On the elasticity and plasticity of dilatant granular materials

\ PERGAMON

Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397

On the elasticity and plasticity of dilatant granular materials S[A[ Nixon\ H[W[ Chandler Department of En`ineerin`\ Fraser Noble Buildin`\ University of Aberdeen\ Aberdeen\ AB13 2UE\ U[K[ Received 12 January 0887

Abstract We present a model which simulates the response of a granular material "made up of non! plastic grains# to small cyclic shear strain[ The model is based upon a dissipation function representing energy dissipated as plastic deformation and a quadratic dilatancy rule describing the volume response which produces kinematic hardening[ Elastic deformation is included in the model through a function describing the rate at which it is stored as deformation occurs[ Two sets of experimental data "one for a shear test\ the other for a circular loading test# are used to evaluate the accuracy of the model[ For the shear test\ both the shear strainÐshear stress curve and the shear strainÐvolume strain curve are reproduced well[ In the circular loading test\ the principal strain response and volume response are modelled realistically[ Þ 0888 Elsevier Science Ltd[ All rights reserved[ Keywords] B[ Anisotropic material^ Constitutive behaviour^ Cyclic loading^ Elastic!plastic^ Granular material

0[ Introduction The sometimes perplexing mechanical properties of granular materials present the mathematical modeller with some fascinating problems of great importance to process engineering\ manufacturing\ transport\ mining\ civil engineering and agriculture[ In this paper we concentrate on the small strain behaviour of compact assemblies of granules\ previously subject to a history of gentle vibration[ Material in this state dilates when sheared further "Reynolds\ 0774#\ and this\ as we shall see\ has a number of interesting consequences "Pande and Pietruszczak\ 0875^ Wood\ 0871#[ The mod!

 Corresponding author 9911Ð4985:88:, ! see front matter Þ 0888 Elsevier Science Ltd[ All rights reserved PII] S 9 9 1 1 Ð 4 9 8 5 " 8 7 # 9 9 0 9 7 Ð 1

0287 S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397

elling of the stress strain behaviour of granular material divides into two\ those with the purpose of modelling monotonic\ and those which can cope with cyclic loading[ While critical state models "Nova and Wood\ 0868^ Chandler\ 0889# can reproduce quite well the behaviour of material under monotonically increasing loads "they give seemingly appropriate levels of volume change and realistic shear stressÐshear strain curves#\ the accurate modelling of cyclic loading presents several di.culties[ To make progress on this front we return to an earlier model "in particular the Drucker and Prager "0841# model# and then modify it to simulate many of the features seen in cyclic deformation[ The basic assumption made in the DruckerÐPrager formulation for monotonic small strain deformation is that the value of the second deviatoric stress invariant required to yield a granular assembly is linearly dependent on the pressure[ There has\ in the past\ been some di.culty over how to relate such yield functions to the consequent plastic deformation of the material described by ~ow rules "Spencer\ 0871#[ This can be overcome\ however\ by starting with a dissipation function\ describing the rate of energy dissipated by plastic deformation\ and an internal constraint specifying the volume strain rate in terms of the shear strain rate "a dilatancy rule#[ In this way but following slightly di}erent routes\ Chandler "0874\ 0889#\ using the theory of envelopes\ and Houlsby "0881#\ using the approach of Ziegler "0874#\ have both produced physically reasonable ~ow rules and yield func! tions[ In particular the DruckerÐPrager yield function is then the consequence of assuming] "i# A dissipation function that is proportional to the product of the pressure and the square root of the second invariant of plastic deviatoric strain rate[ "ii# That the rate of dilatancy "volume increase# is proportional to the same shear rate invariant[ "The proportionality constant is dependant on the state of the packing of the granules[# Cyclic shear deformation\ however\ exposes new features which are not predicted by such monotonic models[ To appreciate some of this bewildering behaviour imagine a box of sand under an imposed and constant pressure\ "−s#\ subject to a cyclic shear stress t producing a cyclic shear strain g[ Figure 0 shows a typical stress!strain cycle accompanied by the volume change during the same cycle[ A marked Bauschinger e}ect can be seen in the stress!strain response and this is not predicted by the DruckerÐPrager yield criterion[ The most successful modellers "Mroz\ 0856^ Stallebrass and Taylor\ 0886# of cyclic deformation have looked to the construct of kinematic hardening to model this\ and to justify this approach for granular materials Houlsby "0881# has used more re_ned dilation rules incorporating the development of anisotropy[ Another feature of the response is that during each loading cycle there is an almost parabolic increase in volume which is more than recovered during the complete cycle "Fig[ 0b#[ The earlier assumptions that volume change is essentially irreversible\ and that the rate is linear with shear strain rate\ are clearly not borne out by these results[ Also\ each complete shear strain cycle produces an overall reduction in bulk volume "Youd\ 0861#\ as long as =gmax−gmin = is su.ciently small[ This slow but persistent

