Development of A Hyperbolic Constitutive Model For Expanded Polystyrene (EPS) Geofoam Under Triaxial Compression Tests [PDF]

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Geotextiles and Geomembranes 22 (2004) 223–237

Development of a hyperbolic constitutive model for expanded polystyrene (EPS) geofoam under triaxial compression tests Byung Sik Chuna, Hae-Sik Limb, Myung Sagongc,*, Kyungmin Kima a Department of Civil Engineering, Hanyang University, Seoul 133-791, South Korea Korea National Housing Corporation, Geotechnical Research Team, 175 Gumi, Bundang, Sungnam, Kyunggi 463-704, South Korea c Track & Civil Engineering Research Department, Korea Railroad Research Institute, 360-1 Woulam, Uiwang, Kyunggi 437-050, South Korea b

Received 7 May 2003; received in revised form 8 November 2003; accepted 18 March 2004

Abstract Triaxial tests were conducted to develop a constitutive model of cellular type expanded polystyrene (EPS) geofoam block under short-term (immediate) loading. Specimens of EPS geofoam with densities of 15, 20, 25, and 30 kg/m3 were tested under confining stresses of 0, 20, 40, and 60 kPa. Test results show that EPS geofoam displays nonlinear major principal stress–strain behavior, which is a function of confining stress and density, and a higher maximum compressive strength with increased density and confining stress. The EPS geofoam exhibits directional deformation along the axis of major principal strain with no dilative shearing, and the generated axial and volumetric strains produce a linear correlation that is a function of density and confining stress. From the test results, a new hyperbolic model simulating the stress–strain behavior of EPS geofoam is proposed. This model includes the major principal stress and strain as well as density and confining stress. Strain-dependent tangent modulus and Poisson’s ratio influenced by the confining stress and density of the EPS geofoam are derived from the model. The model shows the elastic tangent modulus and Poisson’s ratio increase with density and decrease with

*Corresponding author. Fax: +82-31-460-5319. E-mail address: [email protected] (M. Sagong). 0266-1144/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2004.03.005

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confining stress. Compared to previous models, the proposed hyperbolic model is shown to give best fit to the measured test data. r 2004 Elsevier Ltd. All rights reserved. Keywords: EPS geofoam; Hyperbolic constitutive model; Triaxial tests; Tangent modulus; Poisson’s ratio

1. Introduction Many practical techniques are available to improve the engineering properties of soft soil. One of these methods is the use of a light-weight backfill material such as EPS geofoam block. EPS geofoam reduces the total applied pressure on soft ground. Due to its light-weight, stand-alone characteristics, and ease of use, EPS geofoam has been used for many field applications such as a sub-base fill material (Du$skov and Scarpas, 1997; Du$skov, 1997; Riad et al., 2003), backfill material for bridge abutments (Bang, 1995; Abu-Hejleh et al., 2003), and thermal insulation for roads. When EPS geofoam is installed as fill material under a foundation, some deformation of the EPS geofoam will occur because of its primary function as a compressible inclusion. The stress–strain behavior of EPS geofoam for such cases depends on many factors such as density, magnitude of applied load, manufacturing process, and environmental factors. In addition, interactions between geofoam blocks and geofoam-soil further complicate the understanding of the behavior of EPS geofoam (Sheeley and Negussey, 2000). Due to all of these factors, the estimation of the stress–strain behavior of EPS geofoam in the ground is quite difficult and complicated. Typical EPS geofoam displays viscoplastic behavior after the initial linear elastic deformation without apparent rupture, distinguishable slippage, or interaction between particles (Cho, 1992; Preber et al., 1995; Horvath, 1997). Because the material deforms uniaxially along the axis of major principal strain without apparent shear, it is difficult to establish the failure state of the material. Thus, the use the parameters obtained from the initial linear elastic stress–strain behavior of EPS geofoam such as ‘‘elastic limit stress’’ at 1% strain of EPS geofoam, and ‘‘initial tangent modulus’’ measured in a monotonic rapid load test is typically recommended (Horvath, 2001). In addition to the external loading, time dependency (creep and relaxation) is an important characteristic of EPS geofoam (Chun et al., 1996; Horvath, 1997, 1998). The creep behavior of EPS geofoam is a function of applied stress and density of the EPS geofoam. Increased amount of creep is observed under higher applied stresses (Horvath, 1994; Chun et al., 1996). Therefore, with consideration to design, EPS geofoam must be used within an appropriate load range. In practice, the available axial strength of EPS geofoam is restricted to the elastic limit. This prevents longterm creep deformation of EPS geofoam. Another important consideration is the mechanical anisotropic behavior displayed by EPS geofoam. Specimens loaded perpendicular to the direction of fabrication show higher deviator stresses at failure than those loaded parallel to the direction of fabrication (Kutara et al., 1989).

