Gradient plasticity modelling of strain localization in granular … 2004.pdf · 2011-10-12 · INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2004; 28:465–481 (DOI: 10.1002/nag.345)

Gradient plasticity modelling of strain localizationin granular materials

O. Al Hattamleh1, B. Muhunthan1,n,y and H. M. Zbib2

1Civil and Environmental Engineering Department, Washington State University, Pullman, WA 99164, U.S.A.2School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, U.S.A.

SUMMARY

The flow stress in the yield surface of plastic constitutive equation is modified with a higher ordergradient term of the effective plastic strain to model the effect of inhomogeneous deformation in granularmaterials. The gradient constitutive model has been incorporated into the finite element code ABAQUSand used to simulate biaxial shear tests on dry sand. It is shown that the shape of the post-peak segmentof the load displacement curve predicted by the numerical analysis is dependent on the mesh sizewhen gradient term is not used. Use of an appropriate gradient coefficient is shown to correct thisand predict a unique shape of the load displacement curve regardless of the mesh size. The gradientcoefficient required turns out to be approximately inversely proportional to the mesh elemental area. Useof the strain gradient term is found to diffuse the concentration of plastic strains within shear bandresulting in its consistent width. The coefficient of the higher gradient term appears as a function ofthe grain size, the mean confining stress, and the plastic softening modulus. Copyright # 2004 John Wiley& Sons, Ltd.

KEY WORDS: granular material; gradient plasticity; friction; dilatancy; internal length scale; localization;shear band

1. INTRODUCTION

Strain localization is a well-known phenomenon, generally associated with plasticdeformation and rupture in solids, especially in granular materials. During localization,deformation is observed to concentrate in narrow zones called shear bands. Shear bandsand ruptures control the overall pattern of behaviour of a particulate material such as a soilmass in a variety of engineering situations. Examples include foundations and retainingwalls; and soil elements in standard laboratory tests such as the shear box, plain strain,and triaxial apparatus. Consequently, the shear band phenomenon has been studied extensively

Contract/grant sponsor: National Science Foundation; contract/grant number: CMS-0010124

Copyright # 2004 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: B. Muhunthan, Civil and Environmental Engineering Department, Washington State University,Pullman, WA 99164, U.S.A.

over the last two decades by different researchers, experimentally, theoretically, andnumerically.

Experimental observations on shear bands have been made by several investigatorsusing biaxial devices [1–6], simple shear apparatus [7], triaxial devices [8, 9], torsional shearapparatus [10], and a true triaxial device [11]. Parallel theoretical advances on modellingshear band and its characteristics have been made by Bardet [12], Zbib and Aifantis[13–15] and Nemat-Nasser [16]. Different numerical techniques to account for the shearband within a continuum mechanics framework have also been proposed. They include,the softening elasto-plastic model [17–19], hardening elasto-plastic model with enrichedshape functions [20], hardening and softening elasto-plastic formulation with bifurcationanalysis [21], softening elasto-plastic model with higher-order gradients [14, 22, 23],softening polar elasto-plastic model [24], and the softening polar hypoplastic model[25, 26].

Classical continuum models for elasto-plastic deformation of materials are usuallyformulated for homogenously deforming media. Thus, they do not include internal lengthscales. These scales that may define the dimension of the underlying substructure, becomerelevant when deformation is heterogeneous and local strain gradients become intense,consequently, altering the material behaviour. When the deformation fields are highlyheterogeneous, phenomenological constitutive equations that do not consider the effect of highgradients fail to predict deformation patterns that may persist and eventually dominate thedeformation mechanism such as shear bands.

Various methods in which length scales have been introduced to form non-local typecontinuum models are available in the literature. These methods include micropolar theories inwhich rotational degrees of freedom are added to the conventional translational degrees offreedom [27–29]. Many of these studies have shown that a bifurcation analysis coupled with amicropolar theory makes it possible to predict the thickness of shear band as well as itsdirection. Micropolar theories, however, result in non-symmetric stiffness matrix and causeproblems in finite element analysis.

