Stress–strain modeling of EPS geofoam for large-strain applications

ARTICLE IN PRESS

Geotextiles and Geomembranes 24 (2006) 79–90 www.elsevier.com/locate/geotexmem

Stress–strain modeling of EPS geofoam for large-strain applications Hemanta Hazarika Geotechnical and Structural Engineering Department, Port and Airport Research Institute, 3-1-1 Nagase, Yokosuka 239-0826, Japan Received 27 April 2004; received in revised form 12 October 2005; accepted 29 November 2005 Available online 19 January 2006

Abstract Both small- and large-strain applications of expanded polystyrene (EPS) geofoam involve interactions with the surrounding geologic materials. The stress–deformation response of this material, however, differs significantly from those of the adjoining geologic materials. A well-justified constitutive law for EPS is, thus, a prerequisite for reliable solutions for soil-structure interaction problems where such material is used. This paper describes a stress–strain law for EPS geofoam for its large-strain applications based on the incremental theory of plasticity. In the derivation of the constitutive relationship, the geofoam was taken as a von Mises material, and it was assumed that the hardening regime follows a hyperbolic curve. The material parameters of the constitutive model were determined from a series of unconfined compression tests performed on EPS specimens of various sizes, shapes and densities. These parameters are functions of the absolute dimensions of the tested specimens as well as the density of EPS. The validity of the model was confirmed by numerical simulations on the compression testing program of EPS geofoam. r 2005 Elsevier Ltd. All rights reserved. Keywords: Constitutive law; EPS geofoam; Hardening; Modified yield stress; Plasticity

1. Introduction With the rapid change of geosynthetic materials used in construction engineering, expanded polystyrene (EPS) geofoam has firmly established its position as a material to be reckoned with, and has been growing fast for the last forty years. Geofoam usage dates as far back as the 1960s, when a patent for using geofoam as pavement insulation was granted in the USA. Positive experiences with EPS as insulating material led to the next step, the use of it as a lightweight substitute. Since its introduction as a soft ground construction material in Norway in the 1970s and its subsequent application for earth pressure reduction (Nakase, 1974), EPS geofoam has been receiving increasing attention in the civil engineering construction world. It is a material that has been used in many countries as a lightweight and compressible geomaterial. Particularly in Japan, where construction on weak soils is inevitable, the use of EPS has seen a steady increase over the last decade. This, combined with other factors, has contributed to Corresponding author. Tel.: +81 46 844 5058; fax: +81 46 844 0839.

E-mail address: [email protected]. 0266-1144/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2005.11.003

Japan’s position as the leading user of geofoam products. A comprehensive treatment of geofoam products, covering their behavior, applications and design parameters, has been reported by EPS Development Organization (EDO), Tokyo, Japan (EDO, 1992) and Horvath (1995). The widespread popularity of this kind of material is due to its several outstanding characteristics, e.g. lightweight, compressible, self-standing, water resistant, ease of use, etc. EPS, thus, finds a myriad of applications in civil engineering construction. Such applications can be divided broadly into two catergories; small-strain and large-strain applications. Small-strain applications involve the widely used applications of EPS as lightweight fill, in which the lightweight chracteristics of the material are made use of. Large-strain applications are typically limited to its compressible inclusion function (Horvath, 1997) in which the compressibility of EPS (this is not desirable in the small-strain applications) is exploited. The desired constitutive properties of EPS geofoam, thus, vary, depending on each application. EPS products can be manufactured to be stiff or compressible as is needed for a specific functional application. With the use of EPS geofoam now becoming more widespread, the need for material properties research has

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become more evident. Its material properties have been investigated experimentally for many years by the researchers who are involved with the design and application of this geofoam product. Many of the reported large-scale projects, where EPS was used, also resorted to full-scale numerical analysis prior to the execution of those projects. However, the reports lack an adequate description of the constitutive law of EPS geofoam used in such analyses. EPS demonstrates different stress–strain response than those of the adjoining geologic materials. Therefore, in order to analyze the deformation and failure behavior of geotechnical engineering structures where EPS geofoam is used, a mathematically sound constitutive law, that can correctly predict the elasto-plastic behavior of EPS, is important. However, very few researches have been done towards the establishment of a constitutive law that can adequately describe the nonlinear stress–strain behavior of this material for its large-strain applications. This study is a step forward in that direction. 2. Review of the constitutive models of EPS geofoam A constitutive law defines the relationship between the physical quantities such as stress, strain and time. In spite of its widespread usage, studies relating to the constitutive behavior of EPS geofoam have not received much attention in the geotechnical engineering community. While constitutive behavior of geofoam for small-strain applications has been studied for more than 30 years, its counterpart, large-strain applications has been relatively little studied. The constitutive models developed until now for the large-strain applications of EPS geofoam can be broadly classified into two categories depending upon whether the creep (time dependency) is considered or not. In other words, there exist two schools of thought regarding the EPS geofoam constitutive behavior. One is the timeindependent model, which is useful for relatively rapid loading, and thus the influence of creep can be neglected. In this type of models, the emphasis is more on the immediate strain component. Preber et al. (1994) developed a constitutive law based on the test results of triaxial tests and repeated loading test. Included in it were the effects of the EPS density and the confining stress. Poisson’s ratio was taken as a function of the confining stress. A similar approach was adopted by Chun et al. (1998, 2004), and they proposed a hyperbolic constitutive relationship based on the triaxial test of various densities of EPS, in which Poisson’s ratio has multi-linear relation with the density and the confining stress. The model developed by Chun et al. (2004) includes the major principal stress and strain as well as density and confining stress. However, the aforementioned constitutive models are based on the curve fitting method of mathematical functions, and therefore, are highly empirical in nature. For materials such as EPS geofoam, where anisotropy, inhomogeniety, or other complications arise frequently, the simplicity of the

