mazars model

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Numerical aspects of a problem with damage to simulatemechanical behavior of a quasi-brittle material

Isabelle Peyrot *, Pierre Olivier Bouchard, Francois Bay, Fabrice Bernard, Eric Garcia-Diaz

Centre de Mise en Forme des Materiaux, Ecole des Mines de Paris, UMR 7635 CNRS, BP 207, 06904 Sophia-Antipolis, Cedex, France

Ecole des Mines de Douai, 941 avenue Charles Bourseul, BP 938, 59508 Douai Cedex, France

Received 1 February 2006; received in revised form 5 October 2006; accepted 16 January 2007Available online 11 April 2007

Abstract

This paper focuses on numerical problems related to the implementation of the non-local Mazars model in a Finite Element softwareFemcam developed in our laboratory. The model has been successfully applied for quasi-brittle materials. Several mechanical test casesare presented and validate the implementation. A sensitivity study concludes on the effect of the critical equivalent strain and internallength values on the global response of the material. 2007 Elsevier B.V. All rights reserved.

Keywords: Mechanical formulation; Finite element; Damage; Non-local Mazars model; Kill element

1. Introduction

The simulation of non-linear problems for quasi-brittlematerials especially damage and degradation is one ofthe major research tasks in civil engineering. Concretebelongs to a heterogeneous material class whose non-linearbehavior is rather complex. There is a wide literature onexperimental aspects of the mechanical behavior of thesematerials. Constitutive laws aim at describing their macro-scopic behaviors and at modelling their whole non-lineari-ties and irreversibilities. The standard uniaxial test providesuseful information for modelling.

Many theories describe the macroscopic behavior of

quasi-brittle materials. We can quote the plastic fracturingtheory [1], the total strain models [2], or plasticity withdecreasing yield limit[3]. The phenomenological approachuses a continuous internal variable for describing damage.

The concept of damage applied to quasi-brittle materialsbehavior was initially introduced by Kachanov in 1958[4]. Damage mechanics is a theory describing the progres-sive reduction of the mechanical properties of materialdue to initiation, growth and coalescence of microscopiccracks. These internal changes lead to the degradation ofmechanical properties of the material.

The distinction between a healthy and a damaged mate-rial state, at the base of this theory, has led to the conceptofeffective stress, defined by Kachanov. Sdenotes the sec-tion surface of one volume element defined by its normal.The effective resistant surface

eS

eS< S takes into account

the geometrical discontinuities and stress concentrations.

By definition:

E E01D 1

D is a scalar such as 0 6 D 6 1, with D= 0 (undamagedmaterial) and D= 1 (completely damaged material). E isthe effective Young modulus and E0 is the initial Youngmodulus.

The effective stress-concept introduced by Kachanov hasbeen successfully applied to concrete by Mazars [5]. In thispaper, the non-local damage Mazars model is used to

0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2007.01.018

* Corresponding author. Tel.: +33 0 4 93 95 74 05; fax: +33 0 4 92 38 9752.

E-mail addresses: [email protected] (I. Peyrot), [email protected] (P.O. Bouchard), [email protected] (F. Bay),[email protected] (F. Bernard), [email protected] (E.Garcia-Diaz).

www.elsevier.com/locate/commatsci

Computational Materials Science 40 (2007) 327340
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describe the mechanical behavior of a quasi-brittle mate-rial. The material behavior is based on an isotropic elasticdamage constitutive law. We shall first describe the damagemodel. This model has been implemented in the Finite Ele-ment Code Femcam, designed to deal specifically with non-linear material modelling. This code is derived from the

Forge3

code developed by Cemef for forming processesapplications. We shall then describe how this model hasbeen implemented in the code Femcam. Applications ofthis model to a three point bending test and a compressiontest are then presented. Finally we shall study the influenceof some material parameters on the global response of thematerial.

2. Damage modelling

Damage behavior is often studied for compressive stressstates because of their importance at the industrial level.

Behavior in compression is usually characterized using auniaxial compression test on a cylindrical sample. Theloaddisplacement curve obtained for such a test enablesto identify the yield stress in compression rc. We presenthere some results obtained by Ramtani[6] (seeFig. 1).

