A simple elastoplastic hardening constitutive model for EPS geofoam

ARTICLE IN PRESS Geotextiles and Geomembranes 24 (2006) 299–310 www.elsevier.com/locate/geotexmem A simple elastoplastic hardening constitutive model for EPS geofoam Henry Wonga, Chin Jian Leob, a LGM, DGCB (URA CNRS 1652), ENTPE, rue Maurice Audin, 69518 Vaulx en Velin, Cedex, France School of Engineering, University of Western Sydney, Locked Bag 1797 Penrith South DC, Sydney, NSW 1797, Australia b 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 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 Introduction The use of block-moulded expanded polystyrene (EPS) geofoam in geotechnical applications is not new The material has been successfully employed by Norwegian geotechnical engineers since at least the early 1970s (e.g., Refsdal, 1985; Aaboe, 1987) and examples of EPS geofoam applications abound (e.g Abu-Hejleh et al., 2003; Zou et al., 2000; Frydenlund and Aaboe, 1996; Magnan and Serratree, 1989; van Dorp, 1996; William and Snowdon, 1990) In engineering construction, EPS geofoam is increasingly being used as a material of choice to replace other conventional materials in situations where the attributes of EPS can be utilized For instance, as a consequence of its extremely low density (normally ranging from 10 to 35 kg/m3), EPS geofoam makes an ideal backfill material when the self weight of the construction material is a major design consideration (e.g embankment on very soft soil as shown in Fig 1(a)) Another attribute of EPS Corresponding author Fax: +61 98525741 E-mail address: c.leo@uws.edu.au (C.J Leo) 0266-1144/$ - see front matter r 2006 Elsevier Ltd All rights reserved doi:10.1016/j.geotexmem.2006.03.007 geofoam is that it experiences very small or virtually zero lateral expansion such that volume contraction occurs under deviatoric compressive loading, and that it induces significantly lower lateral loading than normal earth pressures (e.g behind a bridge abutment in Fig 1(b)) It is envisaged that these attributes foreshadows the possibility of EPS geofoam being used effectively as a compressible inclusion behind retaining structures The compressible inclusion function of EPS geofoam was thoroughly discussed by Horvath (1997) From an engineering design standpoint, it is of interest to have an efficient constitutive model available as a means to carry out design analysis on a rational and rigorous basis One of the key requirements for such an efficient constitutive model is that the number of independent parameters must be reasonably small, and the model’s parameters can be estimated from routine and non-costly experimental tests Above all, a good model should be able to capture the essential characteristics of the material behaviour in the application for which the model is intended A constitutive model is, therefore, normally developed in the context of the particular area or areas of ARTICLE IN PRESS 300 H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 (a) (b) Fig Examples of EPS applications: (a) lightweight embankment; (b) bridge abutment its intended applications Swart et al (2001) presented a constitutive model of EPS to investigate the non-linear response of a road pavement, utilizing an associated flow rule and a modified version of Gurson flow surface The model was an attempt to provide a unified phenomelogical approach for materials exhibiting strain rate dependent plastic deformations incorporating the effects of strain hardening/softening A specific interest of their research was to simulate cyclic behaviour of pavement structures experienced under traffic loading Developed within the framework of dynamic plasticity, the model is, naturally, quite complicated to apply Hazarika and Okuzono (2004) utilized a non-linear elastoplastic hardening constitutive model of EPS geofoam with von Mises yield criterion to investigate the behaviour of a hybrid interactive system involving soil, structure and EPS geofoam The model ignored possible effects of mean stress and made the assumption that the hardening regime follows a hyperbolic curve The objective of the authors was to develop something simple with the intention of using it specifically to model the interactive behaviour of EPS compressible inclusion behind retaining structures where large strain behaviour is expected Hazarika (2006) later extended the constitutive model to include the size and shape factors of the tested specimen as well as the density of the geofoam, arguing that both the shape and absolute dimensions of test specimens influence the stress–strain behaviour, and hence the material’s compressive strength Chun et al (2004) developed a hyperbolic constitutive model as a function of the density and confining stress, and which is based on the popular hypoelastic Duncan–Chang model (Duncan and Chang, 1970) for soils Though it is shown to fit the measured test data, the model does not contain an explicit yield criterion nor is strain hardening formally incorporated