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Therefore, a higher bearing capacity can be expected if the foam is loaded perpendicular to the direction of fabrication. Numerical modeling has been used for the design and analysis of EPS geofoam (Aytekin, 1997; Murphy, 1997; Du$skov Scarpas, 1997). However, to perform a more precise analysis, a constitutive model accurately describing stress–strain behavior of EPS geofoam is required. Several nonlinear and creep models for EPS geofoam have been proposed, but these models have limitations and cannot adequately describe the short-term stress–strain response of EPS geofoam. In order to develop a constitutive model of a cellular type EPS geofoam block, triaxial compression tests were conducted on EPS geofoam of various densities and subjected to a range of confining stresses. To ensure the reliability of the results, at least two tests were performed for each case. Based on the test results, a new constitutive model that incorporates major principal stress, major principal strain, density, and confining stress for the shortterm (immediate) deformation of EPS geofoam is proposed. The rheological and anisotropic aspects of EPS geofoam were not addressed in this study.

2. Earlier constitutive models of EPS geofoam Several models have been proposed to describe the stress–strain-time behavior of EPS geofoam. These can be divided into two different categories: time-independent stress–strain models (e.g. Preber et al., 1995) and time-dependent (creep) models (e.g. Findley and Khosla, 1956; Findley et al., 1989). The strain of a material showing time-dependent viscoplastic behavior can be expressed as follows: et ¼ ei þ ec ;

ð1Þ

where et is the total strain; ei the immediate strain component; ec the time-dependent strain component. The immediate strain is a time-independent quantity generated under a static or rapid loading condition. General stress–strain behavior for immediate strain can be simulated in two different ways: (1) a linear elastic model using initial tangent modulus or (2) a nonlinear inelastic model. The nonlinear inelastic model is of interest due to its greater accuracy over a larger range of strain. Preber et al. (1995) developed a bilinear model to simulate the undrained triaxial test results of EPS geofoam block; the densities of the tested material were 16, 20, 24, and 32 kg/m3. Each specimen was tested under confining stresses of 0, 21, 41, and 62 kPa. The tested specimens conform to the definition reported in ASTM C 578-92, which specifies the density of rigid cellular polystyrene to be between 12 and 48 kg/m3. The Preber et al. model describes the stress–strain behavior of EPS geofoam using five parameters: the elastic and plastic moduli (Ei ; Ep ), the intersection point of the elastic and plastic ranges (I), the axial strain and stress values at the intersection of the initial elastic, and the plastic tangent line (X0 ; Y0 ). The mathematical form of the model is    Ei e s ¼ ðI þ Ep eÞ 1  exp Ce2  ; ð2Þ I

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  Ei 1 Y0 C¼  2 ln 1  ; IX0 X0 ðI þ EP X0 Þ

ð3Þ

I ¼ ð107 þ 8:9gÞ þ ð0:63  0:062gÞs3 ;

ð4Þ

Ei ¼ ð4180 þ 382:2gÞ þ ð6:2  0:52gÞs3 ;

ð5Þ

Ep ¼ ð104 þ 4:3gÞ þ ð3:6 þ 1:5gÞs3 ;

ð6Þ

Y0 ¼ ð1:4 þ 8:7gÞ þ ð1:1 þ 0:04gÞs3 ;

