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Hypoplasticity InvestigatedParameter Determination and Numerical Simulation

Kambiz Elmi Anaraki

June 28, 2008

Hypoplasticity InvestigatedParameter Determination and Numerical Simulation

Master of Science Thesis

For obtaining the degree of Master of Science in Geotechnical

Engineering at Delft University of Technology

Kambiz Elmi Anaraki

June 28, 2008

Department of Geotechnology· Delft University of Technology

Delft University of Technology

Title : Hypoplasticity InvestigatedParameter Determination and

Numerical Simulation

Author : Kambiz Elmi Anaraki

Date : June 28, 2008Professor : Prof.ir. A.F. van TolSupervisor : Dr.ir. O.M. HeeresSupervisor : Dr.ir. A. FraaijSupervisor : Ir. J. DijkstraReport number : CT/GE/00-00

Postal Address : Section of Geotechnical EngineeringDepartment of GeotechnologyThe Netherlands

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Abstract

Due to the weak soil deposits in the Netherlands, pile foundations consisting of driven dis-placement piles are often used to support the superstructure. The influence of pile installationprocess on the stresses and the soil properties is large. In geotechnical practise the effect ofthe pile installation and the interaction with neighbouring structures is of special interest.These type of geotechnical problems are commonly investigated with the aid of numericalmethods, i.e. the finite element method (FEM). Numerical simulation of the pile installationprocess requires a constitutive model which is able to describe the soil behaviour, taking intoaccount the continuously changing soil properties.In the current study the use of the hypoplasticity constitutive model in an Eulerian frame-work, offered by the commercial package FEAT, is adopted. Hypoplasticity has emerged inthe recent years and is gaining popularity due to its numerous successfull applications. Itis an inelastic (dissipative) and incrementally nonlinear constitutive model, which requiresneither a yield surface nor a decomposition of strain rate into elastic and plastic portions. Itpresents a single tensorial equation holding equally for loading and unloading.This research is formulated to gain more knowledge into the experimental determination ofthe hypoplasticity model parameters, leading to the derivation of a consistent parameter setfor Baskarp sand. The hypoplasticity parameters for Baskarp sand are of special interest suchthat the numerical simulation of the soil behaviour can be compared with the measurementsobtained from two model tests of displacement piles installed in a geotechnical centrifuge. Theoutcome of this thesis will provide an unambiguous procedure to determine the hypoplasticitymodel parameters. The parameter set derived for Baskarp sand can be used to validate andimprove the numerical simulation of the model pile tests performed by Dijkstra et al. (2006).The experimental work starts with the characterisation tests, i.e. grain size distribution, massdensity, maximum and minimum void ratios. Next, a series of oedometric and drained triaxialcompression tests were carried out on loose and dense samples. In addition image processingtechniques are introduced for the determination of the angle of repose.Parameter determination procedures for the hypoplasticity model are evaluated and extended.Subsequently, from the laboratory test results the hypoplastic parameter set for Baskarp sandis derived. The consistency of the parameter set is validated by numerical simulation of oe-dometric and triaxial element tests, where a realistic compressive and shear behaviour was

vi Abstract

observed. A (multi) parametric analysis is performed to evaluate the sensitivity of the ob-tained parameter set. This analysis showed that the compressive and shear behaviour arecontrolled by two different parameter groups. Finally, in order to assess the suitability of thehypoplastic constitutive model in a more complex case where both compression and shearingas well as large deformations are involved, the installation of a displacement pile is numericallysimulated and compared with the experimental results.

Acknowledgements

The writing of this thesis has been one of the most significant academic challenges I have everhad to face. Without the support, patience and guidance of the following people, this studywould not have been completed. It is to them that I owe my deepest gratitude.

• Professor van Tol who undertook to act as my main supervisor despite his many otheracademic and professional commitments. His wisdom, knowledge and commitment tothe highest standards inspired and motivated me.

• My supervisors, Dr. Heeres and Dr. Fraaij. Their knowledge, commitment and wellappreciated comments helped me to improve my work. Thank you for the vital encour-agement and support.

• My friend, roommate and supervisor Jelke Dijkstra, who’s intelligence, encouragement,understanding and assistance constantly motivated me. His commentary was neverdirected to the strangeness of my ideas, always on how to improve the development ofthe idea. He helped me be better at what I did and to expand my skills in differentareas.

• Han Visser, for helping me with the laboratory equipment. He could always spare sometime and help me. Thank you for the support and encouragement.

• All my colleagues at the section of geotechnical engineering at Delft University of Tech-nology who participated in this research project by showing interest and enthusiasm.

Last but not least, would like to thank my family, all my friends and my Niusha who motivatedme to bring this study to the end. Thank you for being so patient and for believing in me.

Delft, University of Technology Kambiz Elmi AnarakiJune 28, 2008

viii Acknowledgements

Table of Contents

Abstract v

Acknowledgements vii

1 INTRODUCTION 1

1-1 Motivation and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1-2 Outline and objective of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 HYPOPLASTICITY 3

2-1 Description of the hypoplasticity model . . . . . . . . . . . . . . . . . . . . . . . 3

2-1-1 Various versions of hypoplastic equation . . . . . . . . . . . . . . . . . . 5

2-2 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2-3 Parameter determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2-3-1 Critical state parameters ϕc and ec0 . . . . . . . . . . . . . . . . . . . . 7

2-3-2 Limit void ratios ed0 and ei0 . . . . . . . . . . . . . . . . . . . . . . . . 7

2-3-3 Stiffness parameters hs and n . . . . . . . . . . . . . . . . . . . . . . . 8

2-3-4 Exponent α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2-3-5 Exponent β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2-3-6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 PARAMETERS 13

3-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3-2 Baskarp sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3-2-1 Grain size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3-2-2 Grain crushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3-2-3 Mass density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3-3 Critical friction angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

x Table of Contents

3-3-1 Proposed test setup to determine the angle of repose . . . . . . . . . . . 18

3-3-2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3-3-3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3-4 Minimum and maximum void ratio . . . . . . . . . . . . . . . . . . . . . . . . . 22

3-4-1 Limit densities according to JGS . . . . . . . . . . . . . . . . . . . . . . 23

3-4-2 Minimum void ratio emin . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3-4-3 Maximum void ratio emax . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3-5 Oedometric test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-5-1 Oedometric response of Baskarp sand . . . . . . . . . . . . . . . . . . . 25

3-5-2 Determination of the granulate hardness hs and exponent n . . . . . . . 27

3-5-3 Proposed method to determine the stiffness parameters hs and n . . . . 27

3-5-4 Proposed method and limit void ratios . . . . . . . . . . . . . . . . . . . 31

3-6 Triaxial compression test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3-6-1 General description triaxial apparatus . . . . . . . . . . . . . . . . . . . . 33

3-6-2 Behaviour of Baskarp sand during drained triaxial testing . . . . . . . . . 34

3-6-3 Determination of exponent α . . . . . . . . . . . . . . . . . . . . . . . . 35

3-6-4 Determination of exponent β . . . . . . . . . . . . . . . . . . . . . . . . 36

3-7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 NUMERICAL SIMULATION OF ELEMENT TESTS 39

4-1 Finite Element Application Technology . . . . . . . . . . . . . . . . . . . . . . . 39

4-1-1 Element tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4-2 Simulation of oedometer tests on Baskarp sand . . . . . . . . . . . . . . . . . . 40

4-3 Simulation of triaxial tests on Baskarp sand . . . . . . . . . . . . . . . . . . . . 41

4-3-1 Drained triaxial test on dense samples . . . . . . . . . . . . . . . . . . . 42

4-3-2 Drained triaxial test on loose samples . . . . . . . . . . . . . . . . . . . 43

4-4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 PARAMETRIC ANALYSIS 45

5-1 Parameter variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5-2 Influence of parameter variation on oedometric test response . . . . . . . . . . . 46

5-3 Multi-parametric analysis for oedometric response . . . . . . . . . . . . . . . . . 50

5-4 Influence of parameter variation on triaxial test response . . . . . . . . . . . . . 51

5-5 Multi-parametric analysis for the response of drained triaxial test . . . . . . . . . 56

5-5-1 Critical friction angle ϕc . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5-5-2 Critical void ratio at zero pressure ec0 . . . . . . . . . . . . . . . . . . . 58

5-5-3 Exponent α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5-5-4 F-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5-6 Influence of the accuracy of the hypoplasticity parameters ϕc and n . . . . . . . 60

5-7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Table of Contents xi

6 SIMULATION OF THE INSTALLATION OF A DISPLACEMENT PILE 63

6-1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6-2 FE-Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6-3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6-4 Comparison with element tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 CONCLUSIONS AND RECOMMENDATIONS 69

7-1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7-2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7-3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A THE CRITICAL FRICTION ANGLE 77

A-1 Image processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A-2 Lens distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

B MINIMUM AND MAXIMUM VOID RATIO 81

B-1 Minimum void ratio procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

B-2 Maximum void ratio procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

C OEDOMETRIC TEST 83

C-1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

C-2 Displacement transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

C-3 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

C-4 Broken glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

C-4-1 Grain size distribution broken glass . . . . . . . . . . . . . . . . . . . . . 86

C-4-2 Oedometric response broken glass . . . . . . . . . . . . . . . . . . . . . 87