We can observe four stages in the material behaviour.We first, have a linear behavior up to 50% of the rc value(stage 1). From 50% to 80% of the rc value, interfacialcracks start to grow and the behavior stops being linear(stage 2). From 80% to 90% ofrc, interfacial cracks startto join, thus leading to macro-cracks initiation (stage 3).In the last stage, there is a fast degradation of mechanical

characteristics which is related to the fast evolution of mac-rocracks (stage 4).Strength in tension is often measured by tests such as

Brazilian splitting tests, three point bending tests, anddirect tensile tests. The results in Fig. 2 clearly displaytwo stages, with a fracture zone much more localised thanin compression. The behavior is linear almost up to thepeak. Tests monitored by sound emission confirm thatalmost no degradation takes place before reaching the peakvalue. In the post-peak stage, material degradation gets

faster.Non-linearity and damage correspond to the initiation

and growth of microscopic cracks which, when the peakload is reached, are located in a material band and endup being organized in macro-cracks. The stress dropsquickly and becomes stable after a certain strain level. Bythe end of the test the specimen stiffness is ten times smallerthan the initial one.

Damage impacts the mechanical behaviour of concretein several ways:

Modification of the elastic behaviour, which results in achange of the mechanical characteristics.

Modification of the plastic behavior (for concrete, these

microstructural changes correspond to decohesion at theinterface of the aggregate or in the mortar paste).

Various damage models of gradually increasing com-plexity have been proposed. They can be divided into twomain groups:

Elastic-brittle, in which irreversible strains arenegligible.

Plastic-brittle, in which permanent irreversible appearafter unloading, in addition to the modification of elasticproperties.

The Mazars model aims at modelling the modificationof the elastic behaviour. In this model, the damage variableis isotropic; it is modelled using a scalar variable whichaffects stiffness.

2.1. The equivalent strain in the Mazars model

Cracks in quasi-brittle materials appear mainly when thematerial is in tension. The Mazars model thus considersonly positive principal strains. This choice is thus well sui-ted for quasi-brittle materials and thus for mortar andconcrete. The expression of the equivalent strain with

respect to positive principal strains is given byFig. 1. Stressstrain curve for a compression test[6].

Fig. 2. Strain curve for a direct tensile test[7].

328 I. Peyrot et al. / Computational Materials Science 40 (2007) 327340


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~effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

i

heii2

r 2

whereheii eijeij

2 and eidenote the principal strain com-

ponents (1 6 i6 3).

2.2. Damage threshold

An evolutive threshold, depending on the damage vari-able, is introduced. Thus for a given damage state D, theform of the loading function is

fe;D ~eKD 3

whereK(D) represents the variable related to the history ofthe damage. Damage D grows when the equivalent strainreaches a threshold K(D) initialized at eD0

If fe;D ~eKD 0 then D D~e

KD ~e

4

2.3. Damage decomposition

Damage defined by Mazars is split into two parts

D abTDT abCDC 5

The parameterbis representative of shear experiments: it isusually considered as constant (b= 1.05). DT and DC arerespectively the tensile and compressive part of the damagevariable D

DT;C 1eD01AT;C

~e

AT;C

expBT;C~eeD0 6

AC, BC, AT, BT are four material parameters. The weightsaT and aCare defined such that

aCX

i

heiiei

~e2 and

aTX

i

heiiei

~e2

7

and the principal strains (ei)i2[1,3] check

eieTieCi 8

For pure tension cases,aC= 0 and Dtends towardsDT; for

pure compression, aT= 0 and D tends towards DC; formixed loads, we have aT+ aC= 1. Let us note that if allprincipal strains are positive or null, aT= 1, aC= 0 andD=DT. On the contrary, if all principal strains are nega-tive, aT= 0, aC= 1 and D= DC. For mixed strain states,the values ofDTand DCdepend on the amplitude of tensileand compressive stresses.

