Rate Effects on the Stress-Strain Behaviour of Eps Geofoam

SOILS AND FOUNDATIONS Japanese Geotechnical Society

Vol. 48, No. 4, 479–494, Aug. 2008

RATE EFFECTS ON THE STRESS-STRAIN BEHAVIOUR OF EPS GEOFOAM G. E. ABDELRAHMANi), SHOHEI KAWABEii), FUMIO TATSUOKAiii) and YOSHIMICHI TSUKAMOTOiv) ABSTRACT The rate-dependency of the stress-strain behavior of EPS (Expanded Polystyrene) geofoam with densities of 19.3 and 28.0 kg/m3 was investigated by performing unconventional unconˆned compression tests. A set of monotonic loading (ML) tests were performed at diŠerent constant values of vertical (axial) strain rate, e· v. The e· v value was stepwise changed many times and several sustained loading (SL) tests were performed during otherwise ML at a constant e· v in other tests. A number of SL tests were performed during global unload and reload cycles to infer the stress-strain relation when e· v＝0. The elastic properties were evaluated by applying minute unload/reload cycles during otherwise ML. The rate-dependent stress-strain behaviour observed in these tests was described by an elasto-viscoplastic model (i.e., a non-linear three-component model), for which the vertical (axial) stress, sv, consists of inviscid and viscous components, sfv and svv, while e· v consists of elastic and irreversible components, e· ev and e· irv. It is shown that the viscous property of EPS geofoam is of Isotach type in that, under the loading conditions where e· irv is always positive, the current svv value is a unique function of instantaneous eirv and e· irv, therefore the strength increases with e· v. This viscous property was quantiˆed based on the test results and incorporated into the model. The rate-dependent stress-strain behaviour, including the creep behaviour, observed in the experiment is simulated very well by the proposed model. In particular, the fact that the creep strain becomes signiˆcant when the sustained sv value becomes larger than the inviscid yield vertical stress is well simulated. Key words: creep deformation, elasto-viscoplastic model, EPS geofoam, loading rate eŠect, non-linear three component model, unconˆned compression test, viscous properties (IGC: D5/D6) plastic properties, of EPS geofoam should be correctly understood and accurately quantiˆed. However, it seems that study on this issue by systematic laboratory stressstrain tests has been rather limited. Several constitutive models to describe the rate-dependent stress-strain behaviour of EPS geofoam have been proposed, as summarized by Chun et al. (2004). Many of these models were developed to predict the creep deformation of EPS geofoam under ˆxed stress conditions (e.g., Findley and Khosla, 1956: Findley et al., 1989; Horvath, 1998). Therefore, they cannot predict the stress-strain-time behaviour when the strain or stress rate changes arbitrarily in the course of loading. Some other models describe the rate-dependent stress-strain behaviour under arbitrary loading conditions, including the creep deformation, of EPS geofoam by using the general time, t, that is deˆned as zero at the start of loading as the basic variable (i.e., the isochronous models: e.g., Missirlis et al., 2004). The isochronous models predict that the current stress-strain state at the same t is the same irrespective of previous loading history since the start of loading. It is known,

INTRODUCTION The stress-strain behaviour of Expanded PolyStyrene (EPS) geofoam is often approximated as linear elastic/ perfectly plastic (e.g., Takahara and Miura, 1998) or nonlinear elasto-plastic (e.g., Hazarika, 2006). However, the behaviour is signiˆcantly rate-dependent and considerable creep deformation takes place, except for the initial rather linear part (e.g., Negussey, 1997; Duskove, 1998). For example, a higher strain rate results in a smaller strain and a higher compressive strength in monotonic loading (ML) compression test and larger creep strains take place when continuing sustained loading longer. Therefore, to accurately predict the long-term residual deformation as well as load-deformation behaviour at diŠerent loading rates of full-scale EPS structures in the ˆeld constructed as foundations to support transient and permanent working loads while allowing limited amounts of instantaneous and residual deformations (e.g., bridge abutments and other transportation structures), the rate eŠects on the stress-strain behaviour, as well as elastoi) ii) iii) iv)

Associate Professor, Civil Engineering Department, Fayoum University, Egypt. Graduate Student, Department of Civil Engineering, Tokyo University of Science, Japan. Professor, ditto (tatsuoka＠rs.noda.tus.ac.jp). Associate Professor, ditto. The manuscript for this paper was received for review on September 19, 2007; approved on April 8, 2008. Written discussions on this paper should be submitted before March 1, 2009 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyoku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month. 479

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however, that the rate-dependent stress-strain behaviour under arbitrary loading conditions of geomaterial (i.e., soil and rock: Tatsuoka et al., 2000, 2001; Di Benedetto et al., 2005; Tatsuoka, 2004, 2008) and polymer geosynthetic reinforcement (Hirakawa et al., 2003; Kongkitkul et al., 2004) become non-objective when predicted based on any isochronous model. It is shown later in this paper that it is also the case with EPS geofoam. To the best of the authors' knowledge, a relevant elasto-viscoplastic model that can describe and predict the stress-strain-time behaviour of EPS geofoam for given arbitrary loading histories has not been proposed. In view of the above, the present study was performed ˆrstly to evaluate the rate-dependency of stress-strain property of EPS geofoam by performing a series of unconventional unconˆned compression tests as follows: 1) A set of continuous monotonic loading (ML) tests at diŠerent constant vertical strain rates (e· v) were performed. 2) e· v was stepwise changed many times and several sustained loading (SL) tests (i.e., creep loading tests) were performed during otherwise ML at a constant e· v. 3) A number of SL tests were performed during global unload and reload cycles to infer the stress-strain relation when e· v＝0. 4) The elastic properties were evaluated by applying minute unload/reload cycles during otherwise ML. Secondly, the rate-dependent stress-strain behaviour of EPS geofoam observed in these tests was analysed as an elasto-viscoplastic material within the framework of a non-linear three-component model that has been known to properly describe the elasto-viscoplastic properties of geomaterial (i.e., unbound and bound soils; Di Benedetto et al., 2002; Tatsuoka et al., 2002, 2008) as well as polymer geosynthetic reinforcement (Hirakawa et al., 2003: Kongkitkul et al., 2004, 2007). Based on this analysis, it is shown that EPS geofoam has a speciˆc type of viscous property, called the Isotach type, for which the current stress is a unique function of instantaneous irreversible strain and its rate under the loading conditions (i.e., when the irreversible strain rate is always increasing). Finally, a wide variety of rate-dependent stress-strain behaviour of EPS geofoam observed in the unconˆned compression tests is simulated by the non-linear threecomponent model. It is shown that, once the model parameters of a given EPS geofoam type are determined by performing some representative tests, the stress-straintime behaviour when subjected to other arbitrary loading histories, including multiple SL loadings at diŠerent load levels, as encountered when incorporated and used in a FEM code to analyze full-scale ˆeld boundary value problems, can be simulated and predicted. The eŠect of conˆning pressure on the elasto-viscoplastic stress-strain property of EPS geofoam is one of the other important issues to be studied and the research on this topic is presently underway. This topic is beyond the scope of this paper.