S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 0288

Fig[ 0[ A comparison between experimental results for simple shear "Nemat!Nasser and Takahashi\ 0873# and theoretical results generated using the constants shown in Table 0 for Material A[

decrease in volume with each complete cycle is also not predicted by simple monotonic models[ There is one more puzzling point and that is at the onset of strain reversal one would expect only elastic deformation\ however the volume decreases immediately even though no plastic deformation is yet taking place[ This indicates a shear!volume coupling even in the elastic response[

0399 S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397

Fig[ 1[ A representation of granule rearrangement showing dilatancy occurring as the grains slide and rotate past each other when a shear strain is applied to the sample[

In order to provide a physical basis to explain these phenomena\ consider the interaction of a few granules "Fig[ 1#[ Any degree of deformation involves a com! bination of granule rotation and either elastic deformation\ or slip\ at the contact patches[ We postulate a reference state in which the initial positions of the granules give a local minimum in overall volume occupied[ Any shear deformation involves particle rotation and so a corresponding increase in volume[ If a small level of shear deformation is reversed\ then one would expect the granules to rotate back almost to their original positions[ This explains the high level of reversibility that is seen in experiments[ Clearly extensive shear deformation will tend to erase the memory of this minimum\ and the rate of volume increase will decrease[ In this case the volume strains will be irreversible[ In this paper we o}er a very simple model for small deformation based upon these ideas which we shall see is capable of simulating experimental results for small strains and gives a yield surface showing kinematic hardening and other associated features[

1[ Preliminaries Consider a compact assembly of granular material occupying a unit cube in a rectangular Cartesian coordinate system with base vectors i0\ i1\ i2[ The components of the Cauchy stress and in_nitesimal strain tensors are sij and eij respectively[ The volumetric strain "e#\ and the components of the deviatoric strain "dij# can be de_ned\ using the summation convention and the Kronecker delta "dij#\ as 0 e  emm ^ dij  eij − edij [ 2

"0#

The components of stress may also be split in a similar way\ into deviatoric "sij# and hydrostatic "s# parts

S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 0390

0 s  smm ^ sij  sij −sdij [ 2

"1#

Throughout the following treatment a dot represents a rate or increment and the components of strain are labelled with superscripts T\ E\ P\ representing in broad terms the total\ elastic and plastic strains[ "The term eE needs careful treatment\ see Section 2[0#[ As we shall see\ the reference state of the model is assumed to be isotropic so the use of invariants will prove to be invaluable[ Particular use is made of the second invariant of the deviatoric strain which will be written as dijdij with the appropriate addition of superscripts and dots as needed[

2[ Material model 2[0[ Volume behaviour Imagine a unit cube of sand which has previously su}ered a small level of vibration which has induced a moderately densely packed state[ This packing is regarded as an isotropic reference state\ where the total strain "eT# is set to zero[ The key to the success of the following model is that the subsequent total volume strain is the sum of three contributions[ "i# The _rst is an elastic term eE[ This is a result of a pressure "−s# acting on the assembly producing elastic compression of the contact patches[ The granules\ at least notionally\ do not rotate at this stage[ This deformation is fully reversible[ "ii# The second is a consequence of granule rotation produced by elastic and plastic shear deformation[ This volume increase is proportional to the product of d ijT d ijT and a constant nR[ Again this is assumed to be fully reversible[ "iii# The third is proportional to another constant nI and the integral with respect to time of zd¾ ijP d¾ ijP [ This is equivalent to the total amount of plastic shear strain that has occurred regardless of sign[ This controls the irrecoverable volume decrease that occurs with each complete strain cycle[ Added together these give eT  eE ¦nR d ijT d ijT ¦nI