mathematical functions begins to disappear under correction factors. For example, in some cases of large-strain applications, where resilient (elasticized) EPS is used, anisotropy becomes significant. Kutara et al.’s (1989) experimental observations also indicate that the direction perpendicular to the fabrication show higher deviator stress at failure than that in the direction of fabrication. Nevertheless, it is important to derive a stress–strain law that is mathematically consistent and physically sound. Especially when EPS is used for large-strain applications such as compressible buffer (Horvath, 1997; Hazarika, 2001; Tsukamoto et al., 2001, 2002), its stiffness becomes an important parameter. Stiffness is governed by the stress level and the duration of the loading, which ideally necessitates the use of visco-plasticity based constitutive law. Time-dependent models that belong to the second group of EPS constitutive models take into account the nonlinear visco-elastic or visco-plastic behavior of EPS. Creep is considered an important factor in such models. The creep of geofoam is a function of applied stress and density. Time-dependent models proposed by Findley and Khosla (1956) and Findley et al. (1989) can deal with the influence of creep of geofoam. The Findley equation has been used since the 1950s as one of the primary mathematical models for the time-dependent mechanical behavior of solid polymeric and non-polymeric materials. The Findley model represents the time-dependent behavior of geofoam by a hyperbolic sine function with dimensionless Findley material parameters, time and applied stress. Horvath (1998) describes the application of such a model to EPSBlock geofoam. Numerical and experimental studies on creep behavior of EPS geofoam have also been studied by various other researchers (Srirajan et al., 2001; Zou and Leo, 2001). Murphy (1997) conducted a numerical study on the influence of geofoam creep on the performance of compressible inclusion. However, for rapid loading behavior, in which creep can be neglected, plasticity-based models suffice for the intended purpose. Such models can be expected to provide greater accuracy over a large-strain range application of EPS such as compressible inclusion. In addition, since EPS exhibits a highly nonlinear stress–strain response, a constitutive law that is based on plasticity models is more suitable than those based on mathematical functions even though the latter are relatively easier to program. Inglis et al. (1996) reported the use of plasticity based EPS constitutive model in their simulation of a basement wall under simulated earthquake loading. Their research was on the numerical simulation of a project involving a rigid retaining wall using FLAC (Itasca, 1995). In the simulations, a double yield model was employed as the constitutive law of EPS geofoam, which was used as a compressible buffer behind the wall. The double yield model has an advantage that the hardening under isotropic compression can be adequately described by using such a model. However, this model does not take into the account

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the density as well as the size- and shape-dependent constitutive behavior of EPS. In this research, an attempt was made towards the development a plasticity-based constitutive model for modeling large-strain behavior of EPS. The model can also take into the account the density dependent characteristics of EPS geofoam. A time-independent model was developed without any explicit consideration to rheological and anisotropic aspects of the EPS geofoam behavior. The emphasis was more on simplicity rather than complexity. The effect of creep was neglected for mere simplicity, and thus the model is valid for immediate strain under rapid (short term) loading conditions. EPS strength demonstrates wide variation, depending on the tested sample size (EDO, 2003; Elragi et al., 2000; Horvath, 1995). Therefore, it is also necessary that the effects of these factors be included in the formulation of geofoam constitutive law. The establishment of such a law will also aid engineers in eliminating the scale effects that arise due to differences in the behavior of a full-size block versus a laboratory-size specimen. In the following sections, the underlying principles in the derivation of such law are described. The model was then calibrated by conducting a numerical simulation, and by comparing the results with the experiments. The experimental data used in the development of the model are presented first. 3. Compression testing of EPS A series of unconfined compression tests was performed on specimens of various sizes and shapes at room temperature. The specimens were prepared from EPS blocks of 2 m
1 m
0.5 m in size with two different nominal densities (16 and 20 kg/m3) supplied by EPS manufactures. The blocks were manufactured using modified expandable polystyrene with 2–3% regrind and 0.8–1% flame retardant. Block molding was accomplished by using pre-puff in steel mold into which steam is injected under vacuum. After removing the blocks from the mold, they were seasoned by allowing them to cool down and to degas any remaining butane and gain their final strength. The seasoning period lasted more than a week. The seasoned blocks were then trimmed to their final dimension (2 m
1 m
0.5 m). A common practice throughout the world is to use 50 mm cubic specimen in the compressive testing of EPS. However, performance observation (Frydenlund and Aaboe, 1996) revealed that, the Young’s modulus values, derived from the tests based on the standard 50 mm cubic specimen are underestimated, resulting in the overestimation of deformation. Further research, however, is necessary to establish a reliable correlation between the Young’s modulus as measured using 50 mm cubes versus the Young’s modulus of full-size blocks as they are used in practice. Because the specimen size has not been standardized yet for EPS testing, any attempt to characterize the constitutive law of EPS should also include factors that

