SMALL-STRAIN STRESS-STRAIN PROPERTIES OF EXPANDED POLYSTYRENE GEOFOAM

Transcription

1 SOILS AND FOUNDATIONS Vol. 48, No. 1, 61 71, Feb Japanese Geotechnical Society SMALL-STRAIN STRESS-STRAIN PROPERTIES OF EXPANDED POLYSTYRENE GEOFOAM G. E. ABDELRAHMAN i),shohei KAWABE ii),yoshimichi TSUKAMOTO iii) and FUMIO TATSUOKA iv) ABSTRACT The small-strain stress-strain properties of expanded polystyrene (EPS) geofoam with densities of about 20 kg/m 3 and 30 kg/m 3 were evaluated by laboratory unconˆned compression tests on specimens of 75 mm in diameter and 150 mm in height. Two series of tests were conducted, which were continuous monotonic loading (ML) tests and ML tests intervened by sustained creep loading and minute cycles of unload and reload. Relatively small vertical and horizontal strains were locally measured by means of a pair of local deformation transducers (LDTs) and a set of three clip gauges, respectively. The paramount importance of measuring local strains in compression tests on EPS to reliably evaluate its stress-strain properties, in particular those at relatively small strains, is demonstrated. The initial modulus, E 0, and Poisson's ratio, n 0, were evaluated from initial stress-strain relations at small strains obtained by these ML tests. The tangent parameters, E tan and n tan, were also evaluated from the ML stress-strain behaviour. The equivalent parameters, E eq and n eq, were evaluated from the stress-strain behaviour during minute cycles of unload and reload. The stress-strain behaviour is essentially linear only at small strains, and it becomes highly non-linear and a signiˆcant drop of stišness occurs as observed in the overall stress-strain behaviour. The Poisson's ratio for inelastic deformation is found to be negative. Key words: expanded polystyrene geofoam, Poisson's ratio, unconˆned compression test, Young's modulus (IGC: D0/E0) INTRODUCTION Expanded polystyrene (EPS) geofoam is a plasticpolymeric material and serves as a light-weight construction material in solving a number of geotechnical problems with slope stabilization, embankments, earth retaining structures, bridge approaches and abutments, buried pipes, railway structures and so on. The density of ordinarily used EPS geofoam ranges from 12 to 35 kg/m 3 while its stišness and strength are comparable to those of relatively stiš clay having a large true or apparent cohesion. It is self-supporting when stacked and has good heat insulation properties. The use of EPS geofoam in geotechnical construction started in Norway in 1960's, and has been adopted worldwide (Frydenlund, 1996; Van Dorp, 1996; Hillmann, 1996; Mohamed, 1996; and others). The ˆrst application in Japan was undertaken in an embankment project in 1985 (Miki, 1996). The experiences in Japan showed that EPS geofoam structures performed also very well under seismic events. During a period of 1993 to 1995, several strong earthquakes struck various regions in Japan, and some damages were incurred on the EPS embankments. However, they were found highly stable during the earthquakes (Hotta et al., 1996). The properties of EPS geofoam have constantly been studied and developed for speciˆc applications, such as thermal, chemical and biological properties and water absorption. However, from a viewpoint of soil - structure interactions, the properties of particular interest are compressive strength and modulus and long-term creep behaviour (Tsukamoto et al., 2002; Hazarika, 2006; and others). The strain levels in EPS geofoam at working loads in full-scale ˆeld cases are usually substantially lower than the strain level at which the stress strain curves become highly nonlinear and the signiˆcant drop of stišness occurs, corresponding to the order of 1.0z or less. On the other hand, it is known that the local axial strain measurement by means of, for example, a pair of local deformation transducer (LDTs: Goto et al., 1991) in laboratory tests is imperative to evaluate reliably the stress-strain properties at such small strains of stiš geomaterials, such as stiš soils and rocks (e.g., Tatsuoka et al., 1999a and 1999b).ThisisbecausetheeŠectsofbeddingerrorsatthe top and bottom ends of the specimen become signiˆcant i) ii) iii) iv) Associate Professor, Civil Engineering Department, Fayoum University, Egypt (gsa00@fayoum.edu.eg). Graduate Student, Department of Civil Engineering, Tokyo University of Science, Japan. Associate Professor, ditto. Professor, ditto. The manuscript for this paper was received for review on March 16, 2007; approved on September 20, Written discussions on this paper should be submitted before September 1, 2008 to the Japanese Geotechnical Society, , Sengoku, Bunkyo-ku, Tokyo , Japan. Upon request the closing date may be extended one month. 61