as required under notions of classical plasticity The model is applicable to unique monotonic loading but would not be considered suitable for problems where complex loading paths and stress-reversal occur The motivation of this paper is to develop a simple elastoplastic hardening constitutive model with the possibility of a fairly wide range of geotechnical applications The model is formulated in a way that ensures the essential characteristics of EPS geofoam, namely, the strain and volumetric response under shearing, can be correctly captured It is suggested that a main appeal of the model is that it is not only simple to calibrate (with six independent parameters) but that it is also relatively easy to incorporate into numerical codes for geotechnical analysis The model is developed within the framework of classical plasticity, with the inclusion of strain hardening in order to correctly describe the post-yield behaviour On the other hand, time dependent viscous behaviours (creep) are not accounted for in the present development Previous researchers (e.g., Chun et al., 2004; Hazarika, 2006) have discussed the context for including visco-elastic or viscoplastic behaviour of EPS, and Horvath (1998) has proposed the application of the Findley model (Findley et al., 1989) to account for visco-elastic time-dependent effects In general, however, the importance of time dependent creep in EPS depends on such factors as the stress or strain level, the temperature and the duration of loading Provided the strains 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-term loading, creep effects can generally be neglected The current model, therefore, excludes application in situations where viscosity effects are important under a combination of high stress (or strain), high temperature and sustained loading for a long period of time Even under cases where the creep may not be entirely negligible, this simple model can still serve as a useful tool for an order-of-magnitude-estimate for the designer to make appraisal of results obtained by other more complex means Investigations on temperature-dependent effects are currently being undertaken and will be addressed in a future publication Also, the model in its current form does not incorporate the factors that account for the specimen 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 tests are described This will be followed by the presentation of the proposed theoretical model and its casting in the triaxial configuration This is not a trivial development on account of the corner-flow condition, since the stress state lies on a singular point of the yield surface where the gradient is not uniquely defined The model will then be validated using the experimental data obtained Triaxial tests The compressive behaviour of EPS geofoam was investigated through a series of triaxial tests conducted at ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 301 Load cell LVDT A/D converter Triaxial cell PC Com1 EPS GDS pressurevolume controller Com2 Fig Set-up of triaxial test system room temperature (23 1C) Shown in Fig is the schematic of the test set-up Cylindrical specimens with a diameter of 50 mm and a height of 50 mm (aspect ratio of 1:1) were cut from a 20 kg/m3 density prismatic EPS block supplied by RMAX Cellular Plastics, an Australian manufacturer of EPS The ends of the EPS specimens were smoothed by sand papering to reduce end friction A 1:1 specimen was considered less likely to give rise to column-type buckling, which was observed in some cases of 2:1 specimens Zou and Leo (1998) observed that provided care was taken to ensure that the end contacts were sufficiently smooth the 1:1 specimens did not result in end restraint problems kPa Deviator Stress (kPa) 2.1 Test specimen 120 100 20 kPa 80 40 kPa 60 60 kPa 40 20 0 10 12 14 16 18 Axial Strain (%) Fig 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 2.2 Test apparatus A 50 kN TRITECH digital loading frame capable of a loading speed ranging from 0.0 to 6.0 mm/min was used in the tests The platen supplied rate was set through a precise thumb wheel selector with an accuracy controlled to better than 1% In this series of experiments, a relatively slow loading rate of 0.4% or 0.2 mm/min was adopted The triaxial compression was performed within a Wykeham Farrance triaxial cell (WF 10201), which can withstand an internal cell pressure up to 1700 kPa The applied loading and the vertical displacement were measured and recorded during the tests by means of a load cell and an LVDT transducer, through an in-house developed A/D data logger and software The load cell and the LVDT were capable of resolutions as small as N and 0.01 mm, respectively A GDS digital pressure/volume controller accurate to volume measurement of mm3 and pressure to kPa was used to control the cell pressure (via the software) and to measure any volume changes in the cell fluid during the test The cell fluid volume change was subsequently used to derive the volume