C-4-3 Crushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

D TRIAXIAL COMPRESSION TEST 89

D-1 Experimental measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D-2 Control software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D-3 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D-4 De-Aired watertank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D-5 Specimen preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D-5-1 Free ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D-5-2 Mould and membrane setting . . . . . . . . . . . . . . . . . . . . . . . . 94

D-5-3 Tamping method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D-5-4 Sealing the specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

D-6 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

D-6-1 Vacuum flushing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

D-6-2 Back pressure saturation . . . . . . . . . . . . . . . . . . . . . . . . . . 96

D-7 Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

D-8 Second B-check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xii Table of Contents

D-9 Consolidated drained test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

D-10 Measurement errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

D-10-1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

D-10-2 Errors in void ratio measurements during sample preparation . . . . . . . 100

D-10-3 Change in void ratio during saturation and consolidation . . . . . . . . . 101

D-10-4 Back pressure line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

E VERIFICATION NUMERICAL SCHEME FEAT 103

E-1 One-dimensional compression test . . . . . . . . . . . . . . . . . . . . . . . . . 103

E-2 Triaxial compression test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

E-3 Biaxial test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

F VERIFICATION PARAMETER DETERMINATION 107

F-1 Hostun RF sand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

F-1-1 Simulation of triaxial tests on Hostun RF sand . . . . . . . . . . . . . . 108

List of Figures

2-1 Development of void ratios ei, ec, ed. . . . . . . . . . . . . . . . . . . . . . . . . 8

2-2 Determination of the exponents n for a selected stress range (Herle 2000). . . . 9

2-3 Influence of β on the oedometric response of loose and dense samples. . . . . . . 11

3-1 Magnified image of Baskarp sand. . . . . . . . . . . . . . . . . . . . . . . . . . 14

3-2 Grain size distribution curves for Baskarp sand obtained by Dry and Wet sieving. 15

3-3 Wet sieving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3-4 Grain size distribution curves of Baskarp sand after performing oedometric tests. 17

3-5 Device for measuring the angle of repose developed by Miura et al. (1997). . . . 18

3-6 Test setup for the determination of angle of repose. . . . . . . . . . . . . . . . . 19

3-7 Progressive average value of the angle of repose for Baskarp sand. . . . . . . . . 19

3-8 Normal distribution of the obtained angles of repose of Baskarp sand. . . . . . . 20

3-9 Factors of influence on the angle of repose. . . . . . . . . . . . . . . . . . . . . 21

3-10 Dependence of ϕc on Dr, and σ3 . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3-11 Mean stress dependency on void ratios. . . . . . . . . . . . . . . . . . . . . . . 22

3-12 JGS mold to determine the limit densities. . . . . . . . . . . . . . . . . . . . . . 23

3-13 Casagrande oedometer apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . 25

3-14 Oedometric response of the specimens of Baskarp sand. . . . . . . . . . . . . . . 26

3-15 Results of the oedometer tests performed on a ø50 mm sample. . . . . . . . . . 28

3-16 Comparison of the results of two test with initial void ratio close to ec0 . . . . . 29

3-17 Results of Method 1 and Method 2 for different validity ranges. . . . . . . . . . 30

3-18 Parameters n and hs for oedometer test O51 for different validity ranges. . . . . 30

3-19 Simulation of the response of specimen O53, O51 and O52 by Method 2. . . . . 31

3-20 ei, ec and ed calculated by Method 2. . . . . . . . . . . . . . . . . . . . . . . . 32

3-21 GDS Triaxial testing system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xiv List of Figures

3-22 Triaxial apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3-23 Results of CD triaxial tests on dense Baskarp sand. T: (q − ε1), B: (εv − ε1). . . 363-24 Results of CD triaxial tests on loose Baskarp sand. T: (q − ε1), B: (εv − ε1). . . 37

4-1 Elements. L: oedometric, M: drained triaxial, R: undrained triaxial. . . . . . . . . 40

4-2 Oedometer element test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4-3 Oedometric tests on specimen O51, O52 and O54. . . . . . . . . . . . . . . . . 41

4-4 Oedometric test on specimen O54. . . . . . . . . . . . . . . . . . . . . . . . . . 41

4-5 Drained triaxial compression element test. . . . . . . . . . . . . . . . . . . . . . 42

4-6 Numerical simulation of triaxial compresison test on dense samples. . . . . . . . 42

4-7 Numerical simulation of triaxial compresison test on loose samples. . . . . . . . . 43

5-1 Influence of the variation of ϕc on the simulated oedometric response. . . . . . . 46

5-2 Influence of the variation of hs on the simulated oedometric response. . . . . . . 46

5-3 Influence of the variation of n on the simulated oedometric response. . . . . . . 47

5-4 Influence of the variation of ed0 on the simulated oedometric response. . . . . . . 47

5-5 Influence of the variation of ec0 on the simulated oedometric response. . . . . . . 48

5-6 Influence of the variation of ei0 on the simulated oedometric response. . . . . . . 48

5-7 Influence of the variation of α on the simulated oedometric response. . . . . . . 49

5-8 Influence of the variation of β on the simulated oedometric response. . . . . . . 49

5-9 Influence of the multi-parametric analysis on simulated oedometric response. . . 50

5-10 Influence of the variation of ϕc on simulated triaxial response. . . . . . . . . . . 51

5-11 Influence of the variation of hs on simulated triaxial response. . . . . . . . . . . 52

5-12 Influence of the variation of n on simulated triaxial response. . . . . . . . . . . . 52

5-13 Influence of the variation of ed0 on simulated triaxial response. . . . . . . . . . . 53

5-14 Influence of the variation of ec0 on simulated triaxial response. . . . . . . . . . . 53

5-15 Influence of the variation of ei0 on simulated triaxial response. . . . . . . . . . . 54

5-16 Influence of the variation of α on simulated triaxial response. . . . . . . . . . . . 55

5-17 Influence of the variation of β on simulated triaxial response. . . . . . . . . . . . 55

5-18 Influence of the multi-parametric analysis on simulated triaxial response. . . . . . 57

5-19 Development of qpeak (Left) and ψ (Right) for variation of ϕc. . . . . . . . . . . 58

5-20 Development of qpeak (Left) and ψ (Right) for variation of ec0. . . . . . . . . . . 58

5-21 Development of qpeak (Left) and ψ (Right) for variation of α. . . . . . . . . . . 59

5-22 e − ps response of specimen O51 for different values of ϕc. . . . . . . . . . . . . 605-23 Stiffness parameters based on different values of ϕc. . . . . . . . . . . . . . . . . 61

6-1 Mesh and the boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 64

6-2 Material flow through the mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6-3 Horizontal and vertical stresses developed during the pile installation process. . . 66

6-4 Shear stresses developed during the installation of a displacement pile. . . . . . . 66

List of Figures xv

6-5 Development of the void ratio under the pile tip. . . . . . . . . . . . . . . . . . 67

6-6 Change in porosity during pile installation . . . . . . . . . . . . . . . . . . . . . 67

6-7 Simulated pile bearing capacity of a displacement pile . . . . . . . . . . . . . . . 68

A-1 Top: Image of the sand heap. Bottom: Cropped image . . . . . . . . . . . . . . 78

A-2 Top: Threshold. Bottom: Calculated angle of repose. . . . . . . . . . . . . . . . 78

A-3 Pincushion distortion (left) and barrel distortion (right). . . . . . . . . . . . . . . 79

A-4 Recognised dots are matched with the grid to determine optimisation parameters. 79

A-5 Lens undistortion optimisation steps. . . . . . . . . . . . . . . . . . . . . . . . . 80

B-1 Equipment to determine minimum void ratio according to JGS standards. . . . . 82

C-1 Results obtained by a calibrated and an uncalibrated oedometer apparatus. . . . 83

C-2 Left: Mitutoya ID-C150B, Right: VJtech BG2111. . . . . . . . . . . . . . . . . . 84

C-3 Calibration results of the displacement transducers. . . . . . . . . . . . . . . . . 85

C-4 Mean value of the calibration results of the displacement transducers. . . . . . . 85

C-5 Grain size distribution curves of broken glass obtained by Dry sieving. . . . . . . 87

C-6 The response of broken glass in an oedometric test. . . . . . . . . . . . . . . . . 87

C-7 Grain size distribution curves of crushed glass obtained by Dry sieving. . . . . . . 88

D-1 Schematic layout of a Digital Pressure Volume Controller. . . . . . . . . . . . . 90

D-2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

D-3 De-central watertank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D-4 Influence of the test conditions for dense Hostun RF sand. . . . . . . . . . . . . 92

D-5 Influence of the test conditions for loose Hostun RF sand. . . . . . . . . . . . . . 93

D-6 Water percolation and back pressure saturation arrangement. . . . . . . . . . . . 95

D-7 Results of CD triaxial tests from GDS1 and GDS2 for σc=100 kPa . . . . . . . . 99

D-8 Results of CD triaxial tests from GDS1 and GDS2 for σc=200 kPa . . . . . . . . 100

E-1 Eulerian and Lagrangian simulation of an oedometeric test. . . . . . . . . . . . . 104

E-2 Simulation of triaxial tests on Hochstetten sand. . . . . . . . . . . . . . . . . . . 104

E-3 Biaxial element test: axial stress versus axial strain. . . . . . . . . . . . . . . . . 105

F-1 Drained triaxial compression test on dense Hostun sand. . . . . . . . . . . . . . 108

xvi List of Figures

List of Tables

2-1 Hypoplastic parameters ec0 and ed0 and the parameters emax and emin. . . . . . 8