2 62 ABDELRAHMAN ET AL. on the axial strains calculated from the displacements of the specimen cap or the loading piston. In addition, in order to reliably evaluate lateral strains of an unsaturated specimen, it is necessary to directly measure lateral deformation of the specimen by means of, for example, clip gauges (e.g., Tatsuoka et al., 1994). Finally, even in a single ML compression test on geomaterials at a ˆxed con- ˆning pressure, the elastic stress-strain parameters such as Young's modulus and Poisson's ratio are not constant, and are mainly ašected by changes in the axial stress. For this reason, these elastic parameters of geomaterials are often evaluated by applying minute unload and reload cycles employed at several levels of vertical stress encountered along ML tests conducted at a constant strain rate. The purpose of such tests was extended to evaluate the entire stress-strain behaviour including the peak strength (e.g., Tatsuoka et al., 1999a and 1999b). However, the systematic study on EPS geofoam based on laboratory stress-strain tests with such local measurements of axial and lateral strains is very limited. In view of the above, the present study was performed aiming at evaluating the small-strain properties of EPS geofoam as well as the overall stress-strain behaviour based on laboratory unconˆned compression tests measuring small axial and lateral strains using local gauges. TESTING DETAILS EPS Geofoam Specimen The types of EPS used in the present study correspond to D-20 and D-30 according to the density category stipulated in the Japanese speciˆcation (Miki, 1996). The precise values of the density are found to be 19.3 and 28.0 kg/m 3. Dozens of cylindrical specimens with 75 mm in diameter and 150 mm in height were trimmed from representative samples by a manufacturer and provided for the present study. EPS geofoam is known to exhibit highly nonlinear mechanical behaviour and density-dependent stressstrain characteristics (e.g., Hazarika and Okuzono, 2004; Chun et al., 2004; Hazarika, 2006; Negussey, 2007). The density appears to be a useful index parameter for EPS geofoam that correlates with most of its mechanical properties such as compression strength, shear strength, tension strength, exural strength, stišness and creep deformation. Therefore, there are well-balanced recommended choices of the density regarding the stišness, strength and durability (Horvath, 1998). Photo 1. Testing apparatus used Testing Equipment and Bedding Errors Photo 1 shows the apparatus with an EPS specimen. In evaluating the small-strain stress-strain properties of EPS geofoam in laboratory compression tests, it is imperative to use an accurately driven loading system. In the present study, the strain-controlled servo-loading system was used (Santucci de Magistris et al., 1999). In addition to a linear variable displacement transducer (LVDT) measuring the `external' axial (vertical) strain from the axial displacement of the loading piston, a pair of local deformation transducers was installed on the opposite sides of respective specimens to measure the `local' axial (vertical) strain, e v, without any bedding errors associated with unexpected minute gaps at both ends of the specimen, as shown in Fig. 1 (LDTs: Goto et al., 1991). Both ends of the LDTs were mounted onto a pair of hinges set on the lateral face of the specimen. The respective hinges were ˆxed to the specimen with a single pin, and these hinges were stabilized with a help of a pair of rubber bands encircling the specimen. The lateral (horizontal) strain, e h, was measured by means of a set of three clip gauges (CG), which were positioned at the heights of 1-in-6, 1-in-2 and 5-in-6 of the total height of the specimen from the bottom, as shown in Fig. 1. Each of the CGs was supported by the two hinges positioned diagonally on the opposite sides of the specimen, which were ˆxed to the specimen with pins and rubber bands. The locally measured axial and lateral strains reported in the present study were those obtained by averaging the readings of a pair of LDTs and a set of three CGs, respectively. A 0.3 mm thick latex rubber membrane was placed between the top end of the specimen and the cap as well as between the bottom end of the specimen and the pedestal. Herein, the surfaces of the cap and the pedestal were smeared with a thin layer of Dow high-vacuum silicon grease for lubrication. The thin layers of silicon grease were pasted on the surfaces of the cap and the pedestal. The thickness of silicon grease was ensured by encircling the cap and pedestal with the sticky tapes of about 50 mm in thickness and pasting the layers of silicon grease within them. During the course of the present study, it was found that the irregularity associated with surface machining at both ends of the specimen may create signiˆcantly inhomogeneous deformation of the specimen and may greatly ašect the measurement of the axial strain. In order to eliminate such errors, a thin layer of