changes of the specimen The tests were carried out over a range of confining pressure from to 60 kPa The samples were initially loaded hydrostatically to the prescribed confining pressure until the volume change was stabilized and then sheared by applying the axial load Throughout the duration of the test, the ‘‘pore pressure’’ drainage valves were kept open to allow the gas in cellular structure of the EPS specimen to dissipate The testing procedures are generally similar to those described by Atmazidis et al (2001) 2.3 Test results Shown in Fig are the experimental plots (full lines) of the deviator stress against the axial stress under confining pressure ranging from to 60 kPa Corresponding to each confining pressure, the initial part of the curve is fairly linear and its slope is defined by the initial modulus E After yielding, the slope reduces and asymptotes quickly to the plastic modulus Ep The response can therefore be reasonably approximated by a bilinear relationship defined by the moduli E and Ep, corresponding to pre- and postyielding, respectively Ep is also markedly smaller than E ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 302 Fig shows the volume change of the EPS specimen (full lines) during the axial loading stage, when the deviator stress was applied In this investigation, as well as in others (e.g Atmazidis et al., 2001; Zou and Leo, 1998), the Poisson ratio was found to be a small positive value during the initial stages of loading when strains were very small, then quickly reducing to virtually zero or a negative value at larger strains (Poisson’s ratio does not strictly apply in the plastic strain) These results are apparent in the shape of EPS specimens after being subjected to compression loading in the picture shown in Fig As a consequence of this, the EPS specimen exhibited contractive volumetric behaviour during the axial loading, since the axial volumetric contraction due to applied loading was not compensated by lateral volumetric expansion The plot in Fig shows the variation of the major principal stress s1 against the minor principal stress s3 when the specimen is at yield In these series of tests, the results showed the major principal stress had a slight negative dependency on the minor principal stress, i.e the major principal stress at yield decreased slightly as the minor principal stress was increased Model presentation Prior to the description of our proposed model, let us first recall some fundamental equations in classic elastoplasticity which will be of use later 3.1 Some fundamental equations in classic elastoplasticity theory For simplicity, all matrix and vector quantities are in bold Positive stresses and strains correspond to compression and fibre shortening For small displacements and small strains, the total strain can be written as the sum of elastic and plastic strains: de ẳ dee ỵ dep , (1) where the vectorial notation r ẳ ẵs11 s22 s33 s12 s23 s13
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; Rị ẳ 0, (2) where R is a variable which defines the current size of the yield surface in order to account for hardening effects The 120 25000 60 kPa 20000 40 kPa 100 20 kPa 15000 σ1 (kPa) Volume Change (mm3) 30000 kPa 10000 5000 80 60 0 10 12 14 16 18 Axial Strain (%) Fig Volume change vs axial strain for various confining stress Full lines represent experimental data while the broken lines are the model results 40 10 20 30 40 50 60 70 σ3 (kPa) Fig Plot of major principal stress vs minor principal stress at yield Fig EPS geofoam at different levels of compression Note the lateral contraction of the specimens under vertical compressive loading ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 stress increment is related to the strain increment through the classical relation: dee ¼ D1  dr, (3) where D is the elastic stiffness matrix and is constant for linear behaviour For clarity, we will use a dot (‘‘’’) operator to denote matrix–matrix and matrix–vector multiplications Onset of (irreversible) plastic strains begins when the yield surface is reached For a hardening behaviour, the yield criterion will depend on an additional variable R which reflects the current level of strength on account of hardening effects Only isotropic behaviour is considered here and R is supposed to be a scalar In addition to the yield function, a flow rule is also needed to describe plastic strains: qg , (4) qr where z is the (positive) plastic multiplier and g the plastic potential The stress increment is still related to the elastic strain increment by (3) During plastic loading, the stress point must remain on the surface of the yield envelope This is known as the consistency condition, which writes: dep ¼ dz qf qf  dr ỵ dR ẳ qr qR For simplicity, we shall denote: df ¼ (5) qf qg ; G¼ (6) qr qr Classically, the hardening modulus H is defined by the relation: F¼ qf dR ¼ Hdz (7) qR There are several ways to define the evolution of R and H, commonly called the hardening behaviour It is supposed that R is related to an internal parameter x, which is itself related in some way to the plastic multiplier z Eq (7) can then be developed to give: H¼