ð7Þ

s3 for 0ps3 p62 kPa; ð8Þ 62 kPa where s is the axial stress (kPa); e the axial strain; X0 the axial strain at the intersection of initial tangent and plastic tangent lines; g the density of EPS geofoam (kg/m3); s3 the confining stress (kPa); and m the Poisson’s ratio. Some coefficients used in the above equations are modified from the original publication to adopt density instead of unit weight in the formulas. All of the parameters are represented as functions of density and confining stress. This bilinear model is a data back-fitted model for a particular EPS geofoam with respect to confining stress and density for immediate strain. However, EPS geofoam blocks behave nonlinearly rather than bilinearly. Therefore, the model is not accurate over a wide range of strain. The second category of the model is the time-dependent or creep model. Findley’s model and the modified Findley’s model fall into this category. Findley and Khosla (1956) proposed a power-law model to describe the creep behavior of a polymer material:  nF t et ¼ ei þ m ; ð9Þ t0   s ei ¼ eiF sinh ; ð10Þ siF   s m ¼ mF sinh ; ð11Þ smF m ¼ 0:2  0:5

where et is the total strain; ei the immediate strain; eiF the dimensionless Findley material parameter; siF ; smF is the Findley material parameters with dimensions of stress; s the applied stress; mF ; nF the dimensionless Findley material parameters. This model represents the time-dependent behavior of EPS geofoam by a hyperbolic sine function with dimensionless Findley material parameters, a dimensionless material parameter, time, and applied stress. Stress and environmental factors such as temperature and water content affect the magnitude of parameters. Some simplified versions of the Findley equation have been proposed, including models developed by Chambers and the French Government Central Road Research Laboratory, the Laboratoire Central Ponts et Chaussees (LCPC) (Horvath, 1998).

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The mathematical form of the Chambers model is similar to that of the original Findley model. Therefore, a brief introduction of the LCPC model is given as follows:   s ð12Þ et ¼ þ atn ; Eti  2:47 s a ¼ 0:00209 ; sy    s n ¼ 0:9 log10 1  ; sy

ð13Þ

ð14Þ

sy ¼ 6:41g  35:2;

ð15Þ

Eti ¼ 479g  2875;

ð16Þ

where Eti is the initial tangent Young’s modulus (kPa); sy the yield strength of EPS geofoam (kPa); g the EPS geofoam density (kg/m3). According to the LCPC model, immediate strain is a function of applied stress and the density of the EPS geofoam block. Although the Findley or modified Findley models can represent the stress–strain–time behavior of EPS geofoam, their usefulness is limited. For example, to define the Findley model parameter for EPS geofoam, the environmental factors must be constant. Obviously environmental factors vary widely in actual field conditions. Therefore, a specific stress–strain–time relationship of the EPS geofoam must be established for each condition. Additionally, Findley or Findley-type models do not consider the influence of confining stress. It is obvious that a more accurate model for predicting the stress– strain–time related behavior of EPS geofoam blocks under actual field conditions is needed.

3. Triaxial compression tests of EPS geofoam EPS geofoam is often used as a backfill material in blocks under moderate confining stress. Therefore, it is necessary to develop a rational model that considers applied confining stress and density. To establish a constitutive model for a cellular type EPS geofoam block, triaxial compression tests were conducted. Unlike the general triaxial test, the specimens were tested in a dry condition, so the generation of pore pressure and the drainage of water from the specimen were not a concern in the tests. Density is a commonly used parameter in practice, and thus was used as an index property for this study. The EPS geofoam with densities of 16–32 kg/m3 are often used in field applications (Horvath, 2001). Therefore, the densities of EPS geofoam used in this study were 15, 20, 25, and 30 kg/m3. These densities follow the practical range of application, conform to ASTM C 578-92, and are similar to those tested by Preber et al. (1995).