g

t

zd¾ ijP d¾ ijP dt[

"2#

9

Appropriate values for the dilatancy constants "material A of Table 0# give volume shear strain relationships shown in Fig[ 0b which mimic those given by Nemat!Nasser and Takahashi "0873#[ Speci_cally nR is positive and nI is negative[ The total volumetric strain rate "e¾T# follows from Eq[ "2#

0391 S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 Table 0 The constants used to generate Figs 0 and 3 using Materials A and B respectively

nI nR m G:s K:s

Material A

Material B

−9[992 2 9[6 19 099

−9[0 09 9[6 19 099

e¾T  e¾E ¦1nR d¾ ijP "d ijP ¦d ijE #¦1nR d¾ ijE "d ijP ¦d ijE #¦nI zd¾ ijP d¾ ijP [

"3#

2[1[ Elastic response Let us _rst consider the case where\ once the reference state has been established\ no plastic deformation has yet taken place[ The total volumetric strain can then be written as eT  eE ¦nR d ijE d ijE \

"4#

and the total shear strain is equal to the elastic shear strain[ The energy stored as elastic deformation\ principally at the contact patches\ can be written in terms of the elastic strains\ the bulk modulus of the material "K#\ and the shear modulus "G# giving 0 UE  Gd ijE d ijE ¦ K"eE # 1 [ 1

"5#

To obtain the stressÐstrain relations from this energy expression we rewrite it in terms of the total strain variables which gives 0 UE  Gd ijE d ijE ¦ K"eT −nR d ijE d ijE # 1 \ 1

"6#

and make use of the established relationships s  1UE :1eT ^ sij  1UE :1d ijE [

"7#

This leads to s  K"eT −nR d ijE d ijE #\ sij  1d ijE "G−nR s#[ These can be inverted to give

"8# "09#

S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 0392

eT 

sij sij s ¦nR \ K 3"G−nR s# 1

"00#

d ijE 

sij [ 1"G−nR s#

"01#

Substitution of realistic values for the constants G\ K\ and nR\ means that the mag! nitude of nRs:G is small compared with unity and so in the elastic region shear induced volume strain is negligible[ However we shall see in the next section that the coupled elastic term can be noticeable once plastic deformation is signi_cant[ 2[2[ Plastic behaviour In order to model plastic deformation and its in~uence on the elastic response we must look at the energy balance for a unit cube of material[ This energy balance consists of three terms] a function describing the rate at which energy is stored in elastic deformation U þ E^ a dissipation function D þ describing the rate at which energy is dissipated by plastic deformation^ and the rate at which energy is provided by the surface tractions "sij e¾ijT # sij e¾ijT −U þ E −D þ  9[

"02#

The rate at which elastic energy is stored is U þ E  1Gd ijE d¾ ijE ¦KeE e¾E \

"03#

and a suitable dissipation function can be written as D þ  zm1 s1 d¾ ijP d¾ ijP \

"04#

where the parameter m is akin to an e}ective friction coe.cient[ The energy balance can then be written as se¾E ¦sij d¾ ijP ¦sij d¾ ijE  F\

"05#

where F is F  −1snR d¾ ijP "d ijP ¦d ijE #−1snR d¾ ijE "d ijP ¦d ijE # −snI zd¾ ijP d¾ ijP ¦zs1 m1 d¾ ijP d¾ ijP ¦1Gd ijE d¾ ijE ¦KeE e¾E [

"06#

F is homogeneous of degree one in e¾E\ d¾ ijE \ and d¾ ijP \ so Euler|s theorem on homogeneous functions indicates that the energy balance is satis_ed if s