TEST METHOD

Specimens: As the details of the test method are reported in the companion paper (Abdelrahman et al., 2008), only a brief summary is herein presented. Two types of EPS geofoam having diŠerent densities (19.3 and 28.0 kg/m3), termed D-20 and D-30, were used. A number of cylindrical specimens (75 mm in diameter and 150 mm in height), which were trimmed from representative block samples by using a thermal wire saw, were provided by a manufacturer. The top and bottom ends of the specimen were well lubricated by arranging a 0.3 mm-thick latex rubber disk smeared with an about 50 mm-thick layer of Dow high-vacuum silicon grease. In the tests performed at the latter stage of the study, to keep the deformation within the specimen as homogeneous as possible by alleviating the eŠects of surface irregularity produced when machining both ends of the specimen, a thin layer of wet soft gypsum pasted between two plastic wrapping sheets was inserted between the lubrication layer at the specimen bottom and the pedestal. As shown later in this paper, the stress-strain relations at small strains from the tests performed at the early stage of the study (i.e., before this treatment method was introduced) are somehow erratic even when based on locally measured vertical and horizontal strains. Unconˆned compression test: Two types of loading systems were used. For strain-controlled tests, a fully automated triaxial apparatus (Fig. 1) was used that consists of a precise gear system for vertical (axial) loading, by which the vertical displacement rate can be kept constant or changed stepwise or gradually by controlling the displacement to an accuracy of the order of 1 mm. By using this, it is also possible to perform sustained loading (SL) at a ˆxed stress state and stress relaxation at a ˆxed vertical strain and to apply minute unload/reload cycles during otherwise monotonic loading (ML) at a constant strain rate. In the respective stress-controlled tests performed in this study, a number of sustained loading tests were performed. These tests used a pneumatic loading system consisting of a double-action Bellofram cylinder to which computer-controlled air-pressure was supplied. In a couple of tests performed at the early stage of the

Fig. 1.

Compression test apparatus

STRESS-STRAIN BEHAVIOUR OF EPS

study, the vertical (axial) strain, ev, of the specimen was measured only externally by using an external displacement gauge a LVDT (i.e., a linear variable diŠerential transducer, Fig. 1). In the other later tests, the vertical compression was measured locally along the specimen side by using a pair of Local Deformation Transducers (LDTs; Goto et al., 1991) to evaluate accurately and sensitively the vertical strain free from eŠects of bedding error (BE). The largest vertical strain measurable with LDTs was about 1.5–2.0z. In the latter stage of the study, the lateral (horizontal) strain, eh, was measured by means of three clip gauges (CGs) positioned at the heights of a sixth, a half and ˆve sixths of the total height of the specimen from the bottom (Fig. 1). The largest lateral strain measurable with CGs was about －1.5z. In the present study, this limit was never reached until the end of the respective tests due to a very low Poisson's ratio at large strains of EPS geofoam (Abdelrahman et al., 2008). Therefore, changes in the cross-sectional area of the specimen evaluated based on the lateral strains measured with CGs were actually very small. The locally measured vertical and lateral strains from these tests reported in this paper were those obtained by averaging the readings of, respectively, a pair of LDTs and three CGs. In a couple of tests performed at the early stage of the present study where CGs were not used, a constant cross-sectional area was assumed when computing the vertical stress. When based on results from similar tests using CGs performed, errors in the vertical stress computed by this simpliˆed method are very small, less than 1z at the largest. LOADING RATE EFFECTS IN THE EXPERIMENT

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Fig. 2. Strain rate eŠects on the ML stress-strain relation, EPS geofoam D-20

before the start of large-scale yielding. 3) The horizontal strain, eh, is signiˆcantly smaller than the vertical strain, ev. For example, in the test at e· v＝ 0.33z/min, eh is only about 0.15z when ev＝10z. Moreover, after the start of large-scale yielding, the increment of eh becomes positive (i.e., the specimen diameter decreases), therefore the tangent Poisson's ratio, nvh＝－[deh/dev](s ＝constant), becomes negative. This peculiar trend of behaviour may be due to some speciˆc yielding mechanism of EPS geofoam. A slight eŠect of e· v on the eh-ev relation is noticeable in that the ratio, deh/dev, increases slightly (i.e., the specimen diameter decreases at a higher rate) with a decrease in e· v after the start of large-scale yielding. However, this rate eŠect tends to decrease with an increase in ev. When based on the fact the eh value itself is very small, we can conclude that the rate eŠect on the ‰ow characteristics is very small, if any. The test results shown above indicate that EPS geofoam is an elasto-viscoplastic material (Cho, 1992; Horvath, 1997). Horvath (2001) recommended the use of the parameters obtained from the initial apparently linear elastic stress-strain behavior of EPS geofoam, such as ``elastic limit stress'' and ``initial tangent modulus'' deˆned at ev＝1z measured by relatively fast ML compression tests, to predict the behaviour of EPS geofoam structures at working loads in design. The test data presented in Fig. 2 indicate that the ``elastic limit stress'' is not a unique value for a given type of EPS geofoam, but it decreases with a decrease in the strain rate. Duskove (1998) argued that, in in-air fast loading tests on EPS geofoam, the air inside the specimen cannot be expelled freely from the inside the specimen, still staying partly inside and resisting against the axial compression. h