3-1 Baskarp sand specimen properties determined by dry and wet sieving. . . . . . . 15

3-2 Mass density of Baskarp sand. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3-3 Minimum void ratio of Baskarp sand according to JGS standard. . . . . . . . . . 24

3-4 Maximum void ratio of Baskarp sand according to JGS standard. . . . . . . . . . 24

3-5 Void ratio dependent hypoplastic parameters for Baskarp sand. . . . . . . . . . . 24

3-6 Oedometric tests performed on Baskarp sand. . . . . . . . . . . . . . . . . . . . 26

3-7 Numerical quantities Figure 3-15D and E. . . . . . . . . . . . . . . . . . . . . . 27

3-8 Calculated values of Figure 3-16 for specimen O51. . . . . . . . . . . . . . . . . 29

3-9 Results of drained triaxial compression tests performed on Baskarp sand. . . . . . 35

3-10 Results of drained triaxial compression tests performed on Baskarp sand. . . . . . 36

3-11 Estimated hypoplastic parameters for Baskarp sand. . . . . . . . . . . . . . . . . 38

4-1 Estimated hypoplastic parameters for Baskarp sand. . . . . . . . . . . . . . . . . 40

5-1 Hypoplastic parameters for the parametric analysis . . . . . . . . . . . . . . . . 45

5-2 Sensitivity of oedometric response to parameter variation. . . . . . . . . . . . . . 49

5-3 Sensitivity of triaxial response to parameter variation. . . . . . . . . . . . . . . . 56

5-4 Investigated sets for the multi-parametric analysis . . . . . . . . . . . . . . . . . 56

5-5 F-test data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5-6 Recalculated stiffness parameters for specimen O51 based on different values of ϕc. 61

5-7 Influential parameters for simulation of oedometric and triaxial tests . . . . . . . 62

6-1 Hypoplastic parameters for Baskarp sand. . . . . . . . . . . . . . . . . . . . . . 65

7-1 Hypoplastic parameters for Baskarp sand. . . . . . . . . . . . . . . . . . . . . . 70

xviii List of Tables

7-2 Influential parameters for compressive and shearing behaviour. . . . . . . . . . . 70

C-1 Technical specification of the displacement transducers. . . . . . . . . . . . . . . 84

C-2 Granulometric properties of broken glass obtained by dry sieving. . . . . . . . . . 86

D-1 Initial void ratios of the specimens prepared by dry compaction method. . . . . . 94

D-2 B-values before and after the consolidation stage. . . . . . . . . . . . . . . . . . 98

E-1 The parameters of the hypoplastic model for the Hochstetten sand . . . . . . . . 103

E-2 The parameters of the hypoplastic model for the biaxial test sand . . . . . . . . 105

F-1 Density parameters of Hostun RF sand determined according to JGS standards. . 107

F-2 Estimated hypoplastic parameters for Hostun RF sand. . . . . . . . . . . . . . . 108

Chapter 1

INTRODUCTION

1-1 Motivation and background

In current practice of geotechnical engineering almost all problems are modelled with finiteelement methods to predict the soil behaviour. Interesting is that due to limitations in thesemethods simplifications are required. These simplifications and applied safety factors in mostcases lead to overly safe designs. One of the geotechnical topics frequently dealt with is theprediction of the pile bearing capacity and effects of the installation on the neighboring piles.During the installation of a displacement pile the soil properties at the pile base and pileshaft are continuously changing. Also the installation involves large deformations. In orderto simulate this problem the constitutive model must be able to describe the soil behaviour,taking into account the continuously changing soil properties.Search for a suitable numerical framework capable of the simulation of this problem hasresulted in the selection of the hypoplastic constitutive equation used in the numerical frame-work FEAT1. Hypoplasticity and in particular the hypoplastic relation proposed by von Wolf-fersdorff (1996) allows the simulation of each desired type of deformation and the stress stateis continuously completely determined. Furthermore, the dependence of the calculated soilstrength on the dilative or contractive behaviour is clearly defined. FEAT is a commerciallyavailable finite element program capable of large strains (Eulerian description).This thesis is related to the recent published article of Dijkstra et al.(2006) concerning thenumerical simulation of the installation of a displacement pile in sand. In his article Dijkstraargues that the observed discrepancy between the predicted and measured base load in hismodel tests is partly due to inaccurate model parameters for the Baskarp sand, used in thetests. The determination of hypoplastic model parameters and optimising the procedures todetermine these parameters, is the main objective of this thesis.

1http://www.feat.nl

2 INTRODUCTION

1-2 Outline and objective of this thesis

The hypoplastic constitutive model has been in development since 1985. In this study theformulation based on the proposal of von Wolffersdorff (1996) is utilised. This formulationcaptures the influence of mean pressure and density along various deformation paths and thesoil behaviour is bounded by asymptotic states including the widely accepted critical state.As stated by von Wolffersdorff, the hypoplastic constitutive relation requires eight parameters:the granular stiffness hs, the critical friction angle φc, the critical void ratio ec0 at zeropressure, the minimum and maximum void ratios ed0 and ei0 at zero pressure and the constantsn, α and β.The procedures to determine these parameters have been published by von Wolffersdorff(1996) and more recently Herle published procedures which improve on those ([Herle, 1997],[Herle, 2000]). However, the same publications show that the validation and determinationof the model parameters are ambiguous. Successful application of the hypoplastic modelis not possible without a reliable procedure to obtain the set of model parameters. Hence,the parameter determination and calibration is an important part of this thesis. The mainobjectives of this thesis are as follows:

• Determination of the hypoplasticity parameter set for Baskarp sand.

• Evaluation of the consistency and the sensitivity of the afore mentioned set.

• Application of the hypoplasticity model for a representative case.

Chapter 2

HYPOPLASTICITY

This chapter provides the theoretical background for the results to be presented in the following

chapters. We start with the description of the hypoplastic constitutive law as it was originally

proposed by Kolymbas and discuss the improvements which were implemented afterwards.

Subsequently, we will discuss the parameters necessary for the hypoplasticity equation used

in this research. Next, parameter determination based on theoretical derivation is explained.

Finally, the parameters are individually elaborated in more detail.

2-1 Description of the hypoplasticity model

The hypoplastic constitutive law describes the deformation behaviour of cohesionless soils,including the nonlinearity and inelasticity. The first version of the hypoplastic constitutivelaw was proposed by Kolymbas (1985). Kolymbas used a single state variable, the currentCauchy stress Ts. Later another state variable the void ratio e was added. The hypoplasticconstitutive equation in general form is given by:

T̊s = F(Ts, e,D) (2-1)

Herein T̊s represents the objective stress rate tensor as a function of the current void ratioe, the Cauchy granulate stress tensor Ts and the stretching tensor of the granular skeletonD. D = (L+L

T

2 ) is the symmetric part of the deformation gradient L =δv(x,t)

δx , v being thevelocity vector of the continuum representing the grain skeleton in a point x.Equation (2-1) forms the basis of the Bauer-Gudehus hypoplastic constitutive equation, whichassumed that the soil is a continuous homogeneous granular body whose state is fully describedby the void ratio e and the stress tensor Ts [Gudehus, 1996]. Bauer (1996) proposed todecompose the function F into two parts:

T̊s = A(Ts, e,D) + B(e,Ts)‖D‖ (2-2)

The first part A(Ts, e,D) is linear in D to represent the particular case where the soilbehaviour is hypoelastic, while the second part B(e,Ts)‖D‖ is nonlinear in D. ‖D‖ stands

4 HYPOPLASTICITY

for the Euclidian norm√

trD2. In order to permit an easier separation and determination ofthe constitutive parameters during the calibration of the hypoplastic constitutive equation,the operators A and B are factorised by introducing two dimensionless factors fd and fedepending only on the void ratio e. Taking into account the requirements concerning theSOM state1 (Sweeping out of memory) and the critical state, Gudehus and Bauer proposedthe following constitutive equation to represent the hypoplastic behaviour of soil:

T̊s = fefb

(

L(T̂s,D) + fdN(T̂s)‖D‖)

(2-3)

The tensorial parts L and N‖D‖ depend on the stretching D and the stress ratio tensor T̂s.T̂s =

Ts

trTsdenotes the so-called granulate stress ratio tensor. It has the same direction of

principal axes as Ts.The hypoplastic equation used in the present study was elaborated by von Wolffersdorff (1996)and is written as

T̊s = fefb1

trT̂2s

(

F 2D + a2tr(T̂sD)T̂s + fdaF (T̂s + T̂∗s)‖D‖

)

(2-4)

where T̂∗s = T̂s − 13I is the deviatoric part of T̂s and I is the unit tensor. Equation (2-4) is a

modification of the relation given by Gudehus and Bauer in which the Matsouka/Nakai limitcondition has been implemented. The coefficients in Equation (2-4) depend on the invariantsof the stress tensor and the void ratio. The factor a is determined by the friction angle ϕc incritical states

a =

√

3

8

(3 − sinϕc)sin ϕc

(2-5)

The factor F 2 is a function of the deviatoric stress ratio tensor T̂∗s

F =

√

1

8tan2 ψ +

(2 − tan2 ψ)2 +

√2 tan ψ cos 3θ

− 12√

2tan ψ (2-6)

where

tan ψ =√

3‖T̂∗s‖, cos 3θ = −√

6trT̂

∗3s

(

trT̂∗2s

)3

2

(2-7)

The factor a and F determine the critical state surface in the stress space. The pycnotropy(Greek for density dependent) factors fd and fe and the barotropy (Greek for pressure de-pendent) factor fb are discussed in the next section.For a detailed discussion of the constitutive equation one is referred to the original papers([Gudehus, 1996], [Bauer, 1996], [Wolffersdorff, 1996]).