Integral and gradient type theories form another class of alternative methods to introducelength scales. In the integral type models [30] the evolution of certain internal variables areexpressed by means of integral equations. Mindlin [31] introduced gradients into linear elasticitymodels to the form of non-local constitutive models. The motivation, significance, andimplications of introducing gradients into plasticity models have been highlighted in the worksof Aifantis and co-workers [13, 32, 33].

The strain gradient theory has proven to be useful to mathematically and physicallymodel the effect of plastic flow heterogeneity within continuum mechanics formulations. This,however, furnishes a new and interesting problem regarding the physical interpretation of theadded material parameters ‘the gradient coefficients’. Although the works of Zbib and Aifantis[13–15] provide a means for indirectly measuring these coefficients, no rigorous analysis hasbeen performed that relates these coefficients to the size and scale of the underlyingmicrostructure.

In this paper, a gradient plasticity model of the form used by Zbib and Aifantis [14] is used.This model incorporates the gradient of effective plastic strain into the yield surface. The modelhas been implemented in a finite element code and used to study the effect of the gradient termon shear band pattern, thickness, and mesh dependency. The study also explores the physicalmeaning of the coefficient of the gradient term in such models.

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2004; 28:465–481

O. AL HATTAMLEH, B. MUHUNTHAN AND H. M. ZBIB466

2. MATHEMATICAL PRELIMINARIES

The deformation gradient Fij of a continuous media undergoing a smooth deformation processis given by the relation

Fij ¼@wiðXj; tÞ

@Xjð1Þ

where Xj is the position vector of a typical particle in the reference undeformed configurationand wi is its corresponding position vector in the current configuration. The velocity gradient Lijis given as Lij ¼ ’FF ikF �1

kj : It can be split into two parts; symmetric and skew-symmetric. Thesymmetric part represents the pure stretching tensor, Dij; and the skew-symmetric partrepresents the spin rate tensor, Wij

Dij ¼ 12ðLij þ LTijÞ; Wij ¼ 1

2ðLij � LTijÞ ð2Þ

The stretching rate Dij can be decomposed as

Dij ¼ Deij þ Dp

ij ð3Þ

where Deij and Dp

ij are the elastic and plastic parts, respectively. Likewise the spin tensor iswritten as

Wij ¼ oij þ W pij ð4Þ

where oij is the spin of microstructure and W pij is the plastic spin.

The governing equation of equilibrium for quasi-static loading conditions [34] is

sij;j ¼ 0 ð5Þ

where sij is the Cauchy stress tensor.The elastic rate of stretching, De

ij is assumed to follow the Hooke’s law:

soij ¼ CeijklD

ekl; Ce

ijkl ¼ G dikdjl þ dildjk þ2n

1� 2ndijdkl

� �ð6Þ

where Ceijkl is the elasticity tensor, G is the shear modulus, n is the Poisson’s ratio. soij is the

corotational rate of Cauchy stress tensor defined with respect to frame rotating with thematerial:

soij ¼ ’ssij � oikskj þ sikokj ð7Þ

where the central difference integration of the rate of the spin Doij is related to the increment ofrotation DRij as [35]

DRij ¼ dij � 12Doij

� ��1 dij þ 12Doij

� �ð7aÞ

Dij can be split into a volumetric strain ðDÞ; and a deviatoric strain rate dij as

D ¼ Dmm; dij ¼ Dij � 13Ddij ð8Þ

Likewise the components of the stress may be split into hydrostatic ðpÞ and deviatoricðSijÞ parts:

p ¼ 13 skk ; Sij ¼ sij � pdij ð9Þ


GRADIENT PLASTICITY MODELLING 467

3. PLASTICITY MODEL

The yield function of the granular material is assumed to follow the form

f ¼ qþ k ð10Þ

where k is the flow stress and q is the effective shear stress, defined as

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi32SijSij

qð11Þ

Note that in conventional plasticity theories k is assumed to be equal to pm where m is thecoefficient of friction.

