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Effects of loading rate on viscoplastic properties of polymer geosynthetics and its constitutive modeling.

INTRODUCTION

Polymer geosynthetics are increasingly used as reinforcements in many civil, geotechnical, and traffic engineering including retaining structures, roads, railroads, and airfields. The viscosity is viewed as a significant property of polymeric materials. Loading rate effects, creep deformation, and stress relaxation of polymer geosynthetics are the inherent responses due to their viscous properties. Some of these characteristics have been studied through experimental tests. In the last decades, many works were focused on the creep deformation and stress relaxation of polymer geosynthetics using conventional testing methods (1-9). Recently, the viscous properties of polymer geosynthetics were investigated based on the tensile test considering the loading rate. Bathurst and Cai (10) performed a series of constant slain rate and load-controlled cyclic loading tests with five frequencies on high-density polyethylene (HDPE) and polyester (PET) geogrids to investigate the influence of constant loading rate. Ling et al. (11) performed displacement-controlled cyclic loading tests at a constant strain rate of 10%/min. Sawicki and Kazimierowicz-Frankowska (12) conducted a set of tests on two different geosynthetics, in which two creep or stress relaxation applied during otherwise ML at a constant stain rate, and pointed out that the observed behavior can be designated as the isotach properties. Shinoda and Bathurst (13), (14) earned out a set of short-term in-isolation tensile tests from ML at different constant strain rate and variable strain rate on three geogrids. According to Shinoda and Bathurst (13), (14), the polyolcfm geogrids were shown to exhibit tensile stiffness and ultimate tensile load capacities that increased with strain rate. The group of Tatsuoka (15-19) has engaged in research on the viscous properties of polymer geosynthetics through unconventional tensile tests. Part of the present contribution is devoted to the presentation of experimental results dealing with the effects of loading rate on the load-strain characteristics of three different polymer geosynthetics.

Over the years, several different constitutive models have been proposed to describe the viscous properties of polymer geosynthetics. Onaran and Findley (20) employed a multiple integral functional relationship, in which the kernel functions involve the first, second and third order stress terms to develop a creep model of polyvinyl chloride copolymer. Findley et al. (21) developed another similar model based on a series of multiple integrals, which has been used by, Helwany and Wu (22) to simulate the creep response of polypropylene (PP) geotextiles. Perkins (23) used the equation built up by creep strain rate, which was proposed by Hibbitt et al. (24), to model the creep behavior of geosynthetics. In addition, rheological models consisting of springs and dashpots have been used to simulate the viscoelastic behavior of geosynthetics. The standard linear solid model (25), (26) was the simplest model for predicting the creep behavior of HDPE geogrids. More complex models such as the multiple Kelvin model (27), (28) and the multiple Maxwell model (29) were proposed to analyze the creep behavior and stress relaxation of polymer geosynthetics. Some other constitutive models were developed on the basis of the isochronous concept (8), (30-36). Most of these models were capable of simulating the creep behavior of polymer geosynthetics. However, the experimental results with a wide variety of loading histories, including variable loading rate, creep, and stress relaxation, have not been sufficiently explained by the existing constitutive models.

In this article, the rate-dependent behaviors of polymer geosynthetics due to their viscous properties are described. Also, an elastoviscoplastic nonlinear three-component constitutive model is developed to simulate the viscous properties of polymer geosynthetics. By comparing the simulated results with the experimental data of the polymer geosynthetics presented in this study and those available from the literature, it is shown that the three-component elastoviscoplastic model can simulate well the rate-dependent behaviors of polymer geosynthetics.

The major part of this study, including the experimental part, was performed by the first writer under the guidance of his supervisor, Professor Fumio Tatsuoka, while he was visiting the Geotechnical Engineering Laboratory, Department of Civil Engineering, Tokyo University of Science from 2006 to 2007.

