a simple elastoplastic hardening constitutive model for eps geofoam

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Geotextiles and Geomembranes 24 (2006) 299–310

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A simple elastoplastic hardening constitutive model for EPS geofoam

Henry Wonga, Chin Jian Leob,�

aLGM, DGCB (URA CNRS 1652), ENTPE, 2 rue Maurice Audin, 69518 Vaulx en Velin, Cedex, FrancebSchool of Engineering, University of Western Sydney, Locked Bag 1797 Penrith South DC, Sydney, NSW 1797, Australia

Received 11 November 2005; received in revised form 15 March 2006; accepted 21 March 2006

Available online 22 May 2006

Abstract

Expanded polystyrene (EPS) geofoam is increasingly being used as a construction material of choice in situations where its mechanical

properties—such as its extremely low density, volume contraction under deviatoric compressive loading, and existence of post-yielding

strain hardening—can be exploited. In this paper, a simple elastoplastic hardening constitutive model of EPS geofoam is formulated to

model the mechanistic behaviour of EPS geofoam taking into account the characteristic properties of EPS. The model is based on

experimental results from a series of triaxial tests performed on EPS samples for confining pressure ranging from 0 to 60 kPa at room

temperature (23 1C). Behaviour under higher temperatures is currently under investigation and will be addressed in a future publication.

The model has a total of six independent parameters and can be calibrated from data obtained from triaxial tests. It is shown that the

constitutive model is able to correctly replicate the characteristic behaviour of the EPS geofoam under shearing. The model is relatively

simple to incorporate into numerical codes for geotechnical analysis.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Eps; Constitutive model; Elastoplastic; Strain hardening; Construction; Triaxial tests

1. Introduction

The use of block-moulded expanded polystyrene (EPS)geofoam in geotechnical applications is not new. Thematerial has been successfully employed by Norwegiangeotechnical engineers since at least the early 1970s (e.g.,Refsdal, 1985; Aaboe, 1987) and examples of EPS geofoamapplications abound (e.g. Abu-Hejleh et al., 2003; Zouet al., 2000; Frydenlund and Aaboe, 1996; Magnan andSerratree, 1989; van Dorp, 1996; William and Snowdon,1990). In engineering construction, EPS geofoam isincreasingly being used as a material of choice to replaceother conventional materials in situations where theattributes of EPS can be utilized. For instance, as aconsequence of its extremely low density (normally rangingfrom 10 to 35 kg/m3), EPS geofoam makes an ideal backfillmaterial when the self weight of the construction materialis a major design consideration (e.g. embankment on verysoft soil as shown in Fig. 1(a)). Another attribute of EPS

e front matter r 2006 Elsevier Ltd. All rights reserved.

otexmem.2006.03.007

ing author. Fax: +612 98525741.

ess: [email protected] (C.J. Leo).

geofoam is that it experiences very small or virtually zerolateral expansion such that volume contraction occursunder deviatoric compressive loading, and that it inducessignificantly lower lateral loading than normal earthpressures (e.g. behind a bridge abutment in Fig. 1(b)). Itis envisaged that these attributes foreshadows the possibi-lity of EPS geofoam being used effectively as a compres-sible inclusion behind retaining structures. Thecompressible inclusion function of EPS geofoam wasthoroughly discussed by Horvath (1997).From an engineering design standpoint, it is of interest

to have an efficient constitutive model available as a meansto carry out design analysis on a rational and rigorousbasis. One of the key requirements for such an efficientconstitutive model is that the number of independentparameters must be reasonably small, and the model’sparameters can be estimated from routine and non-costlyexperimental tests. Above all, a good model should be ableto capture the essential characteristics of the materialbehaviour in the application for which the model isintended. A constitutive model is, therefore, normallydeveloped in the context of the particular area or areas of

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mailto:[email protected]

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(a)

(b)

Fig. 1. Examples of EPS applications: (a) lightweight embankment; (b)

bridge abutment.

H. Wong, C.J. Leo / Geotextiles and Geomembranes 24 (2006) 299–310300

its intended applications. Swart et al. (2001) presented aconstitutive model of EPS to investigate the non-linearresponse of a road pavement, utilizing an associated flowrule and a modified version of Gurson flow surface. Themodel was an attempt to provide a unified phenomelogicalapproach for materials exhibiting strain rate dependentplastic deformations incorporating the effects of strainhardening/softening. A specific interest of their researchwas to simulate cyclic behaviour of pavement structuresexperienced under traffic loading. Developed within theframework of dynamic plasticity, the model is, naturally,quite complicated to apply. Hazarika and Okuzono (2004)utilized a non-linear elastoplastic hardening constitutivemodel of EPS geofoam with von Mises yield criterion toinvestigate the behaviour of a hybrid interactive systeminvolving soil, structure and EPS geofoam. The modelignored possible effects of mean stress and made theassumption that the hardening regime follows a hyperboliccurve. The objective of the authors was to developsomething simple with the intention of using it specificallyto model the interactive behaviour of EPS compressibleinclusion behind retaining structures where large strainbehaviour is expected. Hazarika (2006) later extended theconstitutive model to include the size and shape factors ofthe tested specimen as well as the density of the geofoam,arguing that both the shape and absolute dimensions oftest specimens influence the stress–strain behaviour, andhence the material’s compressive strength. Chun et al.(2004) developed a hyperbolic constitutive model as afunction of the density and confining stress, and which isbased on the popular hypoelastic Duncan–Chang model(Duncan and Chang, 1970) for soils. Though it is shown tofit the measured test data, the model does not contain anexplicit yield criterion nor is strain hardening formallyincorporated as required under notions of classicalplasticity. The model is applicable to unique monotonic