3 EPS GEOFOAM 63 of the tests. In testing EPS specimens, relatively larger strain rates were adopted in the past literature, such as 10 z/min for 5 cm long cubic specimens, as summarized by Negussey (2007). On the other hand, the axial strain rate nominally adopted in unconˆned compression tests is usually 1z/min. In the present study, a relatively smaller axial strain rate was used to avoid any signiˆcant irregularities in measuring local strains. Deˆnitions of Stress-strain Properties In what follows, all the strain properties are calculated with respect to the initial dimensions of EPS specimens. To study the small-strain stress-strain properties of a given material, the stišness parameters are often evaluated based on the linear isotropic elastic theory. Then, the determinations of any two parameters are only necessary, since the other parameters are correlated with these two parameters thus chosen. In the present study, a combination of the tangent modulus for major principal strain increments taking place in the vertical (axial) direction, E v, and the Poisson's ratio for major and minor principal strain increments taking place in the vertical and horizontal directions, n vh, deˆned as follows were evaluated, Fig. 1. Local deformation transducers and clip gauges on EPS specimen, (a) plan view and (b) elevation view wet soft gypsum pasted between two plastic wrapping sheets was placed between the lubrication layer located at the bottom of the specimen and the pedestal. All of the tests were conducted at a constant room temperature of 209C. Test Series Two series of unconˆned compression tests were carried out on the EPS specimens categorized as D-20 and D-30. In the ˆrst test series, continuous monotonic loading (ML) tests were conducted at an axial strain rate of e v =0.33z/min. In the other test series, combinations of sustained creep loading and ˆve or six minute cycles of unload and reload with a double amplitude of axial stress of Ds v =10 kpa were applied at several levels of vertical stress encountered along the monotonic stress path. The same axial strain rate of e v =0.33z/min was used during the cycles of unload and reload. As shown later, the stress-strain behaviour during unload and reload cycles, which were employed after some amount of creep, is highly reversible. Thus, the determination of the elastic deformation characteristics from the stress-strain behaviour during these minute unload and reload cycles becomes rather reliable. In this test series, two dišerent axial strain rates of 0.33z/min and 0.66z/min were adopted for monotonic portions as well as cyclic portions E v =«Dsv De v $(sh=constant), n vh =«Deh De v$(sh=constant) where De v and De h are the vertical and horizontal strain increments taking place when the vertical (axial) stress changes by an amount of Ds v at constant horizontal (lateral) stress, s h, both being deˆned to take positive when compression occurs. Furthermore, as the stressstrain behaviour of EPS is generally non-linear, a set of Young's moduli, E 0, E tan and E eq, and a set of Poisson's ratios, n 0, n tan and n eq, that are explained below are de- ˆned based on Eq. (1): 1) From the ML test results, the initial modulus and Poisson's ratio, E 0 and n 0, were determined as shown in Fig. 2(a). 2) The tangent modulus and Poisson's ratio, E tan and n tan, at respective stress levels were then deduced as illustrated in Fig. 2(a). 3) The equivalent modulus and Poisson's ratio, E eq and n eq, which were deˆned at minute cycles of unload and reload, were evaluated as shown in Fig. 2(b). The average of the readings from ˆve or six unload and reload cycles at each stress level was obtained and reported in the present study. The issue of anisotropic deformation characteristics of EPS geofoam under general anisotropic stress conditions is beyond the scope of the present study. TEST RESULTS Overall Observations Figures 3 and 4 show the results from two continuous ML unconˆned compression tests on EPS specimens categorized as D-20 and D-30, which are typical of those performed in the present study. In Figs. 3(a) and 4(a), the plots of vertical (axial) stress, s v, against the vertical (axi- (1)

4 64 ABDELRAHMAN ET AL. Fig. 2. Deˆnitions of small-strain stress-strain properties evaluated in the present study, (a) tangent parameters during monotonic loading, (b) e- quivalent parameters during cycles of unload and reload and (c) secant parameters (not used in the present study) Fig. 3. Results of monotonic loading tests on D-20 EPS specimen, (a) Fig. 4. Results of monotonic loading tests on D-30 EPS specimen, (a) s v-e v and (b) e h-e v s v-e v and (b) e h-e v al) strain, e v, are shown. The data with axial strains measured by using a LVDT and a pair of LDTs are indicated with broken lines and continuous lines. The e v-value measured by a LVDT is always greater than that measured by a pair of LDTs, which is due to the bedding errors associated with unexpected gaps and irregularity at

