Unified description of sand behavior

Abstract

In this paper, the mechanical behavior of sand, was systematically described and modeled with a elastoplastic model proposed by Zhang et al. [1]. Without losing the generality of the sand, a specific sand called as Toyoura sand, a typical clean sand found in Japan, has been discussed in detail. In the model, the results of conventional triaxial tests of the sand under different loading and drainage conditions were simulated with a fixed set of material parameters. The model only employs eight parameters among which five parameters are the same as those used in Cam-clay model. Once the parameters are determined with the conventional drained triaxial compression tests and undrained triaxial cyclic loading tests, then they are fixed to uniquely describe the overall mechanical behaviors of the Toyoura sand, without changing the values of the eight parameters irrespective of what kind of the loadings or the drainage conditions may be. The capability of the model is discussed in a theoretical way.

Keywords

constitutive model / sand / stress-induced anisotropy / density / structure

Introduction

Mechanical behavior of clean sand has been investigated for years and so many reports on this topic can be found in literature that it is hard to list them completely within limited pages of references. The reason why so many researchers spend so much effort to learn it is that, the mechanical behaviors of clean sand are dependent not only on shape of particles, angular or round, but also on its density, strain history, and even on degree of structure formed in its deposition [2]. Sand may behave totally differently under different above-mentioned conditions. For instance, failure pattern of undrained sand subjected to cyclic loading may take the form of liquefaction or cyclic mobility according to the density of the sand [3-5].

Tatsuoka et al. [6] reported that cyclic undrained strength of loose sand can be rather easily determined while those for dense sand is much more difficult to determine not only because of its density but also the way of the sample preparation which is related to the structure formed in the preparation. Loose sand subjected to cyclic loading may liquefy under undrained condition but may be compacted to a denser state under drained condition [7]. The strain history, sometime also called as stress-induced anisotropy, has great influence on the behavior of undrained sand [8]. In the works by Ishihara and Okada [9,10], the effect of pre-shearing on the cyclic behavior of sand was carefully investigated and it is concluded that the liquefaction resistance is dependent not only on the magnitude of the pre-shearing but also on its initial direction. As to the influence of stress-induced anisotropy, Oda et al. [11] investigated the influence of orientation in the sedimentation of sand both on the stress-strain relation and the liquefaction resistance with macroscopic viewpoint, and the potentiality of re-liquefaction of sand that had experienced liquefaction previously with microscopic viewpoint. Kato et al. [12] conducted a series of undrained triaxial compression tests on sand with anisotropic consolidation and found that the stress induced anisotropy does not affect the state variables at critical state or steady state but that stress-strain relation will be more contractive with larger anisotropy. Hyodo et al. [13] conducted a series of tests on undrained sand with reversal and non-reversal cyclic loadings and found that non-reversal cyclic loadings may also cause large deformation and that initial deviatoric stress may increase phase transform strength where contractive behavior changes to dilative behavior.

Bidirectional simple shear tests were also conducted to investigate influence of superimposing cyclic shear stress in one direction on undrained behavior of sand subjected to monotonic loading in another direction in the works by Meneses et al. [14] and it is concluded that the magnitude and frequency of superimposing cyclic loading may affect the undrained behavior of sand and that loose sand is more susceptible to a small superposing cyclic loading than dense sand. Verdugo and Ishihara [15] investigated systematically a so-called confining-stress dependency of sand in undrained triaxial test in which the samples with the same void ratio but different confining stresses were tested and it is concluded that steady state is not affected by the initial effective confining stress. Meanwhile the steady state evaluated from drained tests is in good agreement with those evaluated from undrained tests.

Compared with undrained cyclic loading tests, drained cyclic loading tests are much fewer in literature, among which the tests on dense sand under constant mean stress, conducted by Hinokio et al. [16], should be mentioned. In the test, confining pressure of the sand is kept constant and a maximum principal stress ratio (σ₁/σ₃) is cyclically loaded up to 4.0. It is found that dense sand subjected to relative large cyclic shearing will contract to some extent but will not contract further even if the cyclic shearing continues.

It is commonly known that effective stress path is independent of loading path such as conventional triaxial loading, true triaxial loading or plane-strain loading under undrained condition. Under drained condition, however, the loading path may greatly affect the mechanical behavior of sand. In the works by Nakai and Mihara [17]; Nakai and Matsuoka [18], Nakai [19]; Nakai and Hinokio [20], systematic investigation were conducted to make clear the drained behavior of sand subjected to very complicated loading paths with computer-controlled true triaxial loading device designed by the authors. In the researches, influence of intermediate stress is carefully investigated which leads to the establishment of tij models [19].

As to the stress-induced anisotropy of sand, Yamada et al. [21] conducted systematically a series of undrained triaxial conventional tests in which the samples before shearing have already subjected to undrained cyclic loading and had already experienced cyclic mobility process. In his research, a continuous and rapid change of the stress-induced anisotropy during liquefaction has been confirmed, which again proves that the stress-induced anisotropy is a very important factor in modeling the mechanical behavior of sand.

Above reviews just gave a brief description of the experimental researches on the mechanical behavior of sand. Researches related to the modeling of sand can be found in literature, e.g., the work by Oka and Ohno [22], Oka et al. [23]; a soil-water coupling method based on a kinematic hardening elastoplastic model, named as LIQCA [24].