2.4. Mazars model improvement: the non-local model

Fracture in quasi-brittle materials such as concrete is adifficult problem because it induces localization and dis-

continuity in the displacement field. During the test, the

stressstrain curve presents a slope drde< 0. Therefore the

stability criterion proposed by Hill is violated [8]. Pijau-dier-Cabot explains why this behavior leads to importantproblems: the differential equations which govern equilib-rium do not have any more the adapted mathematical form[9].

Several models have been proposed. These techniquesconsist in introducing a characteristic length whichenables to specify the localization zone width while pre-venting possible numerical problems which are dependenton it. The characteristic length can be introduced undervarious formulations: use of non-local theories or gradientbased formulation.

The model we have implemented in our finite elementcode is based on an integral formulation. We denote e theaverage of the equivalent strain in a representative volumesurrounding a point x. It is this variable which will controldamage growth at point x

e 1Vr x

ZV

~esasxds

Vr x

ZV

asxds

asx exp 4jsxj2

l2C

! 9

Vis the volume of the structure, ~eis the equivalent strain atpoint s, and a(s x) is a standardized weight function. lCrepresents the internal material length. lC is proportionalto the smallest size of the damage zone. It is generally as-sumed that the lCvalue is between 3dmax and 5dmax where

dmaxis the maximum size of the inclusion[10]. The evalue which can be considered as the non-local equivalentstrain is the variable that controls the growth of damagein compliance with the following conditions:

fe;K eK 10

3. The elastic damage model description for a mixed

velocitypressure formulation

This section deals with the implementation of the non-local Mazars model in the FE Code Femcam developedat Cemef. A specificity of this code is its mixed velocitypressure formulation.

3.1. The equilibrium equations

3.1.1. The dynamic equilibrium

Let us consider a 3D domain X. The classical equilib-rium equations for a solid subject to a mechanical loadcan be expressed in a local form by

rrqf qc 11

where r denotes the stress field in the solid, f the bodyforces per mass unit, q is the volumic mass and c is the

acceleration. The problem to be solved is expressed in

I. Peyrot et al. / Computational Materials Science 40 (2007) 327340 329


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Femcam using a velocitypressure formulation. Thus thestress field tensor is classically expressed as the sum of itsdeviatoric part s and its spherical part p

r spI 12

Iis the unit tensor and p 13

Trr1.s is evaluated using a

local resolution of the behavior law. The classical equilib-rium equations for a solid submitted to a mechanical loadcan be expressed by

Find v;p such as :

rs rpqf qc

13

3.1.2. The incompressibility condition

In addition to(13), we introduce the incompressibilitycondition on the velocity field v

r v 0 14

3.1.3. The boundary conditionsWe also prescribe some boundary conditions on the

surface oX= oXv[ oXs.

A surface load simp or an imposed velocity vimp can beapplied to Xon its boundary oX

s

s r~n simp on oXs

or

v vimp on oXs

15

where ~n is the normal to the part. One of the boundaries oXvofX can be in contact with a

tool. The contact can be defined with A friction condition:

s r ~n r~n ~n 16

or a non-penetration condition between a slave nodeand a master triangular face. It corresponds to theSignorini conditions.

vvtool ~n 6 0

rn 6 0

vvtool ~nrn 6 0

17

where vtool is the tool velocity and rn r~n~n is thecontact pressure at the normal of the surface of thepart. More details on these techniques can be found in[11].

3.2. Constitutive equations

A classical one step Euler scheme is used to compute thesolution at time t + Dtwhen the solution at t is known. Inthe case of a pure elastic behavior, the Young modulus Eremains constant during simulation. In this case, there isno variation of the shear modulus land the compressibility

v between t and t+ Dt. In the case of an elastic damage

model, the Young modulus E depends on the damagevalue. It thus affects the values ofland vwhich can changebetween timet and time t+ Dt.

The material is assumed to be an elastic damage mate-rial. The elastic strain eel is related to the stress field tensorr through the Hooke law

r CD: eel 18

whereCdenotes the elasticity matrix depending on dam-age D. D denotes the damage which affects the Youngmodulus value E in order to model the material degrada-tion. Using the Hooke law, we thus obtain the expressionof the deviatoric part and the spherical part

s 2le

p vTre

19

with

l

ED

21m with l: shear modulusv ED

312m with v: compressibility modulus

8


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Lower or equal to 1 within the framework of an elasticdamaged material.