1F 1e¾E

sij 

\

1F \ 1d¾ ijE

"07#

"08#

0393 S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397

sij 

1F [ 1d¾ ijP

"19#

Equation "07# then reduces to eE  s:K

"10#

which de_nes the contribution to the volume strain from contact patches between non!rotating granules[ Equation "08# gives d ijE 

sij ¦1snR d ijP 1G−1snR

"11#

which sets the level of elastic shear strain[ Equation "19# gives zd¾ ijP d¾ ijP "sij ¦1snR d ijT # d¾ ijP  −s"m¦nI #

"12#

which is a ~ow rule for the components of the plastic strain increment[ These com! ponents of the plastic strain increment can be eliminated by summing their squares to give "sij ¦1nR sd ijT #"sij ¦1nR sd ijT # "sm¦snI # 1

−0  9\

"13#

which is a yield surface[ In principal stress space the intersection of this surface with the constant pressure plane is a circle\ see Fig[ 2[ The centre of this circle depends upon d ijT \ hence this is a yield function with a type of kinematic hardening[ The total strain can be expressed as the sum of its two components the elastic shear strain eliminated using Eq[ "11#[ This gives a modi_ed yield function in terms of the stress and plastic strain G 1 "sij ¦1nR sd ijP #"sij ¦1nR sd ijP # "G−snR # 1 "sm¦snI # 1

−0  9[

"14#

3[ Comparison with experimental observations Experimental data "Nemat!Nasser and Takahashi\ 0873#\ for the stress and volume behaviour\ from a shear test was used to _nd values of the _ve constants used within the model[ By choosing appropriate values of these constants "Material A in Table 0# we were able to obtain good agreement between the model and the experimental data for both the shear stressÐshear strain behaviour\ and the shear strainÐvolume strain response "Fig[ 0#[ A more complex experiment to model is a circular loading test[ In this an initial axial compression is followed by forcing the three principal stresses to follow a circular path\ while keeping the pressure constant[ Part of this stress path is shown in Figs 2a

S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 0394

Fig[ 2[ The stress path "a# and strain path "b# projected onto the constant pressure or volume plane in principal space\ showing how the yield surface develops with strain\ for a material with no elastic deformation[

and b[ The intersection of the yield surface with the constant pressure plane gives a yield circle of centre 1snR d ijT [ This intersects the stress path at the current stress state A[ Note that for a rigid plastic material the tangent to the strain path at this point is a vector which is normal to the yield circle at A[ Figure 3 shows experimental "Bianchini et al[\ 0875# and simulated results based on our Material B "see Table 0#[ All the basic features of the deformation can be reproduced[ 4[ Discussion and conclusions The model presented here incorporates\ into a continuum description\ some of the real microstructural features of granular materials[ In particular\ we introduce the

0395 S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397

Fig[ 3[ The principal strains and volume strain plotted against the angle of stress rotation\ f "see Fig[ 2#\ for Material B in Table 0[

idea that granules\ at least approximately\ retrace their individual displacements and rotations when gross deformation of the assembly is reversed[ This explains the high level of reversibility seen in the volume changes induced by a cyclic shear strain

S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 0396

imposed on the material[ Small deviations from this reversibility seem to produce small incremental reductions in the volume of the assembly\ which after many cycles can accumulate to give a much denser assembly[ It has long been known that volume changes\ along with frictional dissipation\ give a pressure sensitive yield surface[ However the partial reversibility incorporated here produces a type of kinematic hardening\ and consequently gives a physical basis for the Bauschinger e}ect seen in experiments on granular assemblies[ This strain induced anisotropy arises from a basic formulation which is entirely isotropic[ A special feature of the model is that the elastic response of the granules has been included in the dilation rule[ This not only avoids the prediction of jumps in the volume shear strain response seen in the results of Houlsby "0881#\ but gives some pointers to the e}ects of prior plastic deformation on elastic wave speeds "Hicher\ 0885#[ Perhaps the major achievement\ considering the model has only _ve adjustable parameters\ is the ability to reproduce the results of the circular test shown in Fig[ 3 which has been di.cult to achieve in the past without large numbers of adjustable parameters[ The model at present has some limitations[ At large and very small levels of shear strain the model does not give realistic results[ At large strains the model over predicts the volume increase\ though the inclusion of a critical state may reduce this\ allowing nR and nI to change with the density of the packing[ At the other end of the scale\ small strains merely produce reversible deformation\ and this does not appear to be the case in practice[ It is di.cult to see\ at the moment\ how this can be easily modelled[ Although there is still much to do\ this work achieves a great deal with such a simple basis and will form a foundation for further work incorporating plastically deforming granules[