In recent years, research on constitutive model for soils has been developing very quickly. Some works in particular are worthy of mention in advance. The concept of “subloading” was proposed by Hashiguchi and Ueno [25], Hashiguchi [26,27], which made it possible to describe the overconsolidation of soils easily and efficiently. The concept of “superloading”, proposed by Asaoka et al. [2], Asaoka et al. [28], and Asaoka et al. [29], together with the concept of subloading, make it possible not only to describe overconsolidation, but also to explain the effect of the soil structure commonly observed in naturally deposited soils, which is one of the main reasons why soils may differ greatly from place to place. By combining the concepts of subloading and superloading, it is possible for the first time to describe the mechanical behavior of clay and sand with different densities and different structures within the same framework of a constitutive model. The proposed model is based on the Cam-clay model [30] and the modified Cam-clay model [31]. Therefore, the physical meaning of the model is easy to understand and relatively few parameters need to be employed in spite of its ability to deal with different soils using the same model.

Zhang et al. [1] proposed a new constitutive model for sand in which in addition to the concept of superloading related to the soil skeleton structure [2] and the concept of subloading related to the density [25], the authors introduced a new approach to describe the stress-induced anisotropy. As a matter of fact, since the first introduction of the concept of anisotropy [32], it was extended to the subloading model by Ref. [33] using the expression of rotating hardening. Asaoka et al. [34] also emphasized the importance of stress-induced anisotropy in SYS Cam-clay model. Zhang et al. [1] pointed out that the change of density or overconsolidation is contributed not only by plastic stretching and elastic unloading, but also by the stress-induced anisotropy. It is a commonly accepted concept that soils will loss its overconsolidation whenever plastic stretching occurs and gain its overconsolidation whenever the soil is in elastic unloading process. From the viewpoint of Zhang et al. [1], however, the soil will gain its overconsolidation even in the case when the plastic stretching occurs. Moreover, a natural limitation for the development of the stress-induced anisotropy was defined. Based on the model, mechanical behavior of fictional sand subjected to different loadings under different drainage conditions were simulated to verify if the model was suitable to describe the general behavior of the sand with one set of definite parameters. Particular attention was paid to the description of the sand subjected to cyclic loading under undrained conditions, that is, for loose sand, liquefaction happens without transition from contractive state to dilative state; for medium dense sand, cyclic mobility occurs while for dense sand, liquefaction will not occur.

Apart from the tij concept of Nakai and Mihara [17], Yao et al. [35] and Wan et al. [36] also discussed the influence of intermediate stress and proposed a concept of transform stress (TS) by which it is also possible to describe this dependency. The advantage of TS concept is that the constitutive model established in p-q stress space can be easily extended to true triaxial stress space without change any material parameters, nor need to add any material parameter. In this paper, with a rather ambitious goal, the authors try to use the model of Zhang et al. [1], to describe the overall mechanical behaviors of Toyoura Sand, the typical clean sand, in a unified way. In other words, all the behavior of Toyoura Sand, no matter what loading and drainage conditions may be, its mechanical behavior is described with a fixed set of material parameters. The model proposed by Zhang et al. [1] is firstly given in an infinitesimal strain level. Then the influence of intermediate principal stress is taken into consideration by adopting the TS concept. The most important thing that should be emphasized here is that the eight material parameters involved in the model, will be constant no matter what kind of loading or drainage conditions may be.

Constitutive model of sand

The model proposed here, is based on the concepts of subloading and superloading as described in the work by Asaoka et al. [34]. Here we give just a brief description of the yield surfaces shown in Fig. 1.

The similarity ratio of the superloading yield surface to normal yield surface R* and the similarity ratio of the superloading yield surface to subloading yield surface R are the same as those in the work by Asaoka et al. [34], namely,

(1)

(2)

(3){\displaystyle \frac{\overline{q}}{\overline{p}}}={\displaystyle \frac{\tilde{q}}{\tilde{p}}}={\displaystyle \frac{q}{p}}(\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{y}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}),

where \left(p,q\right),\left(\tilde{p},\tilde{q}\right)\mathrm{a}\mathrm{n}\mathrm{d}\left(\overline{p},\overline{q}\right) represent the present stress state, the corresponding normally consolidated stress state and the structured stress state at p-q effective stress space respectively, which is only related to a conventional triaxial stress ({\sigma }_{\mathrm{22}}={\sigma }_{\mathrm{33}}, q={\sigma }_{\mathrm{11}}-{\sigma }_{\mathrm{33}}). It should be pointed out that the stress tensor is described in effective stress space throughout this paper. The normally yield surface is given in the following form as:

(4)f=\ln {\displaystyle \frac{{\sigma }_m}{{\sigma }_{m0}^{\ast }}}+\ln {\displaystyle \frac{M^2-{\varsigma }^2+{\eta }^{\ast 2}}{M^2-{\varsigma }^2}}+\ln R^{\ast }-\ln R-{\displaystyle \frac{{ϵ}_v^p}{C_p}}=0.

The variables involved in Eqs. (1), (2) and (4) are defined in general stress state as:

(5){\eta }^{\ast }=\sqrt{{\displaystyle \frac{3}{2}}{\widehat{\eta }}_{ij}{\widehat{\eta }}_{ij},}{\widehat{\eta }}_{ij}={\eta }_{ij}-{\beta }_{ij},{\eta }_{ij}={\displaystyle \frac{S_{ij}}{{\sigma }_m}},{\sigma }_m=p={\sigma }_{ii}/3,

(6)\eta =\sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\eta }_{ij}{\eta }_{ij},}\varsigma =\sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{\beta }_{ij}{\beta }_{ij},}

where S_ij is the deviatoric stress tensor, β_ij is the anisotropic stress tensor, and σ_ij is the Cauchy effective stress tensor and is assumed to be positive in compression. The definition for the surface is different from the one proposed by Asaoka et al. [2] in its CSL. In the work by Asaoka et al. [2], the CSL, which defines a border line to distinguish dilation and compression, is defined by the value of M_a(M_a^{\mathrm{2}}=M^{\mathrm{2}}+{\zeta }^{\mathrm{2}}), in which ζ is the magnitude of the stress-induced anisotropy, as shown in Eq. (6). It is clear that the gradient of the CSL changes with the development of anisotropy. In reality, however, the assumption that the gradient of the CSL keeps constant seems more reasonable. The physical evidence of this assumption can be found in many literatures, e.g., the work by Hyodo et al. [13] and Kato et al. [12]. Therefore, in present model, the gradient of the CSL is assumed to be constant. It is also very clear from both the definition and Fig. 2 that the flat ratio of the elliptical yield surface changes with the value of anisotropy.