3.2.2. Spherical part

In the same way that above, it can be shown that at time

t+D

t

ptDtvtDtvt

pt vtDtDtr v 26

3.3. The weak formulation

We only consider the case of a quasi-static analysis. Theinertia terms are thus neglected. We use the constitutiveequations to rewrite the equation of behavior and the equa-tion of incompressibility of the elastic damage strain part inthe following way:

divs rp 0

div v _pv _v

v2p 0

( 27This constitutes the strong formulation of the problem. Inorder to solve this problem, let us consider its weak formu-lation by introducing the following virtual fields:

V v; v 2 fH1Xg3

vvtool n 6 0 on oXv

( )

V0 v; v2 fH1Xg3

vn 6 0 on oXv

( )PL2X

28

Thanks to the Virtual Work Principe (VWP), we get theweak velocitypressure formulation associated to themechanical problem

Find v;p 2V PRXsv: _evdX

RXpr vdX

RoXsimpv

dS 0RXpr v _p

v _v

v2pdX 0

8v;p 2V0 P

8>: 29

3.4. Discretization

Let us consider a finite element discretization of the 3Ddomain X

X [eXe; e2 E N 30

3.4.1. Choice of the finite element

We use hereP1+/P1elements[12]. These are linear tetra-hedral elements. + means we use an additional degree offreedom for velocity interpolation, at the centre of the ele-ment. This additional degree of freedom enables to complywith the Brezzi-Babuska compatibility relation between

discretization spaces for velocity and pressure [13]; the

associated shape function is the bubble function. Velocityand pressure are interpolated linearly on the element andthe degrees of freedom are located at each node of the ele-ment. The pressure field is linear and continuous (seeFig. 3).

3.4.2. Discrete weak formulation

We considerVh Vand Ph P. TheBhspace stand forthe bubble function discretization

Vh vh; vh2 C

0X3 and

vhjXe 2 P1Xe3; 8e2 E

( )Ph

ph;ph2 C0X and

phjXe 2P1Xe; 8e2 E

( )

Bh

bh; bh2 C0X3;

bh 0 on oXe; 8e2 E;

bh=Xei 2 P1Xei 3; i2 1; . . . ; 4

8>:9>=>;

31

(C0(X))3 is the space of continuous functions on the fieldX2 R3 P1Xe

3 denotes the space of linear functions

on the element Xe R3.The velocity field w interpolated on element P1+/P1 is

decomposed

wh vhbh;

where

wh 2 Wh Vh Bh

32

wherevhis the linear component of the velocity field and bhthe bubble-related component. Let Nbnoe and Nbelt berespectively the number of nodes and of elements associ-ated to the space triangulation on X. Consequently, thevelocity field can be expressed at any point x of spacethrough the finite element approximation

whx Xnbnoek1

NlkxVkXnbeltk1

Nbj xBj 33

where wh2Wh.Nlkk1; ... ; nbnoe2 Vh are related to interpo-

lation of the linear element associated the node k andNbj j1; ... ; nbelt2Bh, the bubble function associated withthe element j.

Pressure can be expressed at any point x by

phx Xnbnoe

k1

NlkxPk 34

Fig. 3. Representation of the P1+/P1 element.



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The strain ei is updated at each time step. The stressdeviator s and the tangent modulus Lare then calculated.The tangent modulus is used in the NewtonRaphson algo-rithm. The loading is then incremented. Damage can beweakly or strongly coupled to the material behavior.

3.5.1. Weak couplingDamage is evaluated at the beginning of the time step

using strains and stresses computed at the previous timestep. The damage value D calculated at the end of the pre-vious time step affects the Young modulus Efor the follow-ing time step.

