𝝈𝒏


1 -1 0 -1.2

2 0 -1 -1.2

3 1 0 1.2

4 0 1 1.2

5 -1 -1 -1.2

6 -1 -2 -1.2

7 -1 -4 -1.2

8 -1 -8 -1.2

9 -1 -12 -1.2

10 -2 -1 -2.4

11 -4 -1 -4.8

12 -8 -1 -9.6

13 -12 -1 -14.4

14 -4 1 -4.8

15 -2 1 -2.4

16 -1 1 -1.2

17 1 -4 1.2

18 1 -2 1.2

19 1 -1 1.2

20 1 1 1.2

21 1 2 1.2

22 2 1 2.4

F-16 Appendix F |

Table F-16 Load factors for 𝝉 = 𝟏.𝟑𝝈𝒏


1 -1 0 -1.3

2 0 -1 -1.3

3 1 0 1.3

4 0 1 1.3

5 -1 -1 -1.3

6 -1 -2 -1.3

7 -1 -4 -1.3

8 -1 -8 -1.3

9 -1 -12 -1.3

10 -2 -1 -2.6

11 -4 -1 -5.2

12 -8 -1 -10.4

13 -12 -1 -15.6

14 -4 1 -5.2

15 -2 1 -2.6

16 -1 1 -1.3

17 1 -4 1.3

18 1 -2 1.3

19 1 -1 1.3

20 1 1 1.3

21 1 2 1.3

22 2 1 2.6

F-17 Appendix F |

Table F-17 Load factors for 𝝉 = 𝟏.𝟒𝝈𝒏


1 -1 0 -1.4

2 0 -1 -1.4

3 1 0 1.4

4 0 1 1.4

5 -1 -1 -1.4

6 -1 -2 -1.4

7 -1 -4 -1.4

8 -1 -8 -1.4

9 -1 -12 -1.4

10 -2 -1 -2.8

11 -4 -1 -5.6

12 -8 -1 -11.2

13 -12 -1 -16.8

14 -4 1 -5.6

15 -2 1 -2.8

16 -1 1 -1.4

17 1 -4 1.4

18 1 -2 1.4

19 1 -1 1.4

20 1 1 1.4

21 1 2 1.4

22 2 1 2.8

F-18 Appendix F |

Table F-18 Load factors for 𝝉 = 𝟏.𝟓𝝈𝒏


1 -1 0 -1.5

2 0 -1 -1.5

3 1 0 1.5

4 0 1 1.5

5 -1 -1 -1.5

6 -1 -2 -1.5

7 -1 -4 -1.5

8 -1 -8 -1.5

9 -1 -12 -1.5

10 -2 -1 -3

11 -4 -1 -6

12 -8 -1 -12

13 -12 -1 -18

14 -4 1 -6

15 -2 1 -3

16 -1 1 -1.5

17 1 -4 1.5

18 1 -2 1.5

19 1 -1 1.5

20 1 1 1.5

21 1 2 1.5

22 2 1 3

APPENDIX-G

Material configurations and case numberings

G-1 Appendix G |

Table G-1 Case numbering for different unit strengths, mortar strengths and height to thickness ratios

Mortar Strength

Mortar number 𝒉𝒖/𝒕𝒋 Unit Strength Unit Number Case Name

5.0 1

5.0

3 1 M1.0H/T5.0U01 5.0 1 6 2 M1.0H/T5.0U02 5.0 1 10 3 M1.0H/T5.0U03 5.0 1 14 4 M1.0H/T5.0U04 5.0 1 16 5 M1.0H/T5.0U05 5.0 1 18 6 M1.0H/T5.0U06 5.0 1 25 7 M1.0H/T5.0U07 5.0 1 30 8 M1.0H/T5.0U08 5.0 1 50 9 M1.0H/T5.0U09 5.0 1 100 10 M1.0H/T5.0U10 5.0 1