The larger the stress-induced anisotropy ζ is, the larger the eccentric ratio of the ellipse will be. In Eq. (4), C_p is expressed as:

(7)C_p={\displaystyle \frac{\lambda -k}{\mathrm{1}+e_{\mathrm{0}}}},

where, λ and k represent the compression and the swelling index respectively.

Figure 3 shows the changes in the subloading yielding surface in the cyclic mobility stage, from which it is clear that not only the size of the surface but also the axis of elliptical yield surface change according to the states of present stress and the stress-induced anisotropy.

An associated flow rule is employed in the present model, namely,

(8)d{ϵ}_{ij}^p=\mathrm{\Lambda }{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}.

The consistency equation for the subloading yield surface can then be given as:

(9)\mathrm{d}f=\mathrm{0}\Rightarrow {\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}\mathrm{d}{\sigma }_{ij}+{\displaystyle \frac{\partial f}{\partial {\beta }_{ij}}}\mathrm{d}{\beta }_{ij}+{\displaystyle \frac{\mathrm{1}}{R^{\ast }}}\mathrm{d}{R}^{\ast }-{\displaystyle \frac{\mathrm{1}}{R}}\mathrm{d}R-{\displaystyle \frac{\mathrm{1}}{C_p}}\mathrm{d}{ϵ}_v^p=\mathrm{0.}

The following differentials are useful in deriving the constitutive equation for the stress tensor and the stain tensor:

(10)\Rightarrow {\displaystyle \frac{\partial ({\eta }^{\ast \mathrm{2}})}{\partial {\sigma }_m}}=-{\displaystyle \frac{\mathrm{3}{\widehat{\eta }}_{ij}{\eta }_{ij}}{{\sigma }_m}},{\displaystyle \frac{\partial ({\eta }^{\ast \mathrm{2}})}{\partial S_{ij}}}={\displaystyle \frac{\mathrm{3}{\widehat{\eta }}_{ij}}{{\sigma }_m}},{\displaystyle \frac{\partial ({\eta }^{\ast \mathrm{2}})}{\partial {\beta }_{ij}}}=-\mathrm{3}{\widehat{\eta }}_{ij},{\displaystyle \frac{\partial ({\mathit{\varsigma }}^{\mathrm{2}})}{\partial {\beta }_{ij}}}=\mathrm{3}{\beta }_{ij}.

Based on Eqs. (10) and (4), it is easy to obtain the following relations:

(11){\displaystyle \frac{\partial f}{\partial {\sigma }_m}}={\displaystyle \frac{\mathrm{1}}{{\sigma }_m}}+{\displaystyle \frac{{\displaystyle \frac{\partial ({\eta }^{\ast \mathrm{2}})}{\partial {\sigma }_m}}}{M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}}}={\displaystyle \frac{M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}-\mathrm{3}{\widehat{\eta }}_{ij}{\eta }_{ij}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}}={\displaystyle \frac{M^{\mathrm{2}}-{\eta }^{\mathrm{2}}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}},

(12){\displaystyle \frac{\partial f}{\partial S_{ij}}}={\displaystyle \frac{\mathrm{3}{\widehat{\eta }}_{ij}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}},

(13){\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}={\displaystyle \frac{\partial f}{\partial S_{ij}}}+{\displaystyle \frac{\partial f}{\partial {\sigma }_m}}{\displaystyle \frac{{\delta }_{ij}}{\mathrm{3}}}={\displaystyle \frac{\mathrm{3}{\widehat{\eta }}_{ij}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}}+{\displaystyle \frac{M^{\mathrm{2}}-{\eta }^{\mathrm{2}}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}}·{\displaystyle \frac{{\delta }_{ij}}{\mathrm{3}}},

(14)$$\begin{gathered} {\displaystyle \frac{\partial f}{\partial {\beta }_{ij}}}={\displaystyle \frac{-{\displaystyle \frac{\partial \left({\varsigma }^2\right)}{\partial {\beta }_{ij}}}+{\displaystyle \frac{\partial \left({\eta }^{\ast 2}\right)}{\partial {\beta }_{ij}}}}{M^2-{\varsigma }^2+{\eta }^{\ast 2}}}-{\displaystyle \frac{-{\displaystyle \frac{\partial \left({\varsigma }^2\right)}{\partial {\beta }_{ij}}}}{M^2-{\varsigma }^2}} \\ ={\displaystyle \frac{-3{\beta }_{ij}-3{\widehat{\eta }}_{ij}}{M^2-{\varsigma }^2+{\eta }^{\ast 2}}}+{\displaystyle \frac{3{\beta }_{ij}}{M^2-{\varsigma }^2}} \\ ={\displaystyle \frac{3\left(-M^2{\widehat{\eta }}_{ij}+{\eta }^{\ast 2}{\beta }_{ij}+{\varsigma }^2{\eta }_{ij}\right)}{\left(M^2-{\varsigma }^2+{\eta }^{\ast 2}\right)\left(M^2-{\varsigma }^2\right)}}\mathrm{.} \end{gathered}$$

From Eq. (11), it is clear that the CSL, defined by the condition in which \partial f/\partial {\sigma }_{ii}=\mathrm{0}, always satisfies the relation η = M, implying that the CSL, as the threshold between plastic compression and plastic expansion, does not move with the changes in the anisotropy.