loading but would not be considered suitable for problemswhere complex loading paths and stress-reversal occur.The motivation of this paper is to develop a simple

elastoplastic hardening constitutive model with the possi-bility of a fairly wide range of geotechnical applications.The model is formulated in a way that ensures the essentialcharacteristics of EPS geofoam, namely, the strain andvolumetric response under shearing, can be correctlycaptured. It is suggested that a main appeal of the modelis that it is not only simple to calibrate (with sixindependent parameters) but that it is also relatively easyto incorporate into numerical codes for geotechnicalanalysis. The model is developed within the framework ofclassical plasticity, with the inclusion of strain hardening inorder to correctly describe the post-yield behaviour. On theother hand, time dependent viscous behaviours (creep) arenot accounted for in the present development. Previousresearchers (e.g., Chun et al., 2004; Hazarika, 2006) havediscussed the context for including visco-elastic or visco-plastic behaviour of EPS, and Horvath (1998) has proposedthe application of the Findley model (Findley et al., 1989) toaccount for visco-elastic time-dependent effects. In general,however, the importance of time dependent creep in EPSdepends on such factors as the stress or strain level, thetemperature and the duration of loading. Provided thestrains are not exceptionally large (i.e. before the stress–strain curve turns sigmoidal and rapid hardening occurs(Hazarika, 2006)), then under normal operating temperature(typically less than 30 1C) and under a relatively short-termloading, creep effects can generally be neglected. The currentmodel, therefore, excludes application in situations whereviscosity effects are important under a combination of highstress (or strain), high temperature and sustained loading fora long period of time. Even under cases where the creep maynot be entirely negligible, this simple model can still serve asa useful tool for an order-of-magnitude-estimate for thedesigner to make appraisal of results obtained by other morecomplex means. Investigations on temperature-dependenteffects are currently being undertaken and will be addressedin a future publication. Also, the model in its current formdoes not incorporate the factors that account for thespecimen size and shape as proposed by Hazarika (2006)but these can be included in future with no difficulty.This paper is organized as follows. In the following

paragraph experimental results obtained from triaxial testsare described. This will be followed by the presentation ofthe proposed theoretical model and its casting in thetriaxial configuration. This is not a trivial development onaccount of the corner-flow condition, since the stress statelies on a singular point of the yield surface where thegradient is not uniquely defined. The model will then bevalidated using the experimental data obtained.

2. Triaxial tests

The compressive behaviour of EPS geofoam wasinvestigated through a series of triaxial tests conducted at

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GDS pressure-volume controller

PC

A/Dconverter

Com2

Com1EPS

LoadcellLVDT

Triaxialcell

Fig. 2. Set-up of triaxial test system.

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16 18

Axial Strain (%)

Dev

iato

r S

tres

s (k

Pa)

0 kPa

40 kPa

20 kPa

60 kPa

Fig. 3. Deviator stress vs. axial strain for various confining stress. The

experimental curves are shown in full lines. The model results are

represented in broken lines.

H. Wong, C.J. Leo / Geotextiles and Geomembranes 24 (2006) 299–310 301

room temperature (23 1C). Shown in Fig. 2 is the schematicof the test set-up.

2.1. Test specimen

Cylindrical specimens with a diameter of 50mm anda height of 50mm (aspect ratio of 1:1) were cut from a20 kg/m3 density prismatic EPS block supplied by RMAXCellular Plastics, an Australian manufacturer of EPS.The ends of the EPS specimens were smoothed bysand papering to reduce end friction. A 1:1 specimen wasconsidered less likely to give rise to column-type buckling,which was observed in some cases of 2:1 specimens. Zouand Leo (1998) observed that provided care was taken toensure that the end contacts were sufficiently smooth the1:1 specimens did not result in end restraint problems.

2.2. Test apparatus

A 50 kN TRITECH digital loading frame capable of aloading speed ranging from 0.0 to 6.0mm/min was used inthe tests. The platen supplied rate was set through a precisethumb wheel selector with an accuracy controlled to betterthan 1%. In this series of experiments, a relatively slowloading rate of 0.4% or 0.2mm/min was adopted. Thetriaxial compression was performed within a WykehamFarrance triaxial cell (WF 10201), which can withstand aninternal cell pressure up to 1700 kPa.

The applied loading and the vertical displacement weremeasured and recorded during the tests by means of a loadcell and an LVDT transducer, through an in-housedeveloped A/D data logger and software. The load celland the LVDT were capable of resolutions as small as 1Nand 0.01mm, respectively.

A GDS digital pressure/volume controller accurate tovolume measurement of 1mm3 and pressure to 1 kPa wasused to control the cell pressure (via the software) and tomeasure any volume changes in the cell fluid during thetest. The cell fluid volume change was subsequently used toderive the volume changes of the specimen.

The tests were carried out over a range of confiningpressure from 0 to 60 kPa. The samples were initiallyloaded hydrostatically to the prescribed confining pressureuntil the volume change was stabilized and then sheared byapplying the axial load. Throughout the duration of thetest, the ‘‘pore pressure’’ drainage valves were kept open toallow the gas in cellular structure of the EPS specimento dissipate. The testing procedures are generally similar tothose described by Atmazidis et al. (2001).