ML tests at diŠerent strain rates: Figure 2 shows the results from a set of continuous ML unconˆned compression tests on EPS geofoam D-20 at ˆve diŠerent constant vertical strain rates, e· v＝ 0.0046, 0.047, 0.33, 3.3 and 33 z/min (i.e., vertical compression rates＝0.0069, 0.07, 0.5, 5 and 50 mm/min). The largest diŠerence among these strain rates is by a factor of about 7,200. The horizontal strain was measured in three tests. The following trends of behaviour may be seen from Fig. 2: 1) The strength increases signiˆcantly with an increase in e· v. Yet, it is di‹cult to accurately quantify the strain rate eŠects on the stress-strain behaviour based on only these test results because of an inevitable variance in the material properties among the diŠerent specimens used in these tests. 2) The strain rate eŠect on the stress-strain behaviour suddenly becomes very signiˆcant when the vertical strain, ev, becomes larger than about 1z, at which the large-scale yielding starts taking place. Large-scale yielding is deˆned as the start of the occurrence of signiˆcant inelastic (or irreversible or visco-plastic) strains. It is likely that the large-scale yielding is due to the collapse of EPS bead particles and/or distinguishable slippage at the interface in between. On the other hand, the loading rate eŠects are insigniˆcant and the stress-strain behaviour is rather linear and reversible

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Fig. 4. Stress-strain curves from a ML test with step changes in the strain rate and SL stages of EPS geofoam D-30: a) whole range, and b) initial stage (erratic due to end surface irregularity)

Fig. 3. Stress-strain curves from a ML test with step changes in the strain rate and SL stages of EPS geofoam D-20: a) whole range, b) close-up and c) initial stage

Then, somewhat stiŠer behavior is obtained compared with slower loading tests in which air can more freely escape from the inside of the specimen during compression. It is possible that a slightly higher decreasing rate of the specimen diameter at a lower vertical strain rate after the start of large-scale yielding seen in Fig. 2 (and also Figs. 3(a) and 3(b) shown below) is due partly to this mechanism. However, this mechanism is di‹cult to explain why the loading rate eŠect suddenly becomes signiˆcant when the large-scale yielding starts. Furthermore, even if the above-mentioned air-exhausting mechanism is relevant, this mechanism is very di‹cult to quantify and formulate for constitutive modeling. In the present study, irrespective of true mechanism, the loading rate eŠects observed in the experiment are simulated by dealing with EPS as an elasto-viscoplastic material. ML with step changes in the strain rate and SL stages: Figures 3 and 4 show the results from two strain-controlled tests performed on EPS geofoam D-20 and D-30. In these tests, the vertical strain rate, e· v, was stepwise changed many times to evaluate the viscous properties both before and after the start of large-scale yielding. The diŠerent e· v values are denoted by ratios to the basic vertical strain rate, e· 0＝0.33z/min. Two times of SL tests, each for two hours with the D-20 specimen (Fig. 3) or one hour with the D-30 specimen (Fig. 4), were performed before the start of large-scale yielding. In the test on D-20 (Fig. 3), although the measure to remove the eŠects of surface irregularity of the specimen ends was taken, the

STRESS-STRAIN BEHAVIOUR OF EPS

eŠects of bedding error on the externally measured vertical strains are signiˆcant (Fig. 3(c)). As the test on the D-30 specimen was performed before the introduction of this measure, vertical strains measured locally with a pair of LDTs less than about 0.5z are erratic, so not reliable (Fig. 4(b)). Although the horizontal strain was measured in both tests, its reliable data could not be obtained due to malfunction of the CGs in the test on the D-30 specimen. In the test on the D-20 specimen, the recording of the top CG stopped before ev became 3z (Fig. 3(a)). As may be seen from Fig. 3(a), however, the records of the other two CGs (at the middle level and the bottom) are very similar until the end of the test. The eŠects of bedding error on externally measured vertical strains averaged for the whole specimen height increase with a decrease in the specimen size. Therefore, the stress and externally measured strain relation exhibits apparently softer behaviour with a decrease in the specimen size (e.g., Hazarika, 2006). This apparent size eŠect due to the bedding error is pointed out by Abdelrahman et al. (2008) and this eŠect may become negligible in full-scale cases in the ˆeld. In the present study, it is assumed that the relationships between the vertical stress and the locally measured vertical strain, both averaged for the whole specimen, are material properties, independent of specimen size. Moreover, in case a measure to keep the stress state in the specimen as homogeneous as possible, as employed in the present study, is not taken and axial strains are measured externally and including signiˆcant bedding error eŠects, we may observe signiˆcant apparent eŠects of specimen shape. In the present study, no eŠects of specimen shape on the measured stress-strain properties are taken into account following the ordinary concept of material stress-strain testing. Figure 5 compares the sv-ev curves from a set of continuous ML tests at diŠerent constant e· v values (Fig. 2) with the one from the ML test with many step changes in e· v (Fig. 3(a)) of EPS geofoam D-20. The following trends of behaviour may be seen from Figs. 3, 4 and 5: 1) The sv-ev relation exhibits large-scale yielding at ev

Fig. 5. Comparison of the stress-strain relations from ML tests at diŠerent constant strain rates (Fig. 2) and a ML test with step changes in the strain rate (Fig. 3(a)), D-20