t Pre-multiplication by Ft ¼ qf =qr , and on account of (5) and (7): dz ẳ Ft  D  de , H ỵ Ft  D  G (10) where notation (6) has been used Substitution of (10) into (9) leads to:   D  G  Ft  D dr ¼ D  (11)  de ¼ Dep  de, H þ Ft  D  G where Dep is the classical tangent stiffness matrix 3.2 The EPS model In constructing the simple analytical model, it will be supposed the elastic behaviour is linear and isotropic, defined by two rheological constants E and n For the plastic behaviour, based on experimental observations, the following yield function will be adopted which is formally identical to the well-known Mohr–Coulomb criterion: f ðr; RÞ ¼ s1  K p s3  R ¼ 0, (12) where Kp is a rheological constant For an elastic-perfectly plastic behaviour, R would be constant and becomes another rheological parameter, known as the unconfined compressive strength However, for a strain-hardening behaviour developed here, R is variable and is to be identified with the hardening variable in the preceding paragraph and it will be made to depend on the current plastic strain This hardening behaviour will be addressed after the flow rule At present, note the important departure from the classic Mohr–Coulomb model in that triaxial tests suggest that Kp is generally close to zero, and can even be slightly negative in some tests Hence, in the present context, (12) is taken to be a phenomenological relation, and Kp is no longer related to any non-negatively defined internal friction angle Fig below shows a typical section of the yield envelope in the deviatoric plane (or the p-plane) Justifications are briefly presented in Appendix B To define the plastic strain rate, the following form for the flow potential g is adopted: grị ẳ s1  Ks3 , qf dR dx qR dx dz (13) (8) The relations R ¼ R(x) and x ¼ x(z) define completely the hardening behaviour adopted, which is generally a key element of the specific model developed In developing the simple analytical model, we will suppose x to be an invariant of the plastic strain Hence, the strength level depends on the plastic strain, and the resulting model is called a strain-hardening model This point will be further developed in the next paragraph Substitution of (3) and (4) into (1), then pre-multiplying by D gives: D  de ẳ dr ỵ dzD  303 qg qr (9) x2 A A B 60° F B x3 O E C F O C D E D (a) (b) Fig Projection of the yield surface on the p-plane (deviatoric plane) for slightly (a) positive and (b) negative Kp ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 304 where s1 4s2 4s3 are the ordered principal stresses, while K40 is a rheological constant which governs the dilatancy behaviour When the stress state lies at one of the corners (‘‘corner flow’’ condition), the flow direction will apparently be undefined In fact, it has been shown in the past that even under such cases, the problem is still well defined (see for example, Wong and Simionescu, 1996; Berest, 1989) Numerical computations without recourse to special algorithmes still work, with the stress state oscillating from one face to another of the yield surface For negative values of Kp, the yield surface will appear to be non-convex and may lead to unwanted results for a general loading paths Strictly speaking, the Mohr–Coulomb model should only be applied to trixial loading paths where all subsequent developments will remain valid The Drucker–Prager criterion, which gives identical results for trixial stress paths (Appendix C), can in this case be used since it does not have singularities nor non-convexity problems For the hardening law, it is supposed that R varies linearly with the internal variable x: Rxị ẳ R0 þ bx (14) In our model, x is identified as the equivalent deviatoric plastic strain tensor, defined by: Z t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p x ¼ eq ¼ depq ; depq ¼ 12dep ịt  dep , dep ẳ dep  13depv I, 15ị  t where I ẳ 1 0 and depv ¼ It  dep is the volumetric plastic strain Attention is now fixed onto the case of triaxial tests—the only configuration treated in this paper In this ideal geometric configuration, a fully analytical model will be developed, which will allow an easy validation using triaxial test results 3.3 Analytical model for EPS behaviour under triaxial tests In the system of principal directions (Fig 8), only the diagonal terms of the stress and strain tensors are non-zero For mathematical convenience, we revert to the vectorial notation for the following:  t  t  t r ¼ s1 s2 s3 ; e ¼ e1 e2 e3 ; I ¼ 1 , (16) x1 σ1 σ2 σ3 x2 Fig Triaxial test configuration x3 with s2 ¼ s3 and e2 ¼ e3 The elastic behaviour is dened by l ỵ 2m l l l ỵ 2m l dr ẳ D  de; D ¼ l (17) 5, l l l ỵ 2m where l and m are the classical Lame´’s constants, related to E and n by l¼ nE ; ỵ nị1  2nị mẳ E 21 ỵ nị (18) On account of the corner ow condition: s14s2 ¼ s3, we use, instead of (6):  qf 12 qf 13 ỵ (19) Fẳ with f ij ¼ si  K p sj  R, qr qr to evaluate the consistency condition: 9 >t > ds1 > > = qf < = < qf df ẳ Ft  dr ỵ  ds3 ỵ dR ẳ K p =2 dR ẳ 0, > qR : ds > ; qR : K =2 > ; > p (20) while the symmetry of plastic strain rates dep2 ¼ dep3 implies the following two equations:  qg12 qg13 ỵ (21) Gẳ with gij ¼ si  Ksj , qr qr dep ¼ dz G ¼ dz > < > = K=2 > ; : K=2 > (22) The constant K governs the plastic volumetric strain since: depv ¼ dep1 ỵ dep2 ỵ dep3 ẳ  Kị dz, (23) where positive volumetric strain (Ko1) implies contractancy and negative volumetric strain (K41) dilatancy Combining (15), (22) and (23), it can easily be shown that: dx ¼ depq ¼ H¼ K ỵ2 p dz, qf dR dx bK ỵ 2ị p ẳ qR dx qz (24) (25) The tangent matrix Dep ¼ D  Dp , where Dp ¼ ðD  G  Ft  Dị=H ỵ Ft  D  Gị can now be evaluated analytically in this particular case After some lengthy calculations which can be performed by Mathematica or Maple, the following explicit results are found:  1 D  G  Ft  D de ¼ Sep  dr; Sep ¼ D  , (26) H ỵ Ft  D  G where the tangent flexibility matrix Sep is the inverse of the tangent stiffness matrix Dep It admits the following ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 explicit expression: Sep ¼ lỵm l=2 l=2 p l=2 l ỵ m l=2 ỵ 2bK ỵ 2ị m3l ỵ 2mị l=2 l=2 l ỵ m 2K p 2K p 2K K p K K p K 6 ð27Þ 2K K p K K pK Eqs (17) and (26) completely describe the triaxial behaviour of an EPS sample under monotonic loading Suppose an EPS sample is subject to the following classical triaxial stress path shown in Fig We will take point Q to be the reference point (at the end of an isotropic prestressing) for strain measurements It can then be shown that the stress and strain behaviour under triaxial tests is basically bilinear (Fig 10) The elastic–plastic stiffnesses me and mp, as well as the volumetric deformation ratios mve and  t mvp can now be determined Since dr ¼ ds1 0 , the elastic behaviour yields:  qs1 m3l ỵ 2mị ẳ E, (28) me ẳ ẳ ẳ lỵm qe1 QA Se11  mve ẳ qev qe1 ẳ QA Se11 ỵ S e21 þ S e31 m ¼  2n ¼ lþm S e11 (29) The first yield stress (point A) depends on the pre consolidation pressure s0, hence: s1y ¼ K p s0 ỵ R0 (30) 2=3 Q A B σ1 σ0 O Fig Triaxial stress-path σ1 σ1y For elastic–plastic behaviour beyond point A, similar calculations give: pffiffiffi  32 ỵ Kịbm3l ỵ 2mị qs1 ẳ ẳ p mp ẳ qe1 AB Sep ỵ Kịbl þ mÞ þ 6mð3l þ 2mÞ 11 1 pffiffi ẳ ẳ , 31ị 1 ỵ EỵH E A b2ỵKị  ep Sep ỵ S ep qev 21 ỵ S 31 ẳ 11 qe1 AB S ep 11 p 32 ỵ Kịb ỵ 61  Kị3l ỵ 2mị ẳ m p 32 ỵ Kịbl ỵ mị ỵ 6m3l ỵ 2mị mvp ẳ Experimental determination of the constitutive constants This model presented above depends on six rheological parameters (i.e constitutive constants): E; n (or equivalently l, m) : define the isotropic elastic behaviour, 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 independent parameters Note also that the preconsolidation stress s0— which appears in the first yield stress s1y—is an ‘‘external’’ parameter, controllable by the experimental operator, and is not a constitutive constant A means to determine these constants can be the following The elastic stiffnesses me and mve can be directly read off from an experimental curve to determine the elastic constants E and n The constants R0 and Kp which define the first yield stress can be determined graphically by plotting the first yield stress s1y against the preconsolidation stress s0 Given that s1y ¼ R0+Kps0, therefore Kp and R0 are simply the slope and the y-intercept in such a graph From the plastic stiffness mp and the Young’s modulus E previously εv B B mp σ1y−σ0 A B A mvp me σ0 Q ε1 me Q ð32Þ Note that under triaxial conditions, Drucker–Prager yield criterion will give similar results as the Mohr–Coulomb criterion (see Appendix C) Their results differ however under general stress states Only experimental results can decide which criterion is more precise σ1−σ3 mp 305 ε1 mve Q Fig 10 Triaxial behaviour of EPS as predicted by the model ε1 ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 306 determined, the hardening mudulus H can be calculated using (31) For b and K, observe firstly that from the expression of the plastic stiffness mp, we can deduce: p p bK ỵ 2ị ẳ ẳ 3H (33) 1=mp  1=E Hence, the product b(K+2) is known Now, using the plastic volumetric relation on mvp, we can deduce an expression of (1K), hence that of K:   H lỵm H ỵ1 ỵ (34) K ẳ  mvp m 3l ỵ 2m 3l ỵ 2m Experimentally, it has been observed that axial stress increments induce little lateral strains Moreover, parametric studies show that Poisson’s ratio has little influence on the volumetric strain variation We therefore take: v¼0 (35) This particular value implies ee3 ¼ (i.e zero radial elastic strain) in triaxial compression under constant lateral confining pressure Hence we have eevol ¼ ee1 (elastic volumetric strain equals the elastic axial strain) and mve ¼  2n ¼ Under this hypothesis, the expression of K above simplifies to: 