2.3. Test results

Shown in Fig. 3 are the experimental plots (full lines) ofthe deviator stress against the axial stress under confiningpressure ranging from 0 to 60 kPa. Corresponding to eachconfining pressure, the initial part of the curve is fairlylinear and its slope is defined by the initial modulus E.After yielding, the slope reduces and asymptotes quickly tothe plastic modulus Ep. The response can therefore bereasonably approximated by a bilinear relationship definedby the moduli E and Ep, corresponding to pre- and post-yielding, respectively. Ep is also markedly smaller than E.

ARTICLE IN PRESSH. Wong, C.J. Leo / Geotextiles and Geomembranes 24 (2006) 299–310302

Fig. 4 shows the volume change of the EPS specimen(full lines) during the axial loading stage, when the deviatorstress was applied. In this investigation, as well as in others(e.g. Atmazidis et al., 2001; Zou and Leo, 1998), thePoisson ratio was found to be a small positive value duringthe initial stages of loading when strains were very small,then quickly reducing to virtually zero or a negative valueat larger strains (Poisson’s ratio does not strictly apply inthe plastic strain). These results are apparent in the shapeof EPS specimens after being subjected to compressionloading in the picture shown in Fig. 5. As a consequence ofthis, the EPS specimen exhibited contractive volumetricbehaviour during the axial loading, since the axialvolumetric contraction due to applied loading was notcompensated by lateral volumetric expansion.

The plot in Fig. 6 shows the variation of the majorprincipal stress s1 against the minor principal stress s3when the specimen is at yield. In these series of tests, theresults showed the major principal stress had a slightnegative dependency on the minor principal stress, i.e. themajor principal stress at yield decreased slightly as theminor principal stress was increased.

0

5000

10000

15000

20000

25000

30000

0 2 4 6 8 10 12 14 16 18

Axial Strain (%)

Vo

lum

e C

han

ge

(mm

3 )

60 kPa

40 kPa

20 kPa

0 kPa

Fig. 4. Volume change vs. axial strain for various confining stress. Full

lines represent experimental data while the broken lines are the model

results.

Fig. 5. EPS geofoam at different levels of compression. Note the latera

3. Model presentation

Prior to the description of our proposed model, let usfirst recall some fundamental equations in classic elasto-plasticity which will be of use later.

3.1. Some fundamental equations in classic elastoplasticity

theory

For simplicity, all matrix and vector quantities are inbold. Positive stresses and strains correspond to compres-sion and fibre shortening. For small displacements andsmall strains, the total strain can be written as the sum ofelastic and plastic strains:

de ¼ dee þ dep, (1)

where the vectorial notation r ¼ s11 s22 s33 s12 s23 s13½ �t

and e ¼ [e11e22e33e12e23e13]t has been used. The material

behaviour is linear elastic, with zero irreversible strain(dep ¼ 0), when the stress tensor lies inside a yield surface,defined by a yield function f:

f r;Rð Þ ¼ 0, (2)

where R is a variable which defines the current size of theyield surface in order to account for hardening effects. The

l contraction of the specimens under vertical compressive loading.

40

60

80

100

120

0 10 20 30 40 50 60 70

σ3 (kPa)

σ 1 (

kPa)

Fig. 6. Plot of major principal stress vs. minor principal stress at yield.

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(a)

x2

x3

A

C

F

D

B

EO

60°

A

C

F

D

B

E

O

(b)

Fig. 7. Projection of the yield surface on the p-plane (deviatoric plane) forslightly (a) positive and (b) negative Kp.


stress increment is related to the strain increment throughthe classical relation:

dee ¼ D�1 � dr, (3)

where D is the elastic stiffness matrix and is constant forlinear behaviour. For clarity, we will use a dot (‘‘�’’)operator to denote matrix–matrix and matrix–vectormultiplications. Onset of (irreversible) plastic strains beginswhen the yield surface is reached. For a hardeningbehaviour, the yield criterion will depend on an additionalvariable R which reflects the current level of strength onaccount of hardening effects. Only isotropic behaviour isconsidered here and R is supposed to be a scalar. Inaddition to the yield function, a flow rule is also needed todescribe plastic strains:

dep ¼ dzqg

qr, (4)

where z is the (positive) plastic multiplier and g theplastic potential. The stress increment is still related tothe elastic strain increment by (3). During plastic loading,the stress point must remain on the surface of the yieldenvelope. This is known as the consistency condition,which writes:

df ¼qf

qr� drþ

qf

qRdR ¼ 0. (5)

For simplicity, we shall denote:

F ¼qf

qr; G ¼

qg

qr. (6)

Classically, the hardening modulus H is defined by therelation:

qf

qRdR ¼ �Hdz. (7)

There are several ways to define the evolution of R and H,commonly called the hardening behaviour. It is supposedthat R is related to an internal parameter x, which is itselfrelated in some way to the plastic multiplier z. Eq. (7) canthen be developed to give:

H ¼ �qf

qR

dR

dx

dx

dz. (8)

The relations R ¼ R(x) and x ¼ x(z) define completely thehardening behaviour adopted, which is generally a keyelement of the specific model developed. In developing thesimple analytical model, we will suppose x to be aninvariant of the plastic strain. Hence, the strength leveldepends on the plastic strain, and the resulting model iscalled a strain-hardening model. This point will be furtherdeveloped in the next paragraph.

Substitution of (3) and (4) into (1), then pre-multiplyingby D gives:

D � de ¼ drþ dzD �qg

qr. (9)

Pre-multiplication by Ft ¼ qf =qr� �t

, and on account of (5)and (7):

dz ¼Ft �D � de

H þ Ft �D �G, (10)

where notation (6) has been used. Substitution of (10) into(9) leads to:

dr ¼ D�D �G � Ft �D

H þ Ft �D �G

� �� de ¼ Dep � de, (11)

where Dep is the classical tangent stiffness matrix.