5 EPS GEOFOAM 65 Fig. 5. Results of minute cyclic loading tests on D-20 EPS specimen, (a) s v-e v,(b)e h-e v,(a?) close-upviewofs v-e v,(b?) close-upviewofe h-e v both ends of the EPS specimens. The axial strains measured by using a pair of LDTs are found to give a sound basis for axial strain measurements at small strains. The limit of the level of axial strain measured by means of LDTs is 1 to 2z. After such axial strain levels, the axial strains measured by a LVDT are only available and found to be useful in discerning the stress strain behaviour of EPS specimens at large strain levels. Figures 3(b) and 4(b) present the relationships between the horizontal strain, e h, and the vertical strain, e v.itis noteworthy that, at the beginning of axial compression, the lateral expansion is observed. However, the EPS specimens start laterally contracting during axial compression when the vertical (axial) strain, e v, reaches 1 to 2 z. In fact, the diameters of the specimens measured after the tests were smaller than those measured before the tests. It seems that this trend of behaviour results from the fact that the EPS specimens are comprised of an assemblage of expanded polystyrene beads. It is likely that the beads exhibit large volume contraction when they are subjected to yielding. Figures 5(a) and 5(b) show the results of the uncon- ˆned compression test on the EPS specimen categorized as D-20, in which creep loading was sustained for 60 minutes and 5 cycles of unload and reload were then applied at several levels of vertical (axial) stress, s v.the time-dependent response of EPS against creep loading is beyond the scope of the present study. The plots of s v against e v are shown in Fig. 5(a), while the plots of e h against e v are shown in Fig. 5(b). In this test, the plastic wrapping sheets and gypsum were not used yet, leading to a signiˆcantly erratic relationship between s v and the `external' vertical strain measured by a LVDT at small strains. The corresponding relationship between e h and e v is also erratic. The detailed relations at a particular level of s v around 60 kpa are presented in Figs. 5(a?) and5(b?). It may be seen that, because of the relatively large creep strains that have taken place immediately before the cycles of unload and reload, the stress-strain behaviour during these cycles is highly reversible, which makes the evaluation of elastic deformation characteristics rather reliable. It is noteworthy in Fig. 5(b?) that the EPS specimen tends to be laterally compressed during sustained creep loading under axial compression denoted as phase (1). It is to be noted that, even before the start of large-scale yielding that is clearly observed in the entire stress-strain behaviour during ML tests, the strains taking place during creep loading are totally inelastic (or irreversible or visco-plastic). Herein and hereafter, the term of ``large-scale yielding''

6 66 ABDELRAHMAN ET AL. Fig. 6. Results of minute cyclic loading tests on D-30 EPS specimen, (a) s v-e v,(b)e h-e v,(a?) close-upviewofs v-e v,(b?) close-upviewofe h-e v is used to represent the phenomenon in which the stress strain curves of EPS specimens become highly nonlinear and the signiˆcant drop of stišness occurs. Likely for this reason, this trend of behaviour during creep loading is in accordance with the one that can be observed obviously after large-scale yielding (Figs. 3(b) and 4(b)). On the other hand, during cycles of unload and reload denoted as phase (2), the portions of unload are found to result in lateral compression under axial extension, while the portions of reload are found to result in lateral expansion under axial compression, in accordance with the general trend observed at similar levels of s v located below the start of large-scale yielding, where the large part of the strain is elastic. These results indicate that the characteristics of inelastic deformation that takes place at any stress levels, as those taking place during creep loading, exhibit negative Poisson's ratios, compared with positive Poisson's ratios of the elastic deformation characteristics. Figures 6(a) and 6(b) show the results of another unconˆned compression test on the EPS specimen categorized as D-30, in which creep loading was sustained for 30 minutes and 6 cycles of unload and reload were then applied at several levels of s v. The observations