3.5.2. Strong coupling

Once the strain has been computed, a correspondingdamageD is evaluated at the same time. The Young mod-ulus E is then affected. This coupling does not imply animmediate mechanical balance of the problem. The strainsare then corrected within the NewtonRaphson loop. Oncethe equilibrium has been reached, the loading is incre-mented and the process continues.

The tangent modulus is defined as following:

Lijkl16i;j;k;l63 osij

o_ekl40

By developing the expression, we notice that the tangentmodulus is affected by an additional term within the frame-work of a strong coupling. At time t + Dt

LtDtijkl 2~l0oD

tDt

o~etDto~etDt

oetDtkl

oetDtkl

o_etDtkletDtij 2Dt~l

tDto _etDtij

o_etDtkl

41

where ~l0 E021m

denotes the initial Lame and ~ltDt E21m

the reactualized one. Hence, we can obtain

LtDtijkl 1 2

with

1 Dt ~l0

~ltDtoDtDt

o~etDto~etDt

oetDtkl

stDtij

2 2Dt~ltDt o_etDt

ij

o_etDtkl

8>:42

As the term (42-1) is non-linear with respect to thestrains, the bubble tangent modulus remains identicalto the one calculated within the framework of a purelyelastic problem (linear part of the tangent modulus). Theassumption that part (42-1) is negligible leads us to use atangent modulus identical to the pure elastic case. Thisassumption is justified as long as it does not preventconvergence.

4. Choice of the model

4.1. Local versus non-local model

We wish to underline here the interest of the non-local

model. A 3D tensile specimen with a centred hole has been

simulated using both non-local and local Mazars model.Dimensions of the specimen are 200 mm 200 mm 2.5 mm. Due to symmetry only a quarter of the part is dis-

cretized as we see in Fig. 5.The behavior law is an elastic damage law. We have

used the following arbitrary parameters for these simula-tions (Table 1).

We study here the mesh dependency of results using thelocal and non-local Mazars model.Fig. 6shows the threedifferent meshes used for this study. The Kill elementmethod. This method consists in deleting elements whichreach a critical damage value. When the equivalent strain~e reaches ~eCritT , the Kill element method is used.

We test here the effect of the non-local model with anintegral formulation on the material response. In Fig. 7

we have plotted at location A the damage evolution versusdisplacement predicted by the Mazars model in its localversion (Fig. 7a) and the non-local version (Fig. 7b).

This tensile test shall physically induce crack initiationof the sample at point A. The local model gives a damagewhich evolves with different velocity to 1 for h= 10 mmand h = 2.5 mm. However we can see that the local modelgives a damage response for h = 5 mm which does not con-verge to 1. How can we explain this behavior? The work-piece first damages at point A. Then a crack initiates inthe workpiece but not at pointA. The local model is thusresponsible of this non-physical behavior. Consequentlydamage is distributed differently and stagnates (D0.5at the end of the simulation). Fig. 8compares the loaddis-placement curves obtained using different mesh refinements

100 mm

2.5 mm

50 mm

A

Fig. 5. Tensile specimen.

Table 1Material parameters

Parameters E(GPa) t AC

30 0.2 1.4

BC AT BT

1700 0.8 20,000

eD0 b lC(mm)

1.10

4

1.05 15



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for a classical damage model (Fig. 8a) and for the non-localdamage model (Fig. 8b) with an integral formulation.

In the case of the local Mazars model, damage evolutionat point A (Fig. 5) differs according to the mesh size h.Moreover we can see that the loaddisplacement curve isunstable in the damaged part. On the contrary, the non-local model shows its effectiveness whatever the meshrefinement:

Damage evolution in A is insensitive with respect to the

mesh size.

The loaddisplacement curve is more stable in the dam-aged part.

Damage increase is more important with a finer mesh.Mesh dependency of the results is clearly less sensitivefor the non-local approach.

Mesh dependency is thus clearly lower for the non-localapproach.

4.2. Weak coupling versus strong coupling

The influence of coupling on the material response isinvestigated here. In order to simplify interpretation ofresults, we work on a single element submitted to tension

(Fig. 9). Three symmetries are imposed on the three faces

Fig. 6. Finite element meshes with different mesh sizes h.