Evolution rule for stress-induced anisotropic stress tensor β_ij

Different from the works by Hashiguchi and Chen [33] and Asaoka et al. [34], the following evolution rule for the anisotropic stress tensor is defined as:

(15)\mathrm{d}{\beta }_{ij}={\displaystyle \frac{M}{C_p}}{b}_r\left(M-\varsigma \right)\mathrm{d}{ϵ}_d^p{\displaystyle \frac{{\widehat{\eta }}_{ij}}{\Vert {\widehat{\eta }}_{ij}\Vert }}=\sqrt{{\displaystyle \frac{3}{2}}{\displaystyle \frac{M}{C_p}}{b}_r\left(M-\varsigma \right)}\mathrm{d}{ϵ}_d^p{\displaystyle \frac{{\eta }_{ij}}{{\eta }^{*}}},

in which, an artificial limitation on the development of anisotropy originally proposed by Hashiguchi and Chen [33] is no longer necessary, this is because firstly it more or less lacks physical evidence and secondly the stress-induced anisotropic stress tensor β_ij also represents the stress history that the soil experienced and it will not exceed the CSL, which provides us with a natural physical limitation ζ<M, that can be easily accepted. From Eq. (15), it is known that anisotropy will stop its development at the time when ζ approaches to M, which also satisfies the need to avoid the singularity in Eq. (4) at the point ζ = M.

The plastic component of deviatoric strain tensor can be calculated as follow:

(16)\mathrm{d}{e}_{ij}^p=\mathrm{d}{ϵ}_{ij}^p-\mathrm{d}{ϵ}_v^p/\mathrm{3}=\mathrm{\Lambda }{\displaystyle \frac{\mathrm{d}f}{\mathrm{d}{S}_{ij}}},

while the plastic deviatoric strain can be calculated as:

(17)\mathrm{d}{ϵ}_d^p=\sqrt{{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}\mathrm{d}{e}_{ij}^p\mathrm{d}{e}_{ij}^p}=\mathrm{\Lambda }\sqrt{{\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{\displaystyle \frac{\mathrm{d}f}{\mathrm{d}{S}_{ij}}}{\displaystyle \frac{\mathrm{d}f}{\mathrm{d}{S}_{ij}}}}=\mathrm{\Lambda }{\displaystyle \frac{\mathrm{2}{\eta }^{*}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}}){\sigma }_m}}.

Substituting Eq. (17) into Eq. (15), the evolution rule for the anisotropic stress tensor can be rewritten as:

(18)\mathrm{d}{\beta }_{ij}=\mathrm{\Lambda }{\displaystyle \frac{\sqrt{\mathrm{6}}M{b}_r(M-\varsigma ){\widehat{\eta }}_{ij}}{C_p(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}}.

Applying Eqs. (14) and (18), it is easy to calculate the increment in anisotropy as follows:

(19)$$\begin{gathered} {\displaystyle \frac{\partial f}{\partial {\beta }_{ij}}}\mathrm{d}{\beta }_{ij}=\mathrm{\Lambda }{\displaystyle \frac{\sqrt{\mathrm{6}}M{b}_r(M-\varsigma ){\eta }^{\ast \mathrm{2}}\left(-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}{\widehat{\eta }}_{ij}{\beta }_{ij}+\mathrm{2}{\varsigma }^{\mathrm{2}}\right)}{C_p{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}})}^{\mathrm{2}}(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}){\sigma }_m}} \\ =\mathrm{\Lambda }{\displaystyle \frac{\sqrt{\mathrm{6}}M{b}_r(M-\varsigma ){\eta }^{\ast \mathrm{2}}(-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}{\eta }_{ij}{\beta }_{ij})}{C_p{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}})}^{\mathrm{2}}(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}){\sigma }_m}}. \end{gathered}$$

The increment in anisotropy, which plays a very important role in the evolution rule for the overconsolidation, will be discussed in detail later. From Eq. (19), it is clear that if {\eta }_{ij}{\beta }_{ij}\le \mathrm{0}, which means the angle between the deviatoric stress tensor and the anisotropic tensor is larger than 90°, then (\partial f/\partial {\beta }_{ij})\mathrm{d}{\beta }_{ij} will always be less than zero. In Eq. (19), the following relation is used.

(20)\left(-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}{\widehat{\eta }}_{ij}{\beta }_{ij}+\mathrm{2}{\varsigma }^{\mathrm{2}}\right)=[-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}({\eta }_{ij}-{\beta }_{ij}){\beta }_{ij}+\mathrm{2}{\varsigma }^{\mathrm{2}}]=[-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}{\eta }_{ij}{\beta }_{ij}-\mathrm{3}{\beta }_{ij}{\beta }_{ij}+\mathrm{2}{\varsigma }^{\mathrm{2}}]=[-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}{\eta }_{ij}{\beta }_{ij}](\because -\mathrm{3}{\beta }_{ij}{\beta }_{ij}+\mathrm{2}{\varsigma }^{\mathrm{2}}=\mathrm{0})\mathrm{.}

Evolution rule for degree of structure R^*

The following evolution rule for degree of structure R^*, which is the same as in the one proposed by Asaoka et al. [34], is adopted:

(21)\mathrm{d}{R}^{\ast }=U^{\ast }\mathrm{d}{ϵ}_d^p,

where,

(22)

in which a is a parameter that controls the rate of the collapse of the structure during shearing. From the definition, it is clear that the structure of a soil will never be regained once it has been lost. This seems natural because, based on the physical process, the structure of a soil is accumulated during the sedimentary process over a long period time and it would not be easy to regain it within a short period of time without any chemical processes. Substituting Eqs. (17) and (22) into Eq. (21), the rate of R^* can be evaluated as:

(23)\mathrm{d}{R}^{\ast }=\mathrm{\Lambda }{\displaystyle \frac{\mathrm{2}aM{R}^{\ast }(\mathrm{1}-R^{\ast }){\eta }^{\ast }}{C_p(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{\ast \mathrm{2}}){\sigma }_m}}.