1It is well known that the stress paths resulting from proportional strain paths tend to be also proportional.While the beginning of the stress path is influenced by the initial stresses and initial soil density, the memoryof the soil disappears during monotonic deformation. This phenomena is called SOM state.

2adapts the deviatoric yield curve to Matsuoka-Nakai

2-2 Model parameters 5

2-1-1 Various versions of hypoplastic equation

The version of hypoplasticity used in the present study was elaborated by von Wolffersdorff(1996), but there are various versions of hypoplasticity available. The original version ofhypoplasticity as proposed by Kolymbas (1985) was quite limited in its applicability. It wasonly recommended for cohesiveless granular materials consisting of not too soft grains. Alsoeffects like cyclic loading and creep were not included. The present version of hypoplastic-ity can cover the above mentioned limitations. Shortcomings with respect to the simula-tion of cyclic loading is overcome by the introduction of the intergranular strain tensor δ[Niemunis and Herle, 1997] and with visco-hypoplasticity the rate effect for clays and organicsoils is taken into account [Gudehus, 2004].For the present study the behaviour of the soil is modelled with the hypoplastic constitutivelaw in the version of von Wolffersdorff. This suffices since here only granular soil is considered.

2-2 Model parameters

The void ratio involved in the constitutive function is determined from the mass balanceequation

e̊ = (1 + e)trD (2-8)

The void ratio ec in the critical state and the corresponding mean pressure −trTs3 (trTs < 0for compression) are assumed to be connected by the relation

ecec0

= exp

[

−(−trTs

hs

)n]

(2-9)

where ec0, hs and n are material constants. Since the usually assumed logarithmic relationshipbetween the critical void ratio and the mean pressure is known to fail at pressures higherthan about 1 MPa, Been et al. (1991), relation (2-9) was proposed by Bauer (1996) to morerealistically describe the critical void ratio for high as well as low pressures.Besides the critical void ratio, two other characteristic void ratios are specified as functionsof the mean pressure: the minimal void ratio, ed, and the void ratio in the loosest state, ei.The pressure dependence of these void ratios is postulated in the same form as for the criticalstate void ratio:

eiei0

=eded0

= exp

[

−(−trTs

hs

)n]

(2-10)

with the corresponding reference values ei0, ed0 for zero pressure.The transition to the critical state, the peak friction angle and the dilative behaviour iscontrolled by the pycnotropy factor fd. This is also the only parameter that allows the stressrate tensor T̊s to vanish. By definition, the critical state is reached when the factor fd isequal to 1. fd is given by

fd =

(

e − edec − ed

)α

(2-11)

where α is a material parameter.The factor fe controls the influence of the void ratio e on the incremental stiffness. Since thestiffness of the granular material increases when the void ratio decreases (i.e. the soil becomes

6 HYPOPLASTICITY

denser), Bauer (1996) proposed to link the density factor fe to the ratio between the voidratio ec and the current void ratio e. fe is given by

fe =(ec

e

)β(2-12)

where β is a material parameter.The barotropic factor fb was introduced to take into account the increase of the stiffnessconsecutive to an increase of the mean stress. fb is directly determined from the consistencyrequirement that the simulation of a perfect isotropic compression must provide the sameexponential relationship between the current void ratio e and the mean pressure −trTs3 asassumed in the Equation (2-10). fb is defined as

fb =hsn

(

1 + eiei

) (

ei0ec0

)β (−trTshs

)1−n [

3 + a2 −√

3a

(

ei0 − ed0ec0 − ed0

)α]−1(2-13)

Finally fs is defined as the product of the density factor fe and the barotropic factor fb(unified as they are physically non-separable). fs can be written as

fs =hsn

(

1 + eiei

)

(eie

)β(−trTs

hs

)1−n [

3 + a2 −√

3a

(

ei0 − ed0ec0 − ed0

)α]−1(2-14)

by utilising Equation (2-9).The required parameters of the hypoplasticity constitutive model then become: the granularstiffness hs, the critical friction angle ϕc, the critical void ratio at zero pressure ec0, the voidratio at the maximum density at zero pressure ed0, the maximum void ratio at zero pressureei0 and the exponents n, α and β.

2-3 Parameter determination

The parameter determination methodology is discussed in different publications con-cerning hypoplasticity ([Bauer, 1996], [Wolffersdorff, 1996], [Herle and Gudehus, 1999],[Herle, 2000]). The methodology discussed here is mainly based on the publication by Herle(2000).In case of no rotation of principal stress axes, the objective stress rate T̊s corresponds to thetime derivative of Ṫs. Then it is sufficient to consider a compression of a cylindrical samplewith T1 < T2 = T3 (Tij = 0 for i 6= j, compressive stresses and strains are negative) todetermine the hypoplastic parameters. The general hypoplastic equation simplifies to

Ṫ1 = fs(T1 + 2T2)

2

T 21 + 2T22

[

D1 + a2

(

T1D1 + 2T2D2(T1 + 2T2)2

)

T1 + fda

3

(

5T1 − 2T2T1 + 2T2

)

√

D21 + 2D22

]

(2-15)

Ṫ2 = fs(T1 + 2T2)

2

T 21 + 2T22

[

D2 + a2

(

T1D1 + 2T2D2(T1 + 2T2)2

)

T2 + fda

3

(

4T1 − T2T1 + 2T2

)

√

D21 + 2D22

]

(2-16)

(subscript s is omitted).In the remaining part of this chapter the method to determine these parameters are brieflydiscussed. A detailed description and elaboration will be presented in the subsequent sections.

2-3 Parameter determination 7

2-3-1 Critical state parameters ϕc and ec0

If the direction of deformation D‖D‖ is kept constant and there is no volume change (̊e = 0 =⇒trD = 0), the stress tensor asymptotically approaches a certain value which depends on theinitial stresses, density and the direction of deformation. In terms of soil mechanics, the stateof the material approaches a critical state ([Schofield and Wroth, 1968], [Been et al., 1991])defined as Ṫs = 0, trD = 0 (D 6= 0). Considering Equations (2-15) and (2-16) in a criticalstate implies

Ṫj = 0, D1 + 2D2 = 0 (D1 6= 0), e = ec and fd = 1

For a standard triaxial compression test in which a cylindrical sample is compressed axiallyat a constant lateral stress yields (Ṫ2 = 0). By using Ṫ1 +2Ṫ2 = 0 and substituting Equations(2-15) and (2-16), the following relation is obtained

a(T1 − T2) −√

3

2(T1 + 2T2) = 0 (2-17)

Inserting the definition of the critical friction angle

sin ϕc =

(

T1 − T2T1 + T2

)

c

(2-18)

one obtains a relation between a and ϕc

a =

√3(3 − sinϕc)2√

2 sinϕc(2-19)

A simple estimation of ϕc can be obtained from the angle of repose of a dry granular materialif cohesive forces are negligible [Miura et al., 1997].The critical void ratio ec0 is the second parameter related to the critical state. It is definedat zero pressure, hence no direct measurement is possible. Nevertheless, a change of ec withthe mean pressure has been the topic of many experimental studies enabling an extrapolationto zero pressure. Alternatively, ec0 can be taken equal to emax corresponding to the looseststate.

2-3-2 Limit void ratios ed0 and ei0

The parameter ed0 denotes the minimum void ratio at zero pressure. There are differentpossibilities to densify the granular material in order to reach the most dense state. Gen-erally it is assumed that a granular material is best densified by means of cyclic shearingwith small amplitude under constant pressure (see e.g. [Youd, 1972]). Numerous attemptshave been undertaken to develop methods to determine the minimum void ratio which aremore widely applicable (see e.g. [Cresswell et al., 1999], [Ishihara and Cubrinovski, 2002],[Muszynski, 2006]). The relation between the minimum void ratio ed and the minimum voidratio at zero pressure ed0 is given by Equation (2-10). Extrapolation is possible after thestiffness parameters n and hs are determined. Alternatively, ed0 can be taken equal to emin.A comparison of ed0 with emin reveals that these values are very close to each other. In fact it

8 HYPOPLASTICITY

is observed that the values for the emax and emin determined according to the standards of theJapanese Geotechnical Society [JGS, 1996] are almost identical to the hypoplastic parametersfor respectively ec0 and ed0, see Table 2-1. Hence the relation ed0 ≈ emin is assumed.

Table 2-1: Comparison between the hypoplastic parameters ec0 and ed0 and the correspondingparameters emax and emin according to JGS standard.

Sand ec0 ed0 emax emin Reference2

Toyoura 0.98 0.61 0.97681 0.61311 42(6):65-78Leighton Buzzard 0.79 0.49 0.79 0.49 38(3):163-179Ticino 0.94 0.59 0.96 0.59 38(3):163-179Monterey 0.83 0.54 0.83 0.54 36(1):39-50

1 Mean value for the properties of the standard Toyoura sand as defined in 13 independent studies

[Ishihara and Cubrinovski, 2002].2 Reference: Soils and Foundations, Volume(number): pp.