The plastic stretching dpij is defined by means of a potential function Q as

dpij ¼ ’gg@Q@sij

ð12Þ

where ’ggp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi23dpijd

pij

q� �is the effective plastic strain rate. The function for Q is assumed to be

Q ¼ qþ pb gpð Þ; gp ¼Z

’ggp dt ð13Þ

where bðgpÞ is the dilatancy coefficient relating volumetric plastic strain to effective plastic shearstrain as

Dp ¼ b gpð Þ’ggp ð14Þ

Substituting Equation (13) into Equation (12) and combining with Equation (14) leads to

Dpij ¼

3Sij2q

þb3dij

� �’ggp ð15Þ

3.1. Gradient plasticity model

Several researchers have proposed gradient modifications to existing elasto-plastic formulationto account for localization. Zbib and Aifantis [14] have presented a detailed discussion on theirrelative merits. The gradient modification here of the conventional plasticity model involves themodification of the flow stress k (Equation (10)). Following Zbib and Aifantis [14] k is assumedto be

k ¼ jp jmðgpÞ � c1r2gp ð16Þ

where c1 is a gradient related coefficient. Note that in the above form higher order gradients arenot included as their contribution was found to be minimal. Furthermore, because of theisotropic assumption and spherical symmetry of the volume integral of the representativevolume element around the point of interest in the granular material domain, the modificationdoes not involve odd order gradients [14]. With the incorporation of the gradient term the yieldfunction becomes

f ¼ qþ jp jmðgpÞ � c1r2gp ð17Þ



Utilizing the yield and consistency conditions f ¼ 0; ’ff ¼ 0; the effective plastic strain rate couldbe evaluated as

’ggp ¼ðð3Sij=qÞ þ amdijÞDij � ðc1=GÞr2’ggp

3þ jp jðht=GÞ þ abmþ ðc01=GÞr2gp

ð18Þ

where ht ¼ @m=@gp; is the strain hardening/softening modulus and c01 ¼ dc1=dgp; substitutingEquation (18) into Equation (14) and combining it with Equations (3) and (6) and letting c01 ¼ 0for simplicity leads to

’ssij ¼ CijklDkl � ’ssgij ð19Þ

where

Cijkl ¼ Ceijkl � Cp

ijkl ð19aÞ

’ssgij ¼c1r2 ’ggp

H3Sijq

þ abdij

� �ð19bÞ

H ¼ 3þ jpjhtGþ abm; a ¼

KG

ð19cÞ

The resultant plastic stiffness tensor has the following form:

Cpijkl ¼

GH

3Sikq

þ abdik

� �3Sjlq

þ amdjl

� �ð19dÞ

3.1.1. Evolution of friction and dilatancy. The mobilized friction coefficient m is assumed to be afunction of the effective plastic strain as

mðgpÞ ¼ mcv þ x1 gp þm0 � mcv

x1

� �expð�x2gpÞ ð20Þ

where mcv is the internal friction at constant volume, m0 is the initial internal friction, and x1; x2are fitting parameters. Similar models relating m to mcv have been proposed by other researchers[36, 37].

Following the work of Taylor [38] the dilatancy is expressed as (see also Reference [23])

bðgpÞ ¼ mðgpÞ � mcv ð21Þ

3.1.2. Evaluation of gradient term. Special care needs to be exercised in evaluating theLaplacian term in Equation (19b) at each stage of the deformation. For the 2D case, theLaplacian term can be written as

r2 ’ggp ¼ ’ggpxx þ ’ggpyy ð22Þ

where ’ggpxx; ’ggpyy are the derivative of effective plastic shear strain rate with respect to both x and y

co-ordinates, respectively. Assuming that the point of interest is in the centre of four equallyspaced adjacent nodes as shown in Figure 1(a), the Taylor series approximation of thederivatives in Equation (22) can be derived as

’ggpxxðx; yÞ ¼1

h2½’ggpðxþ h; yÞ � 2’ggpðx; yÞ þ ’ggpðx� h; yÞ� ð22aÞ



’ggpyyðx; yÞ ¼1

h2½’ggpðx; y þ kÞ � 2’ggpðx; yÞ þ ’ggpðx; y þ kÞ� ð22bÞ

The above equations are useful for conditions of uniform mesh and deformation that do notlocalize. Since the analysis here involves high geometric non-linearities and localization, adifferent configuration with unequally spaced nodes as shown in Figure 1(b) is required. For thisarrangement, Taylor series approximation of the derivative become

r2’ggpjO ¼2

ðcaþ a2Þ’ggpjA þ ’ggpjC � ’ggpjO 1þ

ac

� �h iþ

2

ðbd þ b2Þ’ggpjB þ ’ggpjD � ’ggpjO 1þ

bd

� �� ð23Þ

Following Zbib [39] the Laplacian term is assumed to be zero along the boundary elements.Note that this assumption is valid only for the case of second order gradient term as used here.For additional gradients the flux term along the boundary may be non-zero and need to beevaluated appropriately.