Evolution rule for degree of overconsolidation R

In the present model, the changing rate of overconsolidation is assumed to be controlled by two factors, namely, the plastic component of stretching that was employed as the only factor in the work by Asaoka et al. [34], and the increment in anisotropy, in other words,

(24)\mathrm{d}R=U\Vert \mathrm{d}{ϵ}_{ij}^p\Vert +R{\displaystyle \frac{\eta }{M}}{\displaystyle \frac{\partial f}{\partial {\beta }_{ij}}}\mathrm{d}{\beta }_{ij},

where

(25)\Vert \mathrm{d}{ϵ}_{ij}^p\Vert =\sqrt{\mathrm{d}{ϵ}_{ij}^p\mathrm{d}{ϵ}_{ij}^p}=\mathrm{\Lambda }\sqrt{{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}}.

By the definition of dβ_ij in Eq. (15), it is known that dβ_ij is proportional to the norm of the plastic strain tensor \Vert \mathrm{d}{ϵ}_{ij}^p\Vert. U is given by the following relation as:

(26)U=-{\displaystyle \frac{mM}{C_p}}\left({\displaystyle \frac{{\left({\sigma }_m/{\sigma }_{m0}\right)}^2}{{\left({\sigma }_m/{\sigma }_{m0}\right)}^2+1}}\right)\ln R\left(p_0=98.0\mathrm{k}\mathrm{P}\mathrm{a},\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\right).

The reason to using the term {({\sigma }_m/{\sigma }_{m\mathrm{0}})}^{\mathrm{2}} is that in the region close to zero point, the effective mean stress is very small and the change of the effective stress is also very slow compared with the quick development of plastic strain. Therefore it is necessary to slow down the changing rate of overconsolidation, that is, to let U be a small value.

In Step (I) shown in both Figs. 3 and 4 where only the plastic shear strain component is developing (no plastic volumetric strain has developed), the shear strain developed very quickly while the stress remains unmoved. The second part of Eq. (24) is adopted to assure the quick acquisition of overconsolidation (a reduction in R) in the period from (H) to (I) shown in Figure 4. It is clear from Eq. (24) that, apart from elastic unloading, the acquisition of overconsolidation (a reduction in R) is impossible in the plastic loading process if only the first part of Eq. (24) is used, as in the works by Asaoka et al. [34].

In Fig. 4 (b), the changing path of the anisotropic stress with the mean effective stress is shown, from which it is known that the first term in Eq. (24) functions well in regions far from the cyclic loading region, while the second term in Eq. (24) functions well within the cyclic loading region. Without introducing the second term, it is impossible to accumulate overconsolidation, and therefore, impossible to describe the cyclic mobility. The reason is explained as follow:

In the process from Step (H) to Step (I) shown in Figs. 4(a) and (d), R changes very quickly while the effective stress does not move at all (see Fig. 3(I)); the plastic volumetric strain rate is exactly zero, and the degree of structure is almost equal to one due to the complete loss of the structure. By substituting the above conditions into Eq. (9), we can immediately obtain the formula for the second term in Eq. (24). Therefore, it is reasonable to say that the second term in Eq. (24) has a strict mathematical and physical meaning.

From Eq. (19), it is clear that if {\eta }_{ij}{\beta }_{ij}\le \mathrm{0}, which means the angle between the deviatoric stress tensor and the anisotropic tensor is larger than 90°, (\partial f/\partial {\beta }_{ij})\mathit{d}{\beta }_{ij} will always be less than zero. This implies that the overconsolidation is accumulated during the plastic loading process, which is totally different from any other constitutive models that always assume that the overconsolidation is only accumulated in elastic unloading process.

Using Eqs. (8) and (13), it is easy to obtain the relation as

(27)\Vert \mathrm{d}{ϵ}_{ij}^p\Vert =\sqrt{\mathrm{d}{ϵ}_{ij}^p\mathrm{d}{ϵ}_{ij}^p}=\mathrm{\Lambda }\sqrt{{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}}=\mathrm{\Lambda }{\displaystyle \frac{\sqrt{\mathrm{6}{\eta }^{*\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{(M^{\mathrm{2}}-{\eta }^{\mathrm{2}})}^{\mathrm{2}}}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}}){\sigma }_m}}.

Substituting Eqs. (26) and (27) into Eq. (24), we obtain:

(28)\mathrm{d}R=\mathrm{\Lambda }{\displaystyle \frac{-mM\ln R\sqrt{\mathrm{6}{\eta }^{*\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{(M^{\mathrm{2}}-{\eta }^{\mathrm{2}})}^{\mathrm{2}}}}{C_p(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}}){\sigma }_m}}\left[{\displaystyle \frac{{({\sigma }_m/{\sigma }_{m\mathrm{0}})}^{\mathrm{2}}}{{({\sigma }_m/{\sigma }_{m\mathrm{0}})}^{\mathrm{2}}+\mathrm{1}}}\right]+R{\displaystyle \frac{\eta }{M}}{\displaystyle \frac{\partial f}{\partial {\beta }_{ij}}}\mathrm{d}{\beta }_{ij}.

The plastic volumetric strain rate can be evaluated as:

(29)\mathrm{d}{ϵ}_v^p=\mathrm{\Lambda }{\displaystyle \frac{\partial f}{\partial {\sigma }_m}}=\mathrm{\Lambda }{\displaystyle \frac{M^{\mathrm{2}}-{\eta }^{\mathrm{2}}}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}}){\sigma }_m}}.