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account for the specimen size and shape. In this research, specimens of various sizes were tested in a series of uniaxial compression tests. Specimens of different sizes and shapes were prepared from EPS blocks of nominal densities 16 kg/m3 (D-16) and 20 kg/m3 (D-20). The dimensions of specimens were as follows: (a) 100 mm
100 mm
100 mm, (b) standard 50 mm
50 mm
50 mm, (c) 50 mm
50 mm
100 mm, (d) f50 mm
100 mm and (e) 100 mm
100 mm
50 mm. The test samples were prepared by cutting the original block (2 m
1 m
0.5 m) using hot wire cutter. Special cautions were taken to ensure that the loading surfaces of the specimens were as smooth as possible to alleviate the effect of surface irregularities on the test results. Special large area loading plates were fixed to the top and bottom loading plates of an automated uniaxial compression device to apply uniform pressure over the specimen. It is worthwhile mentioning here that a strain rate of 10% per minute is considered the de facto standard worldwide. However, due to limitations of the testing machine used in this study, only a single digit strain rate (o10%/min) could be imparted to the loading unit. The experiments were conducted at various strain rates (ranging from 1%/min to 9%/min). For the sake of brevity, however, the results from only the higher strain rate (i.e. 9%/min) are reported here. Interested readers are referred to the publications elsewhere (Hazarika, 2001; Hazarika et al., 2001), where further details can be found. The stress-deformation response of the test specimens of nominal densities 16 kg/m3 (unit weight 0.1670.01 kN/m3) þ0:015 and 20 kg/m3 (unit weight 0:200:01 kN=m3 ) are shown in Fig. 1. It can be seen that the stress–strain response is highly nonlinear and the elastic zone is very small. It is, however, important to note that the shape of EPS stress–strain curve turns sigmoidal if straining is continued to higher range (beyond 60% as shown in Fig. 2). Therefore, it is necessary that the stress–strain behavior be obtained at least up to the 60% strain level. However, due to limitation of the present experimental setup, tests beyond the 30% strain levels could not be performed. Fig. 1 reveals that, depending on the size and shape, there is a significant difference in the compressive behavior as plastic deformation occurs, particularly when the material enters into the large deformation range. Special attention needs to be paid to the commonly used 50 mm cube specimens, which underestimate the response compared to the 100 mm cube specimens. Therefore, in order to confirm the compression behavior of EPS geofoam to be used in any project, recommendations for using larger specimens assume importance. The use of large specimen size in the compression testing of EPS was reported in Duskov (1997) and has been studied extensively by Elragi et al. (2000). Defining the stress corresponding to 10% of the compressive strain as the compressive strength, the strengths of the tested specimens were determined and the strengths are compared in Fig. 3. This figure reveals

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82 160

150 Density D-16 (16 kg/m3) Density D-20 (20 kg/m3)

120 100

Compressive Strength (kPa)

Compressive Stress (kPa)

140

80 60 50*50*50 100*100*100

40

100*100*50 50*50*100

20

100

50

φ=50; H=100

0

0

5

10

15

20

25

30

Axial Strain (%)

(a) 180

100

φ 50×100

Specimen Shape and Size (mm)

80

Fig. 3. Comparisons of compressive strengths. 50*50*50

60

100*100*100

40

50*50*100 100*100*50

20 0

(b)

100×100×50

120

50×50×100

50×50×50

Compressive Stress (kPa)

140

100×100×100

0

160

φ=50; H=100

0

5

10

15

20

25

30

Axial Strain (%)

Fig. 1. Stress–strain behavior of different tested specimens: (a) density D-16 (16 kg/m3) and (b) density D-20 (20 kg/m3).

clearly that the 50 mm cubic specimen renders the least compressive strength and the 100 mm cubic specimen gives the highest compressive strength. It should also be observed that the larger the contact area of compression, the higher the strength. This characteristic of the EPS geofoam again demonstrates the effect of specimen size on its constitutive properties. Based on the experimental results reported here (for 100 mm
100 mm
100 mm specimen loading at 9%/min strain rate) and other experimental data reported elsewhere (Takada et al., 2002) for the same specimen size with nominal density of 12 kg/m3, the elastic modulus and density relationship can be represented as shown in Fig. 4. The slope of the initial linear-elastic portion of the stress–strain curve was taken as the elastic modulus. In this context, it is worth mentioning that the strain rate has a profound influence on the elastic modulus and the compressive strength of EPS. Higher strain rates result in higher modulus and compressive strength. During the course of a test, when the strain rate is changed, the stress–strain curve shifts up or down accordingly. Curve fitting of the test data (for 9%/min strain rate) yields the following modulus–density relationship with the modulus, E (MPa) and the density, r (kg/m3). E ¼ 0:41r  2:8

Fig. 2. Stress–strain behavior of 21 kg/m3 block-molded EPS specimen under rapid, strain-controlled and unconfined compression test (From Horvath, 1995).