3.2. The EPS model

In constructing the simple analytical model, it will besupposed the elastic behaviour is linear and isotropic,defined by two rheological constants E and n. For theplastic behaviour, based on experimental observations, thefollowing yield function will be adopted which is formallyidentical to the well-known Mohr–Coulomb criterion:

f ðr;RÞ ¼ s1 � Kps3 � R ¼ 0, (12)

where Kp is a rheological constant. For an elastic-perfectlyplastic behaviour, R would be constant and becomesanother rheological parameter, known as the unconfinedcompressive strength. However, for a strain-hardeningbehaviour developed here, R is variable and is to beidentified with the hardening variable in the precedingparagraph and it will be made to depend on the currentplastic strain. This hardening behaviour will be addressedafter the flow rule. At present, note the importantdeparture from the classic Mohr–Coulomb model in thattriaxial tests suggest that Kp is generally close to zero, andcan even be slightly negative in some tests. Hence, in thepresent context, (12) is taken to be a phenomenologicalrelation, and Kp is no longer related to any non-negativelydefined internal friction angle. Fig. 7 below shows a typicalsection of the yield envelope in the deviatoric plane (or thep-plane). Justifications are briefly presented in Appendix B.To define the plastic strain rate, the following form for theflow potential g is adopted:

gðrÞ ¼ s1 � Ks3, (13)


where s14s24s3 are the ordered principal stresses, whileK40 is a rheological constant which governs the dilatancybehaviour. When the stress state lies at one of the corners(‘‘corner flow’’ condition), the flow direction will appar-ently be undefined. In fact, it has been shown in the pastthat even under such cases, the problem is still well defined(see for example, Wong and Simionescu, 1996; Berest,1989). Numerical computations without recourse to specialalgorithmes still work, with the stress state oscillating fromone face to another of the yield surface. For negative valuesof Kp, the yield surface will appear to be non-convex andmay lead to unwanted results for a general loading paths.Strictly speaking, the Mohr–Coulomb model should onlybe applied to trixial loading paths where all subsequentdevelopments will remain valid. The Drucker–Pragercriterion, which gives identical results for trixial stresspaths (Appendix C), can in this case be used since it doesnot have singularities nor non-convexity problems.

For the hardening law, it is supposed that R varieslinearly with the internal variable x:

RðxÞ ¼ R0 þ bx. (14)

In our model, x is identified as the equivalent deviatoricplastic strain tensor, defined by:

x ¼ epq ¼Z t

0

depq ; depq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 de

pð Þt� dep

q,

dep ¼ dep � 13depv I, ð15Þ

where I ¼ 1 1 1 0 0 0� �t

and depv ¼ It � dep is thevolumetric plastic strain.

Attention is now fixed onto the case of triaxial tests—theonly configuration treated in this paper. In this idealgeometric configuration, a fully analytical model will bedeveloped, which will allow an easy validation usingtriaxial test results.

3.3. Analytical model for EPS behaviour under triaxial tests

In the system of principal directions (Fig. 8), only thediagonal terms of the stress and strain tensors are non-zero.For mathematical convenience, we revert to the vectorialnotation for the following:

r ¼ s1 s2 s3� �t

; e ¼ e1 e2 e3� �t

; I ¼ 1 1 1� �t

,

(16)

σ1

σ3σ2

x1

x2x3

Fig. 8. Triaxial test configuration.

with s2 ¼ s3 and e2 ¼ e3. The elastic behaviour is definedby

dr ¼ D � de; D ¼

lþ 2m l l

l lþ 2m l

l l lþ 2m

264

375, (17)

where l and m are the classical Lame’s constants, related toE and n by

l ¼nE

1þ nð Þ 1� 2nð Þ; m ¼

E

2 1þ nð Þ. (18)

On account of the corner flow condition: s14s2 ¼ s3, weuse, instead of (6):

F ¼1

2

qf 12

qrþ

qf 13

qr

� with f ij ¼ si � Kpsj � R, (19)

to evaluate the consistency condition:

df ¼ Ft � drþqf

qRdR ¼

1

�Kp=2

�Kp=2

8><>:

9>=>;

t

�

ds1ds3ds3

8><>:

9>=>;þ

qf

qRdR ¼ 0,

(20)

while the symmetry of plastic strain rates dep2 ¼ dep3 impliesthe following two equations:

G ¼1

2

qg12

qrþ

qg13

qr

� with gij ¼ si � Ksj, (21)

dep ¼ dz G ¼ dz

1

�K=2

�K=2

8><>:

9>=>;. (22)

The constant K governs the plastic volumetric strain since:

depv ¼ dep1 þ dep2 þ dep3 ¼ ð1� KÞ dz, (23)

where positive volumetric strain (Ko1) implies contrac-tancy and negative volumetric strain (K41) dilatancy.Combining (15), (22) and (23), it can easily be shown that:

dx ¼ depq ¼K þ 2

2ffiffiffi3p dz, (24)

H ¼ �qf

qR

dR

dx

dx

qz¼

bðK þ 2Þ

2ffiffiffi3p . (25)

The tangent matrix Dep ¼ D�Dp, where Dp ¼ ðD �G � Ft �

DÞ=ðH þ Ft �D �GÞ can now be evaluated analytically inthis particular case. After some lengthy calculations whichcan be performed by Mathematica or Maple, the followingexplicit results are found:

de ¼ Sep � dr; Sep ¼ D�D �G � Ft �D

H þ Ft �D �G

� ��1, (26)

where the tangent flexibility matrix Sep is the inverseof the tangent stiffness matrix Dep. It admits the following