3.1.3. Basis of the selection of length scale values. In order to identify the physical attributes ofthe coefficient of the gradient term c1 (Equation (16)), the effective plastic shear strain rateEquation (18) can be split into two terms; a first term that is not contributed by the length scale,’ggpo; (local term) and a second term that is contributed by the length scale, ’ggpg (non-local term).Assuming for simplicity c01 to be zero

’ggp ¼ ’ggpo þ ’ggpg ð24Þ

where

’ggPo ¼ðð3Sij=qÞ þ amdijÞDij

3þ jp jðht=GÞ þ abmð24aÞ

and

’ggpg ¼�c1r2 ’ggp

3Gþ jp jht þ abmGð24bÞ

hh

k

k

(x+h,y)(x-h,y)

(x,y-k)

(x,y+k)

O(x,y)ac

d

b

B(x,y+b)

A(x+a,y)C(x-c,y)

D(x,y-d)

O(x,y)

Uniform Mesh Deformed Mesh(a) (b)

Figure 1. (a) Points in Equation (22); and (b) neighbouring points A, B, C, D of mesh pointO and notation in Equation (23).



It can be seen that ’ggpg is the effective plastic shear strain rate that gets contributions from localrate of plastic stretching of the surrounding integration points through the Laplacian as shownin Figure 2.

Let ’ggp be the average of effective plastic shear strain rate within the grains in a small but finitevolume, V ; of REV surrounding material at material point wi: (Note that for 2D as in this casethe REV becomes representative elemental area (REA) and i is summed from 1 to 2). The non-local of form of the rate of plastic stretching can be expressed formally by [40, 41, 33]

’ggp ¼ ð1=V ÞZv’ggpðwi þ xiÞ dV ð25Þ

Taylor’s series expansion of the function ’ggpðwi þ xiÞ around wi is given by

’ggpðwi þ xiÞ ¼ ’ggpðwiÞ þ@j ’ggp

@wjðwiÞxj þ

1

2!

@2jkð’ggpÞ

@2ðwjwkÞðwiÞxjxk þ � � � ð26Þ

Substituting Equation (26) into Equation (25) results in

’ggp ¼ ð1=V ÞZv

’ggpðwiÞ þ@j ’ggp

@wjðwiÞxj þ

1

2!

@2jkð’ggpÞ

@2ðwjwkÞðwiÞxjxk þ � � �

!dV ð27Þ

For a spherical REV (or in the present case, circular REA) of radius R the integration can becarried out using polar co-ordinates within the range rð04r4RÞ and yð04y42pÞ:Note that R isa multiple of the average grain diameter within an REV. The resulting relation between theaverage and local effective plastic shear strain rate can be shown as [42]

’ggp ¼ ’ggpi þ1

pR2

Z R

0

Z 2p

0

@i’ggp

@wirni þ

1

2!

@2ijð’ggpÞ

@2ðwiwjÞr2ninj þ � � �

" #dr r dy ð28aÞ

’ggp ¼ ’ggpi þ1

pR2

R3

3

@i ’ggp

@wi

Z 2p

0

ni dyþR4

8

@2ij’ggp

@wiwj

Z 2p

0

ninj dyþ � � �

" #ð28bÞ

Note that ni denotes the outward unit normal of the circle r ¼ R:

Figure 2. Location of ’ggpo in representative volume element and its location withinfinite element discretization mesh.