 H K ẳ  mvp ỵ (36) E plane) is shown in Fig 7(b) for the case when Kp is slightly negative The results of the deviatoric axial stress versus the axial strain from the model for confining pressure from to 60 kPa are plotted in broken lines in Fig It is observed that the model is able to reproduce the experimental behaviour under shearing for the full range of confining pressure, but perhaps a little better at the lower confining pressures The replication is still good at 60 kPa at the higher strain values, though there is some deviation from experimental data, because of the presence 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 are very similar for all confining pressures) The model gives a volume change which corresponds more to the average volume change for the confining stresses but in general, the trend of the shear-contractive volumetric behaviour of EPS geofoam is correctly modelled To improve the volumetric strain prediction, we can include the dependence of contractancy/dilatancy on the confining stress Fig 11 shows the variation of K against s3 deduced from experimental results in Fig 3, as well as a linear approximation This linear approximation leads to an improved prediction on the volumetric strain, as shown in Fig 12 0.1 Model verification -0.3 -0.5 -0.7 10 20 30 40 50 60
3 (kPa) Fig 11 Dependence of dilatant coefficient K on lateral confinement s3 30000 Volume Change (mm3) We observe that the initial slope is more or less constant for all cases, this condition giving the young’s modulus: E ¼ 3950 kPa We have taken n ¼ as mentionned above The axial stress versus axial strain on the post-yield portion also shows little dependence on the initial isotropic prestress, it’s slope is found to be mvpE110 kPa This also leads to HE113 kPa The variation of the first yield stress s1y leads to the determination of R0 and Kp: R0 ¼ 98 kPa, Kp ¼ 0.17 It remains to determine the constants b and K The experimental curves on the volumetric variation show a slight dependence of K on the lateral pressure This leads to a little complication on the determination of K and b As a first approach, we may take an average value (corresponding to a lateral confinement of 20 kPa) of K ¼ 0.25 and b ¼ 225 kPa To summarize, estimated values of the model parameters for the 20 kg/m3 EPS are presented in Table It is observed that K is a small negative value suggesting that the material is clearly plastically contractant during post-yield The yield surface on the p-plane (deviatoric K -0.1 25000 60 kPa 40 kPa 20000 20 kPa 15000 kPa 10000 5000 0 Table Estimated values of model parameters 10 12 14 16 18 Axial Strain (%) E (kPa) n R0 (kPa) Kp b (kPa) K 3950 98 0.17 225 0.25 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 ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 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 very small confining stress This appears to be consistent with visual observation on stressed samples (observe the incurving lateral surface in Fig 5) Conclusions A simple elastoplastic hardening constitutive model for EPS geofoam is developed and presented in this paper A series of ‘‘drained’’ triaxial tests was carried out at room temperature conditions (23 1C) to study the behaviour of EPS geofoam under shear based upon which the constitutive model was developed The model has been shown to correctly predict the response of the material under shearing in terms of deviatoric stress-axial strain behaviour and the shear-contraction behaviour It is suggested that this model will be useful in modelling the behaviour of EPS geofoam in a wide variety of geotechnical applications The behaviour at high operating temperatures, stress levels and long-term loading situations, where time dependent creep behaviour may become more important, is not addressed in the present study Investigations are currently being undertaken to study temperature effects on EPS and the findings will be presented in a future publication Thus the applicability of this model is currently restricted to normal operating temperature (typically less than 30 1C) and strain levels below the ‘‘sigmoidal’’ strain (i.e before rapid strain hardening occurs), and for loading of relatively short duration In its current form the model has not included the size and shape factors of the test specimens as suggested by Hazarika (2006) but these can be included later on to improve its applicability Acknowledgements The authors gratefully acknowledge the generous donation of the EPS material used in the experimental tests by RMAX Cellular Plastics, Australia Appendix A Reduced matrix calculations in the case of triaxial test To take the maximum advantage of the symmetry in a triaxial test, the stresses and strains, as well as the gradients F and G can be represented by two-components vectors: ( ) ( ) ( ) ds1 de1 dr ¼ ; de ¼ ; F¼ , K p ds3 de3 ( ) Gẳ A:1ị K=2 307 The gradient F is defined in such a way that the scalar product Ft ds remains unchanged The elastic stiffness matrix now becomes: " # " # k ỵ 4G 2k  4G l ỵ 2m 2l 3 Dẳ ẳ (A.2) k  2G 2k ỵ 2G l 2l ỵ mÞ 3 It can be verified that all the previous formulae remain valid In particular: 8