0

0.2

0.4

0.6

0.8

1

0 0.002 0.004 0.006 0.008 0.01

DamageD

Displacement (mm) (*-1)

h = 2.5 mm

h = 5 mm

h = 10 mm

0

0.2

0.4

0.6

0.8

1

0 0.002 0.004 0.006 0.008 0.01

DamageD


h = 2.5 mm

a

bh = 5 mm

h = 10 mm

Fig. 7. Damage evolution: (a) local model and (b) non-local model.



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of the tetrahedron in order to prevent any rigid bodymotion.

The behavior law is an elastic damage law.Table 2givesthe material parameters.

Here the influence of the time step on the material global

response is tested. Fig. 10 shows the loaddisplacement

curves for each time step using a weak coupling(Fig. 10a) or a strong coupling (Fig. 10b).

It can be noted that results are less dependent on timefor strong coupling. However a strong coupling leads tohigher computation times. It can become unacceptablefor large multimaterial cases. In this case, it can thus bepreferable to use a weak coupling in association with an

adaptive time step.

4.3. Adaptive time step

In order to improve the convergence of the Mazarsmodel without decreasing the time step over the total calcu-lation time we have developed a specific procedure in theFemcam software, enabling time step adaptation withrespect to damage evolution.

In Mazars model case, dtnis computed using the follow-ing formulae:

dtn

MindtFEMCAM; a~e2calcb~ecalc c if 0 6 ~ecalc 6 eD0

with a 1e2d0

; b 2aed0; c 1

MindtFEMCAM; a~ecalcb if ~ecalc> eD0

with a 1eCrited0

; b aed0

8>>>>>>>: 43

~ecalc corresponds to the value on the element for which theequivalent strain is the nearest to eD0; eD0is the equivalentstrain threshold in the Mazars model and ~eCrit is the equiv-alent strain beyond which macro cracks are initiated. It en-ables to reduce time step when equivalent strain is close to

the equivalent strain threshold.Fig. 11shows the evolution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.002 0.004 0.006 0.008 0.01 0.012

Load(kN)(*-1)


h= 2.5mm

h= 5 mm

h= 10mm

h= 2.5mm

h= 5 mm

h= 10mm

0 0.002 0.004 0.006 0.008 0.01

Lo

ad(kN)(*-1)


Fig. 8. Loaddisplacement curves: (a) local model and (b) non-localmodel.

Fig. 9. Single element submitted to a tensile test.


Parameters E(GPa) t AC BC

30 0.2 1.4 1700

AT BT eD0 b

0.8 20,000 104 1.05



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of the time step versus the equivalent strain for eD0= 104

and~eCrit 103.Fig. 11 shows that the time step decreases when the

equivalent strain~ecalcbecomes close to the equivalent strainthresholdeD0. When~ecalcis equal toeD0there is a minimumvalue for the global time step. If~ecalcexceedseD0, time stepincreases linearly until it reaches a maximal value. Theloaddisplacement curves for the classical and adaptivetime step cases are compared (Fig. 12). The initial time stepof the adaptive time step case is 1 s. The behavior law is anelastic damage law. The parameters used for these simula-tions are the same than inTable 2.

We see that adaptive time step gives good results; themaximal load is as accurate as when using a very small con-stant time step. CPU time results with an adaptive time

step and a small time step are also compared in Table 3.

The Adaptive time step procedure is almost threetimes faster than the classical time step one.

4.4. Discussion

First the correlation between the model and the meshsize is established. It is shown that a non-local model leads

to results independent on the mesh size. Furthermore for

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.002 0.004 0.006 0.008 0.01 0.012

Load(kN)(*-1)

Displacement (*-1)

dt= 1.5 s/step

dt = 1 s/step

dt= 0.5 s/step

dt= 0.25 s/step

dt= 0.1 s/step

dt= 0.05 s/step

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Load(kN)(*-1)

Displacement (*-1)

dt= 1.5 s/step

dt = 1 s/step

dt= 0.5 s/step

dt= 0.25 s/step

dt= 0.1 s/step

dt= 0.05 s/step

Fig. 10. (a) Weak coupling and (b) strong coupling.