SinceR 2p0 ni dy ¼ 0 and

R 2p0 ninj dy ¼ pdij; and setting ’ggpi ¼ ’ggpo results in the following

relation:

’ggp ¼ ’ggpo þR2

8

@2ij ’ggp

@wiwj

" #ð29Þ

’ggp ¼ ’ggpo þR2

8r2’ggp ð30Þ

Comparison of Equations (30) and (24) and since the particle size distribution is wellrepresented by D50; setting 2R ¼ D50; an initial estimate of the gradient term, c1; would be

c1 ¼D2

50

32ð3Gþ jp jht þ KbmÞ ð31Þ

4. EXPERIMENTAL DATA AND MODEL CONFIGURATION

The gradient plasticity model has been implemented into the ABAQUS [34] standard finiteelement code as a special user material UMAT subroutine. The model has been used to studythe characteristics of strain localization and shear band initiation of some biaxial shear tests ondry sand reported by Han and Drescher [2].

The biaxial apparatus used a prismatic specimen of 40 mm width, 80 mm length, and 140 mmheight. The specimen was enclosed between two rigid walls 80 mm apart and placed on a platen,which rested on a linear bearing. The linear bearing provided kinematic freedom for theformation of shear bands with the lower specimen portion sliding horizontally. The apparatuswas placed inside a pressure chamber and the specimen was subjected to a confining pressureand kinematically or statically controlled axial load.

Biaxial tests were performed on coarse, poorly graded Ottawa sand with rounded particles ofmean grain diameter D50 ¼ 0:72 mm: Homogenous dense specimens, with an initial porosity ofn0 ¼ 0:32–0.33, were prepared. All tests were performed with displacement controlled axialloading. Additional details of the test device and measurements of relevant parameters areprovided in Reference [2].

Let s1; s2 denote the axial and lateral stresses and let e1; e2 are axial and lateral strains, and lett ¼ 1

2js1 � s2j; s ¼ 1

2ðs1 þ s2Þ; g ¼ je1 � e2j; e ¼ ðe1 þ e2Þ; denote measures of shear stress, mean

stress, shear strain, and volumetric strain, respectively. Using the pre-shear band experimentaldata the stress ratio ðt=sÞ versus shear strain was plotted. The stress ratio ðt=sÞ is related to themobilized angle of shearing resistance as t=s ¼ sin jmob: The mobilized friction coefficient,which defined as m ¼ sin jmob can be related to s1 and s2 from Mohr circle representation ofstresses as m ¼ js1 � s2j=ðs1 þ s2Þ: The data for mobilized friction, volumetric strain and shearstrain reported by Han and Drescher [2] is shown in Figure 3. It is re-plotted as m versus g andfitted using the proposed evolution equation (Equation (20)) as shown in Figure 4. It can beeseen that the model is able to capture the experimental data well. Note by definition, themobilized friction coefficient m ¼ 0 initially, since s1 ¼ s2 and it increases from this value to 0.69at the peak. However, to obtain a smooth curve fit an initial value of m ¼ 0:01 was found to benecessary.



5. SIMULATION OF SHEAR BAND INITIATION

The biaxial test has been modelled using a plane strain finite element analysis. The modelconfiguration and the discretization of the mesh are shown in Figure 5. All analyses were carriedout using a four-node plane strain element with reduced integration, CPE4R. The total numbers

0.00

0.20

0.40

0.60

0.80

0 0.02 0.04 0.06 0.08

Shear strain, γ

stre

ss r

atio

, t/s

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

Vol

umet

ric S

trai

n, ε

t/s= µ

ε

Figure 3. Mobilized friction and volumetric strain vs shear strain prior to shear band formation [2].

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Equivalent Plastic Strain

µ, β

µβµ, Experimental Data

Figure 4. Fitted mobilized friction evolution equation from experimental data of Han and Drescher [2].



of mesh were varied in order to study the effect of strain localization and shear band on meshsensitivity. The lower boundary of the mesh is assumed to be rigid.