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For the purposes of this study, an EPS geofoam block was cut into a 100 mm thick plate. Cylindrical specimens with a diameter of 50 mm and a height of 100 mm were then cut from the EPS geofoam plate. The specimens were extracted by using a specially designed hollow cylindrical cutter having the same inside diameter and height as the specimens. The cutter was carefully screwed into the EPS geofoam block at a slow rate to prevent any irregularities from forming on the surface of the specimens. The cutter was then released in the opposite direction. This process nearly eliminated the generation of surface irregularities often observed when a hot wire is used for this purpose and produced a very smooth surface on the specimens. The entire processes of extracting a specimen from the EPS geofoam plate took about 10–20 min. The confining stresses applied to the EPS geofoam specimens were 0, 20, 40, and 60 kPa. The axial load was applied perpendicular to the direction of fabrication with a loading rate of 1 mm/min (1%/min of strain rate). The typical monotonic straincontrolled loading rate adopted for EPS geofoam is about 10% of strain per minute (Elragi et al., 2000; Horvath, 2001). Obviously, the loading rate used in this study is slower than that used in earlier studies. The average required loading time for each specimen was about 10 min. In an earlier study conducted by Chun et al. (1996), specimens with a density of 12 kg/m3 and an applied stress of 90 kPa did not show any creep deformation after 4 days of loading. The test results showed that the increase of applied stress increases the possibility of creep deformation. Since the applied stress in this test is less than 90 kPa, no significant creep effect was expected during the test. Axial and volumetric strains were measured to calculate Poisson’s ratio. The volumetric strain was measured by the water drained from the triaxial cell during loading. It was assumed that the volume of drained water from the triaxial cell was equal to the deformed volume of EPS geofoam. The water that was drained from the cell was automatically replaced to ensure a constant confining stress. Fig. 1 shows the nonlinear principal stress–strain behavior of EPS geofoam of different density and under different confining stress conditions. The data show that a higher EPS geofoam density results in a higher compressive strength and initial tangent elastic modulus. It can also be observed that the higher the confining stress, the higher the ultimate compressive strength. However, it is evident that the effect of confining stress is small. In the elastic range, the stress–strain behavior of specimens with the same density is similar despite the different confining stress levels. Thus, density is the major influence on the stress–strain behavior of EPS geofoam when the confining stress is relatively low. The ultimate compressive strengths of the specimens in this test are higher than those of Preber et al. (1995). The ultimate compressive strengths of EPS geofoam reported by Preber et al. (1995) were about 80 kPa for the specimen with a density of 24 kg/m3 under 21 kPa of confining stress and 160 kPa for 32 kg/m3 under 41 kPa of confining stress. Because there are no details available on the actual testing procedures, specimen dimensions, and specimen preparation from the Preber et al. (1995) study, it is not clear what caused the differences with respect to this study. Volumetric and major principal strains were measured and are presented in Fig. 2. In this figure, the volumetric strain of the EPS geofoam is correlated linearly with

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Fig. 1. Stress–strain behavior of EPS geofoam of various density and confining stress conditions: (a) density=15 kg/m3, (b) density=20 kg/m3, (c) density=25 kg/m3 and (d) density=30 kg/m3.

major principal strain except near the origin of the 15 kg/m3 density specimen with no confining stress. It is thought that irregular contact between the specimens and the loading plate and a possible seating error may have produced the deviated volumetric and major principal strain relationship near the origin. Especially under the zero confining stress condition, minor folds formed on the confining membrane may have affected the measurement of the volume change of the specimens. Regardless of the deviation, Fig. 2 shows a trend similar to those reported by Hamada and Yamanouchi (1989). Their study showed a constant volume change rate that is dependent on the density of the EPS geofoam and the lateral pressure applied. The various slopes of the plots in Fig. 2 indicate the different values of Poisson’s ratio for given densities of EPS geofoam and the applied confining stresses. As the deviator stress increased, the EPS geofoam deformed more in the major principal stress direction, and displayed little lateral deformation.

4. Development of a constitutive model for EPS geofoam 4.1. Development of hyperbolic model from the tests The hyperbolic model has been used to describe stress–strain behavior of various soil materials (Kondner, 1963; Duncan and Chang, 1970; Kulhawy and Duncan,

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Volumetric Strain ( σv)

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Fig. 2. Volumetric and major principal strains behavior of EPS geofoam of various density and confining stress conditions: (a) density=15 kg/m3, (b) density=20 kg/m3, (c) density=25 kg/m3 and (d) density=30 kg/m3.