similar to those on Fig. 5 can also be made. Tangent Modulus and Poisson's Ratio during Monotonic Loading From the data shown in Figs. 3 and 4, the tangent modulus, E tan, and Poisson's ratio, n tan, were obtained and plotted against e v,asshowninfigs.7and8. Figures 7(a) and 7(b) show the values of E tan and n tan plotted against e v, respectively, from the test on the EPS specimen categorized as D-20. The data based on the axial strains measured by a LVDT and LDTs are indicated with broken lines and continuous lines. It is found that both E tan and n tan are kept nearly constant up to a vertical strain of about e v =0.1z, after which both parameters degrade signiˆcantly with an increase in e v. The ešects of bedding errors on the E tan-values obtained from the axial strains measured by a LVDT are signiˆcant. It is to be noted that, despite that it is to a lesser extent, it is also the case with the n tan-values obtained from the axial strains measured by a LVDT. Figures 8(a) and 8(b) show similar results of the EPS specimen categorized as D-30. The observations similar to those on Fig. 7 can also be made. It may be seen from Figs. 7(b) and 8(b) that the n tanvalue becomes negative as the axial strain exceeds about 1z, corresponding to the behaviour of the EPS specimen after the start of signiˆcant large-scale yielding as observed in the overall stress-strain behaviour (Figs. 3 and

7 EPS GEOFOAM 67 Fig. 7. Plots of tangent parameters E tan and n tan against vertical strain Fig. 8. Plots of tangent parameters E tan and n tan against vertical strain e v on D-20 EPS specimen, (a) E tan-e v and (b) n tan-e v e v on D-30 EPS specimen, (a) E tan-e v and (b) n tan-e v 4). Initial Modulus and Poisson's Ratio From the initial portions of the relations determined based on the axial strains measured by using a pair of LDTs as described above, the initial modulus, E 0,and Poisson's ratio, n 0, are determined accordingly as shown in Figs. 7 and 8. As the most important factor that controls the mechanical properties of EPS is its density, the values of E 0 and n 0 thus determined have been plotted against the density of EPS in Figs. 9(a) and 9(b). In Fig. 9(a), the values of the initial modulus, E 0, obtained from the present study are denoted with two open circles, while the solid circles denote the results from various previous studies (Duskov, 1997; Elragi et al., 2000; Eriksson and Trank, 1991; Horvath, 1995; Miki, 1996; Van Dorp, 1988; Negussey, 2007). In all the unconˆned compression tests performed in these previous studies, the axial strains were measured externally. The E 0-values obtained from the present study are found to be similar to those obtained from the unconˆned compression tests using large cubic specimens of 60 cm in length, which were reported by Elragi et al. (2000) and Duskov (1997). On the other hand, the E 0-values from the present study are substantially larger than those from the unconˆned compression tests using small cubic specimens of 50 mm in length. Despite dišerent testing techniques used among the previous studies and the present study, among the possible reasons for this variance, the in uence of bedding errors at the ends of the specimen is the most likely major reason. This is because, due to the ešects of bedding errors, the Young's modulus evaluated from the externally measured axial strains becomes smaller than the true value as seen in Figs. 7(a) and 8(a), and its extent increases with a decrease in the specimen size. It is likely that the ešects of bedding errors on the externally measured axial strains become negligible with su ciently larger specimens such as 60 cm long cubic specimens. These results indicate that the global deformation of a full-scale EPS structure in the ˆeld would be substantially overestimated if it were estimated based on the stišness obtained from externally measured axial strains in unconˆned compression tests on small specimens such as 50 mm long cubic specimens. Although it is less important, the ešect of non-linearity that is likely to be included in the stišness values of these smallcubicspecimensof50mminlengthisanotherreason. This is because it is likely that the secant modulus, E sec, (see Fig. 2(c)) was evaluated at strains at which the stress-strain behaviour was already noticeably non-linear in the unconˆned compression tests on the small cubic specimens of 50 mm in length in the previous studies. Considering that EPS specimens are rather uniformly comprised of an assemblage of expanded polystyrene beads, the systematic specimen size ešects on the Young's