RATE-DEPENDENT BEHAVIOR OF POLYMER GEOSYNTHETICS

Most of the previous experiments reported in the literature arc those performed using only cither continuous ML or creep. In this section, the experimental results obtained from a series of unconventional tensile tests on three types of polymer geosynthetics are discussed. The test setup is illustrated in Fig. 1. The shapes and basic characteristics of the tested polymer geosynthetics arc shown in Fig. 2 and Table 1, respectively.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]
TABLE 1. Basic characterstics of the tested polymer geosynthetics.

Geosynthetic             Fiber material             Nominal tensile
                                                      strength (a)
                                                       (kN/m) at
                                                      strain rate
                                                      of 1.0%/min

      A       Nonwoven, needle punched. PP               157.0
              geotextile reinforced with PET yarns

      B       HDPE geogrid                                50.0

      C       PET geogrid                                 39.2

(a) Values provided by the manufacturers.


Loading Rate Effects Due to Viscous Properties

Figure 3a-c shows the experimental results obtained from complex loading histories rather than ML at a constant strain rate for Geosynthetics A, B, and C, respectively. In these tests, the strain rate was changed step-by-step several times with or without a pair of creep and stress relaxation tests during otherwise ML at a constant strain rate. The tensile load-strain curves obtained from ML at a constant strain rate are also presented in Fig. 3a and c as references.

[FIGURE 3 OMITTED]

It can be seen in Fig. 3 that the effects of strain rate on the prerupture tensile load-strain behavior and the rupture strength arc significant, namely, the tensile load-strain characteristics become stiffer with increasing of strain rate during ML. Significant creep deformation and stress relaxation can also be observed. Geosynthelic B (HDPE gcogrid) exhibits the highest creep deformation rate among those geosynthetics.

Despite what are mentioned earlier, all the tested geosynthetics display very high stiffness immediately after a step increase/decrease in the strain rate during otherwise ML at a constant strain rate. With further stretching after having exhibited clear yielding, the tensile load-strain relationship tends to rejoin the original one that was obtained by continuous ML at a constant strain rate equal to the changed one. A similar behavior is observed when ML was restarted at a constant strain rate following a creep or stress relaxation stage.

In addition, for Geosynthetic C (PET geogrid), the tensile load-strain curve exhibits an obvious load-overshooting on a step increase in the strain rate during otherwise ML or after a creep or stress relaxation stage as shown in Fig. 3c. On the other hand, a phenomenon of load-undershooting takes place upon a step decrease in the strain rate during otherwise ML. For the other types of polymer geosynthetics, the observed load-overshooting and undershooting do not occur.

These trends of behavior should be attributed to the viscous properties of the tested geosynthetics, and cannot be interpreted by the isochronous concept. For most of the geosynthetics, such as Geosynthetic A (PP geotextile and PET yarn), Geosynthetic B (HDPE geogrid) in the present study, PP geogrid (14), Geosynthetic W (PP and PET geotextile), and Geosynthetic P (polyamide geotextile) (12), this rate-dependent viscous property is defined as the isotach viscosity. The aforementioned trends indicate that the viscous properties could vary for different types of geosynthetics. For Geosynthetic C (PET geogrid), the load-overshooting and undershooting phenomenon can be described by the TESRA (Temporary Effect of Strain Rate and Acceleration) viscosity, which means the effects of strain rate and its rate (i.e., strain acceleration) on viscous load are temporary, and the viscous load is also controlled by the recent strain history. Meanwhile, the trend obtained from continuous ML at a constant strain rate exhibits the isotach viscosity. Therefore, this special load-strain behavior is defined as the combined viscosity which consists of an isotach component and a TESRA component. In essence, isotach, TESRA, and combined viscosity can be attributed to the combined effects of strain rate and strain acceleration, which will be introduced in the later sections. All these observed rate-dependent behaviors should be taken into account when developing the constitutive model.