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equal to about 1z, which is consistent with the data presented in Fig. 2. During the subsequent loading, upon a step increase in e· v, the sv value increases very sharply, and vice versa. As explained later in this paper, the stiŠness immediately after a step increase in e· v is close to the elastic modulus. 2) The jump in the sv value that takes place upon the respective step changes in e· v is rather persistent as long as ML continues at constant e· v. It may be seen from Fig. 5 that, when we allow some variance in the data among the diŠerent specimens, the segment of sv-ev curve at respective e· v values in the ML test with many step changes in e· v is consistent with the one from the continuous ML test at the same e· v. These results indicate that, under the loading condition (i.e., when e· v is kept always positive), the current sv value is uniquely controlled by the instantaneous values of ev and e· v (more rigorously by the irreversible vertical strain, eirv, and its rate, e· irv) and the strength increases with an increase in e· v. This rate-dependency of stress-strain behaviour is called the Isotach viscosity (e.g., Di Benedetto et al., 2002; Tatsuoka et al., 2002). 3) The eŠect of step change in e· v on the sv-ev behaviour is insigniˆcant before the start of large-scale yielding when compared with after the start of large-scale yielding. This limited loading rate eŠect in the preyielding regime is consistent with the data presented in Fig. 2 and also with some previous studies (Negussey, 1996; Duskove, 1998; Horvath, 1998). Despite the above, the sv-ev behaviour in the pre-yielding regime is not perfectly elastic as evident from small but noticeable creep deformation that takes place at the respective SL stages (Figs. 3(c) and 4(b)). 4) The horizontal strain, eh, is signiˆcantly smaller than the vertical strain, ev and, after the start of large-scale yielding, the increment of eh becomes positive (i.e., compressive). These trends of behaviour are consistent with those seen from Fig. 2. In Fig. 3(a), eh increases suddenly upon a decrease in the vertical strain rate, e· v, and vice versa ( see also Fig. 3(b)). This trend is essentially due to an elastic response of the material: i.e., a positive (i.e., compressive) elastic increment, Deeh, takes place suddenly (i.e., the specimen diameter decreases suddenly) by a sudden negative (i.e., extensive) elastic vertical strain increment, Deev, caused by a sudden decrease in the vertical stress that takes place upon a step decease in e· v and vice versa. On the other hand, the tangential slope of the eh-ev relation, deh/ dev, during ML at the respective constant e· v values tends to increase slightly (i.e., the specimen diameter tends to decrease at a slightly higher rate) with a decease in e· v. Correspondingly, eh increases slightly (i.e., the specimen diameter decreases) with time during SL (Fig. 3(c)). These trends are consistent with the one seen from Fig. 2. However, this strain rate eŠect is very small as the eh value itself is very small and this trend becomes weaker as the strain increases in the post-yielding regime. During ML at the respective constant e· v values after the start of large-scale yield-

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Fig. 6. Non-linear three-component model (Di Benedetto et al., 2002; Tatsuoka et al., 2002)

ing, the sv value is rather constant, therefore, the ratio, deh/dev, is nearly the same as the irreversible strain-increment ratios, deirh/deirv (i.e., the ‰ow rate). Then, we can conclude that the eŠects of strain rate on the ‰ow characteristics are insigniˆcant, if any. Paths 1 and 2 as well as points A and B indicated in Figs. 3(a) and 4(a) are explained later in this paper. NON-LINEAR THREE-COMPONENT MODEL

Model description: In this section, the elasto-viscoplastic stress-strain property of EPS geofoam described above is modeled within the framework of a non-linear three-component model (Fig. 6; Di Benedetto and Tatsuoka, 1997; Di Benedetto et al., 2001a, b, 2002; 2005; Tatsuoka et al., 2000, 2001, 2002, 2008; Tatsuoka, 2005). Applying this model (Fig. 6) to the unconˆned compression test conditions in the present study, the vertical stress, sv, and the vertical strain, ev are used as the stress and strain parameters, s and e. The vertical stress, sv, is decomposed into the inviscid (non-viscous) component, sfv, and the viscous component, svv, while the vertical strain rate, e· v, into the elastic component, e· ev, and the irreversible (or inelastic or visco-plastic) component, e· irv. For the test results described in this paper, the current irreversible vertical strain, eirv, was obtained by integrating the increment, deirv＝dev－deev, for a given loading history, where dev is the measured (or a given) vertical strain increment; and deev is the elastic strain increment obtained as dsv/Eeq(sv), where Eeq(sv) is the current elastic Young's modulus, which is a function of sv (explained later), and dsv is a given vertical stress increment. The elastic vertical strain, eev, was obtained by integrating the increment, deev. It is assumed that, whatever strain (or stress) rates take place during a given time history of irreversible strain (eirv) under the loading conditions, there exists a unique sfv-eirv relation (called the reference relation). On the other hand, diŠerent sfv-eirv relations are formed for loading, unloading, reloading and so on. It is to be noted that the reference relation can also be expressed in terms of sfv-ev relation because the sfv-eev relation is independent of loading history. DiŠerent viscosity types: Tatsuoka et al. (2008) reported that diŠerent trends of rate-dependent stress-strain behavior had been observed in a number of laboratory

Fig. 7. a) DiŠerent stress-strain curves for ML at a strain rate of 10 e· 0 for four diŠerent viscosity types of geomaterial when assuming the same curve at a strain rate of e0 (Tatsuoka et al., 2007) and b) stress-strain behaviour including creep deformation in the case of Isotach viscosity

stress-strain tests on diŠerent types of geomaterial. In these tests, the viscous properties were evaluated by stepwise changing of the strain rate, e· , and performing SL tests during otherwise ML at a constant e· and also by performing ML tests at diŠerent constant values of e· as performed in the present study. They showed that these various rate-dependent stress-strain behaviours can be categorized into four basic types (i.e., Isotach, Combined, TESRA and Positive & Negative, illustrated in Fig. 7(a)). In this ˆgure, it is assumed that, for the diŠerent viscosity types, the stress-strain relation at a certain strain rate, e· 0, is the same while the same positive stress jump, Ds, takes place when the strain rate, e· , is stepwise increased by a factor of 10 from e· 0 to 10 e· 0 at the same strain. The Isotach type is the most classical one (Suklje, 1969). The viscous stress increment, denoted by Dsv, that has developed at a given moment does not decay with an increase in e· ir during subsequent loading. In the speciˆc formulation proposed by Di Benedetto et al. (2002) and Tatsuoka et al. (2002), under the loading conditions, the current viscous stress, sv, is proportional to the current sf value while it is a highly non-linear function of instantaneous irreversible strain rate, e· ir. Therefore, a unique s-e relation is obtained by ML at a given constant e· and