(1)

The curves proposed by other researchers (Duskov, 1997; Elragi et al., 2000; Eriksson and Trank, 1991; Horvath, 1997; Negussey and Sun, 1996; van Dorp, 1988) are also

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20

Elastic Modulus, E (MPa)

necessitates to include the terms that can express these dependencies.

Duskov (1997) van Dorp (1988) Eriksson & Trank (1991) Horvath (1995) Negussey & Sun (1996) Elragi et al. (2000) Author's Tests Results

25

4.1. General formulation for elasto-plastic model Matsuzawa and Hazarika (1996) developed a constitutive model, describing the backfill behavior of a retaining wall problem, based on the incremental theory of plasticity with non-associative flow rule. The general form of constitutive relation obtained from that model, for any geologic material displaying non-linear strain hardening behavior, can be expressed as follows:
 De ðqg=qrÞ  De ðqf =qrÞ e dr ¼ D  0 de (2) h þ ðqf =qrÞ  De ðqg=qrÞ

Elragi et al. (2000) (0.6 m Cubic Specimen)

15 Duskov (1997) (φ=0.15 m, H= 0.3 m) 10

5 E = 0.41
- 2.8 (0.1 m Cubic Specimen)

0 5

10

15

20

25

30

83

35

40

Density, ρ (kg/m3)

Fig. 4. Elastic modulus–density relationship.

Here, r is the stress tensor, e is the strain tensor, De is the elasticity matrix, f is the yield function and g is the plastic potential function. h0 is the normalized hardening modulus (Rudniki and Rice, 1975) given by the following equation: h0 ¼

plotted in the same figure for comparison. It can be observed that the curve obtained by Elragi et al. exhibits the maximum value for the modulus, followed by that of Duskov. Also, the curve obtained by Elragi et al. is steeper than the other curves. Duskov’s test results yield a nonlinear power function for the relationship between the modulus and the density. All the other curves can be expressed as a linear function, and they display no significant differences in the values. The high values obtained by Elragi et al. and also the nonlinear curve obtained by Duskov were due to the differences of the sizes of the specimen used for testing. While Elragi et al. used 0.6 m cubic specimen, Duskov used cylindrical specimen (f ¼ 0:15 m, H ¼ 0:3 m). Duskov (1997) also observed that the elastic modulus values remain constant within the strain range up to 0.5%, beyond which it decreases with the increase of strain as a result of material softening. Concerning Fig. 4, it should be noted that the differences of magnitudes might also be the consequence of the nonuniformities of the specimens, which are from different sources with different manufacturing methods. For example, in Japan the blowing agent used for producing EPS block is butane, whereas in most other countries it is pentane. The influence of these differences (due to different product sources), therefore, also cannot entirely be ruled out. 4. Constitutive modeling The experimental results reviewed and discussed in the previous section reveal that the EPS geofoam displays not only highly non-linear material properties, but also density and shape-dependent characteristics. Therefore, any attempt to frame the constitutive law of EPS geofoam

qk qk þb p p qe q¯e

(3)

where k is the hardening function of the material to be modeled, ep and e¯ p are, respectively, the deviatoric and the volumetric component of the plastic strain ep . b represents the dilatancy of the material. Using Eq. (3), Eq. (2) can be rewritten in the following form:
 De ðm þ bnÞ  De ðqf =qrÞ e dr ¼ D  0 de (4) h þ ðqf =qrÞ  De ðm þ bnÞ where m and n refers to the unit normal vectors in the direction of deviatoric and volumetric stresses respectively. 4.2. Formulation for EPS geofoam constitutive law EPS undergoes tremendous compression when used as compressible buffer (Hazarika, 2001; Horvath, 1997; Tsukamoto et al., 2002). Since, EPS geofoam contains 98% air by volume, the initial volume change of EPS geofoam is due entirely to the expulsion of air from the cells. The contribution from the plastic volumetric component of strain is, thus, negligible for such material. On the other hand, EPS is a kind of material, which is free from dilatancy, a common phenomenon observed in most other geologic materials. Therefore, it can be said that the deviatoric plastic strain, ep and the volumetric plastic strain, e¯ p are uncoupled. Thus, omitting the term representing the dilatancy, the above equation leads to the following equation:
 De m  De ðqf =qrÞ dr ¼ De  de (5) ðqk=qep Þ þ ðqf =qrÞ  De m Taking EPS as a von Mises type material, and assuming the hardening to be isotropic, the yield function f can be

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84

4.4. Generalized constitutive law

expressed as f ðr; kðep ÞÞ ¼ s  k

(6)

Here s ¼ sij sij is the deviatoric stress tensor. The hardening function k can be assumed to be of hyperbolic form: ep a þ bep

(7)

Here a and b are the parameters that can be determined from the experimental data of unconfined compressive testing of the EPS geofoam blocks.