The parameter ei0 denotes the maximum void ratio at zero pressure. In theory this situationis reached during an isotropic consolidation of a grain suspension in a gravity free space. Sinceit is impossible to determine this value experimentally, it requires some approximation. Forwell-graded granular materials the relation ei0emax = 1.15 is assumed.

2-3-3 Stiffness parameters hs and n

The parameter hs denotes the granulate hardness and is used as a reference pressure. Thepressure sensitivity of a grain skeleton, i.e. a non-proportional increase of the incrementalstiffness with increasing mean granulate pressure ps, is taken into account by the exponentn. The logarithmic relation between the void ratios and ps (ps =

−trTs3 ) normalised by the

Figure 2-1: Relation between of void ratios ei, ec, ed and the ratiopshs

in logarithmic scale (Herleand Gudehus 1999).

granulate hardness parameter is depicted in Figure 2-1. The relation observed in this figureis given by the Equations (2-9) and (2-10). In a similar fashion, for a particular void ratio ep


yields

ep = ep0exp

[

−(

3pshs

)n]

(2-20)

The value ep0 for a proportional compression (i.e. constant ratio of stress components) isbounded by ec0 ≤ ep0 ≤ ei0. Determination of the parameters is done by performing anisotropic compression test and fitting the obtained compression curve by means of back sub-stitution. It is possible to replace the demanding isotropic compression test by the oedometertest. The proportional stress path is then obtained by assuming a value for the pressurecoefficient K0 [Bauer, 1996]. The value for this stress ratio is often approximated by theJaky’s expression K0 = 1 − sin ϕc. This approximation is only tolerable when an initiallyvery loose sample is prepared, since in that case only a slight underestimation of the stressratio is introduced. This principle is further elaborated by Herle and Gudehus (1999). Theexponent n is calculated by:

n =ln

(

e1λ2e2λ1

)

ln(

ps2ps1

) (2-21)

and subsequently the granulate hardness parameter hs is calculated by

hs = 3ps

(ne

λ

)1

n(2-22)

with λ defined as λ = ∆e∆ln

“

ps2ps1

” . The value obtained by Equation (2-22) is valid within the

pressure range ps1 ≤ ps ≤ ps2, see Figure 2-2. It should be emphasised that the values for hs

Figure 2-2: Determination of the exponents n for a selected stress range (Herle 2000).

and n are only valid within this validity range (see [Herle and Gudehus, 1999]). Occurrenceof grain crushing at higher pressures changes the granulometric properties, thus the values ofhs and n will change. Applying the parameters obtained for a lower pressure range will causean underestimation of n.

10 HYPOPLASTICITY

2-3-4 Exponent α

Parameter α was introduced as an exponent in the definition of density factor fd that controlsthe evolution of the soil behaviour toward the critical state. To determine α it is sufficient toconsider a triaxial test at peak state. At peak state the axial stress rate vanishes (Ṫ1 = 0).Exponent α can be obtained by substituting this condition in Equation (2-15). In terms ofstress and stretch components α is given by

α =

ln

(

3(D1+a2T 21 D1+2a2T1T2D2)

a(5T1−2T2)√

D21+2D2

2

)

ln(

e−edec−ed

) (2-23)

This equation can be rewritten as

α =

ln

(

6(2+Kp)

2+a2Kp(Kp−1−tan νp)a(2+Kp)(5Kp−2)

√4+2(1+tan νp)

)

ln(

e−edec−ed

) (2-24)

with peak ratios Kp =T1T2

=(

1+sin ϕ1−sin ϕ

)

, sin ϕp =(

T1−T2T1+T2

)

pand tan νp =

(

−D1+2D2D1)

p

in which ν is the dilatancy angle.

2-3-5 Exponent β

Parameter β enters as an exponent in the definition of the stiffness factor fe, Equation (2-12),introduced to take into account the stiffness increase consecutive to a soil densification. Fordense material, i.e. e ≪ ei, parameter β has a large influence on the development of the voidratio. This behaviour is depicted in Figure 2-3. The definition of the incremental stiffnessmodulus is given in Equation (2-25).

E =Ṫ1D1

(2-25)

For a measured E corresponding to a particular pressure, density and direction of stretching,exponent β can be calculated from Equations (2-15) and (2-16) respectively. In case of anisotropic compression Equation (2-15) reduces to

Ṫ1 = fs

(

3 + a2 − fda√

3)

D1 (2-26)

and can be rewritten as

E =Ṫ1D1

= fs

(

3 + a2 − fda√

3)

(2-27)

The determination of β is simplified by considering the ratio of the stiffness moduli at twodifferent void ratios, but at the same pressure. This causes the elimination of the influence


Figure 2-3: Influence of exponent β on the oedometric response of loose (Left) and dense (Right)samples.

of mean effective pressure ps, which is due to the consistency requirements of the barotropicfactor fb. Stiffness moduli ratio is given by

E2E1

=

(

e1e2

)β 3 + a2 − fd2a√

3

3 + a2 − fd1a√

3(2-28)

hence β becomes

β =ln

[

E2E1

(

3+a2−fd1a√

3

3+a2−fd2a√

3

)]

ln(

e1e2

) (2-29)

Calculating β from the less demanding oedometric test is also possible. The relation isobtained from Equation (2-29) and is given by

β =ln

[

E2E1

(

m1−n1fd1m2−n2fd2

)]

ln(

e1e2

) (2-30)

with mi = (2 + K0i)2 + a2 and ni = a(2 + K0i)(5 − 2K0i)/3

K0i indicates the pressure coefficient at two different void ratios, but at the same pressureand is best determined using an Soft oedometer [Kolymbas and Bauer, 1993]. In case thatconventional oedometric tests are performed, the value of K0i can again be approximated byJaky’s expression: K0 = 1 − sinϕp but in this case instead of the critical friction angle thepeak friction angle (e.g. from triaxial testing) of loose and dense sample is used. For naturalsands the value of β ≈ 1.

2-3-6 Summary

In the previous sections the theoretical derivation of the hypoplasticity model parameters arepresented. The parameters are derived by first considering the case of no rotation of prin-cipal stress axes in which the objective stress rate T̊ corresponds to the time derivative of

12 HYPOPLASTICITY

Ṫ (subscript s is omitted). The simplified hypoplastic equation is then given by Equations(2-15) and (2-16).The critical state parameter ϕc is determined by substituting the boundary conditions ofthe critical state defined as Ṫs = 0, trD = 0 (D 6= 0) and the extra conditions relating tothe definition of the hypoplastic equation, namely e = ec and fd = 1 in Equations (2-15)and (2-16). This results in a relation between the factor a and ϕc. Alternatively ϕc can beapproximated from the angle of repose of a dry granular material [Miura et al., 1997].In Section 2-3-2 we discussed the determination of the limit void ratios ec0, ed0 and ei0 andshowed correspondence between these values and the minimum and maximum void ratios asobtained by the JGS standards.The requirement of performing oedometric tests to determine the granulate hardness param-eter hs and exponent n is discussed in Section 2-3-3 and later in Section 2-3-5 in order todetermine β. The stiffness parameters hs and n are determined by considering the void ratiodevelopment of a loose sample under increasing mean effective pressure ps in a certain pres-sure range.Exponent α controls the evolution of the soil behaviour towards the critical state and is de-termined by considering the behaviour of a dense sample in a triaxial test at peak state, i.e.Ṫ1 = 0.Exponent β enters as an exponent in the definition of the stiffness factor fb. β is determinedby comparing the stiffness moduli ratio of dense specimens in an oedometric test with twodifferent void ratios (loose/dense), but at the same effective mean pressure.The performed laboratory tests to determine the hypoplasticity parameters and the resultsare presented in Chapter 3.

Chapter 3

PARAMETERS

This chapter discusses the required tests to derive values of the hypoplasticity parameters,

which were theoretically derived in previous chapter. We start with the description of Baskarp

sand for which the parameters will be determined. After the characterisation of this material,

the tests to determine the hypoplastic parameters will follow. Methods are proposed to deter-

mine the critical friction angle and the limit void ratios and series of oedometric and triaxial

tests have been conducted. Based on the results the hypoplastic parameters for the Baskarp

sand are determined.

3-1 Introduction

In Chapter 2 the hypoplastic constitutive equation and its model parameters are discussed.The hypoplastic parameters for the version of hypoplasticity used in this study were discussedin Section 2-3. By considering certain boundary conditions, the hypoplasticity constitutiveequation could be simplified, which enabled us to determine individual parameters based onsimple laboratory tests. In this chapter the required tests to derive values for the hypoplas-ticity parameters are described and for Baskarp sand the values for the parameter set aredetermined.The objective of this thesis is to determine the hypoplastic parameter set for the Baskarpsand. In the remainder of this chapter different laboratory tests are performed, which arerequired to determine these parameters.Baskarp sand in characterised by performing simple laboratory tests to determine the grainsize distribution curves and the mass density. The critical friction angle ϕc can be approx-imated from the angle of repose of dry granular material. Section 3-3 discusses a method,which utilises image processing techniques to determine this angle.In Section 2-3-2 we found that the void ratio related parameters of the hypoplastic model,i.e. ec0 and ed0, can be approximated by the values of minimum and maximum void ratiosobtained by following the JGS procedures. In Section 3-4 we will discuss these proceduresand subsequently these parameters for Baskarp sand are determined. To determine the stiff-ness parameters hs and n, series of oedometric tests on dense and loose specimens have been

14 PARAMETERS

conducted. Two different methods are elaborated to determine hs and n, see Section 3-5.Finally parameters α and β are determined based on the results of the drained triaxial tests.