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explicit expression:

Sep ¼1

mð3lþ 2mÞ

lþ m �l=2 �l=2

�l=2 lþ m �l=2

�l=2 �l=2 lþ m

2664

3775þ

ffiffiffi3p

2b K þ 2ð Þ

�

4 �2Kp �2Kp

�2K KpK KpK

�2K KpK KpK

2664

3775. ð27Þ

Eqs. (17) and (26) completely describe the triaxialbehaviour of an EPS sample under monotonic loading.Suppose an EPS sample is subject to the following classicaltriaxial stress path shown in Fig. 9. We will take point Q tobe the reference point (at the end of an isotropicprestressing) for strain measurements. It can then beshown that the stress and strain behaviour under triaxialtests is basically bilinear (Fig. 10).

The elastic–plastic stiffnesses me and mp, as well as thevolumetric deformation ratios mve and mvp can now bedetermined. Since dr ¼ ds1 0 0

� �t, the elastic beha-

viour yields:

me ¼qs1qe1

� QA

¼1

Se11

¼m 3lþ 2mð Þ

lþ m¼ E, (28)

mve ¼qev

qe1

� QA

¼Se11 þ Se

21 þ Se31

Se11

¼m

lþ m¼ 1� 2n.

(29)

The first yield stress (point A) depends on the preconsolidation pressure s0, hence:

s1y ¼ Kps0 þ R0. (30)

σ1

ε1

σ0

σ1y

me

mpA

Q

B

σ1−σ3

σ1y−σ0

me

mpA

Q

Fig. 10. Triaxial behaviour of EP

σ1

σ2=σ3

σ0

σ0

BQ

O

A

Fig. 9. Triaxial stress-path.

For elastic–plastic behaviour beyond point A, similarcalculations give:

mp ¼qs1qe1

� AB

¼1

Sep11

¼

ffiffiffi3pð2þ KÞbm 3lþ 2mð Þffiffiffi

3pð2þ KÞb lþ mð Þ þ 6m 3lþ 2mð Þ

¼1

1Eþ 2

ffiffi3p

bð2þKÞ

¼1

1Eþ 1

H

, ð31Þ

mvp ¼qev

qe1

� AB

¼Sep11 þ S

ep21 þ S

ep31

Sep11

¼ m

ffiffiffi3pð2þ KÞbþ 6ð1� KÞ 3lþ 2mð Þffiffiffi


. ð32Þ

Note that under triaxial conditions, Drucker–Prager yieldcriterion will give similar results as the Mohr–Coulombcriterion (see Appendix C). Their results differ howeverunder general stress states. Only experimental results candecide which criterion is more precise.

4. Experimental determination of the constitutive constants

This model presented above depends on six rheologicalparameters (i.e. constitutive constants):

E; n (or equivalently l, m) : define the isotropic elasticbehaviour,

Kp ; R0 : define the first yield stress s1y,b : defines the hardening rate,K : defines the flow rule and the dilatancy behaviour.Note that owing to Eq. (25) relating the hardening

parameter H to b and K, only two of them are independentparameters. Note also that the preconsolidation stress s0—which appears in the first yield stress s1y—is an ‘‘external’’parameter, controllable by the experimental operator, andis not a constitutive constant. A means to determine theseconstants can be the following. The elastic stiffnesses me

and mve can be directly read off from an experimental curveto determine the elastic constants E and n. The constantsR0 and Kp which define the first yield stress can bedetermined graphically by plotting the first yield stress s1y

against the preconsolidation stress s0. Given thats1y ¼ R0+Kps0, therefore Kp and R0 are simply the slopeand the y-intercept in such a graph. From the plasticstiffness mp and the Young’s modulus E previously

εv

ε1

mve

mvp

Q

A

B

ε1

B

S as predicted by the model.

ARTICLE IN PRESS

5000

10000

15000

20000

25000

30000

lum

e C

han

ge

(mm

3 )

60 kPa

40 kPa

20 kPa

0 kPa

-0.7

-0.5

-0.3

-0.1

0.1

0 10 20 30 40 50 60

�3 (kPa)

K

Fig. 11. Dependence of dilatant coefficient K on lateral confinement s3.

H. Wong, C.J. Leo / Geotextiles and Geomembranes 24 (2006) 299–310306

determined, the hardening mudulus H can be calculatedusing (31). For b and K, observe firstly that from theexpression of the plastic stiffness mp, we can deduce:

bðK þ 2Þ ¼2ffiffiffi3p

1=mp � 1=E¼ 2

ffiffiffi3p

H. (33)

Hence, the product b(K+2) is known. Now, using theplastic volumetric relation on mvp, we can deduce anexpression of (1�K), hence that of K:

K ¼ 1�mvpH

mlþ m

3lþ 2mþ 1

� �þ

H

3lþ 2m. (34)

Experimentally, it has been observed that axial stressincrements induce little lateral strains. Moreover, para-metric studies show that Poisson’s ratio has little influenceon the volumetric strain variation. We therefore take:

v ¼ 0 (35)

This particular value implies ee3 ¼ 0 (i.e. zero radial elasticstrain) in triaxial compression under constant lateral confin-ing pressure. Hence we have eevol ¼ ee1 (elastic volumetricstrain equals the elastic axial strain) and mve ¼ 1� 2n ¼ 1.