8 68 ABDELRAHMAN ET AL. Fig. 9. Plots of initial parameters E 0 and n 0 against EPS density, (a) E 0 -density of EPS and (b) n 0-density of EPS Fig. 10. Plots of equivalent parameters E eq and n eq against vertical stress s v on D-20 EPS specimen, (a) E eq-s v and (b) n eq-s v modulus obtained from global axial strains are unlikely. The initial Poisson's ratios, n 0, obtained from the present study are plotted against the density of EPS with two open circles in Fig. 9(b). The data from various previous studies (Duskov, 1998; Negussey and Sun, 1996; Ooe et al., 1996; Sanders, 1996; Yamanaka et al., 1996) are also shown with solid circles, most of the data being in a range of n 0 between 0.07 and 0.1. On the other hand, the n 0-values from the present study are 0.12 and 0.17 for the EPS specimens categorized as D-20 and D-30. In Figs. 7(b) and 8(b), the trend of the values obtained from the present study, being about the upper bound of the values from the previous studies, may also be explained by the ešects of bedding errors on the n 0-valuesasseeninFigs. 7(b) and 8(b), since the values from the previous studies were those obtained from externally measured axial strains. Equivalent Modulus and Poisson's Ratio during Minute Cyclic Loading It is known that the unload and reload cycles of axial stress give stišness values greater than those at initial virgin loading with many materials including soils. It is also known that the vertical Young's modulus for major principal strain increments acting in the vertical direction de- ˆned in Eq. (1) for unbound granular materials (i.e., sands and gravels) increases with an increase in the vertical (axial) stress, s v, during triaxial compression conducted at a ˆxed conˆning pressure, s h, unless the stress ratio, s v/s h, is close to the peak value (e.g., Hoque and Tatsuoka, 1998; Tatsuoka et al., 1999a and 1999b). Therefore, it is of interest to plot the E eq-values of the EPS specimens against the respective values of s v at which these E eq - values were measured. Figure 10(a) shows such plots for the EPS specimen of D-20, and Fig. 11(a) for the EPS specimen of D-30. The n eq-values are also plotted against s v in Figs. 10(b) and 11(b). The following trends of behaviour may be seen from Fig. 10(a): 1) By deˆnition, when the E tan-s v relation exhibits essentially linear and elastic behaviour, the E eq-value is the same as the E tan-value. On the other hand, in a stress range where the E tan-s v relation exhibits noticeably non-linear and inelastic behaviour, the E eq-value is consistently greater than the E tan-value at the same level of vertical stress, s v. Despite the above outcome, the dišerence between the values of E eq and E tan is not very signiˆcant, given by a factor of less than 1.3 within the range of vertical stress at which these E eq-values were obtained. 2) The E eq-value starts decreasing signiˆcantly with an increase in the vertical stress, s v,whens v reaches 20 to 30 kpa, in a manner similar to that of the E tan-value. It is possible that the fabrics deteriorate due to irreversi-

9 EPS GEOFOAM 69 Fig. 11. Plots of equivalent parameters E eq and n eq against vertical stress s v on D-30 EPS specimen, (a) E eq-s v and (b) n eq-s v Fig. 12. Relations between equivalent parameters E eq and n eq and vertical stress s v,(a)e eq-s v and (b) n eq-s v ble straining associated with an increase in s v.this trend of behaviour with EPS is opposite to that with unbound granular materials. 3) The in uence of the vertical strain rate e v,onthee eqvalue is negligible within the limit of the test conditions in the present study. Because of a small range of e v (only a factor of two), no deˆnite conclusions in this respect can be derived from the present study. The following trends of behaviour may be seen from Fig. 10(b): 1) The n eq -value is noticeably greater than the n tan -value at the same level of vertical stress, s v. 2) The n eq-value decreases noticeably with an increase in e v. However, any distinct conclusion in this respect cannot be derived from the present study due to a limited amount of the data. 3) As the vertical stress, s v, reaches 20 to 30 kpa, the n eqvalue also starts reducing signiˆcantly in a manner similar to the n tan-value. Unlike the n tan-value, negative values of n eq were not observed within the limit of test conditions in the present study. Figures 11(a) and 11(b) show the values of E eq and n eq plotted against s v fortheepsspecimenofd-30.the same observations as those derived from Fig. 10 can be basically made. The E eq-values from the present study are summarized in Fig. 12(a), plotted against s v in a full-log plot. It may be seen in Fig. 12(a) that the following linear relation ˆts the data: E eq =a (s v /s o ) b (2) where a and b are constants, and s o is the reference stress taken as 1 kpa in the present study. Eqation (2) is relevant only for some limited range of s v.thelowestapplicable limit of s v is obtained by substituting the initial modulus, E 0, into the parameter of E eq ineq.