0

0.2

0.4

0.6

0.8

1

0 0.0002 0.0004 0.0006 0.0008 0.001

Timestep

(s)

Equivalent strain

Fig. 11. Evolution of the time step with respect to the evaluatedequivalent strain~ecalc.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012

Load(kN)(*-1)


Small Time Step

Adaptive Time step N1

Adaptive Time step N2

Fig. 12. Comparison of loaddisplacement curves in function of the typeof time step.

Table 3

CPU time for the two methodsVery small time step Adaptive time step

Time (s) 150.0 51.8



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strong coupling, the model is completely independent oftime discretization. For the weak coupling a small time stepis necessary. However, for matters of convergency andCPU time, the weak coupling procedure will be used. Inorder to get the most accurate maximum load value, weshall use an adaptive time step allows to remain accurate

and to reduce the CPU time.

5. Applications

The constitutive law is now going to be validated on athree-point bending test and on a compression test. Wehave used the following parameters (see Table 4).

~eCritT and ~eCritC are respectively the equivalent strain

beyond which macro cracks are initiated in compressionand in tension. All tests are performed using a constant dis-placement rate prescribed on the upper loading plate.

5.1. Three point bending test

The three point bending test is widely used to determinefracture mechanics properties.Fig. 13shows the geometryand an example of initial mesh.

The contact between the upper tool and the workpiece isbilateral sticking. The contact between the lower tool andthe workpiece is frictional. The Kill element method isused when damage exceeds a prescribed value. The internallength is equal to 15 mm (a commonly used value for mor-tar). Damage iso-values for the local model simulation are

displayed inFig. 14.The macrocrack is initiated at the location where theequivalent strain ~e reaches the critical equivalent strain~eCritT . It corresponds also to the location where the maximalprincipal stress r1 reaches the critical tensile strength rt.

Associated to the Kill element method, a non-localmodel may give unaccurate results. The average area(based on the characteristics length) for a node close tothe crack, may include a node located on the other sideof the crack, which is non-realistic. This problem is showninFig. 15.

To underline the effects of this interaction, we compareloaddeflection curves for three different models:

The local model (LM). The non-local model (NLM). The evolutive non-local model (ENLM): A non local

approach is used at the beginning of the simulationand stops being used once damage has reached a criticalvalue in the sample.

Fig. 13. Mesh of the 40 40 160 mm three point bending test; 7394

nodes; 36,261 elements.

Fig. 14. Evolution of damage for different values of displacement of the

superior tool.


Parameters E(GPa) t AC

30 0.2 1.4

BC AT BT

457.08 1.29 25316.5

eD0 ~eCritC ~e

CritT

1.52.103 1.83.104

b

1.05



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Fig. 16 gives loaddeflection curves for these four

different techniques to simulate damage in quasi-brittlematerials.Fig. 16underlines efficiency of the NLM and the ENLM

methods. The maximal load is reached and the smooth partof the curve is well described. We also notice the effect ofthe localization on the peak load reached. Each test is per-formed on a standard PC (Intel Xeon Machine, 1.70 GHz,512 Mo RAM). We compare the different CPU times (seeTable 5).

Compared to the ENLM method, CPU time is smallerthan for the NLM method but almost fifteen times higherthan for the LM method. And even if some optimizationswould enable to decrease CPU time, this is not acceptable.

In this way a non-local approach with an implicit formula-tion is going to be implemented to improve the CPU time.

5.2. Compression analysis

A compression test on a quasi-brittle material has beenperformed. The contact between the tools and the work-

piece is frictional. We model only one quarter of the part thanks to symmetry. We use the evolutive non-localmodel (ENLM) with an internal length equal to 15 mm.Fig. 17 shows damage evolution.

Damage does not take place uniformly over the wholecylinder due to the frictional nature of contact between

the upper tool and the workpiece. Hence, we do notobserve the X-shaped diagonal shear bands which wewould observe in the case of a bilateral sticking contact.To improve this model, we have to consider a statisticaldistribution of voids and micro cracks in the workpiece

Fig. 15. A 2D case where Kill element method and non-local modelleads to a problem of determination.