3-2 Baskarp sand

This Yellow-orange fine-grained sand was deposited a few miles north of Jönköping (Sweden)under melting of the icecap ca. 10.000 years ago. The sand consists of more than 90% quartz(SiO2), the remaining part consists of feldspar (about 8%) and other secondary minerals. Thegrains are classified as angular to sub-angular, see Figure 3-1. Baskarp sand is characterisedas an uniform sand and has a D50 of approximately 140 µm.In this section classification tests and modified laboratory tests are performed to determinesome characteristic parameters of this sand.

Figure 3-1: Magnified image of Baskarp sand.

3-2-1 Grain size distribution

The procedures to determine the grain size distribution are published by the British StandardInstitute (BSI) and the American Society for Testing of Materials (ASTM). Here, modifiedprocedures are applied to obtain the required grain size distribution curves.Before sieving, the sand specimen was thoroughly mixed to achieve an uniform mixture. Sinceless then 5 per mill (31 g in a batch of 6.4 kg) of the grains were larger then 500 µm, thisportion has been removed prior to sieving. Furthermore the minimum grain size is limited to63 µm. Hence, only sieves between 500 µm and 63 µm have been utilised in the tests. Thismade it possible to obtain a quite detailed grain size distribution curve.The tests have been performed using sieves with different diameters (200 mm and 300 mm).In order to assess the consistency of the results two batches are dry sieved and two wet sieved.The results for each sieving are listed in Table 3-1. The sieving curves are depicted in Figure3-2. The grain size distribution curves obtained during the performed tests show deviationnear the point of average grain size. During some of the tests (dry) a gap-graded curve wasobserved since the 125 µm and 106 µm sieves became clogged. As a consequence wet sievingthe samples became inevitable. Wet sieving is also preferred when a large portion of thesample consists out of fine grains. The test results obtained by wet sieving show a better

3-2 Baskarp sand 15

Table 3-1: Baskarp sand specimen properties determined by dry and wet sieving.

Specimen Name Cu Cc D10 D50 D60 Percentage Lost

[-] [-] [µm] [µm] [µm] [%]

D1 1.35 0.89 108 135 146 1.10D2 1.32 0.90 110 135 145 0.51

W1 1.62 1.05 89 133 144 1.38W2 1.58 1.04 92 137 145 1.50W3 1.54 1.06 94 136 144 1.52

Cu is the coefficient of uniformity(Cu = D60/D10), Cc is the coefficient of curvature (Cc = D230/D10D60).

63 75 90 106 125 150 175 212 250 300 350 425 5000

10

20

30

40

50

60

70

80

90

100

Grain size (µm)

Per

cent

fine

r by

mas

s (%

)

Grain size distribution Baskarp sand (DRY sieving)

63 75 90 106 125 150 175 212 250 300 350 425 5000

10

20

30

40

50

60

70

80

90

100

Grain size (µm)

Per

cent

fine

r by

mas

s (%

)

Grain size distribution Baskarp sand (WET sieving)

W1

W2

W3

D1

D2

Figure 3-2: Grain size distribution curves for Baskarp sand obtained by Dry and Wet sieving.

and more logic distribution at grain sizes between 150 µm and 63 µm. Due to the limitedavailable sieves in the required range and the capacity of the vibrator, see Figure 3-3, it wasnot possible to obtain more detailed curves.

16 PARAMETERS

Figure 3-3: Wet sieving.

3-2-2 Grain crushing

Crushing of the particles leads to a change in the granulometric properties of the sand. Thisvariation in the granulometric properties is not included in the hypoplasticity constitutiveequation. It is recommended to determine the hypoplasticity parameters for different stressranges because the parameters for the stress range where no crushing occurs differ fromthe parameters for the stress range where severe crushing is observed. Since in this studyparameters for a large stress range are investigated, it is decided to determine the grain sizedistribution curve after performing oedometric tests in order to investigate the influence ofaxial compression on the grain particles.In Figure 3-4 the grain size distribution curves before (denoted with W1-W3) and afterperforming oedometric tests (CR) are plotted. The curves show almost the same distribution.It is concluded that for the investigated stress range the amount of crushing in compressivestress path is negligible and does not have an effect on the granulometric properties of theBaskarp sand.

3-2 Baskarp sand 17

63 75 90 106 125 150 175 212 250 300 350 425 5000

10

20

30

40

50

60

70

80

90

100

Grain size (µm)

Per

cent

fine

r by

mas

s (%

)

Grain size distribution Baskarp sand (Crushing)

W1

W2

W3

CR

Figure 3-4: Grain size distribution curves of Baskarp sand after performing oedometric tests.

3-2-3 Mass density

The mass density of the Baskarp sand has been obtained by pouring specified amount ofsand in a 1000 ml (± 5 ml) cylinder graduate filled with water. After ”de-airing” the volumeincrease is determined. The test results are listed in Table 3-2. It should be mentioned that

Table 3-2: Mass density of Baskarp sand.

T1 T2 T3 T4 T5 MEAN SD

ρd [t/m3] 2.645 2.642 2.644 2.648 2.644 2.645 0.002

the tests were performed in a non-climatised environment, hence temperature variation duringthe tests could not be prevented. Still, the tests show a small variation in the results. Forthe mass density of dry Baskarp sand ρd=2.645 [t/m

3] is assumed1.

1ρd=2.647 [t/m3] (Deltares)

18 PARAMETERS

3-3 Critical friction angle

The critical friction angle ϕc can be determined from triaxial, simple shear, direct shear testor could be approximated by the angle of repose of a loose sample. Here the latter approachis chosen due to its simplicity and reproducibility. The angle of repose is the angle of a pileof cohesionless soil formed by slowly pouring the material.A detailed study on the measurement of the angle of repose of sand was published by Miuraet al. (1997). In this paper some series of test were conducted on plates with differentroughness and the influence of size, material pouring rate and boundary friction effects wereinvestigated.The goal is to achieve a sand heap at the loosest state. This state is obtained when a funnelfilled with dry sand is lifted vertically and slowly to form a sand heap [JGS, 1996].Miura et al. (1997) also proposed a device for measuring the angle of repose, see Figure 3-5.Here a different measuring method is proposed to determine ϕc. The proposed method whichmakes use of the image processing techniques will be briefly explained and subsequently theresults are discussed. More details about the image processing and the method to calculateϕc is elaborated in Appendix A.

Figure 3-5: Device for measuring the angle of repose developed by Miura et al. (1997).

3-3-1 Proposed test setup to determine the angle of repose

The proposed method involves the determination of the angle of repose using image processingtechniques. The setup used for this test is depicted in Figure 3-6.First a sand heap is formed by gently pouring sand on a plate through a funnel. The funnelwith a diameter of 12 mm (according to JGS 1996) is attached to an arm which can bevertically moved by a stepper motor. A program has been written to control the displacementand the speed of the stepper motor. The speed limitation should be in accordance to therequirements concerning the vanishing pouring height of the sand from the funnel. The sandis poured on a plate, which is equipped with a cylindrical roller-bearing. This enables theplate to rotate. Hence, it is possible to capture images of the sand heap from different angles.

3-3 Critical friction angle 19

Figure 3-6: Test setup for the determination of angle of repose.

3-3-2 Results

The described method for creating the sand heap and processing the images, see Appendix A,has been performed three times. Each sand heap has been photographed 16 times by rotatingthe plate by an angle of 22.5◦. For each image the angle of repose has been calculated. Foreach photographed sand heap (H1-H3) the average value of the angle of repose based on thenumber of images is depicted in Figure 3-7. When a few angles are taken into account the

0 2 4 6 8 10 12 14 1629.8

29.9

30

30.1

30.2

30.3

30.4

30.5

Number of images

φ rep

[° ]

H1.H2.H3.

Figure 3-7: Progressive average value of the angle of repose for Baskarp sand.

curves (H1-H3) show a large deviation. It can be observed that this deviation decreases byincreasing number of images. The approximated normal distribution of the obtained data (48angles of repose) is depicted in Figure 3-8.The mean value of the angles of repose amounts µ=30.014◦ with a standard deviation ofσ=0.347.

20 PARAMETERS

29 29.2 29.4 29.6 29.8 30 30.2 30.4 30.6 30.8 310

0.2

0.4

0.6

0.8

1

1.2

Data

Den

sity

normal distribution

angle of repose

Figure 3-8: Normal distribution of the obtained angles of repose of Baskarp sand.

3-3-3 Discussion

According to Miura et al. (1997) the following factors influence the angle of repose formedby pouring granular material on a plate with a funnel:

• The angle of repose φrep increases with reduction of the pouring rate of the granularmaterial; this tendency means that the largest φrep is attained in the static state.

• The angle φrep increases with the roughness of the base plate.

• The angle of repose φrep decreases with an increase in the amount of material.