Substituting Eqs. (19), (23), (28) and (29) into Eq. (9), the positive valuable Λ can then be determined as:

(30)\mathrm{\Lambda }={\displaystyle \frac{{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}\mathrm{d}{\sigma }_{ij}}{{\displaystyle \frac{\mathrm{1}}{C_p(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}}){\sigma }_m}}[M_s^{\mathrm{2}}-{\eta }^{\mathrm{2}}]}},

where

(31)M_s^{\mathrm{2}}=M^{\mathrm{2}}-{\displaystyle \frac{mM\ln R}{R}}\left[{\displaystyle \frac{{({\sigma }_m/{\sigma }_{m\mathrm{0}})}^{\mathrm{2}}}{{({\sigma }_m/{\sigma }_{m\mathrm{0}})}^{\mathrm{2}}+\mathrm{1}}}\right]\sqrt{\mathrm{6}{\eta }^{*\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{(M^{\mathrm{2}}-{\eta }^{\mathrm{2}})}^{\mathrm{2}}}-\mathrm{2}aM(\mathrm{1}-R^{*}){\eta }^{*}-\left(\mathrm{1}-{\displaystyle \frac{\eta }{M}}\right){\displaystyle \frac{\sqrt{\mathrm{6}}M{b}_r(M-\varsigma ){\eta }^{*\mathrm{2}}(-\mathrm{2}{M}^{\mathrm{2}}+\mathrm{3}{\eta }_{ij}{\beta }_{ij})}{(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}})(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}})}}.

If the incremental strain tensor is divided into elastic and plastic components, then the following relation can be obtained:

(32)$$\begin{gathered} \mathrm{d}{ϵ}_{kl}=\mathrm{d}{ϵ}_{kl}^e+\mathrm{d}{ϵ}_{kl}^p,\mathrm{d}{\sigma }_{ij}=E_{ijkl}\mathrm{d}{ϵ}_{kl}^e, \\ \mathrm{d}{\sigma }_{ij}=E_{ijkl}d{ϵ}_{kl}-\mathrm{\Lambda }{E}_{ijkl}{\displaystyle \frac{\partial f}{\partial {\sigma }_{kl}}}. \end{gathered}$$

Substituting Eqs. (19), (23), (28), (29), (31) and (32) into Eq. (9), it is possible to obtain another expression for the following positive valuable Λ:

(33)\mathrm{\Lambda }={\displaystyle \frac{{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}{E}_{ijkl}d{ϵ}_{kl}}{h_p+{\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}{E}_{ijkl}{\displaystyle \frac{\partial f}{\partial {\tilde{\sigma }}_{kl}}}}},

where

(34)h_p={\displaystyle \frac{\mathrm{1}}{C_p(M^{\mathrm{2}}-{\varsigma }^{\mathrm{2}}+{\eta }^{*\mathrm{2}}){\sigma }_m}}[M_s^{\mathrm{2}}-{\eta }^{\mathrm{2}}].

The loading criteria are given as:

(35)

In most cases, the denominator is positive, therefore, \mathrm{\Lambda }>\mathrm{0} is equivalent to the following relation:

(36){\displaystyle \frac{\partial f}{\partial {\sigma }_{ij}}}{E}_{ijkl}\mathrm{d}{ϵ}_{kl}>\mathrm{0},

which is the same as the way proposed by Hashiguchi [27,37], or Zienkiewicz and Taylor [38]. It should be pointed out that by adopting the subloading concept, the plastic strain may occur in the area within the normal yielding surface according to the above loading criteria, which is different from classical plastic models.

Equations (1)-(36) are discussed in conventional triaxial stress space. If the intermediate stress dependency is taken into consideration, then the TS concept proposed by Yao et al. [35] is adopted. Because most of the equations are in the same form as those in the conventional triaxial stress space, the detailed description about this model in TS space is omitted. Reader can refer to the works by Yao et al. [35] and Wan et al. [36]. In the following chapters, all the calculations are carried out in TS space.

Determination of material parameters

Among the eight parameters involved in the model, five parameters, M, N, λ, κ, and ν are the same as in the Cam-clay model. The other three parameters and their functions are listed below,

a: parameter that controls the collapse rate of structure,

m: parameter that controls the losing rate of overconsolidation,

b_r: parameter that controls the developing rate of stress-induced anisotropy.

These three parameters have clear physical meanings and can be determined by undrained triaxial cyclic loading tests and drained triaxial compression tests. Parameter m can be determined based on the losing rate of overconsolidation of soil sample in conventional triaxial compression tests. Parameter a can be determined based on the collapse rate of the structure of the soil sample formed in its natural depositary process, as shown in the work by Asaoka et al. [34]. Parameter b_r can be determined based on the developing rate of the stress-induced anisotropy of the soil sample in undrained triaxial cyclic loading tests. The higher the developing rate is, the larger the parameter b_r will be. Detailed description about the physical meaning and the determination of the structure, the overconsolidation and the stress-induced anisotropy will be given later.

Undrained triaxial cyclic loading test and its simulation

In this section, the material parameters of Toyoura Sand are determined in above-mentioned tests. The test conditions are listed in Table 1. Cyclic loading with sine wave is applied with 0.01 Hz under confining stress of 98 kPa and the sand samples are prepared with the method of sedimentation within water. Loose specimen was prepared by depositing the saturated sand slowly in de-aired water using a funnel with an opening of 3 mm. Medium dense specimen was prepared by pouring the saturated sand into a mold in several layers, each of which is compacted with a 6 mm-diameter rod in prescribed number of times.