Quantification of Viscous Properties

The amount of load jump, [DELTA]T, taking place by the step changes in the strain rate, could represent the quantity of viscous properties of the tested geosynthetics. Figure 4a and b illustrates the definition of load jump [DELTA]T on a step increase and decrease in the strain rate, respectively. It can be seen in Fig. 3 that the load jump [DELTA]T caused by each step change in the strain rate was always proportional to the instantaneous tensile load T where the step change in the strain rate was made. Figure 5 shows the relationships between the ratio of [DELTA]T to the instantaneous value of T at which the strain rate was changed and the logarithm of the ratio of the stain rate before and after a step change obtained from the data presented in Fig. 3. It can be seen that the following linear relationship is relevant to the range of the strain rate examined.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[[DELTA]T/T] = [beta] * 1g ([[[epsilon].sub.after.sup.ir]/[[epsilon].sub.before.sup.ir]]) (1)

where, the parameter [beta] is the rate-sensitivity coefficient; and [[epsilon].sub.before.sup.ir] and [[epsilon].sub.after.sup.ir] are the irreversible strain rates before and after a step change in the strain rate.

The following observations can be noted from Fig. 5. The ratio [DELTA]T/T is essentially independent of the value of T, which validates that the value of [DELTA]T is always proportional to the instantaneous value of T. Besides, [DELTA]T/T increases linearly with an increase in the logarithm of the ratio of irreversible strain rates after and before a step change, which means the viscous load component changes in a nonlinear manner with the changes in the strain rate, unlike the Newtonian viscosity. Therefore, the rate-sensitivity coefficient [beta] can represent the characteristics of viscous property. For different geosynthetics, the values of [beta] are different. The slope [beta] will be used in the simulation to determine the model parameters.

MODEL DESCRIPTION

To simulate the load-strain-time behavior of polymer geosynthetics presented in this study, an elastoviscoplastic constitutive model, which is described in a nonlinear three-component model framework, has been developed. In this three-component model (see Fig. 6), the tensile load, T, consists of inviscid and viscous components, [T.sup.f] and [T.sup.v], while the strain rate, [epsilon], consists of elastic and irreversible components, [[epsilon].sup.e] and [[epsilon].sup.ir], which can be expressed as:

[FIGURE 6 OMITTED]

[epsilon] = [[epsilon].sup.e] + [[epsilon].sup.ir] (2)

T = [T.sup.f] + [T.sup.v] (3)

[[epsilon].sup.e] is obtained by the hypoelastic model (37), (38) with the elastic modulus, [k.sub.eq](T), which is a function of instantaneous tensile load, T. In addition, loading and unloading are defined by the occurrence of positive and negative values of [[epsilon].sup.ir], not the occurrence of load rate, T. Therefore, even if T is negative immediately after a step decrease in the strain rate or at a load relaxation stage, the geosynthetic is under loading conditions as long as [[epsilon].sup.ir] is positive. [T.sup.f] is a rate-independent [T.sup.f] - [[epsilon].sup.ir] relationship, which is a unique function of if in the case of ML. The basic variable for the viscous load [T.sup.v] is not "the general time, t", for which it is impossible to define the origin in an objective way. [T.sup.v] is a function of [[epsilon].sup.ir], [[epsilon].sup.ir], and the strain history parameter, [h.sub.s]. As explained previously, most geosynthetics are of the isotach viscosity, while, Geosynthetic C (PET geogrid) exhibits the combined visocosty. For different viscosity types, the details of viscous load component are explained in the following sections.

Isotach Viscosity

In ML case, the isotach viscous load is a unique function of the instantaneous value of [[epsilon].sup.ir] and its rate [[epsilon].sup.ir]. On the basis of the fact that the load jump, [DELTA]T, on a step change in the strain rate is always proportional to the instantaneous value of T, as described earlier, the isotach viscous load can be expressed as:

[T.sup.v] = [T.sub.iso.sup.v] ([[epsilon].sip.ir], [[epsilon].sup.ir]) = [T.sup.f] ([[[epsilon].sup.ir]) * [g.sub.v] ([[epsilon].sup.ir]) (4)

[T.sup.v] is described in the framework of the Newtonian viscosity as following:

[[sigma].sup.v] = [eta]' ([[epsilon].sup.ir], [[epsilon].sup.ir]) . [[epsilon].sup.ir]; [eta]' ([[epsilon].sup.ir], [[epsilon].sup.ir]) = [[[[sigma].sup.f] ([[epsilon].sup.ir]) . [g.sub.v] ([[epsilon].sup.ir])][[epsilon].sup.ir]]] (5)

For polymer geosynthetics, the parameter [eta]'([[epsilon].sup.ir], [[epsilon].sup.ir]) is not a constant, unlike the Newtonian viscosity. By replacing [T.sup.v] with Eq. 4.. then Eq. 3 can be expressed as:

T = [T.sup.f] ([[epsilon].sup.ir]) . [1 + [g.sub.v] ([[epsilon].sup.ir])] (6)

where [g.sub.v]([[epsilon].sup.ir]) is the viscosity function. According to Di Benedetto et al. (39) and Tatsuoka et al. (40), it can be expressed as for geomaterials:

[g.sub.v] ([[epsilon].sup.ir]) = [alpha] * {1 - exp [[1 - [(|[[epsilon].sup.ir]|/[[epsilon].sub.r.sup.ir]] + 1).sup.m]]} ([greater than or equal to] 0) (7)

where [alpha], m, and [[epsilon].sub.r.sup.ir] are the positive parameters. The experimental results presented in Fig. 3 indicate that this viscous function is also applicable to the polymer geosynthetics. According to Eq. 7, [g.sub.v]([[epsilon].sup.ir]) is a highly nonlinear function of the absolute value of if with [g.sub.v] (0) = 0 and [g.sub.v] ([infinity]) = [alpha] (a positive constant). The viscous component, V, increases as [alpha] increases under otherwise the same conditions, while it increases as [[epsilon].sub.[GAMMA].sup.ir] decreases. The jump, [DELTA][T.sup.v], for a given ratio between the irreversible strain rates after and before a step change, increases as m increases.

TESRA Viscosity

In Fig. 3c, it can be observed that the Geosynthetic C (PET geogrid) exhibits a noticeable load-overshooting and undershooting behavior. Namely, on the restart of ML at a constant strain rate following a creep or stress relaxation stage, or on a step increase in the strain rate, the viscous component, [T.sup.v], increases sharply and then decays with an increase in the strain after clear yielding. And an opposite behavior occurs immediately after a step decrease in the strain rate exhibiting load-undershooting. Di Benedetto et al. (39) and Tatsuoka et al. (40) pointed out that decay of the viscous load component described previously can be accurately modeled by the following TESRA viscosity component:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

d[T.sub.iso.sup.v] = d{[T.sup.f] * [g.sub.v]([[epsilon].sup.ir])} = [[[partial derivative][T.sup.f]/[partial derivative][[epsilon].sup.ir]] * [g.sub.v]([[epsilon].sup.ir]) + [T.sup.f] * [[[partial derivative][g.sub.v]([[epsilon].sup.ir])]/[[partial derivative][[epsilon].sup.ir]]] * [[epsilon].sup.ir]/[[epsilon].sup.ir]]] * d[[epsilon].sup.ir] (9)

where [[epsilon].sub.l.sup.ir] is the initial irreversible strain at [T.sup.v] = 0; [[d[T.sub.iso.sup.v]].sub.([tau])] is the developed viscous stress increment at [[epsilon].sup.ir] = [tau]; and [r.sub.d] is the decay parameter that is positive less than unity, therefore, the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] decays with the strain difference [[epsilon].sup.ir] - [tau] (not with time). In this way, the current value of [T.sub.TESRA.sup.v] (when [[epsilon].sup.ir] = [[epsilon].sup.ir]) depends on the history of [[epsilon].sup.ir]. When [r.sub.d] = 1.0, [T.sub.TESRA.sup.v] (Eq. 8) becomes the same as [T.sub.iso.sup.v] (Eq. 4). In Eq. 9, the term ([[[partial derivative][T.sup.f]]/[[partial derivative][[epsilon].sup.ir]]]) * [g.sub.v]([[epsilon].sup.ir]) * d[[epsilon].sup.ir] represents the component of d[T.sup.v] due to the effect of the irreversible strain increment d[[epsilon].sup.ir], which is observed from continuous ML at a constant [[epsilon].sup.ir] (i.e., the effects of irreversible strain rate); and the term [T.sup.f] * ([[[partial derivative][g.sub.v]([[epsilon].sup.ir]]/[[partial derivative][[epsilon].sup.ir]]]) * d[[epsilon].sup.ir] = [T.sup.f] * ([[[partial derivative][g.sub.v]([[epsilon].sup.ir])]/[[partial derivative][[epsilon].sup.ir]]]) * ([[[epsilon].sup.ir]/[[epsilon].sup.ir]]) * d[[epsilon].sup.ir] represents the component of d[T.sup.v] due to the effects of the increment of irreversible strain rate d[[epsilon].sup.ir] applied at a certain [[epsilon].sup.ir] (i.e., the effects of irreversible strain acceleration).