The pouring rate during the tests has been adjusted such that a vanishing pouring height isguaranteed. Since no hand movement is involved we can assume that this will not influenceφrep. The angle of repose as determined by the aforementioned method is influenced by thethe roughness of the base plate and the amount of the material. Miura et al. have investigatedthis influence for Toyoura sand (TO-SAND), Soma sand (SO-SAND) and two different kindof glass beads with different uniform grain sizes (GB-A, GB-B). The effect of the increase inthe amount of material and base roughness for these material is given in Figure 3-9.For the two investigated sand types (subangular grain shape) the amount of the increase inthe angle of repose φrep by the increase of the roughness of the base plate is less than 1 degreeand the reduction of φrep due to the increase of the amount of the material is approximately 2degrees. During the performed tests in this study the influence of the base roughness and theamount of poured material are not further investigated. Based on the experimental resultsof Miura et al. we can conclude that atbest the proposed method can estimate the angle ofrepose with an accuracy of ± 1◦, which is a remarkable accuracy in comparison with otheralternatives.The critical friction angle can also be determined from the triaxial test data performed onloose specimen. This angle is highly influenced by the confining pressure and the initialdensity of the specimen. For Karlsruhe sand this relation has been depicted in Figure 3-10.For a loose prepared specimen at different confining pressures the critical friction angle ϕcvaries between 29 and 35 degrees. The triaxial tests are usually performed using rough end

3-3 Critical friction angle 21

Figure 3-9: Left: Increase of the angle of repose φrep with roughness of the base plate. Right:Reduction of the angle of repose φrep with an increase in the amount of material.

platens. Barrelling is a direct consequence of such imperfect boundary conditions, a factwhich makes the results questionable. Furthermore, the critical friction angle for a triaxialtest is in most cases determined from the triaxial response at 10% axial strain, to minimisethe effects due to barrelling. For loose samples the peak stress is often reached at an axialstrain higher then 10%. More detail about the effect of the sample preparation regarding theend platens is given in Section D-5-1.

Figure 3-10: Left: Dependence of ϕc on Dr, and σ3. Right: Dependence of friction angle ϕc onconfining pressure for loose Karlsruhe sand.

The proposed method to determine φrep is easy to perform and the results are reproducible, seeSection 3-3-2. The obtained value for φrep with an accuracy of ± 1◦ is a good approximationof the critical friction angle ϕc and should be treated as such. Only by performing numericalsimulations, the validity and consistency of this parameter can be assessed.

22 PARAMETERS

3-4 Minimum and maximum void ratio

The maximum and the minimum density values (or void ratios) represent the upper andlower density boundaries for a given soil specimen. The inclusion of limit density data ofsand is common in a variety of soil mechanics research. For example, limit density data areamong others included in the specimen property description in such research on sand involvingaging [Baxter and Mitchell, 2004], liquefaction [Kokusho et al., 2004], and measurements ofcritical state parameters [Santamarina and Cho, 2001]. In this section we concentrate on theimportance of the maximum and the minimum void ratio with respect to the hypoplasticconstitutive equation.As mentioned in Section 2-1, the hypoplastic constitutive equation expresses the objective rateof stress tensor T̊s as a function of the void ratio e, the stress tensor Ts, and the strain ratetensor of the granulate skeleton D̊s. Three of the eight hypoplastic parameters are directlyrelated to the void ratio of the specimen, namely the maximum void ratio for a stress-freestate ei0, the minimum void ratio for a stress-free state ed0, and the void ratio at the criticalstate for a stress-free state ec0. Gudehus (1996) postulates that the minimum, maximum, andcritical void ratios decrease proportionally with the mean effective stress, see Equation (3-1).

eiei0

=eded0

=ecec0

= exp

[

−(

3pshs

)n]

(3-1)

The void ratios reach limit values ei0, ec0 and ed0 at vanishing effective mean pressure, andthey approach zero for very high mean pressure, see Figure (3-11).

Figure 3-11: Mean stress dependency on void ratios.

Many methods are currently used to determine limit densities of sands. Conventional meth-ods to determine these densities are given by the ASTM. The common ASTM methods fordetermining the maximum densities are the vibrating table method (ASTM D4253-00) andthe modified Proctor test (ASTM D1557-00). The method to determine the minimum den-sity includes gently pouring the sand through a funnel into a standard Proctor mold (ASTMD4254-00). Beside these conventional methods many alternate methods of obtaining limitdensities exist. Among many researchers, Muszynski (2006) has suggested simplified meth-ods based on the conventional methods which uses smaller specimen sizes and requires less

3-4 Minimum and maximum void ratio 23

time to obtain limit densities. Due to the variety of the methods and the varying physicalproperties of sand it is not possible to determine which method is the best. In most casesseveral methods are used to achieve the most accurate estimate of these values.With regard to determination of the void ratio dependent hypoplastic parameters ei0, ec0and ed0, no method is explicitly suggested by the authors. However it is suggested that thevalues for ec0 and ed0 are very comparable with respectively emax and emin. In fact in Section2-3-2 it was observed that the methods described by the Japanese Geotechnical Society (JGS1996), hereafter referred to as JGS method, are the most appropriate methods to determineec0 and ed0.

3-4-1 Limit densities according to JGS

The JGS method involves the determination of the limit densities using a smaller mold,see Figure 3-12, similar to a scaled down version of a Proctor mold. This mold has aninner diameter of 6 cm and is 4 cm tall (without the collar). This results in a volume of113.1 cm3. The mold is made of stainless steel with a thickness of 8 mm. The minimum

Figure 3-12: JGS mold to determine the limit densities.

density is determined by gently pouring sand in the mold through a funnel. The maximumdensity is found using a tapping procedure to densify the specimen. The procedures for thedetermination of the minimum and the maximum void ratios according to JGS are describedin Appendix B. In following sections the obtained results based on these procedures arediscussed.

3-4-2 Minimum void ratio emin

The minimum void ratio is determined following the procedure given in Section B-1. Theresults are summarised in Table 3-3. Since this is a time consuming test only three testshave been performed. Still a very consistent result has been obtained with a small standarddeviation. Following the reasoning discussed in Section 2-3-2, namely ed0 ≈ emin, the meanvalue is taken as a best approximation of the void ratio at maximum density at zero pressureed0.

24 PARAMETERS

Table 3-3: Minimum void ratio of Baskarp sand according to JGS standard.

T1 T2 T3 MEAN SD

emin [-] 0.547 0.550 0.549 0.548 0.002

3-4-3 Maximum void ratio emax

The critical void ratio at zero pressure ec0 can be approximated by emax obtained by the JGSprocedures, see also Table 2-1. The procedures to determine the maximum void ratio hasbeen discussed in Section B-2. The obtained results are listed in Table 3-4.

Table 3-4: Maximum void ratio of Baskarp sand according to JGS standard.

T1 T2 T3 T4 T5 MEAN SD

emax [-] 0.930 0.927 0.926 0.931 0.933 0.929 0.003

For well-graded granular materials the relation ei0emax = 1.15 is assumed. This relation is usedto obtain a first approximation of the maximum void ratio at zero pressure ei0. The voidratio parameters of the hypoplasticity model for Baskarp sand are given in Table 3-5.

Table 3-5: Void ratio dependent hypoplastic parameters for Baskarp sand.

Material ed0 ec0 ei0[-] [-] [-]

Baskarp 0.548 0.929 1.08

3-5 Oedometric test 25

3-5 Oedometric test

The stiffness parameters hs and n describe the material stiffness. The parameters are deter-mined by performing an isotropic compression test or an oedometric test by approximatingthe K0 value based on the friction angel, see Section 2-3-3. In this study the latter is adoptedto determine the hypoplastic stiffness parameters.The Casagrande oedometer test is most widely used. The apparatus, see Figure 3-13, consistsof a cell which can be placed in a loading frame and loaded vertically. In the cell the soilsample is laterally restrained by a steel ring, which incorporates a cutting shoe used duringspecimen preparation. Normally the top and bottom of the specimen are placed in contactwith porous discs, such that drainage of the specimen takes place in the vertical directionwhen vertical stress is applied. The oedometer tests in this study are performed on specimensof dry sand. Since the response for a large stress range is desired and to minimise the effectof the porous discs, it is decided to make alternative loading caps made of stainless steel forboth the 37.4 mm ring and the 50 mm ring.

Figure 3-13: Casagrande oedometer apparatus.

More detailed discussion on the calibration of the oedometer apparatus and sample prepara-tion is given in Appendix C.

3-5-1 Oedometric response of Baskarp sand

A total of eight oedometric compression tests were performed on samples of Baskarp sandwith an initial void ratio ranging between 0.82 and 0.66. The tests were performed on 37.4mm and 50 mm diameter samples. The mean effective stress ps at the end of the tests rangedfrom 3.68 MPa to 6.58 MPa depending on the sample diameter. In order to calculate themean effective stress, Jaky’s expression for the earth pressure coefficient was assumed with afriction angle of 30 ◦. Table 3-6 gives a summary of the performed oedometric tests. Figure3-14 presents the oedometric response of the specimens.The development of void ratio is plotted against the mean effective stress (e-p plot). For dense

26 PARAMETERS

Table 3-6: Oedometric tests performed on Baskarp sand.