Figure 5 shows experimental results of undrained triaxial cyclic loading test with three different amplitude of cyclic loading ratios q/2p₀ (= 0.15, 0.20 and 0.25), in which p is mean effective stress and q is stress difference. It is seen from the test results that because the samples are medium dense sand, cyclic mobility does happen in all three tests and that the larger the cyclic loading ratio is, the faster the samples come into the cyclic mobility loop, an typical behavior that can be found in any literature available.

Figure 6 shows the theoretical simulation of the undrained triaxial cyclic loading test with different cyclic loading ratios. The effective stress paths and the stress-strain relations predicted by the model coincides qualitatively with the test results, while the cyclic number necessary for causing cyclic mobility is less than the test results.

In the simulation, the unique values of the material parameters of Toyoura Sand, listed in Table 2, are determined with above cyclic loading tests and drained compression tests [20]. Detailed method to determine the values of these parameters can be found in the work by Ye [39]. It should be pointed out that all the eight parameters of Toyoura Sand will be the same throughout the paper.

Compared to the material parameters, it is rather difficult to determine the initial conditions of state parameters for sand because the values are dependent not only on present stress state but also on its history. The initial conditions of the state parameters of Toyoura Sand are listed in Table 3. Initial anisotropy ζ₀ is assumed to be zero, which means that the sand is isotropic at the beginning of shearing. Structure R^* which influenced by the process in depositing the sand, usually collapses quickly during shearing and never recover. Therefore initial degree of structure R_{\mathrm{0}}^{*} is assumed to be a relatively large value R_{\mathrm{0}}^{*}=\mathrm{0.75} for the medium dense sand. The initial degree of overconsolidation 1/R₀ is determined to be 70 according to its initial void ratio.

Drained triaxial compression test and its simulation

Simulation of drained triaxial compression test under constant mean principal stress is conducted to determine the material parameters. In the test [20], the initial void ratio of loose sand is e₀ = 0.81 and medium dense sand is e₀ = 0.666.

It is, however, very difficult to identify the reference void ratio N in a small confining stress condition. Therefore, by extending the e-ln p relation to small stress range, the reference void ratio N of Toyoura Sand is determined to be 0.87. Combined with the undrained triaxial cyclic loading test in previous section, the material parameters of Toyoura Sand are fully determined and listed also in Table 2. The initial values of the state parameters for the drained triaxial compression test, however, are different from the cyclic loading test and listed in Table 4.

Figure 7 shows the comparison of experimental and theoretical results of triaxial compression test with different densities. The test results of loose sand, as shown in Fig. 7(a), are reproduced quite well quantitatively. The results of dense sand, as shown in Fig. 7(b), are reproduced relatively well before peak strength while in the residual state, a discrepancy between the test and the prediction exists. On the whole, however, the model can describe the behaviors of the sand to some extent in these tests.

Influence of density

The behavior of sand is known to be dependent on its density. In order to verify the influence of the initial density on the behavior of sand, we now consider numerically a set of sands with different densities which is prepared from a very loose sand using the same method proposed by Asaoka [7]. Table 5 lists the initial conditions of the loose sand before compaction. From the initial values of R, R^*, and ζ, it is understood that the sand is originally normally consolidated and highly structured loose sand without stress-induced anisotropy and has a very large void ratio. In preparing the set of sand samples with different densities, the very loose sand is compacted by a small vibration load along vertical direction with an amplitude of 2.3 kPa under a small confining pressure of 10 kPa. After the compaction, these sands with different densities are isotropically consolidated to a prescribed confining pressure of 196 kPa. The set of sands with eight different densities are prepared by different numbers of vibrating compaction, as shown in Fig. 8. Table 6 lists the state variables of these sands after they are compacted and Table 7 lists the state variables of these sands after they are isotropically consolidated to the confining pressure of 196 kPa.

By using the material parameters listed in Table 2 and the initial values of the state variables for the sands with different densities listed in Table 7, various kinds of triaxial tests under drained/undrained conditions subjected to monotonic and cyclic loadings, are calculated systematically in the following sections.

Simulation of sand with different densities subjected to undrained/drained cyclic loading

The eight sands with different densities listed in Table 7 are simulated in cyclic loading tests with confining stress of 196 kPa. The amplitude of the cyclic loading in shear stress ratio (q/2p₀) is 0.12.

Figure 9 shows the stress paths and stress-strain relations of the sands with different densities in undrained tests. It is clear from the figures that very loose sands ([1] and [2]) generate a large failure strain along the path directly towards the zero effective stress state without transition from contractive state to dilative state. For relatively loose sands ([3] and [4]), they also generate large failure strain at last but transition from contractive state to dilative state can be observed. For medium dense sands ([5]–[7]), however, cyclic mobility occurs and the strain increases gradually to a relatively larger scale. On the other hand, the dense sand ([8]) only generates a small amount of strain and never shows cyclic mobility. Therefore, the mechanical behavior of sand subjected to undrained cyclic loading can be uniquely and properly described by the constitutive model under the condition that all the material parameters are kept constant.

Figure 10 shows the simulated results of loose sand ([2]) subjected to cyclic loading with a amplitude of 60 kPa under drained condition. Other simulate conditions are the same as aforementioned undrained test. As the cyclic loading goes on, the loose sand is compacted and experiences a large contractive volumetric stain.

The above-discussed simulations of undrained/drained cyclic loading tests in Figs. 9 and 10 show the same phenomenon pointed out by Asaoka [7] that the consolidation or the liquefaction of sand due to cyclic loading is just dependent on the drainage condition of the test. If the loading is conducted under drained condition, then the consolidation of sand will happen; while under undrained condition, the liquefaction will occur, which is totally coincident with the reality.

Another important thing that should be pointed out, is the phenomenon observed in shaking table tests [40] that after the excessive pore water pressure dissipates, the once-liquefied soils, that are usually denser than their original state, may liquefy again if subjected to another strong motions. The present model can describe this phenomenon naturally because it is clear from the figure that the liquefaction of the sand is dependent on its density. Once liquefied soils may get denser after dissipation of pore water pressure but may liquefy again if the density is not dense enough.