1972). The mathematical form of the model is represented by the deviator stress and the major principal strain terms with material-dependent constants. Engineering properties such as stress-dependent tangent modulus can be obtained from the derivative of the stress–strain relationship. The modulus has been applied for numerical analyses of geotechnical structures (Duncan and Chang, 1970; Kulhawy and Duncan, 1972). Another elastic property, Poisson’s ratio, was initially assumed to be constant (Duncan and Chang, 1970), but subsequently modified to be a function of confining stress (Kulhawy and Duncan, 1972) for numerical analyses. The stress–strain behavior of EPS geofoam is closely linked to its density and confining stress as shown in the previous section. From the experimental results, a constitutive model for EPS geofoam can be represented as follows: s1 ¼

aeb1ð%Þ c þ eb1ð%Þ

;

ð17Þ

where s1 is the major principal stress (kPa); e1% the major principal strain (in percent) and a; b; c are functions of confining stress and density as given in the following: a ¼ 60:955 þ 9:843g þ 0:339s3 ;

ð18Þ

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b ¼ 1:135 þ 0:042g  0:008s3 ;

ð19Þ

c ¼ 0:437 þ 0:102g  0:002g2 þ 0:011s3  0:00039gs3 ;

ð20Þ

where g is the density of EPS geofoam (kg/m3) and s3 the confining stress (kPa). It can be seen that the mathematical form of the proposed model is different from the typical hyperbolic function represented by the deviator stress and the major principal strain terms with material-dependent parameters. The part of difference between the models may be due to the different deformation processes of EPS geofoam and soil. As shown in Fig. 2, EPS geofoam deforms mostly in major principal strain direction with relatively small lateral deformations. A soil, however, undergoes contraction or dilation processes affected by the soil properties, confining stress, and deviator stress. Therefore, the stress–strain response of EPS geofoam can be effectively represented in terms of major principal stress and strain. A sensitivity study of the parameters a, b, and c was performed. The result shows that the most sensitive parameter to the behavior of EPS geofoam is the parameter ‘‘a’’. Parameter ‘‘a’’ produces a higher ultimate compressive strength and elastic modulus of EPS geofoam than do the other parameters. Parameter ‘‘a’’, derived from Eq. (17), is the asymptotic value when the major principal strain goes to infinity. Density is the controlling factor for parameter ‘‘b’’. As parameter ‘‘b’’ increases, ultimate strength increases and the elastic modulus decreases. Parameter ‘‘c’’ shows the opposite effect compared to parameter ‘‘a’’. An increase in parameter ‘‘c’’ decreases the ultimate strength and the elastic tangent modulus value. 4.2. Estimation of tangent modulus and Poisson’s ratio The main feature of the proposed hyperbolic model in this study is that the model can represent the stress–strain behavior of EPS geofoam as a function of an applied confining stress and density of the material. In addition, it is also possible to obtain strain dependent tangent modulus and Poisson’s ratio value. The strain-dependent tangent modulus is obtained through the derivative of Eq. (17) with respect to strain. Thus, the modulus is represented as a function of confining stress, density, and strain. The equation for tangent modulus is Et ¼

abceb1 ds 1ð%Þ ¼ 2b  100; de1ð%Þ =100 e1ð%Þ þ 2ceb1ð%Þ þ c2

ð21Þ

where Et is the tangent modulus (kPa); e1% the axial strain (in percent); a; b; c the parameters shown in Eqs. (18)–(20). Since the strain used for Eqs. (17) and (21) is a percentage, a coefficient of 100 is required. The tangent modulus in Eq. (21) can be employed for numerical analysis with successive increments of loading where variation of the modulus is expected. Elastic tangent modulus at 1% of strain in Eq. (21) increases with density and decreases with confining stress.