Name ei ea ps Sample preparation method

[-] [-] [MPa]

O371 0.800 0.724 6.58 Dry compactionO373 0.733 0.682 6.41 Dry compactionO374 0.777 0.706 6.58 Dry compactionO51 0.823 0.763 3.68 Dry compactionO52 0.657 0.625 3.68 Dry compactionO53 - 0.908 3.68 Moist compactedO54 0.784 0.728 3.68 Dry compactionO55 0.817 0.755 3.68 Dry compactionei = void ratio after compaction, ea = void ratio after oedometric compression

100

101

102

103

104

0.5

0.6

0.7

0.8

0.9

1

1.1

ps [kPa]

e [−

]

Void ratio vs. Effective mean stress

O371, ei=0.80

O373, ei= 0.733

O374, ei= 0.777

O51, ei= 0.823

O52, ei= 0.657

O54, ei=0.784

O55, ei=0.817

O53

Figure 3-14: Oedometric response of the specimens of Baskarp sand.

specimen the compression curve is more or less linear. The non-linearity increases for looserprepared specimen. Preparing loose samples in an oedometer ring is very challenging. Thisis due to the height of the oedometer ring. Striking off the surface of the specimen will cause


certain amount of compaction of the specimen. Two important oedometric tests which willoften be referred to in the remainder of this chapter are O51 with highest initial void ratio andO52, which has the lowest initial void ratio. Specimen O53 has been moist compacted. Theinitial void ratio is in this case irrelevant due to the existence of macro voids in the specimen.However the result of this test will lead to the approximation of the maximum void ratio atzero pressure ei0, see Section 3-5-4.

3-5-2 Determination of the granulate hardness hs and exponent n

The calculation method to determine the stiffness parameters hs and n has already beendiscussed in section 2-3-3. Parameters will be determined by considering the oedometric re-sponse of specimen O51, which has the highest initial void ratio. The response has again beendepicted in Figure 3-15A.To determine the stiffness parameters for different validity ranges the response curve is ap-proximated by a polynomial, see Figure 3-15B. A continuous function is then obtained whichsimplifies the calculation of the quantities required by Equations (2-21) and (2-22). Figure3-15C. shows the value of λ = ∆e

∆ ln“

ps2ps1

” for the whole range.

For the calculation of exponent n for a given validity range, i.e. range between ps1 and ps2,the corresponding void ratios and λ are calculated using the obtained polynomial and itsderivative respectively. Subsequently granulate hardness given by Equation (2-22) is calcu-lated. The results for two different validity ranges are depicted in Figures 3-15D and 3-15E.The numerical values are given in Table 3-7.

Table 3-7: Numerical quantities Figure 3-15D and E.

Validity range ps1 ps2 e1 e2 λ1 λ2 n hs[kPa-kPa] [kPa] [kPa] [-] [-] [-] [-] [-] [GPa]

100-500 100 500 0.811 0.797 6.72 e-3 1.1 e-2 0.314 ≈ 30100-1000 100 1000 0.811 0.788 6.72 e-3 1.4 e-2 0.332 ≈ 20

For a small validity range, the granulate hardness is almost constant. By enlarging the rangesmall variation of the parameter in the specified range is introduced. This behaviour is bettervisible for larger validity ranges, see Figure 3-16.The stiffness parameters are coupled since it is the combination of these parameters thatdescribes the void ratio development with respect to the initial void ratio and mean granulatestress, see e.g. Equation (2-20). Separate calculation as required by this calculation procedurecauses a large scatter in the results, see Table 3-7. Next we will discuss an alternative methodto determine hs and n as a pair.

3-5-3 Proposed method to determine the stiffness parameters hs and n

In the previous section we discussed the method to calculate the stiffness parameters accordingto [Herle and Gudehus, 1999]. In my opinion the goal of this method, hereafter referred toas Method 1, is to find the parameters of e.g. Equation (2-20) which are suitable to describethe oedometric response of a loose specimen. Here an alternative method, hereafter referredto as Method 2, to determine the stiffness parameters is proposed and elaborated.

28 PARAMETERS

100

101

102

103

104

0.76

0.78

0.8

0.82

0.84

ps [kPa]

e [−

]

Void ratio vs. Mean Granulate Pressure

0 2 4 6 8 100.76

0.78

0.8

0.82

0.84

log(ps) [−]

e [−

]

Fitting data

0 1 2 3 4 5 6 7 8 9−0.04

−0.02

0

0.02

log(ps) [−]

λ

λ vs. log(ps)

0 100 200 300 400 5000

1

2

3

4x 10

7

ps [kPa]

h s

hs [validity range 100−500 kPa]

0 200 400 600 800 10000

1

2

3x 10

7

ps [kPa]

h s


O51 [n=0.314] O51 [n=0.332]

O51, ei=0.823

A. B.

C.

D. E.

Figure 3-15: Results of the oedometer tests performed on a ø50 mm sample.

For a particular initial void ratio ep, the development of void ratio with respect to the meangranulate stress is given by

epep0

= exp

[

−(

3pshs

)n]

(3-2)

This equation can be solved numerically, e.g. by Maple2. In order to calculate the twounknowns, namely hs and n, only the ratio

epep0

at two points, which are the boundaries of

the validity range, are required. The validity ranges as indicated in Figure 3-16 are againcalculated by the proposed method. The results are summarised in Table 3-8.Figure 3-17 shows the response of specimen O51 and the calculated response by Equation(3-2) for different validity ranges according to the parameters determined by both methods.The proposed method shows a better fit of the oedometric response and is very accurate in

2www.maplesoft.com


100

101

102

103

104

0.7

0.75

0.8

0.85

0.9

ps [kPa]

e [−

]


0 2 4 6 8 100.7

0.75

0.8

0.85

0.9

log(ps) [−]

e [−

]

Fitting data

0 50 100 150 2000

2

4x 10

7

ps [kPa]

h s


0 100 200 300 400 5000

1

2

3

4x 10

7

ps [kPa]

h s


0 500 1000 1500 20000

5

10x 10

6

ps [kPa]

h s


0 500 1000 1500 2000 2500 3000 35000

1

2

3x 10

6

ps [kPa]

h s


O371

O51

O371 [n=0.406]

O51 [n=0.314]

O371 [n=0.338]

O51 [n=0.314]

O371 [n=0.423]

O51 [n=0.388]

O371 [n=0.528]

O51 [n=0.504]

A. B.

E. F.

C. D.

Figure 3-16: Comparison of the results of two test with initial void ratio close to ec0

Table 3-8: Calculated values of Figure 3-16 for specimen O51.

Method 1 Method 2

Validity range n hs n hs[kPa-kPa] [-] [GPa] [-] [GPa]

20-200 0.314 ≈ 33 0.642 0.240200-2000 0.388 ≈ 7 0.43 4.595100-500 0.314 ≈ 30 0.486 1.780500-3500 0.504 ≈ 2 0.432 4.310

the particular validity range for which the parameters are determined. Figure 3-17 also showsthat the parameters are not sufficiently precise to cover a large range of stress. In fact in most

30 PARAMETERS

100

101

102

103

104

0.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

ps [kPa]

e [−

]


100

101

102

103

104

0.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

ps [kPa]

e [−

]Void ratio vs. Mean Granulate Pressure

O51

[20−200]

[200−2000]

[100−500]

[500−3500]

O51

[20−200]

[200−2000]

[100−500]

[500−3500]

B.A.

Figure 3-17: Calculated response for oedometer test O51 by Method 1 (A) and Method 2 (B)for different validity ranges.

publications concerning hypoplasticity, it is advised to determine the parameters n and hs fora particular stress range of interest ([Herle, 1997], [Herle and Gudehus, 1999]). An importantargue for this is that the granulometric parameters of the sample for high stress ranges maychange, e.g. due to grain crushing.Figure 3-18 shows the response calculated by Method 2 for three different validity ranges.This figure argues that the error is merely caused by the deficiency of the Equation (3-2) toaccurately describe the response3.

100

101

102

103

104

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

ps [kPa]

e [−

]


0 500 1000 1500 2000 2500 3000 3500 40000.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

ps [kPa]

e [−

]


O51VR [10−3500] kPa, n=0.7, h

s=425 MPa

VR [50−3500] kPa, n=0.478, hs=2390 MPa

VR [100−3500] kPa, n=0.424, hs=4030 MPa

Figure 3-18: Determined parameters n and hs for oedometer test O51 for different validityranges.

3This argue is true since for Baskarp sand the grain crushing does not play a role for higher stress ranges.


3-5-4 Proposed method and limit void ratios

In previous Section 3-5-3 we discussed Method 2 and its usage to determine the stiffnessparameters. Here an attempt is made to develop approximations for the development ofthe void ratio parameters ei0, ec0 and ed0. In Section 3-4-1 we determined these parametersbased on the values of the minimum and maximum void ratio determined according to theJGS standards. The development of these parameters with respect to the mean granulatepressure are required for the determination of the exponents α and β, which will be discussedin Sections 3-6-3 and 3-6-4 respectively.Method 2 is used to fit the results of the oedometric tests of specimen O53 (closest to ei0),O51 (closest to ec0) and O52 (closest to ed0), see Figure 3-19. For O53 the parameters aredetermined for the stress range, where a realistic response is observed, i.e. 350-3600 kPa. Theresults are extrapolated to a maximum ps of 100 MPa. The extrapolation of the response ofO53 to zero pressure is very interesting. The obtained value by this extrapolation approachesthe value of ei0, which is approximately 1.08. The best way to evaluate the development of the

100

101

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ps [kPa]

e [−

]


O53; ei=1.08

O53−APPRO51; e

i=0.823

051−APPR052; e

i=0.657

052−APPR

Figure 3-19: Simulation of the response of specimen O53, O51 and O52 by Method 2.

critical void ratio ec and the void ratio at the maximum density ed is to prepare samples whichhave an initial void ratio equal to ec0 and ed0 respectively. In practic

hypoplasticity investigated parameter determination and ...2008) hypoplasti… · the...

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