The data presented in Fig. 3c indicate that, for Geosynthetic C (PET geogrid), the decaying rate of tensile load increases as the strain increases. To realistically simulate this trend of viscous property, a new equation, which is more general than Eq. 8, is developed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the decay function; r([[epsilon].sup.ir]) is the parameter that decreases with an increase in [[epsilon].sup.ir], and it can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The viscous property represented by Eq. 10 will herein be called the general TESRA viscosity. When r([[epsilon].sup.ir]) [equivalent to] [r.sub.i] [equivalent to] [r.sub.f] is constant and lower than unity, Eq. 10 returns to Eq. 8.

Combined Viscosity

As mentioned earlier, Geosynthetic C (PET geogrid) exhibits the isotach viscosity in continuous ML tests which were carried out at different strain rates and during each lest the strain rate was kept constant. The load-strain relationships deviate from each other and the deviation becomes larger as the strain increases. Geosynthetic C also has the feature of the general TESRA viscosity described earlier. To account for these two features of viscosity, the following more general expression is proposed:

[T.sup.v]([[epsilon].sup.ir], [[epsilon].sup.ir], [h.sub.s]) = [[lambda].sup.v] * [T.sub.iso.sup.v] ([[epsilon].sup.ir], [[epsilon].sup.ir]) + (1 - [[lambda].sup.v]) * [T.sub.TESRA.sup.v]([[epsilon].sup.ir], [[epsilon].sup.ir], [h.sub.s]) (12)

where [[lambda].sup.v] is the material constant between zero and unity. When [[lambda].sup.v] is equal to 1.0 and 0.0, Eq. 12 becomes to Eq. 4 (the isotach viscosity) and Eq. 10 (the general TESRA viscosity), respectively. By using a value of [[lambda].sup.v] between 0.0 and 1.0, Eq. 12 can explain the trends of behavior of Geosynthetic C presented in Fig. 3c.

IDENTIFICATION OF MODEL PARAMETERS

The parameters of the nonlinear three-component model used in the simulation are determined in the following sections.

Elastic Property

The elastic strain rate is calculated as:

[[epsilon].sup.e] = [T/[k.sub.eq]](T) (13)

where [k.sub.eq](T) is the equivalent elastic stiffness, which is a function of the tensile load, T, obtained from unload/ reload tests with small load amplitude on the corresponding type of geosynthetic. More details about elastic property of geosynthelics are reported in Hirakawa et al. (15).

Rate-independent Inviscid Load Component

In the three-component model, it is assumed that there is a rate-independent [T.sup.f] - [[epsilon].sup.ir] relationship, which can be determined by the curve fitting method using any appropriate function (e.g., polynomial, exponential, allometric). The following polynomial function is employed to express the [T.sup.f] - [[epsilon].sup.ir] relationship in this study.

[T.sup.f] = [10.summation over (i=1)][a.sub.i] * [([[epsilon].sup.ir]).sup.[i-1]] (14)

where [a.sub.i] is the coefficient for term i, which is determined so that Eq. 14 could best fit the corresponding inferred load-strain relationship in ML at zero strain rate that is extrapolated from the test results. It is to be noted that the load and strain state ultimately reaches the [T.sup.f] - [[epsilon].sup.ir] relationship during creep or stress relaxation stage in the case of the isotach viscosity.