In order to solve the non-linear governing equation numerically UMAT uses an implicitintegration predictor–corrector algorithm. Accordingly, for the time interval ½tn; tnþ1�;corotational rate of Cauchy stress tensor snþ1

ij is numerically predicted using

snþ1ij ¼ DRiksnklDRlj þ Ce

ijklðDekl � DepklÞ ð32Þ

where snij is the known stress state of previous step, DRij is the increment of the rotation tensor,and Deij is the strain increment. snþ1

ij ¼ DRiksnklDRlj þ CeijklDe

ekl can be recognized as the elastic

trial stress [43, 44].Further to account for the gradient stress sgij appearing in Equation (19), the corrector term of

the stresses due to plastic flow is modified as

snþ1ij ¼ snþ1

ij � CeijklDe

pkl �

c1r2Dgp

H3Sijq

þ abdij

� �ð32aÞ

Note that the addition of the last term on the r.h.s of Equation (32a) implies that the effect ofgradient term enters the global stiffness matrix implicitly.

In the first phase, a confining pressure was applied to consolidate the specimen isotropically.Thereafter, axial compression was applied by increasing the vertical displacement on the top ofthe specimen. Based on the characteristics of the sand and the test conditions, the followingmaterial constants were used in the gradient plastic model: stiffness modulus ðEÞ ¼ 200 MN=m2;

∆

(a) (b)

Figure 5. Model configuration: (a) first stage: all around confining pressure; and (b) second stage: like firststage with addition of pre-describe displacement.



Poisson ratio ðnÞ ¼ 0:2; initial friction ðmoÞ ¼ 0:010; constant volume friction ðmcvÞ ¼ 0:612;x1 ¼ 25:00 and x2 ¼ �55:00:

In order to initiate shear band, reduced value constant volume friction ðmcvÞ was assigned toone of the elements within central portion of the FE mesh. Figure 6 shows the shape of thedeformed meshes with localization and the appearance of shear band. Note that since the lowerboundary is constrained in all directions the shear band can also initiate from its corners. Theinclination of the resulting shear band was measured to be 55� 28 relative to the horizontal axisdepending on the number of total meshes (Figure 6). The experimentally measured angle wasfound to vary between 55 and 598 [2]. It lies in the range between y ¼ ðp=4Þ þ c=2 predicted byRoscoe [45] and y ¼ ðp=4Þ þ 1=4ðcþ fÞ predicted by Vardoulakis [1] and Arthur et al. [46]where c is the dilation angle measured at localization. Figure 7 shows the location of the initialweak elements as well as the progressive development of the shear band in 10� 35 mesh.

The effect of the coefficient of strain gradient on the force displacement curve with a fixedmesh size ð10� 35Þ is shown in Figure 8. It can be seen that the use of negative values of c1 leadsto decreased strain softening effect. It appears that the use of a very small negative value maylead to elimination of post-peak strain softening in granular materials. It is noted that theopposite is true in the case of positive values of c1: Therefore, it appears that the addition of thegradient term in the yield function leads to hardening of the material particularly in the regionof material softening thereby contributing to its overall strength. This means that in this modelnon-local effects result in strengthening of the material, since the gradient dependent stressproduces effectively an increase of the local confining pressure.

Figure 6. Deformed mesh with shear band at D ¼ 0:014 m with c1 ¼ 5� 10�3 Nðl ¼ 7:75 mmÞ: (a) 10� 35mesh; (b) 20� 40 mesh; and (c) 20� 70 mesh.



6. INTERNAL LENGTH SCALE AND NUMERICAL SHEAR BAND THICKNESS

In order to give a physical attribute to the gradient coefficient, a material length scale can bedefined as (see also Reference [47])

l ¼

ffiffiffiffiffic1G

rð33Þ

The thickness of the shear band is controlled by the length scale coefficient, l: CombiningEquations (31) and (33) results in

l ¼D50

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

23þ jp j

htGþ abm

� �sð34Þ

It can be seen that l depends on material characteristics D50; confining pressure, and elasticmodulus G:

Depending on the mesh size a different value of l was needed to obtain a consistent thicknessof the shear band. For example, the measured shear band thickness was found to be 10:25 mmfor the model when the number of elements is 350 with l ¼ �36:56 mm; 800 with l ¼ 0:0 mm and

Figure 7. Development of shear band for 10� 35 mesh at c1 ¼ 5� 10�3 Nðl ¼ 7:75 mmÞ:



1400 with l ¼ þ7:75 mm: This shows that in order to avoid mesh-size dependency, it isimportant to include the effect of internal length scale by means of the gradient term. Thecomputed thickness of the shear band ð10:25 mmÞ is 14� D50: Past researchers have observedthe thickness of shear band within granular materials to vary between 5 and 20 times the meangrain diameter D50 [1, 8].