Under this hypothesis, the expression of K abovesimplifies to:

K ¼ 1�mvp

� �1þ

H

E

� . (36)

5. Model verification

We observe that the initial slope is more or less constantfor all cases, this condition giving the young’s modulus:E ¼ 3950kPa. We have taken n ¼ 0 as mentionned above.The axial stress versus axial strain on the post-yield portionalso shows little dependence on the initial isotropic prestress,it’s slope is found to be mvpE110kPa. This also leads toHE113kPa. The variation of the first yield stress s1y leads tothe determination of R0 and Kp: R0 ¼ 98kPa, Kp ¼ �0.17. Itremains to determine the constants b and K. The experi-mental curves on the volumetric variation show a slightdependence of K on the lateral pressure. This leads to a littlecomplication on the determination of K and b. As a firstapproach, we may take an average value (corresponding to alateral confinement of 20kPa) of K ¼ �0.25 andb ¼ 225kPa. To summarize, estimated values of the modelparameters for the 20kg/m3 EPS are presented in Table 1.

It is observed that K is a small negative value suggestingthat the material is clearly plastically contractant duringpost-yield. The yield surface on the p-plane (deviatoric

Table 1

Estimated values of model parameters

E (kPa) n R0 (kPa) Kp b (kPa) K

3950 0 98 �0.17 225 �0.25

plane) is shown in Fig. 7(b) for the case when Kp is slightlynegative.The results of the deviatoric axial stress versus the axial

strain from the model for confining pressure from 0 to 60kPaare plotted in broken lines in Fig. 3. It is observed that themodel is able to reproduce the experimental behaviour undershearing for the full range of confining pressure, but perhapsa little better at the lower confining pressures. The replicationis still good at 60kPa at the higher strain values, though thereis some deviation from experimental data, because of thepresence of non-linearity, at small strains.In Fig. 4, the volume changes under shearing predicted

by the model are again shown in broken line (the results arevery similar for all confining pressures). The model gives avolume change which corresponds more to the averagevolume change for the confining stresses but in general, thetrend of the shear-contractive volumetric behaviour of EPSgeofoam is correctly modelled.To improve the volumetric strain prediction, we can

include the dependence of contractancy/dilatancy on theconfining stress. Fig. 11 shows the variation of K against s3deduced from experimental results in Fig. 3, as well as alinear approximation. This linear approximation leads toan improved prediction on the volumetric strain, as shownin Fig. 12.

00 10 12 14 16 18

Axial Strain (%)

Vo

8642

Fig. 12. Volume change vs. axial strain for various confining stress. Full

lines represent experimental data while the broken lines are the model

results. The theoretical predictions account for the stress dependence of

the dilatation coefficient K, contrary to Fig. 4.

ARTICLE IN PRESSH. Wong, C.J. Leo / Geotextiles and Geomembranes 24 (2006) 299–310 307

Finally, for n ¼ 0, we have dee3 ¼ 0, hence de3 ¼ dep3 ¼�K dz=2 according to Eq. (22), with dz40. In other words,during a triaxial compression, radial contraction (resp.expansion) will occur for Ko0 (resp. K40). From Fig. 11,it appears that radial contraction is the rule except for verysmall confining stress. This appears to be consistent withvisual observation on stressed samples (observe the in-curving lateral surface in Fig. 5).

6. Conclusions

A simple elastoplastic hardening constitutive model forEPS geofoam is developed and presented in this paper. Aseries of ‘‘drained’’ triaxial tests was carried out at roomtemperature conditions (23 1C) to study the behaviour ofEPS geofoam under shear based upon which the constitu-tive model was developed. The model has been shown tocorrectly predict the response of the material undershearing in terms of deviatoric stress-axial strain behaviourand the shear-contraction behaviour. It is suggested thatthis model will be useful in modelling the behaviour of EPSgeofoam in a wide variety of geotechnical applications. Thebehaviour at high operating temperatures, stress levels andlong-term loading situations, where time dependent creepbehaviour may become more important, is not addressed inthe present study. Investigations are currently beingundertaken to study temperature effects on EPS and thefindings will be presented in a future publication. Thus theapplicability of this model is currently restricted to normaloperating temperature (typically less than 30 1C) and strainlevels below the ‘‘sigmoidal’’ strain (i.e. before rapid strainhardening occurs), and for loading of relatively shortduration. In its current form the model has not includedthe size and shape factors of the test specimens as suggestedby Hazarika (2006) but these can be included later on toimprove its applicability.

Acknowledgements

The authors gratefully acknowledge the generous dona-tion of the EPS material used in the experimental tests byRMAX Cellular Plastics, Australia.

Appendix A. Reduced matrix calculations in the case of

triaxial test

To take the maximum advantage of the symmetryin a triaxial test, the stresses and strains, as well as thegradients F and G can be represented by two-componentsvectors:

dr ¼ds1

ds3

( ); de ¼

de1

de3

( ); F ¼

1

�Kp

( ),

G ¼1

�K=2

( ). ðA:1Þ

The gradient F is defined in such a way that the scalarproduct Ftds remains unchanged. The elastic stiffnessmatrix now becomes:

D ¼lþ 2m 2l

l 2ðlþ mÞ

" #¼

k þ 4G3

2k � 4G3

k � 2G3

2k þ 2G3

" #. (A.2)

It can be verified that all the previous formulae remainvalid. In particular:

de1 ¼

D�1� �

11� ds1 if f ðr;RðzÞÞo0 or dfo0

i:e: elastic behaviour or unloading;