Rate-dependent Viscous Load Component

Equation I could be rewritten as:

[[DELTA]T/T] = [beta] * 1g([[[epsilon].sub.after.sup.ir]/[[epsilon].sub.before.sup.ir]]) = b * 1n([[[epsilon].sub.after.sup.ir]/[[epsilon].sub.before.sup.ir]]) (15)

where b = [[beta]/ln] 10, and its incremental form is:

[dT/T] = d[ln[([[epsilon].sup.ir]).sup.b]] (16)

In which, dT is defined for a fixed value of [[epsilon].sup.ir], therefore, [T.sup.f] is a fixed value according to Eq. 14 and dT = d[T.sup.v] = [T.sup.f] * d[[g.sub.v] ([[epsilon].sup.ir])]. Then, in the case of isotach viscosity, by replacing T with Eq. 6, Eq. 16 can be rewritten as:

[dT/T] = [[[T.sup.f] * d[[g.sub.v]([[epsilon].sup.ir])]]/[[T.sup.f] * [1 + [g.sub.v]([[epsilon].sup.ir])]]] = [[d[[g.sub.v]([[epsilon].sup.ir])]]/[1 + [g.sub.v]([[epsilon].sup.ir])]] = d[ln[([[epsilon].sup.ir]).sup.b]] (17)

d[ln[gamma](1 + [g.sub.v]([[epsilon].sup.ir]))] = d[ln[([[epsilon].sup.ir]).sup.b]] (18)

By integrating Eq. 18 with respect to [[epsilon].sup.ir], it can be obtained:

1 + [g.sub.v]([[epsilon].sup.ir]) = [c.sub.v] * [([[epsilon].sup.ir]).sup.b] (19)

where [c.sub.v] is a constant. Eq. 19 is also essentially valid for the TESRA viscosity and the combined viscosity. As shown in Fig. 7, the viscosity function (Eq. 7) should be defined so that the linear part for a range of [[epsilon].sup.ir], for which Eq. 19 is derived, has a slope of b = [[beta]/ln] 10. It is noted that this linear portion is valid only for a range of [[epsilon].sup.ir] between a certain lower limit and a certain upper limit. [[epsilon].sup.ir] examined in the concerned tests is located in this range. A relevant value should be assumed for parameter [alpha], which represents the upper bound of [g.sub.v] ([[epsilon].sup.ir]) when [[epsilon].sup.ir] is infinitive. A parameter m is then obtained by trial and error. The parameter [[epsilon].sub.r.sup.ir] in Eq. 7 is the value of [[epsilon].sup.ir] where the log(1 + [g.sub.v] ([[epsilon].sup.ir])) and log ([[epsilon].sup.ir]) relation becomes non-linear as [[epsilon].sup.ir]decreases.

[FIGURE 7 OMITTED]

The parameters [r.sub.i], [r.sub.f], c, and n of the decay function (Eq. 11) as well as the combined parameter [[lambda].sup.v] in Eq. 12 are determined empirically so that all the observed trends of rate-dependent behavior can be well simulated. For Geosynthetic C (PET geogrid), the value [[lambda].sup.v] is determined to be 0.8 by referring to the observed separation between the tensile load-strain curves obtained by continuous ML at different constant strain rates.

MODEL ASSESSMENT AND VALIDATION

As Eqs. 8 and 10 have an integration term for the full previous loading history starting from the initial state, it is not realistic to use them in the elastoviscoplastic numerical analysis. Taking advantages of the nature of the power law of the decay function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the following incremental equation obtained from Eq. 10 is accurate enough for a sufficiently small irreversible strain increment [DELTA][[epsilon].sup.ir]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

And the following equation is obtained from Eq. 4:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

where the terms, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are known at the current step (when [[epsilon].sup.ir] = [[epsilon].sup.ir]); while the terms, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [DELTA][T.sub.iso.sup.v], can be obtained from the given values of [[epsilon].sup.ir] as well as [DELTA][[epsilon].sup.ir] and [DELTA]t (a given time increment). Therefore, the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] at the next step can be determined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The simulations presented below are performed on the basis of Eqs. 20-22.