4.3. Definition of yield stress A typical stress–strain response of EPS geofoam is shown in Fig. 5(a). The figure also indicates the original method (Magnan and Serratrice, 1989) of determining the yield (plastic) stress ðsy Þ of such materials based on the initial tangent line and the plastic tangent line as shown. Horvath (1995) coined this plastic stress as the yield stress of EPS. This yield stress is rather a fictitious one, as it is not on the actual stress–strain curve. If this is taken as the yield stress, depending on the material tested, it may deviate far away from the actual experimental behavior, and consequently the curve shown in the dashed line will be yielded. This will also lead to overestimation of the elastic strain. In order to make up for these deficiencies, the point lying on the actual curve vertically down is selected as yield stress and is referred to here as modified yield stress (marked sM y in the figure). Taking sM y as the yield stress, the hardening function is redefined as shown in Fig. 5(b). Under associated flow rule ð f ¼ gÞ assumption, for Mises-type materials, the slope of this graph will represent the hardening modulus appeared in Eq. (2), which is the plastic tangent modulus E pt . This, however, will not be the case under non-associative flow assumption ð f agÞ.

h0 ¼

að1 þ ðq¯e p =qep ÞÞ ða þ bep Þ2

(8)

The term q¯e p =qep in the above equation represents nothing but the dilatancy, which can be neglected for geofoam type material. Thus, the hardening modulus of Eq. (8) reduces to a h0 ¼ (9) ða þ bep Þ2 In addition, for Mises-type materials, qf =qr ¼ s. Therefore, the constitutive relation of Eq. (5) takes the final form as follows: " # De m  De s e dr ¼ D  de (10) ða=ða þ bep Þ2 Þ þ s  De m The above equation indicates that, knowing the parameters a and b from the experimental results, the stress–deformation behavior of the materials can be predicted quite easily. On the other hand, the results of experimental investigations described in the preceding section confirm that the parameters that determine the EPS constitutive law depend to a great extent on the density of EPS used, and also on the absolute dimensions of the tested specimen. In his monograph on EPS geofoam, Horvath (1995) observes that, since no standardization of EPS testing exists until now even for basic aspects such as the specimen size and strain rate, etc., the relevant variables need to be stated for the interpretation of test data. Thus, the shape parameters also need to be included in the formulation so that the derived constitutive law can best describe the trend irrespective of these non-material parameters. The parameters a and b of the hardening modulus in Eq. (9) were, therefore, determined by making them functions of the σy = Yield stress of geofoam (Horvath, 1995)

σ y − σ yM

σyM = Modified yield stress a, b = Material parameters to be determined

Axial Stress, σ

Initial tangent

(a)

No modification

σy

σyM

Plastic tangent

Axial Strain, ε

Axial Stress, σ

k¼

Using Eqs. (3) and (7), the following equation can be obtained:

(b)

κ ( ε p) =

εp a +bε p

Plastic Strain, ε p

Fig. 5. Constitutive behavior and yield stress determination: (a) modified yield stress concept and (b) hardening concept.

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density (r) and the shape and size parameter (ss) as follows: a ¼ f ðr; ssÞ ¼ f 1 ðrÞ þ f 2 ðssÞ

(11)

b ¼ gðr; ssÞ ¼ g1 ðrÞ þ g2 ðssÞ

(12)

4.5. Deformation related material parameters The modified yield stress concept, explained in Fig. 5, was utilized for determining the deformation parameters by adopting the following steps. (1) Plot the stress–strain curve of the experimental results for a particular EPS density and specimen shape, and determine the elastic modulus (E) as explained in Fig. 4. Determine the modified yield stress (sM y ). Calculate the corresponding plastic strain and plot the stress–plastic strain curve. By linearizing the curve, the constants a and b can be determined. (2) Repeat the same procedure for another density and find a and b again. (3) Tabulate the values of a and b for each density and specimen shape (shape factor). (4) Plot the values as a function of density for each shape factor. An example plot from the test results is shown in Fig. 6 for the 100 mm cubic specimen. From this plot a and b can be obtained: a ¼ f 1 ðrÞ and b ¼ g1 (r) (5) Interpolate the results for other densities and the shape factors, and finally obtain them as functions of both in the following forms: a ¼ f ðr; ssÞ ¼ f 1 ðrÞ þ f 2 ðssÞ b ¼ gðr; ssÞ ¼ g1 ðrÞ þ g2 ðssÞ 5. Model validation The developed constitutive model was calibrated by performing two sets of numerical analysis: one for the 0.01

0.04

0.0095

Parameter b

Constant a

0.008

0.025

0.0075

0.02

n ¼ 0:0056r  0:0024

(13)

The issue of Poisson’s ratio is a controversial one as far as EPS geofoam is concerned. While normal polymer foams have a positive Poisson’s ratio, re-entrant polymer foams have a negative Poisson’s ratio (Lakes, 1987). Atmatzidis et al. (2001) obtained the Poisson’s ratio values for EPS geofoam of various densities based on triaxial compression test. These Poisson’s ratio values were computed according to formulations established by Timoshenko and Goodier (1970) in their classic book on the theory of elasticity. The experimental evidence indicates that the average lateral strains during the hydrostatic compression stage are definitely contractive leading to negative Poisson’s ratio. However, the Poisson’s ratio values should not be computed for strains higher than the limit of elastic behavior. The numerical analyses were carried out using a finite element method (FEM) code developed by the author. Parameters shown in Table 2 were determined using the test data reported in Elragi et al. (2000). The hardening parameters a and b in Tables 1 and 2 were obtained by curve fitting of the test results presented here and those reported by Elragi et al. (2000) for only a particular strain rate. It is worthwhile mentioning once again here that the developed model is valid only for the EPS stress–strain behavior up to 60% of the compressive strain (refer to