(2),asseen in Fig. 12(a). At the level of vertical stress, s v,belowthis limit, the stress-strain behaviour is essentially linear and elastic, and therefore the E eq-value becomes essentially thesameasthee 0-value. The n eq-s v relations summarized in a full-log plot are presented in Fig. 12(b). The same expression as Eq. (2) is herein introduced to formulate the reduction of the n eqvalue with an increase in s v, as shown in Fig. 12(b): n eq =a (s v/s o) b (3) where Eq. (3) is relevant again only for some limited range of s v, as seen in Fig. 12(b). At the level of vertical stress, s v, below this limit, the n eq-value becomes essentially the same as the n 0-value. It is to note here that the range of the vertical stress, s v, in which Eqs. (2) and (3) are valid should coincide with each other, as shown in Figs. 12(a) and 12(b). The outcome of the present study implies that even at

10 70 ABDELRAHMAN ET AL. working loads, the small-strain properties of EPS change signiˆcantly, where the equivalent tangent modulus and Poisson's ratio decrease with increasing vertical stress level. It is therefore recommended that the stress level at which EPS materials would undergo in the ˆelds needs to be carefully considered in determining design small-strain parameters of EPS. CONCLUSIONS The stress-strain properties, in particular those at small strains, of EPS geofoam categorized as D-20 and D-30 were evaluated by performing unconˆned compression tests on EPS specimens of 75 mm in diameter and 150 mm in height. The small vertical and horizontal strains were locally measured with a pair of local deformation transducers and a set of three clip gauges. The following conclusions were obtained: 1) The local measurement of axial strains along the lateral surface of the specimen is imperative to accurately evaluate small axial strains. In particular, the values of the initial modulus, E 0, evaluated as above in the present study are comparable only with the data from the previous studies that externally measured axial strains of relatively large specimens. On the other hand, the E 0-values from the present study are substantially larger than the data from the previous studies that externally measured axial strains of relatively small specimens. 2) The local measurement of lateral strains on the lateral surface of the specimen is also necessary to accurately evaluate small lateral strains. 3) The design parameters for the Young's modulus and Poisson' ratio at relatively small strains, say less than about 1z, of EPS geofoam can be evaluated by compression tests on relatively small specimens, as used in the present study, if the axial and lateral strains are measured sensitively and accurately by using relevant local gauges. 4) The tangent modulus, E tan, and Poisson's ratio, n tan, are nearly constant until the vertical strain, e v, becomes about 0.1z and starts reducing signiˆcantly afterwards with an increase in e v. After large-scale yielding starts at e v equal to about 1z, then tan-value becomes negative. The n tan-values observed during creep loading are also negative. It is therefore likely that the Poisson's ratio for inelastic deformation is negative, compared with positive values for elastic deformation. 5) After the stress-strain behaviour becomes non-linear, the values of the equivalent modulus, E eq, evaluated from minute unload and reload cycles applied after creep loading are noticeably greater than the values of E tan when compared at the same level of vertical stress, s v. Except for the initial linear zone, the E eq-values decrease with an increase in s v in a manner similar to the E tan-values. The above is also the case with the Poisson's ratio, n eq. The empirical equations of E eq and n eq expressed as a function of s v were derived. ACKNOWLEDGEMENTS The laboratory stress-strain tests described in the present study were carried out when the ˆrst author stayed at Tokyo University of Science as a visiting Associate Professor. The authors express their sincere appreciationtodr.d.hirakawaandmr.t.kanemaruoftokyo University of Science for their cooperation in carrying out the laboratory tests described in the present study. Thanks are also extended to Mr. K. Chiyoda of JSP Corporation for providing them with the samples of EPS geofoam. NOTATION E: Young's modulus (=Ds v/de v) e h: horizontal (lateral) strain (compression positive) e v : vertical (axial) strain (compression positive) n: Poisson's ratio (=-De h/de v) s v: vertical stress Abbreviated words D-20, D-30: Density category of EPS used in the present study LVDT: linear variable displacement transducer LDT: local deformation transducer CG: clip gauge Subscripts eq: equivalent sec: secant tan: tangent 0: initial REFERENCES 1) Chun, B. S., Lim, H. S., Sagong, M. and Kim, K. (2004) : Development of a hyperbolic constitutive model for expanded polystyrene (EPS) geofoam under triaxial compression tests, Geotextiles and Geomembranes, 22(4), ) Duskov, M. (1997): EPS as a light-weight sub-base material in pavement structures, Ph.D. Thesis, Delft University of Technology. 