The experimental results presented in Fig. 3 and the related literatures (10), (12), (14) are used to calibrate the proposed model and then the results of simulation are compared with the experimental results. The test conditions include continuous ML at different constant strain rates, variable strain rate, creep, and stress relaxation during otherwise ML at a constant strain rate. The model parameters used in the simulation for the various tests are summarized in Table 2. The model parameters for HDPE geogrids are a little different, probably because of the mechanical properties such as fabric and chemical composition.
TABLE 2. Model parameters for simulation of experimental results.

Geosynthetic                        [[lambda].sup.v]  [alpha]    m

Fig. 3        Geosynthetic A (PP          1.0          0.52    0.08
              geotextile and PET
              yarns)

              Geosynthetic B (HDPE        1.0          1.60    0.085
              geogrid)

              Geosynthetic C (PET         0.8          0.70    0.12
              geogrid)

(a)           HDPE geogrid                1.0          2.48    0.12

              PET geogrid                 0.8          0.70    0.12

(b)           Geosynthetic W (PP          1.0          1.98    0.11
              and PET geotextile)

              Geosynthetic P              1.0          1.45    0.07
              (polyamide
              geotextile)

(c)           PP geogrid                  1.0          1.851   0.071

              HDPE geogrid                1.0          1.60    0.085

Geosynthetic                      [[epsilon].sub.r.sup.ir]  [r.sub.i]

Fig. 3        Geosynthetic A (PP  1.0 x [10.sup.-4]            1.0
              geotextile and PET
              yarns)

              Geosynthetic B      3.5 x [10.sup.-4]            1.0
              (HDPE geogrid)

              Geosynthetic C      1.0 x [10.sup.-4]            1.0
              (PET geogrid)

(a)           HDPE geogrid        5.0 x [10.sup.-4]            1.0

              PET geogrid         1.0 x [10.sup.-4]            1.0

(b)           Geosynthetic W (PP  1.0 x [10.sup.-4]            1.0
              and PET
              geotextile)

              Geosynthetic P      1.0 x [10.sup.-4]            1.0
              (polyamide
              geotextile)

(c)           PP geogrid          1.0 x [10.sup.-4]            1.0

              HDPE geogrid        3.5 x [10.sup.-4]            1.0

Geosynthetic                                     [r.sub.f]   c    n

Fig. 3        Geosynthetic A (PP geotextile and    1.0       -    -
              PET yarns)

              Geosynthetic B (HDPE geogrid)        1.0       -    -

              Geosynthetic C (PET geogrid)         0.15     0.4  0.6

(a)           HDPE geogrid                         1.0       -    -

              PET geogrid                          0.15     0.4  0.6

(b)           Geosynthetic W (PP and PET           1.0       -    -
              geotextile)

              Geosynthetic P (polyamide            1.0       -    -
              geotextile)

(c)           PP geogrid                           1.0       -    -

              HDPE geogrid                         1.0       -    -

(a) Bathurst and Cai (10); (b) Sawicki and Kazimierowicz-Frankowska
(12); (c) Shinoda and Bathurst (14).


Figure 8a-c show the simulation of the results from the tests in which the strain rate was changed stepwise with or without a pair of creep and stress relaxation stages during otherwise ML at a constant strain rate, as shown in Fig. 3a-c, for Geosynthetics A (PP geotextile and PET yarns), B (HDPE geogrid), and C (PET geogrid), respectively. The test results are also presented. It can be seen that the proposed model can simulate rather accurately all the viscous aspects of tensile load-strain behavior of the tested polymer geosynthetics subjected to a wide range of loading histories. It is noted that the creep behavior is well simulated using the parameters that were determined from the tensile load-strain behavior upon step changes in the strain rate. This means if the model parameters for a given type of polymer geosynthetic can be determined by relevant tests (such as ML tests with several step changes in the strain rate), the proposed model is able to predict the load-strain-time behavior of polymer geosynthetics under arbitrary loading histories, including long-term creep behavior. This is one of the most important features of the developed three-component model.