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Another elasticity parameter, Poisson’s ratio (n), is generally expressed as   1 ev 1 n¼ ; 2 e1

ð22Þ

where ev is the volumetric strain and e1 the axial strain. According to Preber et al. (1995), Poisson’s ratio of EPS geofoam was expressed as a function of confining stress only. However, Fig. 2 shows the Poisson’s ratio of EPS geofoam varies with not only confining stress but also the density of the material. Thus, density and confining stresses must be considered to accurately estimate Poisson’s ratio. The mathematical form of Poisson’s ratio obtained by trial and error method is as follows: n ¼ 0:0967 þ 0:00308g  0:0023s3 ;

ð23Þ 3

where g is the density of EPS geofoam (kg/m );and s3 the confining stress (kPa). Eq. (23) shows that Poisson’s ratio increases with density and decreases with confining stress, as does the elastic tangent modulus. The distribution of Poisson’s ratio at different confining stresses and densities is represented in Fig. 3. In Fig. 3, Poisson’s ratio is nearly zero when the confining stress is approximately 60 kPa and the density is 15 kg/m3. This shows that EPS geofoam of a specific density exposed to some level of confining stress can display no lateral strain; only major principal strains occur under that condition. Due to this characteristic, EPS geofoam can display negligible lateral stress development in abutments and other retaining structures. A comparison between measured and estimated Poisson’s ratio values shows that the coefficient of correlation is at least 0.9934. Poisson’s ratio used for this comparison was obtained when the major principal strain of EPS geofoam was about 10% for each specimen in Fig. 1. However, as shown in Fig. 2, constant ratios of volumetric and major principal strains representing constant Poisson’s ratios were observed during the tests. This particular characteristic of EPS geofoam is obviously different from that of the soil. Soils showing volumetric expansion, such as dense sand or highly overconsolidated soil under a moderate confining stress condition,

Poisson's Ratio

0.2 0.15 0.1 0.05 0 30 E 25 De PS G ns ity eofo a [k g/ m m3 ]

20 15 0

40

20

60

Pa] ing Stress [k

Confin

Fig. 3. Distribution of Poisson’s ratio as function of density and confining stress.

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Table 1 A comparison between initial tangent modulus and Poisson’s ratio from this study and from previous correlations Density (kg/m3)

Confining stress (0 kPa)

15 Initial tangential Young’s Modulus

Eti Eti Eti Eti

Poisson’s ratio

n (Eq. (27)) n (Eq. (23))

(MPa) (MPa) (MPa) (MPa)

(Eq. (24)) (Eq. (25)) (Eq. (26)) (Eq. (21))

20

3.8 4.3 3.8 3.6

5.4 6.7 6.0 6.6

0.086 0.143

0.114 0.158

25

30

7.5 9.1 8.3 10.1

10.1 11.5 10.5 13.9

0.142 0.173

0.170 0.189

display different ratios of major principal and volumetric strains during compression. Several correlations provided for initial tangent modulus (Eti ) and Poisson’s ratio are summarized below (Horvath, 1995) (density is the only variable in these correlations) Eti ¼ 0:0097g2  0:014g þ 1:8;

ð24Þ

Eti ¼ 0:479g  2:875;

ð25Þ

Eti ¼ 0:45g  3;

ð26Þ

v ¼ 0:0056g þ 0:0024:

ð27Þ

Table 1 shows a comparison between the earlier correlations of initial tangent modulus and Poisson’s ratio and the calculated values from Eqs. (21) and (23). For the calculation of the elastic modulus, a strain of 1% or less is assumed to be in the elastic range. Thus a strain of 1% is applied to Eq. (21) to calculate the initial tangent modulus. In addition, because Eqs. (24)–(27) cannot account for confining stress, no confining stress is applied to Eqs. (21) and (23). This will generate a similar condition between the earlier correlations and the results of this study. A deviation of the estimated initial tangent modulus between earlier correlations and this study increases with the density of the EPS geofoam. Conversely, the difference between the estimated Poisson’s ratios obtained from Eq. (23) and from Eq. (27) decreases with an increase in EPS geofoam density. 4.3. Verification of the proposed model A comparison between the proposed model from this study and the test results is shown in Fig. 4. The open symbols represent the test results, and the estimated stress–strain behavior of EPS geofoam from the model is plotted with a dotted line.

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Fig. 4. A comparison between the hyperbolic model and test results: (a) confining stress=0 kPa, (b) confining stress=20 kPa, (c) confining stress=40 kPa and (d) confining stress=60 kPa.