Table 1 Materials parameters for small-size model (50 mm cube) Parameters

D-16 EPS (nominal density ¼ 16 kg/m3)

D-20 EPS (nominal density ¼ 20 kg/m3)

Elastic modulus, E (MPa) Poisson’s ratio, n a (Eq. (9)) b (Eq. (9))

3.76 0.09 0.0071 0.021

5.4 0.11 0.0061 0.012

Table 2 Materials parameters values for large-size model (0.6 m cube)

0.007

0.006 10

Constant b

0.03

0.0085

0.0065

testing program discussed here, where small-size specimen was used, and the other for the testing program described by Elragi et al. (2000), where relatively large-size specimen was used. The calculations were made using the material parameters shown in Table 1 and 2. In Table 1, the elastic modulus values of EPS were determined based on Eq. (1). Poisson’s ratio (n) was determined as a function of density r (kg/m3) based on the relation proposed by EDO (1992), which is given by the following equation:

0.035

0.009

85

0.015

Parameter a 0.01 12

14

16

18

20

22

Geofoam Density, ρ (kg/m3)

Fig. 6. Material parameters (100
100
100 mm3 sample).

Parameters

D-29 EPS (nominal density ¼ 29 kg/m3)

Elastic modulus, E (MPa) Poisson’s ratio, n a (Eq. (9)) b (Eq. (9))

13.4 0.175 0.0043 0.0022

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5.1. Small-size specimen

160 140

Compressive Stress (kPa)

Fig. 2), beyond which the material strain hardens at a much faster rate resulting in a sigmoidal shape of the stress–strain curve. Modeling the strain range at which the material behavior turns sigmoidal is beyond the scope of this research. This may be one limitation of the model. However, as mentioned elsewhere, the model described here is particularly intended for use in the large-strain application (such as compressible inclusion) of EPS geofoam under immediate loading. In compressible inclusion function of EPS geofoam, where its compressibility is a desired and exploited function, allowing the straining to develop beyond certain level (say 70–80%) will undermine the very purpose of the compressible inclusion, when all the compressibility of the material will be lost. In addition, for evaluating the seismic performance of structures, performance based design (SEAOC, 1995; Steedman, 1998) has been becoming the norm in many countries. Therefore, considering the stress level beyond its required function will have no significance in such analysis and design. Therefore, if any experimental data exist for strain levels up to 60% (before the material stress–strain curve transforms into sigmoidal shape) at any desired strain rates, and the hardening parameters a and b are determined by curve fitting of those data, the present model can be successfully applied to simulate the compressible inclusion function or similar functions of EPS geofoam.

120 100 80 60 40

Experiment

20 0

Model Simulation 0

5

10

(a)

15

25

30

160 140 120 100 80 60 40

Fig. 7 shows the FEM discretization for the case in which a 100 mm cubic specimen was used in the compression testing. In the analysis, the strain rate (9%/min) was simulated by applying a forced displacement at the bottom boundary of the model as shown in the figure. Various

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Model Simulation

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(b)

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Fig. 8. Results of simulations (small-size specimen): (a) density D-16 (16 kg/m3) and (b) density D-20 (20 kg/m3).

100 mm

strain rates can, thus, be taken into the consideration during the analysis itself, applying each strain rate in the form of a forced displacement. Figs. 8(a) and (b) show comparison between computed and experimental results. It can be observed that the model can qualitatively predict the experimental trend in the small deformation range. The assumption of hyperbolic relation may have caused the deviation of the computed results from the experimental ones at the large deformation stage. 5.2. Large-size specimen

Forced Displacement 100 mm Fig. 7. Domain discretization and boundary conditions (small-size specimen).

Test model of Elragi et al. (2000), where a 0.6 m cubic specimen of density 29 kg/m3 (D-29) was tested, was also simulated using the developed constitutive law. Similar to the small-size specimen, the strain rate of 10%/min used in the test was simulated by applying forced displacement at the model boundary.

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Fig. 9a shows the comparisons between experimental and numerically predicted results for the overall stress– strain behavior of the specimen. It was observed that the model could predict the experimental trend fairly well in this case too, provided that the material parameters were determined properly from the test results for the particular specimen size and density. Fig. 9b shows the computed stress–strain results of the elements at the top and the bottom of the specimen model. In the laboratory testing, strains were measured using linear variable displacement transducer (LVDT) at the upper one-third (referred here as top), at the middle and at the lower one third (referred here as bottom) of the specimen. It can be observed that the model qualitatively predicts the experimental trends. The end part and the part adjacent to the loading plate show much greater deformation. The deviation of the computed results from the test results at the large strain plastic range (beyond 8%) is due to the hyperbolic assumption of the hardening function.