STRESS-STRAIN BEHAVIOUR OF EPS

this relation is located consistently above the s-e relation obtained by imaginary ML at zero e· (i.e., the sf-e relation, called the reference stress-strain relation). So, the strength increases with an increase in e· . Generally, coherent geomaterials, such as sedimentary soft rocks and cement-mixed soils, highly interlocked geomaterials, such as highly-compacted well-graded relatively angular unbound gravels, and plastic clays, exhibit this viscosity type in the pre-peak regime (Tatsuoka et al., 2008). Polymer geosynthetic reinforcement also exhibits this type of viscosity (Hirakawa et al., 2003: Kongkitkul et al., 2004, 2007). It is evident from the test results presented in this paper that the viscosity type of EPS geofoam is also Isotach. As illustrated in Fig. 7(b), when a material having the Isotach viscosity is subjected to sustained loading (SL) at a certain stress, s, from point S during otherwise ML at a constant strain rate, the creep deformation continues until the viscous stress component, sv, becomes zero at point E after an inˆnite period of time. With the other three types illustrated in Fig. 7(a), the viscous stress increment, Dsv, that has developed at a given moment decays towards diŠerent residual values during subsequent loading. With the Combined type, the afore-mentioned Dsv decays only partially. So, the strength during ML at constant e· increases with an increase in e· , like the Isotach type. With the TESRA (temporary eŠects of strain rate and strain acceleration) type, Dsv decays eventually to zero, therefore the strength during ML at constant e· is essentially independent of e· . Poorly graded relatively angular sands and gravels exhibit this type of viscosity in the pre-peak regime. With the P&N (positive and negative) type, a positive Dsv decays eventually to a negative value and vice versa. So, the strength during ML at constant e· decreases with an increase in e· . Poorly graded round sands and gravels exhibit this type of viscosity. The details are described in Tatsuoka et al. (2008). About the isochronous models: Based on the fact that the viscous property of EPS geofoam is of the Isotach type, it can be readily seen that the isochronous models are not realistic with EPS geofoam. For example, from point A indicated in Figs. 3(a) and 4(a), we can reach point B via diŠerent paths, such as path 1 (along which e· v is kept high, equal to 20 e· 0) or path 2 (along which e· v is much lower). It is obvious that, despite the same stressstrain state at point B and the same subsequent stressstrain behaviour for the same strain rate (i.e., 20 e· 0), the time increment that has elapsed between points A and B can be largely diŠerent (i.e., not isochronous) among diŠerent loading histories via paths 1 and 2 and others. Similarly, in Fig. 7(b), after the restart of ML at a certain e· v from point A during otherwise SL starting from point S, the stress-strain curve tends to rejoin the original curve for the original strain rate at point B. This trend of behaviour can be seen in Fig. 3(c). It is obvious in the case of Fig. 7(b) that the time increment necessary to move from points S to B by continuous ML is much shorter than the one necessary by SL and then restarting ML (i.e., path Sª Aª B) (i.e., not isochronous), despite that

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the subsequent stress-strain behaviour at the same strain rate after point B is the same irrespective of diŠerent loading histories between points S and B. MODELLING THE STRESS-STRAIN BEHAVIOUR OF EPS GEOFOAM

Isotach viscosity model: The Isotach viscosity for EPS geofoam under the test conditions in the present study can be described as: sv(eirv, e· irv)＝sfv(eirv)＋svv(eirv, e· irv)

(1)

where sv(eirv, e· irv) is the vertical stress, which is a function of the current irreversible vertical strain, eirv, and its rate, e· irv; sfv(eirv) is the inviscid vertical stress component that is a non-linear function of eirv; and svv(eirv, e· irv) is the viscous vertical stress component given by:

svv(eirv, e· irv)＝sfv(eirv)･gv(e· irv)

(2)

where gv(e· irv) is the viscosity function, which is a highly non-linear function of e· ir. Di Benedetto et al. (2002) and Tatsuoka et al. (2002) proposed the following equation for geomaterials:

«

gv(e· irv)＝a･ 1－exp

{ Ø 1－

`e· irv` ＋1 (e· irv)r

» }$ m

(Æ0)

(3)

where `e· irv` is the absolute value of e· irv; and a, m and (e· irv)r are positive material constants. Equation (3) satisˆes the following conditions: 1) gv(e· irv) is always positive irrespective of the sign of e· irv; 2) when e· irv＝0, gv(e· irv)＝0 and dgv (e· irv)/de· irv is a ˆnite value, equal to a･m/e· irv; therefore, numerical simulation is smooth when ML starts from the static state (where e· irv＝0); and 3) gv(e· irv) becomes a ˆnite positive value (equal to a) when e· irv becomes inˆnitive. Equation (3) is also used for EPS geofoam. Reference stress-strain relation: According to the threecomponent model (Fig. 6), the reference stress-strain relation (i.e., the sfv-eirv relation or the sfv-ev relation) is obtained by performing a ML test at an essentially zero strain rate (i.e., at e· v¿e· irv¿0). However, it is practically impossible to perform such ML tests as above. Another way to obtain the reference stress-strain relation is to ˆnd multiple neutral (sv, ev) states where essentially no creep strain takes place that can be reached by decreasing sv from (sv, ev) states under the loading condition (i.e., e· v À0). Figures 8 and 9 show results from such two stresscontrolled tests as above performed on EPS geofoam D-20 and D-30. Several nearly full unload/reload cycles of sv were applied at respectively constant stress rates, s· v, during otherwise primary loading at a constant s· v. During the respective global unload/reload cycles, several SL tests, each lasting for the respectively speciˆed periods, were performed. As a measure to alleviate the eŠects of surface irregularity at the specimen ends was taken in these tests, the eŠects of bedding errors on the externally measured vertical strains are relatively small, in particular, after the start of large-scale yielding, like the test result presented in Fig. 3. The vertical strains were measured only externally because the vertical strain range of