5.7


6.5


7.6


9.0

3 1 M1.0H/T9.0U01 5.0 1 6 2 M1.0H/T9.0U02 5.0 1 10 3 M1.0H/T9.0U03 5.0 1 14 4 M1.0H/T9.0U04 5.0 1 16 5 M1.0H/T9.0U05 5.0 1 18 6 M1.0H/T9.0U06 5.0 1 25 7 M1.0H/T9.0U07 5.0 1 30 8 M1.0H/T9.0U08

G-2 Appendix G |

Mortar Strength

Mortar number 𝒉𝒖/𝒕𝒋 Unit Strength Unit Number Case Number

5.0 1 9.0 50 9 M1.0H/T9.0U09 5.0 1 100 10 M1.0H/T9.0U10 5.0 1

11

3 1 M1.0H/T11U01 5.0 1 6 2 M1.0H/T11U02 5.0 1 10 3 M1.0H/T11U03 5.0 1 14 4 M1.0H/T11U04 5.0 1 16 5 M1.0H/T11U05 5.0 1 18 6 M1.0H/T11U06 5.0 1 25 7 M1.0H/T11U07 5.0 1 30 8 M1.0H/T11U08 5.0 1 50 9 M1.0H/T11U09 5.0 1 100 10 M1.0H/T11U10 5.0 1

19


5.0


5.7


6.5

3 1 M2.0H/T6.5U01 12.0 2 6 2 M2.0H/T6.5U02 12.0 2 10 3 M2.0H/T6.5U03 12.0 2 14 4 M2.0H/T6.5U04 12.0 2 16 5 M2.0H/T6.5U05 12.0 2 18 6 M2.0H/T6.5U06 12.0 2 25 7 M2.0H/T6.5U07 12.0 2 30 8 M2.0H/T6.5U08

G-3 Appendix G |

Mortar Strength

Mortar number 𝒉𝒖/𝒕𝒋 Unit Strength Unit Number M1.0H/T6.5U09

12.0 2 6.5 50 9 M2.0H/T6.5U09 12.0 2 100 10 M2.0H/T6.5U10 12.0 2

7.6


9.0


11


19

3 1 M2.0H/T19U01 12.0 2 6 2 M2.0H/T19U02 12.0 2 10 3 M2.0H/T19U03 12.0 2 14 4 M2.0H/T19U04 12.0 2 16 5 M2.0H/T19U05 12.0 2 18 6 M2.0H/T19U06 12.0 2 25 7 M2.0H/T19U07 12.0 2 30 8 M2.0H/T19U08 12.0 2 50 9 M2.0H/T19U09 12.0 2 100 10 M2.0H/T19U10

APPENDIX-H

Comparison of AS3700 and the models compressive strength predictions

H-1 Appendix H |

Figure H-1 Comparison of the predicted results of the proposed model and AS3700 for

compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 5 and mortar strength = 5 MPa


compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 5.67 and mortar strength = 5 MPa

H-2 Appendix H |

Figure H--3 Comparison of the predicted results of the proposed model and AS3700 for




H-3 Appendix H |





H-4 Appendix H |

Figure H-7 Comparison of the predicted results of the proposed model and AS3700 for compressive strength of masonry with ℎ𝑢/𝑡𝑗 = 19 and mortar strength = 5 MPa



H-5 Appendix H |





H-6 Appendix H |





H-7 Appendix H |





H-8 Appendix H |


compressive strength of masonry with characteristic compressive strength 𝑓′𝑢𝑢 = 3 𝑀𝑀𝑀



H-9 Appendix H |





H-10 Appendix H |





H-11 Appendix H |





H-12 Appendix H |



Figure H-24 Comparison of the predicted results of the proposed model and AS3700 for compressive strength of masonry with characteristic compressive strength 𝑓′𝑢𝑢 = 100 𝑀𝑀𝑀

H-13 Appendix H |





H-14 Appendix H |





H-15 Appendix H |





H-16 Appendix H |





H-17 Appendix H |





development of a transient gradient enhanced non local continuum damage … 4.5 flowchart for the...

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