References Bianchini\ G[\ Puccini\ P[\ Saada\ A[\ 0875[ Test results[ In] Saada\ A[\ Bianchini\ G[ "Eds[#[ Constitutive Equations for Granular Non!Cohesive Soils[ A[A[ Balkema\ pp[ 78Ð86[ Chandler\ H[W[\ 0874[ A plasticity theory without drucker|s postulate\ suitable for granular materials[ J[ Mech[ Phys[ Solids[ 22\ 104Ð115[ Chandler\ H[W[\ 0889[ Homogeneous and localised deformation in granular materials] A mechanistic approach[ Int[ J[ Eng[ Sci[ 17\ 608Ð623[ Drucker\ D[C[\ Prager\ W[\ 0841[ Soil mechanics and plastic analysis in limit design[ Q[ Appl[ Math[ 09\ 046[ Hicher\ P[Y[\ 0885[ Elastic properties of soil[ J[ Geotech[ Engng[ Div[ ASCE 011\ 530Ð536[ Houlsby\ G[T[\ 0881[ Interpretation of dilation as a kinematic constraint[ In] Kolymbas\ D[ "Ed[#[ Pro! ceedings of a Workshop on Modern Approaches to Plasticity[ Horton\ Greece\ 01Ð05 June 0881[ Elsevier Science\ pp[ 08Ð27[ Mroz\ Z[\ 0856[ On the description of anisotropic hardening[ J[ Mech[ Phys[ Solids[ 04\ 052Ð064[ Nemat!Nasser\ S[\ Takahashi\ K[\ 0873[ Liquefaction and fabric of sand[ J[ Geotech[ Engng[ Div[ ASCE 009\ 0180Ð0295[

0397 S[A[ Nixon\ H[W[ Chandler:Journal of the Mechanics and Physics of Solids 36 "0888# 0286Ð0397 Nova\ R[\ Wood\ D[M[\ 0868[ A constitutive model for sand in triaxial compression[ Int[ J[ Num[ Anal[ Meth[ Geomech[ 2\ 144Ð167[ Pande\ G[N[\ Pietruszczak\ S[\ 0875[ A critical look at constitutive models for soils[ In] Dungar\ R[\ Studer\ J[A[ "Eds[#[ Geomechanical Modelling In Engineering Practice[ A[A[ Balkema\ pp[ 258Ð284[ Reynolds\ O[\ 0774[ On the dilatancy of media composed of rigid particles in contact with experimental observations[ Phil[ Mag[ "Series 4# 19\ 358Ð370[ Spencer\ A[J[M[\ 0871[ Deformations of ideal granular materials[ In] Hopkins\ H[G[\ Sewell\ M[J[ "Eds[#[ Mechanics of Solids[ The Rodney Hill 59th Anniversary Volume[ Pergamon Press\ pp[ 596Ð541[ Stallebrass\ S[E[\ Taylor\ R[N[\ 0886[ The development and evaluation of a constitutive model for the prediction of ground movements in overconsolidated clay[ Geotechniq[ 36\ 124Ð142[ Wood\ D[M[\ 0871[ Laboratory investigations of the behaviour of soils under cyclic loading] A review[ In] Pande\ G[N[\ Zienkiewicz\ O[C[ "Eds[#[ Soil Mechanics*Transient and Cyclic Loads[ John Wiley and Sons\ pp[ 402Ð471[ Youd\ T[L[\ 0861[ Compaction of sands by repeated shear straining[ J[ Soil[ Mech[ Found[ Div[ ASCE 87\ 698Ð614[ Ziegler\ H[\ 0874[ An Introduction to Thermomechanics[ NorthÐHolland Publishing Company[