D�1ep

�11� ds1 if f ðr;RðzÞÞ ¼ 0 and df ¼ 0

i:e: plastic loading:

8>>>>>><>>>>>>:

(A.3)

Hand calculations are now possible since the matrices areonly 2� 2. Inversion of the elastic and elastic–plasticstiffness matrices gives:

Se ¼ D�1 ¼1

2m 3lþ 2mð Þ

2 lþ mð Þ �2l

�l lþ 2m

" #, (A.4)

Sep ¼ D�1ep ¼

ffiffi3pð2þKÞb lþmð Þþ6m 3lþ2mð Þffiffi

3pð2þKÞbm 3lþ2mð Þ

�ffiffi3pð2þKÞblþ6Km 3lþ2mð Þ

2ffiffi3pð2þKÞbm 3lþ2mð Þ

2664

�

�

ffiffi3pð2þKÞblþ6Kpm 3lþ2mð Þffiffi

3pð2þKÞbm 3lþ2mð Þffiffi

3pð2þKÞb lþ2mð Þþ6KKpm 3lþ2mð Þ

2ffiffi3pð2þKÞbm 3lþ2mð Þ

3775, ðA:5Þ

which can be decomposed into two parts:

Sep ¼ Se þ Sp ¼1

2m 3lþ 2mð Þ

2 lþ mð Þ �2l

�l lþ 2m

" #

þ

ffiffiffi3p

bð2þ KÞ

2 �2Kp

�K KpK

" #. ðA:6Þ

Note that Sp can also be written in the following form:

Sp ¼2ffiffiffi3p

bð2þ KÞ

1 �Kp

�K=2 KpK=2

" #¼

1

HG � Ft. (A.7)

The slopes of the stress-strain curve are of course identicalto the previous calculations:

me ¼qs1qe1

� e

¼1

Se11

¼m 3lþ 2mð Þ

lþ m¼ E

under elastic behaviour, ðA:8Þ

mp ¼qs1qe1

� p

¼1

Sep11

¼

ffiffiffi3pð2þ KÞbm 3lþ 2mð Þffiffiffi


¼1

1Eþ 2

ffiffi3p

bð2þKÞ

¼1

1Eþ 1

H

. ðA:9Þ


Appendix B. Study of convexity of the Mohr–Coulomb

criterion

We will show briefly in this appendix the calculationswhich prove the shape of the yield surface. We begin bydefining a second system of coordinates (x1, x2, x3) with x1

corresponding to the hydrostatic axis and (x2, x3) to thedeviatoric plane (p-plane) (Fig. B.1).

B.1. p-plan and coordinate transformation

The two systems of coordinates, (s1, s2, s3) and (x1, x2,x3), are related by:

x1

x2

x3

0B@

1CA ¼

1=ffiffiffi3p

1=ffiffiffi3p

1=ffiffiffi3p

2=ffiffiffi6p

�1=ffiffiffi6p

�1=ffiffiffi6p

0 1=ffiffiffi2p

�1=ffiffiffi2p

264

375

s1s2s3

0B@

1CA;

s1s2s3

0B@

1CA ¼

1=ffiffiffi3p

2=ffiffiffi6p

0

1=ffiffiffi3p

�1=ffiffiffi6p

1=ffiffiffi2p

1=ffiffiffi3p

�1=ffiffiffi6p

�1=ffiffiffi2p

264

375

x1

x2

x3

0B@

1CA:

(B.1)

Notice that:

x1 ¼s1 þ s2 þ s3ffiffiffi

3p ¼

I1ffiffiffi3p (B.2)

and that we are only interested in positive (compressive)stresses, so that s1, s2, s3 and I1 are all positive quantities.The p-plane is defined by

s1 þ s2 þ s3 ¼ I1 ¼ffiffiffi3p

x1 ¼ constant. (B.3)

B.2. EPS yield surface

The EPS yield criterion writes:

sI � KpsIII � R ¼ 0 with sI4sII4sIII the ordered

principal stresses. ðB:4Þ

On the face AF (Fig. 5), we have s14s34s2, hence stresseson AF satisfy:

s1 � Kps2 � R ¼ 0. (B.5)

σ1

σ3σ2

x2

x3x1

Fig. B.1. Definition of the coordinates (x1, x2, x3). x1 corresponds to the

hydrostatic axis and (x2, x3) to the deviatoric plane (p-plane).

Substitution of (B.1) into (B.5) yields:

1ffiffiffi3p x1 þ

2ffiffiffi6p x2

� � Kp

1ffiffiffi3p x1 �

1ffiffiffi6p x2 þ

1ffiffiffi2p x3

� � R ¼ 0.

On account of (B.2), we arrive at the equation ofline AF:

AF :Kp þ 2ffiffiffi

6p x2 �

Kpffiffiffi2p x3 ¼ Rþ Kp � 1

� � I1

3, (B.6)

where the coordinates of A can easily be found:

A : xA2 ¼Rþ ðKp � 1ÞI1=3

Kp þ 2� �

=ffiffiffi6p ; xA3 ¼ 0. (B.7)

The equation of OF is simply s1 ¼ s3, giving:

OF : x3 ¼ �ffiffiffi3p

x2. (B.8)

Solving simultaneous (B.6) and (B.8) gives the coordinatesof F:

F : xF2 ¼Rþ ðKp � 1ÞI1=3

Kp þ 2� �

=ffiffiffi6pþ

ffiffiffiffiffiffiffiffi3=2

pKp

¼Rþ ðKp � 1ÞI1=3

4Kp þ 2� �

=ffiffiffi6p ,

xF3 ¼ �ffiffiffi3p

xF2 ¼ �ffiffiffi3p Rþ ðKp � 1ÞI1=3

4Kp þ 2� �

=ffiffiffi6p . ðB:9Þ

On the line FE, we have s34s14s2, hence the yieldcriterion writes s3 � Kps2 � R ¼ 0. Substitution of (B.1)gives:

1ffiffiffi3p x1 �

1ffiffiffi6p x2 �

1ffiffiffi2p x3

� � Kp

1ffiffiffi3p x1 �

1ffiffiffi6p x2 þ

1ffiffiffi2p x3

� � R ¼ 0.