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Fig. 8. Results from a stress-controlled test with full unload/reload cycles including SL stages (each for 15 minutes), D-20: a) time history of vertical stress, b) whole stress-strain curve, c) stress-strain curves at small strains and d) stress-strain curves during primary loading, global unloading and reloading at large strains

the major concern exceeds the capacity of the LDT. The horizontal strain was measured only until ev became about 2z on the D-20 specimen (Fig. 8(c)) to reconˆrm the trends of ‰ow behaviour seen in other tests as shown in Fig. 3. The following trends of behaviour may be seen from these ˆgures: 1) As seen typically from Figs. 9(b) and 9(c), the creep strain by SL for the same period during otherwise primary loading increases with an increase in sv. The increasing rate is particularly high after the start of large-scale yielding (Fig. 9(b)). On the other hand, as seen from Fig. 9(c), the creep strain is very small before the start of large-scale yielding. 2) Figure 8(d) describes the stress-strain relation immediately before and during the last global unload/reload cycle in the test on EPS geofoam D-20. Immediately after sv starts decreasing from the stress-controlled primary loading stage, ev is still increasing until sv becomes about 88 kPa. According to the Isotach three-component model, this is due to that the value of svv is still positive with the (sv, eirv) state being locat-

ed above the reference curve, which results in positive e· irv values. This trend becomes weaker when stress-unloading is started after svv has become much smaller by SL for some period (Figs. 9(d) and (e)). 3) In Fig. 8(d), during the ˆrst SL stage S1-E1 during otherwise stress-unloading, ev increases with time. On the other hand, ev decreases with time during the next SL stage S2-E2. Such a negative creep vertical strain as above is much larger at the next SL stage S3-E3. The same trend as above can be seen in Fig. 9(d): i.e., the positive creep vertical strain at stage (1) changes to a negative one at stage (2). In the data presented in Fig. 9(e), the creep vertical strain is still positive at stage (2) and it becomes negative at stage (3). Therefore, we can anticipate essentially zero creep strain rate if a SL test is performed at a relevant sv between those at these two SL stages where the creep vertical strain is respectively positive and negative (e.g., between stages S1E1 and S2-E2 in Fig. 8(d)). According to the Isotach model, the reference stress-strain relation for loading conditions (i.e., e· vÀ0) passes these neutral stress-

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Fig. 9. Results from a stress-controlled test with full unload/reload cycles including SL stages (each for 30 minutes or one hour), D-30: a) time history of vertical stress, b) whole stress-strain curve, c) stress-strain curves at small strains and d) and e) stress-strain curves during primary loading, global unloading and reloading at large strains

strain states. Then, positive creep vertical strain rates, e· irvÀ0, take place if SL starts from any stress-strain state located above the reference curve ( see Fig. 7(b)). The reference stress-strain relations in terms of sfv-ev relation of EPS geofoam D-20 and D-30 are depicted in Figs. 10(a) and 10(b). The ev values of the reference curve for EPS geofoam D-20 presented in Fig. 10(a) do not include the bedding error (BE) eŠects, while the measured vertical strains do. For this reason, their relevant comparison cannot be made at small strains, before the start of large-scale yielding. A due comparison is made later in this paper. The ev values of the two reference relations (1)

and (2) for EPS geofoam D-30 depicted in Fig. 10(b), respectively, does not include and includes the BE eŠects. The diŠerence between the two reference curves is signiˆcant only at small strains. Reference curve (1), not including the BE eŠects, is used to simulate the results of a test in which the vertical strains not including the BE eŠects were measured by using LDTs (Figs. 19(c) and 19(d)). On the other hand, Reference curve (2), including the BE eŠects, is used to simulate the test results presented in Fig. 10(b) (Fig. 21), in which only vertical strains including BE eŠects were measured by using a LVDT. These reference stress-strain curves presented in Figs.

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Fig. 11. Elastic Young's modulus of EPS geofoam (Abdelrahman et al., 2008)

ds v

fE

eev＝

(5)

eq

Fig. 10.

Reference curves of EPS geofoam: a) D-20 and b) D-30

10(a) and 10(b) were determined so that the following conditions are satisˆed: 1) With EPS geofoam D-20, the reference curve is located slightly lower than the stress-strain relation from the slowest continuous ML test among those depicted in Fig. 2. 2) With EPS geofoam D-20, the reference stress-strain curve is located between SL stages S1-E1 and S2-E2 in Fig. 8(c). With EPS geofoam D-30, similarly, the reference curve is located between SL stages (1) and (2) in Fig. 9(d) and slightly below stage (2) in Fig. 9(e). 3) The experimentally obtained rate-dependent stressstrain relations can be simulated appropriately when based on the determined reference stress-strain curve, as shown later in this paper. The following empirical function was ˆtted to these respective reference relations:

－P4･(eirv)P sfv＝P1＋( P2＋P3･eirv)[1－exp s ir －P6･ev /P2t ]

5

(4)

where P1, P2, P3, P4, P5 and P6 are the constants determined experimentally (which are listed in the respective ˆgures presenting simulations, shown below). Elastic vertical strain: The elastic vertical strains were obtained as:

where Eeq is the equivalent vertical Young's modulus representing the instantaneous elastic characteristics at a given stress state that is obtained by the empirical equations for EPS geofoam D-20 and D-30 presented in Fig. 11. The Eeq values presented in Fig. 11 were evaluated from the relationships between sv and ev (measured with a pair of LDT) during ˆve or six minute unload/reload cycles after SL had been applied for a half or one hour during otherwise ML, as typically shown in Fig. 12. The stress amplitude of sv was 10 kPa and the strain rate during these unload/reload cycles was the same as the one during ML. The details are reported in Abdelrahman et al. (2008). Figures 13(a) and 13(b) present the sv-eev and sv-eirv relations as well as the measured sv-ev relations for EPS geofoam D-20 and D-30 (presented in Figs. 3 and 4). The ev values plotted in these ˆgures were obtained by connecting those at small strain levels measured with a pair of LDTs to subsequent vertical strain increments measured externally. It may be seen from Figs. 13(a) and 13(b) that the stiŠness of sv-ev relation immediately after a step increase in e· v made after the start of large-scale yielding is very high and close to the elastic one at similar stress levels. Correspondingly, the increment of eirv immediately after a step increase in e· v is very small. Furthermore, the irreversible vertical strain before the start of large-scale yielding is very small when compared with those that take place after the start of large-scale yielding. A slightly negative increment of eirv before the start of large-scale yielding of D-30 seen in Fig. 13(b) is unnatural, which is due to errors in the locally measured vertical strains caused by surface irregularity at the specimen ends ( see Fig. 4(b)). The stress-strain relations presented in Figs. 2, 9, 12 and 13 are simulated later in this paper. Rate-sensitivity coe‹cient and viscosity function: The major feature of the viscous property of a given material