 > D1 11  ds1 if f ðr; RðzÞÞo0 or df o0 > > > > > < i:e: elastic behaviour or unloading; de1 ¼ > D1  ds1 if f r; Rzịị ẳ and df ẳ > ep > 11 > > > : i:e: plastic loading: (A.3) Hand calculations are now possible since the matrices are only  Inversion of the elastic and elastic–plastic stiffness matrices gives: " # 2l ỵ mị 2l e 1 S ẳD ẳ , (A.4) l l ỵ 2m 2m3l ỵ 2mị p ep S ẳ D1 ep ẳ6 32ỵKịb p lỵmịỵ6m3lỵ2mị 32ỵKịbm3lỵ2mị p 3lỵ2mị p  322ỵKịblỵ6Km 32ỵKịbm3lỵ2mị   p p 32ỵKịblỵ6K p m3lỵ2mị p 32ỵKịbm3lỵ2mị 7, 32ỵKịblỵ2mịỵ6KK p m3lỵ2mị p 32ỵKịbm3lỵ2mị which can be decomposed into two parts: " # 2l ỵ mị 2l ep e p S ẳS ỵS ẳ 2m3l ỵ 2mị l l ỵ 2m # p " 2K p ỵ b2 þ KÞ K K p K ðA:5Þ ðA:6Þ Note that Sp can also be written in the following form: # pffiffiffi " K p p S ẳ (A.7) ẳ G  Ft b2 ỵ Kị K=2 K p K=2 H The slopes of the stress-strain curve are of course identical to the previous calculations:  e qs1 m3l ỵ 2mị e ẳE m ẳ ẳ e ẳ S11 lỵm qe1 under elastic behaviour, mp ẳ  qs1 qe1 p A:8ị p 32 ỵ Kịbm3l ỵ 2mị p ẳ S ep ỵ Kịbl ỵ mị ỵ 6m3l ỵ 2mị 11 1 p ẳ ẳ A:9ị 1 ỵ ỵ E H ẳ E b2ỵKị ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 308 Appendix B Study of convexity of the Mohr–Coulomb criterion We will show briefly in this appendix the calculations which prove the shape of the yield surface We begin by defining a second system of coordinates (x1, x2, x3) with x1 corresponding to the hydrostatic axis and (x2, x3) to the deviatoric plane (p-plane) (Fig B.1) B.1 p-plan and coordinate transformation The two systems x3), are related by: pffiffiffi x1 1= B x C 2=pffiffi6ffi @ 2A ¼ x3 0 pffiffiffi s1 1= B s C 1=pffiffi3ffi @ 2A ¼ pffiffiffi s3 1= Substitution of (B.1) into (B.5) yields:   1 p x1 ỵ p x2  K p p x1  p x2 ỵ p x3  R ¼ 6 On account of (B.2), we arrive at the equation of line AF: AF : R ỵ K p  1ịI =3
 p ; Kp ỵ = of coordinates, (s1, s2, s3) and (x1, x2, A: pffiffiffi 1= pffiffiffi 1= pffiffiffi 1= pffiffiffi 2= pffiffiffi 1= pffiffiffi 1= The equation of OF is simply s1 ¼ s3, giving: pffiffiffi OF : x3 ¼  3x2 pffiffiffi 30 s1 1= pffiffiffi 7B C 1= 5@ s2 A; pffiffiffi s3 1= 30 x1 pffiffiffi 7B C 1= 5@ x2 A: pffiffiffi x3 1= (B.1) xA2 ¼ F: (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 pffiffiffi s1 þ s2 þ s3 ¼ I ¼ x1 ¼ constant (B.3) xA3 ¼ (B.7) (B.8) Solving simultaneous (B.6) and (B.8) gives the coordinates of F: R ỵ K p  1ịI =3 R ỵ K p  1ịI =3  p , ẳ
 p p 4K p ỵ = K p þ = þ 3=2K p pffiffiffi pffiffiffi R ỵ K p  1ịI =3 ẳ  xF ẳ 
 p B:9ị 4K p ỵ = xF ẳ
xF s1 ỵ s2 ỵ s3 I1 p ẳ pffiffiffi 3 (B.6) where the coordinates of A can easily be found: Notice that: x1 ¼
 I1 Kp ỵ Kp p x2  p x3 ẳ R ỵ K p  , On the line FE, we have s34s14s2, hence the yield criterion writes s3  K p s2  R ¼ Substitution of (B.1) gives:   1 1 1 pffiffiffi x1  pffiffiffi x2  pffiffiffi x3  K p p x1  p x2 ỵ pffiffiffi x3 6 2  R ¼ On account of (2), we get: FE : B.2 EPS yield surface The EPS yield criterion writes: sI  K p sIII  R ¼ with sI 4sII 4sIII the ordered principal stresses ðB:4Þ On the face AF (Fig 5), we have s14s34s2, hence stresses on AF satisfy: s1  K p s2  R ¼ (B.5) σ1 σ2 (B.10) It can be checked that the line p defined by (B.10) intersects ffiffiffi the plane s1 ¼ s3 (i.e x3 ¼  3x2 ) at the same point F defined by (B.9) Point E is determined bypthe ffiffiffi intersection of (B.10) and the plane s1 ¼ s2 (i.e x3 ¼ 3x2 ):
 pffiffiffi R þ ðK p  1ÞI =3 , E : xE2 ẳ Kp ỵ
 p p R ỵ K p  1ịI =3 B:11ị xE3 ẳ 3xE2 ẳ  Kp þ We check that:
 pffiffiffi OE ẳ OA ẳ R ỵ K p  1ịI =3 Kp ỵ x2 x3
 I1 Kp  Kp ỵ p x2  p x3 ẳ R ỵ K p  x1 On line ED, s34s24s1 hence the yield criterion writes s3  K p s1  R ¼ Substitution of (B.1) gives: σ3 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)   1 1 pffiffiffi x1  pffiffiffi x2  pffiffiffi x3  K p pffiffiffi x1 þ pffiffiffi x2  R ¼ 6 ARTICLE IN PRESS H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 similarity with Mohr–Coulomb criterion, we will write the plastic potential in the following form: pffiffiffiffiffiffiffiffi (C.2) 3J  cI gðsÞ ¼ 1c A In the case of triaxial compression tests, with sffi1 4s2 ¼ s3 , pffiffiffiffiffiffiffi it is straight forward to show that 3J ¼ s1  s3 Together with the denition I ẳ s1 ỵ 2s3 , (C.1) can be simplified to: F B O C 309 D E Kp = f sị ẳ s1  Fig B.2 Typical sections of the yield for K p ẳ ỵ 2b a s3  ¼ 0, 1b 1b (C.3) while (C.2) becomes: gðsÞ ¼ s1  On account of (B.2), we get: ED : 
 I1 2K p ỵ 1 p x2 ỵ p x3 ẳ  R ỵ K p  (B.12) Point D is determined by the intersection of (B.12) and the plane s2 ¼ s3 (i.e x3 ¼ 0):
D: xD2 ¼   R ỵ K p  1ịI =3 p 6; xD3 ẳ 2K p ỵ (B.13) We check that:
 p R ỵ K  1ịI =3 p OD ẳ OF ẳ 2K p ỵ To obtain a numerical example, we can take I1 to be a certain percentage of R, say I ¼ 0:1R, and a small negative value for Kp, say K p ¼ 0:1 For comparison purpose, we also draw the case with K p ¼ We arrive at Fig B.2 Appendix C Analytical model with Drucker–Prager criterion Instead of the Mohr–Coulomb yield criterion, we may also choose the Drucker–Prager criterion We will show in this appendix that these two criteria give similar results in the case of triaxial tests Bearing this objective in mind and in order to simplify notation, we can take the following form for the Drucker–Prager yield criterion: pffiffiffiffiffiffiffiffi f sị ẳ (C.1) 3J  bI  a ¼ 0, 1b t Where
 J ¼ s  s is the second stress invariant and s ¼ t I  r I is the deviatoric stress tensor Geometrically, this corresponds to a conical surface, with the symmetry axis coinciding with the hydrostatic axis The apex angle is governed entirely by the constant b, whereas a, together with b, determines the distance separating the cone tip from the origin It would then be natural to adopt a similar form for the plastic potential In order to emphasize the 2c ỵ s3 1c (C.4) Eqs (C.3) and (C.4) are exactly equivalent to their analogous expressions (12) and (13) if the three terms a, b and c are related to Kp, R and K in the following way: Kp ẳ aẳ ỵ 2b ; 1b 3R ; Kp ỵ Rẳ bẳ a ; 1b Kp  ; Kp ỵ Kẳ cẳ 2c ỵ or inversely, 1c (C.5) K 1 K ỵ2 (C.6) To account for hardening effects, a (similar to R) will depend on the plastic strain epq via Eqs (14), (15) and (C.5b) The rest of the derivation is trivial and we would arrive at identical expressions as (28–32) Hence we have shown that under triaxial test conditions (s1 4s2 ¼ s3 ), Drucker–Prager yield criterion can be made to give identical results as Mohr–Coulomb yield criterion with a suitable choice of the constitutive coefficients References Aaboe, R., 1987 13 years of experience with expanded polystyrene as a lightweight fill material in road embankments, Publication No 61 Norwegian Road Research Laboratory, Oslo, Norway, pp 21–27 Abu-Hejleh, N.M., Zornberg, J.G., Elias, V., Watcharamonthein, J., 2003 Design assessment of the founders/meadows GRS abutment structure In: Proceedings of TRB 2003 Annual Meeting CD-ROM Atmatzidis, D.K., Missirlis, E.G., Chrysikos, D.A., 2001 An investigation of EPS Geofoam behaviour in compression, In: EPS Geofoam 2001— Third International Conference, Salt Lake City, Utah, USA, December 10–12, 2001, CD-ROM Conference preprints—http:// geofoam.syr.edu/EPS2001/ Berest, P., 1989 Proble`mes de me´canique associe´s au stockage souterrain Ph.D Thesis, Ecole Nationale Supe´rieure des Mines de Paris (ENSMP) Chun, B.S., Lim, S.H., Sagong, M.S., Kim, K., 2004 Development of a hyperbolic constitutive model for expanded polystyrene (EPS) geofoam under triaxial compression tests Geotextiles and Geomembranes 22, 223–237 Duncan, J.M., Chang, C.-Y., 1970 Nonlinear analysis of stress and strain in soils Journal of the Soil Mechanics and Foundation Division, ASCE 96 (SM5), 1629–1653 Findley, W.N., Lai, J.S., Onaran, K., 1989 Creep and Relaxation of Nonlinear Viscoelastic Materials Dover Publications, New York, USA ARTICLE IN PRESS 310 H Wong, C.J Leo / Geotextiles and Geomembranes 24 (2006) 299–310 Frydenlund, T.E., Aaboe, R., 1996 Expanded polystyrene—the light solution, In: International Symposium on EPS Construction Method, EPS TOKYO’96, Japan, pp 32–46 Hazarika, H., Okuzono, S., 2004 Modeling the behavior of a hybrid interactive system involving soil, structure and EPS geofoam Soils and Foundations 44 (5), 149–162 Hazarika, H., 2006 Stres-strain modeling of EPS geofoam for large-strain 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2001, CD-ROM Conference preprints—http://geofoam.syr edu/EPS2001/ van Dorp, T., 1996 Building on EPS Geofoam in the ‘low-lands’ experiences in The Netherlands, In: International Symposium on EPS Construction Method, EPS TOKYO’96, Japan, pp 60–69 Williams, D., Snowdon, R.A., 1990 A47 Great Yarmouth Western Bypass: performance during the first three years Contactor Report 211, Transport and Road Research Laboratory Wong, H., Simionescu, O., 1996 Closed form solution on the thermoplastic behaviour of a deep tunnel in a thermal-softening material International Journal of Cohesive Frictional Materials 2, 321–337 Zou, Y., Leo, C.J., 1998 Laboratory studies on the engineering properties of expanded polystyrene (EPS) material for geotechnical applications In: Proceedings of the Second International Conference on Ground Improvement Techniques, Singapore 7–9 October, pp 581–588 Zou, Y., Small, J.C., Leo, C.J., 2000 Behaviour of EPS geofoam in model test on pavements Geosynthetics International V (1), 1–22  ... model with the possibility of a fairly wide range of geotechnical applications The model is formulated in a way that ensures the essential characteristics of EPS geofoam, namely, the strain and... is, naturally, quite complicated to apply Hazarika and Okuzono (2004) utilized a non-linear elastoplastic hardening constitutive model of EPS geofoam with von Mises yield criterion to investigate... flow surface The model was an attempt to provide a unified phenomelogical approach for materials exhibiting strain rate dependent plastic deformations incorporating the effects of strain hardening/ softening