In Fig. 10, a comparison is made between the stress– strain plot of the 50 mm cubic specimen (density 16 kg/m3) and that of the lower 50 mm of the 0.6 m cubic specimen (density 29 kg/m3). This figure reveals that the former yields much higher elastic modulus values than the latter. Normally it is the large-size specimens that are expected to give higher elastic modulus values (cf. Fig. 4). Therefore, in order to predict the elastic modulus values precisely, it has to be calculated from the total stress–strain curve shown in Fig. 9(a). This fact was also confirmed by the test results of Elragi et al. (2000). The differences in the compressive strengths are due to the differences in the densities used in the test. Fig. 11 shows the computed and experimental distributions of the vertical strain within the large-size specimen (0.6 m cube). It can be seen that the trend predicted by the calculated results agree reasonably well with those of the 180 Density = 29 kg/m 3

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Fig. 10. Comparisons of local (near the bottom) stress–strain curve.

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Fig. 9. Results of simulations (large-size specimen): (a) total strain and (b) strains at top and bottom.

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Vertical Strain Fig. 11. Strain distribution within the large-size specimen (block density 29 kg/m3).

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experiments. The model, thus, also could explain the nonuniform distribution of vertical strain within an EPS specimen. 5.3. Comparison with other models In order to evaluate the accuracy of the model, prediction performance of the proposed model was also compared with that of the other constitutive models, which were developed for the large-strain and rapid loading applications of EPS geofoam. FEM analyses were performed for the test program reported by Chun et al. (2004). Two sets of calculations were performed for geofoam specimen of density 20 kg/m3 tested at two 200 Preber et al.(1994) Hyperbolic Model (Chun et al., 2004)

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Confining Stress = 20 kPa 0 0

(b)

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4

6

8

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Fig. 12. Comparison with other models (density ¼ 20 kg/m3): (a) unconfined state and (b) confined state.

different confining pressures: 0 and 20 kPa. Figs. 12(a) and (b) show the analyses results for the confining pressures 0 and 20 kPa, respectively. The figures show that the proposed model closely predicts the stress–strain behavior of EPS geofoam tested under different confining pressures. It can also be seen that while hyperbolic model proposed by Chun et al. (2004) predicts the upper bound of the test results, the proposed model predicts the lower bound of the test results. Thus, qualitatively results from this two models do not differ much. The model proposed by Preber et al. (1994), however, deviates substantially from the test results and underestimates the ultimate strength. 6. Summary and conclusions EPS geofoam applications can be divided broadly into two categories; small-strain and large-strain. The desired constitutive (stress–strain–time) properties vary depending on each application. In this paper, a constitutive model for EPS geofoam for its large-strain application under rapid loading is described. Laboratory investigations on the unconfined compressive strength of EPS confirmed that both the shape and the absolute dimensions of test specimens influence the stress–strain behavior, and hence the material’s compressive strength. The stress–strain behavior is highly nonlinear and it depends on the strain range as well as on the strain rate. The constitutive model developed and reviewed in this paper was based on the incremental theory of plasticity. The model incorporates the size and shape factors of the tested specimen as well as the density of geofoam. The constitutive law was derived in a form suitable for implementation in any simulation software (commercial or academic), since it is based on the conventional theory of incremental plasticity. The yield stress of the EPS geofoam was determined such that the law allows for modeling the post yield behavioral strength of the material in a more realistic manner. The strength of the model lies in its simplicity and robustness, as it involves very few material parameters, which can be determined from the conventional testing methods. The model could predict the compressive behavior of the EPS geofoam, if model parameters are properly determined. It also can predict fairly well the non-uniform distribution of vertical strain over the height of a geofoam specimen. The developed constitutive model was successfully applied to the analysis of a soil–structure–EPS interaction system, in which the EPS was used as a compressible buffer for reducing seismic load against a retaining wall (Hazarika and Okuzono, 2004). If the model parameters are determined from proper testing (e.g. large-size block testing), it can also be applied to other problems albeit within a limited strain range. As of now, the model has certain limitations in its applicability. Firstly, the model is valid only for the large-strain applications of EPS geofoam under rapid loading (no creep). Secondly, the model is

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valid only up to the strain range, when the stress–strain curve of EPS geofoam turns sigmoidal. Laboratory investigations on EPS creep indicate that EPS experiences pronounced creep in the large deformation range. However, careful consideration must be given to the potential for creep deformation, which may be exaggerated when laboratory tests are based on the smallsize specimens. Inclusion of the wider strain range as well as the visco-plastic component into the proposed model could greatly improve the capability and the applicability of the model. Acknowledgements This research was initially started with the financial support under the Research Grant for Feasibility Study provided by Maizuru National College of Technology, Kyoto, Japan, while the author was a faculty at the institute. A part of the financial support for this research also came from the Japan Society of the Promotion of Science (JSPS) under the Grant-in-Aid for Scientific Research (Grant no. 13750480). The author gratefully acknowledges the support. Author’s special thanks go to Dr. Juichi Nakazawa, Professor Emeritus, Maizuru National College of Technology, Kyoto, Japan, for allowing the author to use the digital uniaxial testing device of his laboratory. The author is also grateful to Professor Seishi Okuzono and his laboratory staff of Kyushu Sangyo University, Fukuoka, Japan for the invaluable support during the model experiments on EPS geofoam applications. Great appreciation also goes to Sekisui Plastic Co. Ltd., Tokyo, Japan for providing the materials and instruments used during EPS testing. Author’s gratitude also goes to Mr. Yoshihiro Satoh of EPS Development Organization (EDO), Tokyo, Japan, for his helpful suggestions on EPS material behavior and providing with various data on EPS applications. The author is grateful to all the reviewers whose constructive comments helped shaping the final form of this manuscript. Last but not the least, the author offers his heartfelt thanks to Port and Airport Research Institute (PARI), Japan, for the continued support provided for this particular field of research. References Atmatzidis, D.K., Missirlis, E.G., Chrysikos, D.A., 2001. An investigation of EPS geofoam behavior in compression. In: Proceedings of the Third International Conference on EPS–EPS Geofoam 2001, Salt Lake City, USA, CD ROM. Chun, B.S., Lee, S.D., Lim, H.S., Ahn, T.B., Ha, K.H., 1998. Strength deformation characteristics of EPS. In: Proceedings of the Eighth ISOPE Conference, Montreal, Canada, vol. 1, pp. 491–496. Chun, B.S., Lim, H.S., Sagong, M., Kim, K., 2004. Development of a hyperbolic constitutive model for expanded polystyrene (EPS) geofoam under triaxial compression tests. Geotextiles and Geomembranes 22, 223–237. Duskov, M., 1997. EPS as a lightweight sub-base material in pavement structures. Ph.D. Thesis, Delft University of Technology, Holland.