[FIGURE 8 OMITTED]

To demonstrate the applicability of the proposed model to various types of polymer geosynthetics, some previous test results are also simulated. Figure 9a and b show the simulation of the test results from continuous ML at different constant strain rate reported by Bathurst and Cai (10), respectively for HDPE and PET geogrids. Sawicki and Kazimierowicz-Frankowska (12) performed a series of ML tests at a constant strain rate of 0.32%/sec with two intermediate periods of creep or stress relaxation each with duration of 1 hr, on Geosynthetics W (PP and PET geotextile) and P (polyamide geotextile). The simulated results of these tests by the nonlinear three-component model are shown in Figs. 10 and 11 along with the experimental results. Figure 12a and b present the simulation of another two tests (14) performed on PP and HDPE geogrids. Similar to the tests in Fig. 3, in these two tests the strain rate was changed several times during ML at constant strain rate.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

It can be seen in Figs. 9-12 that the proposed elasto-viscoplastic model could simulate the rate-dependent behavior of various types of the polymer geosynthetics with appropriate model parameters. It should be pointed out that the model parameters used for HDPE geogrid in Fig. 12b and PET geogrid in Fig. 9b are the same as those in Fig. 3b and c, respectively. This confirms the important feature of the model described above and validates the proposed model.

CONCLUSIONS

The experimental results showed that all the investigated polymer geosynthetics exhibit rate-dependent behaviors due to viscous properties of the materials. On the step change in loading strain rate, immediately following creep or stress relaxation stage, the stiffness is very high. After exhibiting apparent yielding, with or without a noticeable overshooting in the load, the tensile load-strain relation tends to rejoin the original one that could be obtained by continuous ML. The tensile load-strain relations of all the investigated polymer geosynthetics are found to be highly nonlinear and dependent on both strain and strain rates. Besides, the tensile load-strain relation of PET geogrid is also strain history dependency. A constitutive model which can interpret the elastoviscoplastic tensile load-strain behaviors of polymer geosynthetics is recommended. The capability of the model in simulating the experimental results of tests under arbitrary loading histories (e.g., variable strain rate loading, creep, and stress relaxation) indicates that the irreversible strain rate is reasonably used as internal variable in nonlinear three-component elastoviscoplastic model. The proposed model could also be applied to predict the other experimental results considering loading rate in the literature.

ACKNOWLEDGMENTS

The authors would like to express their sincere thanks to Prof. Fumio Tatsuoka, Department of Civil Engineering, Tokyo University of Science, for his erudite guidance and discussions, and Dr. Daiki Hirakawa from Tokyo University of Science for his help.

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Fang-Le Peng, (1) Fu-Lin Li, (1) Yong Tan, (1) Warat Kongkitkul (2)

(1) Department of Geotechnical Engineering, Tongji University, Shanghai 200092, People's Republic of China

(2) Department of Civil Engineering, King Mongkut's University of Technology Thonburi, Bangkok 10140, Thailand

Correspondence to: Fang-Le Peng; e-mail: pengfanglc@gmail.com

Contract grant sponsor: Ministry of Education, Government, Science and Culture of Japan; contract grant sponsor: NCET; contract grant number: 06-0378; contract grant sponsor: NSFC; contract grant numbers: 50679056 and 40972176; contract gram sponsor: Shuguang Project; contract grant number: 05SG25: contract grant sponsor: Shanghai Leading Academic Discipline Project: contract grant number: B308.

DOI 10.1002/pen.21548

Published online in Wiley InterScience (www.interscience.wiley.com).

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Author:Peng, Fang-Le; Li, Fu-Lin; Tan, Yong; Kongkitkul, Warat
Publication:Polymer Engineering and Science
Article Type:Report
Geographic Code:9THAI
Date:Mar 1, 2010
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