3) Duskov, M. (1998): Materials research on EPS20 and EPS15 under representative conditions in pavement structure, Geotextiles and Geomembranes, 15, ) Elragi, A., Negussey, D. and Kyanka, G. (2000): Sample size ešect on the behavior of EPS geofoam, Proc. Soft Ground Technology Conference, The Netherlands. 5) Eriksson, L. and Trank, R. (1991): Properties of expanded polystyrene, laboratory experiments, Swedish Geotechnical Institute, Sweden. 6) Frydenlund, T. E. and Aaboe, R. (1996): Expanded polystyrene the light solution, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Goto, S., Tatsuoka, F., Shibuya, S., Kim, Y.-S. and Sato, T. (1991): A simple gauge for local strain measurements in the laboratory, Soils and Foundations, 31(1), ) Hazarika, H. and Okuzono, S. (2004): Modeling the behavior of a hybrid interactive system involving soil, structure and EPS geofoam, Soils and Foundations, 44(5), ) Hazarika, H. (2006): Stress-strain modeling of EPS geofoam for large strain applications, Geotextiles and Geomembranes, 24(2),

11 EPS GEOFOAM ) Hillman, R. (1996): Research project on EPS in Germany; material behaviour and full scale model studies, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Hoque, E. and Tatsuoka, F. (1998): Anisotropy in the elastic deformation of granular materials, Soils and Foundations, 38(1), ) Horvath, J. S. (1995): Geofoam Geosynthetic, Horvath Engineering,P.C.,Scarsdale,NewYork,USA. 13) Horvath, J. S. (1998): The compressible inclusion function of EPS geofoam: analysis and design methodologies, Research Report, (CE/GE 98 2), Manhattan College, NY, USA. 14) Hotta, H., Nishi, T. and Kuroda, S. (1996): Report results of assessments of damage to EPS embankments caused by earthquakes, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Miki, H. (1996): EPS construction method in Japan, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Mohamed, E. (1996): History of EPS as embankment ˆll in Malaysia under PIC and its future, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Negussey, D. and Sun, M. C. (1996): Reducing lateral pressure by geofoam (EPS) substitution, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Negussey, D. (2007): Design parameters for EPS geofoam, Soils and Foundations, 47(1), ) Ooe, Y., Matsuda, Y., Tada, S. and Nishikawa, J. (1996): Earth pressure reduction for culverts using EPS, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Sanders, R. L. (1996): United Kingdom design and construction experience with EPS, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Santucci de Magistris, F., Koseki, J., Amaya, M., Hamaya, S., Sato, T. and Tatsuoka, F. (1999): A triaxial testing system to evaluate stress strain behaviour of soils for wide range of strain and strain rate, Geotechnical Testing Journal, ASTM,22(1), ) Tatsuoka, F., Teachavorasinskun, S., Dong, J., Kohata, Y. and Sato, T. (1994): Importance of measuring local strains in cyclic triaxial tests on granular materials, Proc. ASTM Symposium Dynamic Geotechnical TestingII, ASTM, STP 1213, ) Tatsuoka, F., Jardine, R. J., Lo Presti, D., Di Benedetto, H. and Kodaka, T. (1999a): Characterizing the pre-failure deformation properties of geomaterials, Theme Lecture for the Plenary Session No. 1, Proc. 14th ICSMFE, Hamburg, September 1997, 4, ) Tatsuoka, F., Modoni, G., Jiang, G. L., Anh Dan, L. Q., Flora, A., Matsushita, M. and Koseki, J. (1999b): Stress-strain behaviour at small strains of unbound granular materials and its laboratory tests, Keynote Lecture, Proc. Workshop on Modelling and Advanced Testing for Unbound Granular Materials, January 21 and 22, 1999, Lisboa (ed. by Correia), Balkema, ) Tsukamoto, Y., Ishihara, K., Kon, H. and Masuo, T. (2002): Use of compressible expanded polystyrene blocks and geogrids for retaining wall structures, Soils and Foundations, 42(4), ) Van Dorp, T. (1988): Expanded polystyrene foam as light ˆll and foundation material in road structures, International Congress on Expanded Polystyrene, Milan,Italy. 27) Van Dorp, T. (1996): Building on EPS geofoam in the ``Low- Lands'' experiences in the Netherlands, Proc. International Symposium on EPS Construction Method, Tokyo, Japan, ) Yamanaka, O., Onuki, T., Katsurada, H., Kitada, I., Kashima, K., Takamoto, A. and Maruoka, M. (1996): Use of vertical wall-type EPS elevated ˆlling (H=15 m) for bridge abutment back ˆll, Proc. International Symposium on EPS Construction Method, Tokyo, Japan,