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where, referring to Fig. 14, Dsv is the vertical stress jump that takes place upon a stepwise change in the irreversible vertical strain rate from (e· irv)before to (e· irv)after when the vertical stress is equal to sv. According to the three-component model (Fig. 6), this stress jump, Dsv, is the same as the viscous stress increment, Dsvv. Figures 15(a) and (b) present the relationships between

the ratio, Dsv/sv, and log ((e· irv)after/(e· irv)before) for these two types of EPS geofoam obtained from the data presented in Figs. 13(a) and (b). It may be seen that these relations are essentially independent of sv when the strain rate is stepwise changed while they are highly linear. The slope of a given linear relation is deˆned as the rate-sensitivity coe‹cient, b (Eq. (6)), which is equal to 0.123 for EPS geofoam D-20 and 0.114 for EPS geofoam D-30. Considering some scatter of data in Fig. 15, we can conclude that the b values of these two EPS geofoam types are essentially the same. These b values of the EPS geofoam D-20 and D-30 are very similar to those of polymer geosynthetic reinforcements measured by tensile loading

Fig. 12. Typical ML test with SL stages and minute unload/reload cycles, D-20: a) the whole strain range and b) small strain level

Fig. 13. Stress-strain relations with total and decomposed strain components: a) D-20 and b) D-30

can be quantiˆed in terms of the rate-sensitivity coe‹cient, b, which is deˆned as follows under the unconˆned compression test conditions in the present study:

Ø

Dsv (e· irv)after ＝b･log sv (e· irv)before

Fig. 14.

»

(6)

Stress jump upon a step increase and decrease in the irreversible vertical strain rate

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Fig. 15.

Rate-sensitivity coe‹cients, b: a) D-20 and b) D-30

Fig. 16. Rate-sensitivity coe‹cients, b, of representative polymer geosynthetic reinforcements (Hirakawa et al., 2003)

tests shown in Fig. 16. On the other hand, the b value of unbound soils ranges between about 0.015 and about 0.055 while those of young natural and reconstituted saturated clays ranges up to about 0.08 (Tatsuoka et al., 2006). Viscosity function: Figure 17(a) shows the viscosity functions, gv(e· irv) (Eq. (3)), of the two types of EPS geofoam, which were used in the simulations of the measured stress-strain-time behaviour shown below. The parameters m and a of the viscosity functions were determined so that the slope, b, of the apparent linear part of these respective relations (in the full-log plot) becomes the same as b/ln 10, where b is the measured values shown in Figs. 15(a) and (b). The theoretical relation that b＝b/ln 10 is explained in details in Di Benedetto et al. (2002) and Tatsuoka et al. (2002). The parameter (e· irv)r was determined so that the whole aspects of the ratedependency of stress-strain relations observed in the tests on EPS geofoam D-20 and D-30 can be properly simulated based on the respective reference stress-strain curves presented in Figs. 10(a) and (b). In so doing, the viscosity functions of polymer geosynthetic reinforcements, which

exhibit similar Isotach viscosity as EPS geofoam, presented in Fig. 17(b), were referred to. Simulations of unconˆned compression tests: The test results shown above were simulated by incrementally solving Eq. (1) to obtain the time history of either sv(eirv, e· irv) for a given time history of ev in the case of strain-controlled loading, or ev for a given time history of sv in the case of stress-controlled loading (e.g., SL), while satisfying Eqs. (2) and (3), as well as Eq. (5) for elastic strains, based on a speciˆed reference relation (Eq. (4)). The broken curves presented in Fig. 18 are the simulated sv-ev relations of the ML tests at diŠerent vertical strain rates of EPS geofoam D-20 presented in Fig. 2 while the data points are the test data. It may be seen that the general trend of rate-dependency observed in the experiment is well simulated. A discrepancy between the measured and simulated relations seen in this ˆgure can be deemed to be due to an inevitable variance in the material properties among the diŠerent specimens used in these tests. As the vertical strains measured in the tests include the BE eŠects while those in the simulations do not, their rigorous comparison at small strains, before the start of large-scale yielding, cannot be made. Figures 19(a) through (d) compare the simulated sv-ev relations with those measured presented in Fig. 13(a) (EPS geofoam D-20) and Fig. 13(b) (D-30), in which the vertical strain rate was changed stepwise many times. The parameters for the viscous properties and reference stress-strain relation used in these simulations are listed in the respective ˆgures. It may be seen from Figs. 19(a) and (c) that the proposed model can simulate rather accurately the rate-dependency of stress-strain behaviour observed upon stepwise changes in the strain rate (i.e., the Isotach viscosity) made after the start of large-scale yielding. It may be seen from Figs. 19(b) and (d) that the proposed model can also simulate rather accurately the trend that the creep strain is very small when the vertical stress, sv, is smaller than the inviscid yield stress, (sfv)y, at which large-scale yielding starts in the reference stressstrain relation ( see Figs. 10(a) and (b)). The discrepancy between the simulated and measured relations at vertical

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Fig. 17. Viscosity functions determined based on the measured b values: a) EPS geofoam D-20 and D-30 and b) some typical polymer geosynthetic reinforcement (Hirakawa et al., 2003)