The figure shows that the proposed model closely predicts the approximate stress– strain behavior of EPS geofoam tested under various conditions. Furthermore, the proposed hyperbolic model is compared with the earlier models such as Findley, LCPC, and Preber et al. to evaluate the accuracy of the models. The required parameters for the Findley model are adopted from Horvath (1998). For the Findley and LCPC creep models, only the immediate strains are compared in Fig. 5. The parameters adopted from Horvath (1998) are applicable when the density of EPS geofoam is 20 kg/m3 and applied stress is below 50 kPa. The applied Findley material constant (eiF ) is 0.011, and the applied confining stresses for the comparison are 0, 20, 40, and 60 kPa. The bilinear model by Preber et al. (1995), the Findley model, and the proposed hyperbolic model show the nonlinear behavior of EPS geofoam in Fig. 5. The LCPC model, however, shows a linear trend of EPS geofoam. As shown in Fig. 5, the LCPC model and the hyperbolic model show similar stress–strain behavior when the principal stress is less than 100 kPa and the principal strain is less than 2% in the linear elastic range. Among the models, the proposed hyperbolic model shows the best agreement with the test results with respect to the ultimate strength and the location of the transition point between elastic and plastic regions. Within the given range of strain (about 8% maximum), the Findley and Preber et al. models do not approach a limiting asymptotic stress value. From this comparison, it is clear that, when compared with all other available models, the hyperbolic model can most

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Fig. 5. A comparison between hyperbolic model and previous models (density 20 kg/m3): (a) confining pressure=0 kPa, (b) confining pressure=20 kPa, (c) confining pressure=40 kPa and (d) confining pressure=60 kPa.

accurately simulate the stress-immediate strain behavior of the particular EPS geofoam in this study.

5. Conclusions EPS geofoam has been used as a fill material for embankments, sub-base, backfill for retaining walls, and as an insulation material. A series of triaxial tests were carried out to understand the stress-immediate strain behavior of EPS geofoam. From the tests, a nonlinear constitutive model of EPS geofoam as a function of density and confining stress was developed, and model predictions were compared with test results. The conclusions of this study are as follows: (1) The EPS geofoam shows nonlinear stress–strain behavior related to the density and confining stress under triaxial loading conditions. From the test results, the higher the density of EPS geofoam, the higher the ultimate strength. Furthermore, as the confining stress increases, the ultimate strength increases. Additionally, a linear correlation was observed between the volumetric and axial strain, this correlation is a function of density and confining stress. (2) Based upon the test results, the proposed hyperbolic model accurately describes the nonlinear stress–strain behavior of EPS geofoam. The hyperbolic model uses

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the parameters of major principal stress and strain with coefficients a; b; and c related to density and applied confining stress. (3) A parametric study shows that the coefficient ‘‘a’’ represents the asymptotic ultimate strength at the infinite strain. The coefficient ‘‘b’’ is predominantly controlled by density. By increasing the parameter ‘‘b’’ the major principal stress is increased and the elastic modulus is decreased. Coefficient ‘‘c’’ affects the stress–strain relationship of EPS geofoam by lowering the ultimate strength and elastic modulus, as ‘‘c’’ increases. (4) A strain-dependent tangent modulus is derived from the hyperbolic model and a correlation of Poisson’s ratio is proposed. Initial tangent modulus and Poisson’s ratio derived from the model show a good correlation with previous empirical equations. (5) From the comparison of previously proposed models including Findley’s model, the LCPC model, and the Preber et al. model, the hyperbolic model from this study gives the best fit to the test results. The comparison is made only for immediate strain under static loading and no creep. A time-independent constitutive model of EPS geofoam is developed for a static loading condition for a particular type of geofoam material. However, the proposed model may not accurately account for the influence of intermediate stress (s2 ), complex stress paths (loading/unloading), dynamic loading conditions, creep, anisotropy, loading rate, and environmental factors. In addition, since the test results were acquired from a single specific type and size of specimen, material dependency and size effects cannot be considered. Further studies are necessary in order to verify and develop the advanced model.

Acknowledgements The writers are thankful to reviewers for their detailed and constructive comments.

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