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EDO, 1992. Expanded Polystyrene Construction Method. Riko Tosho Publishers, Tokyo, Japan (in Japanese). EDO, 2003. Investigations on structural shape of EPS embankment. Internal Report of CPC Consultant, Tokyo, Japan (in Japanese). Elragi, A., Negussey, D., Kyanka, G., 2000. Sample size effects on the behavior of EPS geofoam. In: Proceedings of the Soft Ground Technology Conference, ASCE Geotechnical Special Publications 112, The Netherlands. Eriksson, L., Trank, R., 1991. Properties of expanded polystyrene— laboratory experiments. Swedish Geotechnical Institute, Linkoping, Sweden. Findley, W.N., Khosla, G., 1956. An equation for tension creep of three unfilled thermoplastics. SPE Journal 12 (12), 20–25. Findley, W.N., Lai, J.S., Onaran, K., 1989. Creep and Relaxation of Nonlinear Viscoelastic Materials. Dover Publication, New York, USA. Frydenlund, T.E., Aaboe, R., 1996. Expanded polystyrene—the lightweight solution. In: Proceedings of the International Symposium on EPS Construction Method (EPS-Tokyo’96), Tokyo, pp. 32–46. Horvath, J.S., 1995. Geofoam Geosynthetic. Horvath Engineering, PC, New York, USA. Horvath, J.S., 1997. The compressible inclusion function of EPS geofoam. Geotextiles and Geomembranes 15 (1–3), 77–120. Horvath, J.S., 1998. Mathematical modeling of the stress–strain–time behavior of geosynthetics using the Findley equation: general theory and application to EPS-block geofoam. Manhattan College Research Report, No. CE/GE-98-3, USA. Hazarika, H., 2001. On the compressible properties of EPS geofoam and its application as compressible inclusion. Research Report, Maizuru National College of Technology, Kyoto, Japan, vol. 36, pp. 50–54. 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., Okuzono, S., Ideno T., 2001. On the material properties of EPS determined from laboratory investigations. In: Proceedings of the Third International Symposium of JSCE, Tokyo, Japan, pp. 177–180. Inglis, D., Macleod, G., Naesgaard, E., Zergoun M., 1996. Basement wall with seismic earth pressures and novel expanded polystyrene foam buffer layer. In: Proceedings of the 10th Annual Symposium on Earth Retention System, Canadian Geotechnical Society, Vancouver, Canada, pp. 1–10. Itasca Consulting Group Institute, 1995. Fast Lagrangian Analyses of Continua (FLAC 3.4). Itasca Consulting Group Institute, MN, USA. Kutara, K., Aoyama, N., Takeuchi, T., Takeichi, O., 1989. Experiments on application of expanded polystyrol for light fill materials. Soils and Foundations 37 (2), 49–54. Lakes, R.S., 1987. Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040. Magnan, J.P., Serratrice, J.F., 1989. Proprie´te´s me´caniques du polystyre`ne expanse´ pour ses applications en remblai routier. Bulletin liaison Laboratoire Ponts et Chausse´es, LCPC 164, 25–31. Matsuzawa, H., Hazarika, H., 1996. Analyses of active earth pressure against rigid retaining wall subjected to different modes of movement. Soils and Foundations 36 (3), 51–65. Murphy, G.P., 1997. The influence of geofoam creep on the performance of compressible inclusion. Geotextiles and Geomembranes 15 (1–3), 121–131. Nakase, A., 1974. New technologies in the field of soil engineering. Kouwan, No. 2, pp. 21–25, (in Japanese). Negussey, D., Sun, M.C., 1996. Reducing lateral earth pressure by geofoam (EPS) substitution. In: Proceedings of the International Symposium on EPS Construction Method (EPS-Tokyo’96), Tokyo, Japan, pp. 202–211. Preber, T., Bang, S., Chung, Y., Cho, Y., 1994. Behavior of expanded polystyrene blocks. Transportation Research Record, No. 1462, pp. 36–46.

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