On account of (2), we get:

FE :Kp � 1ffiffiffi

6p x2 �

Kp þ 1ffiffiffi2p x3 ¼ Rþ Kp � 1

� � I1

3. (B.10)

It can be checked that the line defined by (B.10) intersectsthe plane s1 ¼ s3 (i.e. x3 ¼ �

ffiffiffi3p

x2) at the same point F

defined by (B.9). Point E is determined by the intersectionof (B.10) and the plane s1 ¼ s2 (i.e. x3 ¼

ffiffiffi3p

x2):

E : xE2 ¼Rþ ðKp � 1ÞI1=3� �

Kp þ 2

ffiffiffi6p

2,

xE3 ¼ffiffiffi3p

xE2 ¼ �Rþ ðKp � 1ÞI1=3� �

Kp þ 2

3ffiffiffi2p

2. ðB:11Þ

We check that:

OE�� ¼ OA

�� ¼ Rþ ðKp � 1ÞI1=3� �

Kp þ 2

ffiffiffi6p

.

On line ED, s34s24s1 hence the yield criterion writess3 � Kps1 � R ¼ 0. Substitution of (B.1) gives:

1ffiffiffi3p x1 �

1ffiffiffi6p x2 �

1ffiffiffi2p x3

� � Kp

1ffiffiffi3p x1 þ

2ffiffiffi6p x2

� � R ¼ 0.

ARTICLE IN PRESS

A

C

F

D

B

E

O

Kp = 0

Fig. B.2. Typical sections of the yield for Kp ¼ 0.


On account of (B.2), we get:

ED :2Kp þ 1ffiffiffi

6p x2 þ

1ffiffiffi2p x3 ¼ � Rþ Kp � 1

� � I1

3

� .

(B.12)

Point D is determined by the intersection of (B.12) andthe plane s2 ¼ s3 (i.e. x3 ¼ 0):

D : xD2 ¼ �Rþ ðKp � 1ÞI1=3� �

2Kp þ 1

ffiffiffi6p

; xD3 ¼ 0. (B.13)

We check that:

OD�� ¼ OF

�� ¼ Rþ ðKp � 1ÞI1=3� �

2Kp þ 1

ffiffiffi6p

.

To obtain a numerical example, we can take I1 to be acertain percentage of R, say I1 ¼ 0:1R, and a smallnegative value for Kp, say Kp ¼ �0:1. For comparisonpurpose, we also draw the case with Kp ¼ 0. We arrive atFig. B.2.

Appendix C. Analytical model with Drucker–Prager

criterion

Instead of the Mohr–Coulomb yield criterion, we mayalso choose the Drucker–Prager criterion. We will show inthis appendix that these two criteria give similar results inthe case of triaxial tests. Bearing this objective in mind andin order to simplify notation, we can take the followingform for the Drucker–Prager yield criterion:

f sð Þ ¼1

1� b

ffiffiffiffiffiffiffiffi3J2

p� bI1 � a

�¼ 0, (C.1)

Where J2 ¼12st � s is the second stress invariant and s ¼

13It � r� �

I is the deviatoric stress tensor. Geometrically, thiscorresponds to a conical surface, with the symmetry axiscoinciding with the hydrostatic axis. The apex angle isgoverned entirely by the constant b, whereas a, togetherwith b, determines the distance separating the cone tip fromthe origin. It would then be natural to adopt a similar formfor the plastic potential. In order to emphasize the

similarity with Mohr–Coulomb criterion, we will writethe plastic potential in the following form:

g sð Þ ¼1

1� c


p� cI1

�. (C.2)

In the case of triaxial compression tests, with s14s2 ¼ s3,it is straight forward to show that


p¼ s1 � s3.

Together with the definition I1 ¼ s1 þ 2s3, (C.1) can besimplified to:

f sð Þ ¼ s1 �1þ 2b

1� bs3 �

a

1� b¼ 0, (C.3)

while (C.2) becomes:

g sð Þ ¼ s1 �2cþ 1

1� cs3. (C.4)

Eqs. (C.3) and (C.4) are exactly equivalent to theiranalogous expressions (12) and (13) if the three terms a,b and c are related to Kp, R and K in the following way:

Kp ¼1þ 2b

1� b; R ¼

a

1� b; K ¼

2cþ 1

1� cor inversely,

(C.5)

a ¼3R

Kp þ 2; b ¼

Kp � 1

Kp þ 2; c ¼

K � 1

K þ 2. (C.6)

To account for hardening effects, a (similar to R) willdepend on the plastic strain epq via Eqs. (14), (15) and(C.5b). The rest of the derivation is trivial and we wouldarrive at identical expressions as (28–32). Hence we haveshown that under triaxial test conditions (s14s2 ¼ s3),Drucker–Prager yield criterion can be made to giveidentical results as Mohr–Coulomb yield criterion with asuitable choice of the constitutive coefficients.

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a simple elastoplastic hardening constitutive model for eps geofoam

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