Fig. 16. Loaddeflection curves for different models.

Table 5CPU time for several models (7394 nodes; 36,261 elements)

NLM ENLM

CPU TIME 4 h 42 mn 9 s 3 h 40 mn 13 s

Fig. 17. Evolution of damage for different values of displacement of theupper tool.

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5

Load(kN)

Displacement (mm)

Experiment

Numerical result

Fig. 18. Loaddeflection curve for a compressive test.



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before the loading starts. These defects orient preferentially

damage and macrocracks during the compression test.Fig. 18shows simulation results are in good agreement

with experimental observations on a mortar after 7 days.The load reaches about 285 kN which corresponds to aglobal strength of 29.9 MPa. It highlights the necessity todeterminate material parameters with a good accuracy.

6. Sensitivity analysis

6.1. Sensitivity to the internal length

The maximal load reached depends on the value of theinternal length. We have used here the three point bendingtest as a way to evaluate the effect of this internal length onthe loaddisplacement curve. We use the ENLM method(the characteristic length is equal to 3 mm).

Fig. 19underlines the effect of the choice of the internallengthlC. WhenlCis small with respect to the mesh size, weare close to the effect of a local model. When lC increases,

we do not converge to a unique solution. It is therefore nec-essary to adapt parameters identification in function of theinternal length lC.

6.2. Sensitivity to the critical value of equivalent strain

Usually macro cracks appear before damage becomescomplete (D= 1). We test here the influence of ~eCritT onthe global response of the elastic damage material. The testis a three point bending test. The model used is a localmodel but we keep the mesh size constant.

Fig. 20underlines the fact that our model is dependenton this value. It is hence necessary to identify with a greataccuracy the parameter ~eCrit to describe as well as possibledamage and cracks evolution.

7. Conclusions

We have developed and implemented in our finite ele-

ment software Femcam a model for quasi brittle materialslike mortar under uniaxial tests. The specificity of this codeis its velocitypressure formulation. We assume quasi-brit-tle materials to have an elastic damage behavior. We havechosen to model it with a non-local version of the Mazarsmodel. The damage model can be implemented through aweak or a strong coupling. Whereas accuracy is lower thanfor a strong coupling, we can adapt automatically the timestep without losing accuracy. The non-local model hasshown its advantages. Compared to a classical local model,the main drawback of the non-local model is its high CPUtime. We are currently working on the optimization of the

procedure devoted to the research of the neighbours ofeach element. We also work on the development of theimplicit formulation.

The strategy which has been developed in this paper hasproved its efficiency on examples of quasi-brittle materialsunder tension test or compression test. These tests under-line the effect of a non-local model compared to a classicallocal model.

Furthermore we have focused on the effect of the valueof the internal length and of the critical value of the equiv-alent strain on the global response. As the effect is relativelyimportant on the post peak part of the loaddisplacementcurve (we have seen that this value changes the globalbehavior of the material), it will be very important in thefuture to identify this parameter with respect to experimen-tal data.

Femcam can also handle multi-domain cases. This willenable us to deal in the future with a mesoscopic scaledescription. Thus it will enable us to take into accountthe heterogeneous aspect of a quasi-brittle material asconcrete.

Acknowledgement

This work has been supported by Atilh. The authors

would like to express their appreciation for this support.

Fig. 19. Loaddisplacement curve for a three point bending test.

Fig. 20. Loaddisplacement curve for a compression test for different

critical equivalent strain.



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[9] G. Pijaudier-Cabot, Z.P. Bazant, Non local damage theory, ASCE J.Eng. Mech. 113 (1997) 15121533.

[10] Z.P. Bazant, Crack band theory for fracture of concrete, Mater.Struct. 16 (94) (1983) 155177.

[11] E. Pichelin, K. Mocellin, L. Fourment, J.-L. Chenot, an applicationof a master-slave algorithm between deformable bodies in formingprocesses, Eur. J. Finite Elem. 10 (8) (2001) 857880.

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