Yoshida et al. [3] observed that large strain gradients are present within a shear zone. This isalso seen from Figure 9. Notice, however, that the use of strain gradient enables the diffusion of

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016Displacement (m)

For

ce (

kN)

c1=+1E-1

c1=+5E-2

c1=+0E-0

c1=-5E-2

c1=-1.E-1

Figure 8. Effect of gradient coefficient in force displacement curves for 10� 35 mesh.

0.0

0.2

0.4

0.6

0.8

1.0

-20 -15 -10 -5 0 5 10 15 20

l=16.67

l=0.000

Equ

ival

ent

Pla

stic

Str

ain,

γp

l =0.00µ ml =25.82 µm

(a) (c)(b)

25.82

X

γp

Distance, x

Figure 9. Effect of length scale on defusing the equivalent plastic strain concentration within shear bandfor 800 elements: (a) contour plot of equivalent plastic strain ðl ¼ 0 mmÞ; (b) distribution of equivalent

plastic strain a cross shear band; and (c) contour plot of equivalent plastic strain ðl ¼ 25:82 mmÞ:



the concentration of plastic strains and results in a consistent width of the shear band(Figure 9b).

The shape of the load displacement curve predicted by the numerical analysis is dependent onthe mesh size when gradient term is not used (Figure 10). However, as shown in Figure 11, it ispossible to predict a unique shape of the load displacement curve with the use of an appropriategradient coefficient depending on the mesh size. The results show that a finer mesh requires theuse of a finer length scale. Moreover, the gradient coefficient turns out to be approximatelyinversely proportional to the elemental area.

The dependency of the maximum predicted force on the gradient term is illustrated in Figure12. It shows the variation of maximum force as a function of c1: It can be seen that themaximum value of the force is not dependent on c1 for finer mesh. Thus there is no need to use afiner mesh than the minimum size required obtaining consistent results. This minimum size isdependent on confining pressure as well as particle size (Equation (34)). For the problem here20� 70 mesh appears to the minimum size required

7. CONCLUSIONS

This study demonstrates that incorporating a higher order gradient in terms of effective plasticshear strain into the flow stress of the yield surface is effective in capturing the post-localizationbehaviour of granular materials. The gradient constitutive model has been incorporated into afinite element code and was used to simulate the biaxial shear test results on dry sand. Thepredicted shear band thickness was found to be in a good agreement with experimental results.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Displacement (m)

For

ce (

kN)

10x35

20x40

20x70

Figure 10. Force displacement curves for 10� 35; 20� 40; and 20� 70 meshwithout the effect of length scale.



The gradient term was found to act as a stabilizer and to eliminate the mesh size dependency onpredicted stress strain behaviour in the post localization regime where material exhibits strainsoftening.

The study also interpreted the physical meaning of the coefficient of the higher gradientterm c1: It has been shown that it is a function of the grain size ðD50Þ; the mean stress p; andthe softening modulus ht: Use of a material length scale in terms of c1 and elastic modulus G

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Displacement (m)

For

ce (

kN)

20x40, l=-0.00

20x70, l=+7.75

10x35, l=-36.56

Figure 11. Effect of length scale on the softening of force displacement curve for different mesh size andlength scale values (l values are in mm).

2.00

2.10

2.20

2.30

2.40

2.50

2.60

2.70

2.80

2.90

3.00

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15

c1 (N)

Fm

ax (

kN)

10x35

20x70

20x40

Figure 12. Maximum force as a function of gradient coefficient, c1:



showed that the thickness of the shear band is also controlled by the grain size, confiningpressure, elastic modulus, and softening modulus.

ACKNOWLEDGEMENT

The study presented in this paper was sponsored by the National Science Foundation under the GrantCMS-0010124 to Washington State University. This support is gratefully acknowledged.

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