Fig. 18. Simulation of ML stress-strain relations at diŠerent vertical strain rates, EPS geofoam D-20

strains less than about 0.8z in Fig. 19(d) is due to the eŠects of surface singularity at the specimen ends on the measured vertical strains, which could not be removed even by using LDTs ( see Fig. 4(b)). These eŠects have been removed in the reference curve by referring to the results of similar tests in which reliable small local strains were measured by using LDTs ( see Fig. 6(a) of Abdelrahman et al., 2008). Figure 20 shows the simulation of the measured relationships between sv and ev (LDT) before the start of large scale yielding of EPS geofoam D-20 presented in Fig. 12(b). In this simulation, minute unload/reload cycles were ignored. In this test, several times of SL stages (for one hour at each stage) were performed. Due to an inevitable variance among diŠerent specimens, the refer-

ence relation could be slightly diŠerent among diŠerent specimens of the same type of EPS geofoam. To examine whether the proposed model can simulate the rate-dependency of the measured stress-strain relation at small strains presented in Fig. 20, another more relevant reference relation, which is slightly diŠerent from the one presented in Fig. 19(a), was redeˆned. It may also be seen from Fig. 20 that the trend that the creep strain is very small at small strains is well simulated. Figures 21(a) and (b) compare the measured stressstrain relation presented in Fig. 9 with its simulation. The simulation was performed ignoring the global unload/reload cycles. This simpliˆcation has no direct eŠects on the simulated time histories of creep strain at the respective SL stages applied during otherwise primary ML shown below. This is because, due to the nature of Isotach viscosity, the stress-strain behaviour for given strain and its rate is independent of precedent strain history while the initial strain and strain rate at the start of the respective SL stages are rather independent of precedent strain history. The measured and simulated ev values presented in this ˆgure include BE eŠects, because the local axial strain was not measured by using LDTs in this test. Therefore, reference curve (2) presented in Fig. 10(b), of which the ev values include BE eŠects, is used in this simulation to examine the relevance of the simulation. It is to be noted that the BE eŠects are noticeable only at small strains before the start of large-scale yielding (Fig. 21(b)), while they are not signiˆcant on large strains after the start of large-scale yielding (Fig. 21(a)). It may be seen that the creep vertical strain, Dev, that takes place at SL stages lasting for one hour when sv is lower than the inviscid yield stress, (sfv)y, during otherwise

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Fig. 19. Simulations of the rate-dependent stress-strain relations of EPS geofoam: a) and b) D-20 (Fig. 13(a)); and c) and d) D-30 (Fig. 13(b)) (small ev values measured with LDTs connected to subsequent ev increments measured externally)

Fig. 20.

Simulation of a ML test with SL stages; D-20 (Fig. 12(b))

the primary loading is very small (Fig. 21(b)), while it becomes very large after sv becomes larger than (sfv)y and increases at a very high rate with an increase in sv (Fig. 21(a)). This trend of behaviour can also be seen from Fig. 21(c), which compares the measured and simulated time histories of creep vertical strain at these SL stages. The creep vertical strains at the ˆrst two SL stages when sv is 45 kPa and 65 kPa are very small, therefore, they are not

visible in the strain scale of this ˆgure. It may be seen from Figs. 21(a), (b) and (c) that the measured ratedependent stress-strain behaviour is well simulated. In summary, it is shown above that the rate-dependent stress-strain behaviour of EPS geofoam when subjected to a wide variety of loading histories can be simulated very well by the Isotach model that has been developed within the framework of a non-linear three-component model (Fig. 6). An essential and important diŠerence between the present simulations and a simple curve ˆtting procedure is that, in the present simulations, the various rate-dependent stress-strain relations observed along arbitrary loading histories, including ML at diŠerent constant strain rates, step changes in the strain or stress rate, multiple SL stages at increasing ˆxed stress states and others are simulated in a consistent manner using the same set of the model parameters. Therefore, once the model parameters are determined based on relevant test results and their analysis, a FEM code that incorporates this model can be used for numerical analysis of boundary value problems in which the stress and strain activated in EPS blocks change arbitrarily with time. However, more study will be necessary to examine whether the model proposed in this paper can be applied to loading conditions more general than those employed in the present study, such as those at diŠerent conˆning pres-

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2.

3.

4.

5.

493

ofoam become signiˆcant after the start of large-scale yielding. On the other hand, at small strains before the start of large-scale yielding, the rate eŠects are insigniˆcant. The stress-strain property of EPS geofoam can be modeled by treating this material as an elasto-viscoplastic one and its viscous property is categorized into the Isotach type: i.e., the current stress-strain state is controlled uniquely by the instantaneous strain rate while the strength increases with an increase in the strain rate. Therefore, the inviscid yield stress, which is deˆned as the yield stress in the inviscid stress–strain relation (i.e., the reference relation), has important engineering implications. That is, the stress-strain relation is rather linear elastic and the creep strain is small as long as the stress level is lower than the inviscid yield stress. A method to evaluate the inviscid yield stress is suggested. The rate-sensitivity coe‹cient, b, represents the major feature of the viscous property of a given material. The b value of the two types EPS geofoam tested in the present study is about 0.12. This value is very close to those of polymer geosynthetic reinforcement. The rate-dependent stress-strain behaviour of EPS geofoam observed by step changes in the strain rate and sustained loading applied during otherwise ML at constant strain rate as well as by ML tests at diŠerent constant strain rates can be well simulated by the Isotach non-linear three-component model using a single set of model parameters for the respective EPS geofoam types. The lateral strain, eh, is signiˆcantly smaller than the vertical strain, ev, and the increment of eh becomes positive (i.e., the specimen diameter decreases) after the start of large-scale yielding. The rate eŠect on the ratio of irreversible horizontal and vertical strains increments, deirh/deirv, is not signiˆcant, showing that the eŠect of strain rate on the ‰ow characteristics is small, if any.

ACKNOWLEDGEMENTS

Fig. 21. Simulation of the test results presented in Fig. 9: a) stressstrain relation, b) close-upped relation and c) time histories of creep vertical strain at the last three SL stages excluding those by SL at sv ＝45 kPa and 65 kPa

sures, axial strains larger than about 15z, at strain rates lower than 0.0046z/min and higher than 33z/min. CONCLUSIONS The following conclusions can be derived from the test results and their model simulations presented above: 1. Rate eŠects on the stress-strain behaviour of EPS ge-

The laboratory stress-strain tests described in the present study were carried out when the ˆrst author stayed at Tokyo University of Science as Visiting Associate Professor. The authors express their sincere appreciation to Dr. D. Hirakawa and Mr. T. Kanemaru of Tokyo 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. This work was supported by the Grant-in-Aid for Scientiˆc Research (B), KANENHI No. 18360231, for the ˆscal years 2006 and 2007 of the Ministry of Education, Culture, Sports, Science and Technology, the Japanese Government.

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