Simulation of sand with different densities subjected to undrained/drained monotonic loading

The relationships between stress, strain and void ratio under undrained/drained triaxial compression tests on the sands listed in Table 7 are simulated in this section.

Under undrained conditions, three different types of typical stress-strain relations can be observed in the simulation, as shown in Fig. 11. For loose sands [1] and [2], the sands reach peak strength in small strain level and then collapse and flow rapidly toward the origin of the stress space, showing a typical strain-hardening/softening and contractive behavior. For medium dense sands [3] to [6], the stiffness of the sands decreases abruptly in certain strain level where a typical transition from contractive to dilative occurs. Dense sands [7] and [8], however, only show strain hardening. In Fig. 11(c), traces of e-lnp′ states during shearing are plotted for all sands, showing that all sands finally move towards the CSL, and that even for a very loose sand, transition process from contractive to dilative would occur. For dense sand, dilation will be predominant and the stress, that is needed to shear the sands to CSL, will be extremely large.

Figure 12 shows simulated stress-strain-dilatancy relations of the sands with different densities listed in Table 7 in drained triaxial compression tests with constant confining stresses. It is known that dense sands show the typical strain hardening-strain softening and the dilation while loose soils only show the strain hardening along with monotonic contraction. The transition from contractive state to dilative state is just dependent on the density of the sand. It is known from Fig. 12(c) that all sands approach to the same point in e-p space at the critical state and have the same void ratio, irrespective of the different initial densities at the beginning of shearing, because they are originated from the same sand.

The above simulated density-dependent behavior of the sands in drained/undrained triaxial compression are well-known to the researches and have already been confirmed in laboratory tests that it is not necessary to give any comparison between the test and the simulation.

Confining-stress dependency of sand in undrained monotonic loading test

Verdugo and Ishihara [15] reported their experimental results of Toyoura Sand, in which undrained triaxial compression tests on the sands with the same void ratio but different confining pressures were conducted under very high confining pressures (up to 3 MPa). The test results in Fig. 13 show that under the same void ratio, if a confining stress is large, the sand behaves like a loose sand, while if the confining stress is small; the sand behaves like a dense sand. Such a phenomenon is called as “confining-stress dependency of sand”, originally defined in the research by Ishihara [41]. Nakai [42] also reported the same phenomenon in his tests on silica sands.

In the tests, three groups of sands were considered, each of which has the same void ratio, while different group has different void ratio. Therefore, in the simulation it is necessary to adjust the density for all the sands before shearing. The initial values of the void ratios are set to be equal to 0.78, 0.70 and 0.65 respectively and are listed in Table 8. It is known from Fig. 14 that the simulated results on the whole coincide well with the test results quantitatively and qualitatively.

It is also known for the simulation that the mechanical behavior of sands with the same density but different confining stresses can also be reproduced uniquely with one set of the same material parameters in all different conditions.

Sand subjected to drained cyclic loading

In this section, the behaviors of dense sand subjected to drained cyclic loading under constant mean effective stress are simulated. The confining pressure of the sand is 196 kPa and cyclic loading condition is that the mean effective stress is kept constant and a maximum principal stress ratio (σ₁/ σ₃) is loaded to 4. Figure 15 shows the test results by Hinokio [16], in which the stress-strain curves are plotted in terms of effective stress ratio σ₁/σ₃, a dimensionless normalized stress. The volumetric strain shows dilatancy at the very beginning under cyclic loading and then turns to compression until it reaches a steady state at which the compression almost stops, as shown in Fig. 16(b). For deviatoric stress-strain relation, at the beginning, it shows a relatively large loop, as the cyclic loading number increases, however, the stiffness of the sand grows up and the stress-strain relation comes into an almost fixed loop as shown Fig. 16(c). In the simulation, the initial conditions of the dense sand are shown in Table 9. In determining the initial conditions of the sand, it is assumed that the sand is well remolded with extremely low structure and relatively high overconsolidation. As can be seen in Fig. 16, the overall characteristics of the sand predicted by the present model, for instance, the changes in dilatancy and stress-strain relations, agree qualitatively well with the test results, but showing a slight over-estimation of volume strain.

It should be emphasized here that in the simulation, volumetric compression also stopped automatically after certain cycles of loadings, which agrees well with the experimental results. The reason why the model can describe this behavior is quite simple. Taking a look at Figs. 16(d) and (e), in which the changes of stress-induced anisotropy and overconsolidation are plotted, it is easy to find out that during plastic loading the degree of overconsolidation sometime may even increase, not always the case in which overconsolidation only develops in elastic unloading process.

From Fig. 16(f), it is also known that the degree of overconsolidation also shows an increasing tendency along with the increase of compressive volumetric strain, which is a very comprehensive and natural behavior of a sand in reality. One of the most important features of the model is that the changing rate of overconsolidation, or density, is assumed to be controlled by two factors, one is the plastic component of stretching and the other is the increment of stress-induced anisotropy. The physical meaning of the first part is very familiar to the readers and the second part, is just used to consider the influence of stress-induced anisotropy which is usually dependent on the roundness of soil particles and their orientation of deposition. It is known that the stress-induced anisotropy is largely dependent on the shape of sand particle, that is, the more angular the particle is, the more easily the anisotropy will develop. It can be seen in the process of cyclic mobility that, when the stress state passes through the original, which means the change of the loading direction (from compression to extension or vice versa), or on the returning point of cyclic loading, the volumetric strain rate always change its direction dramatically (from dilatant to compressive or vice versa). The only reason that can be regarded as this dramatic change is the re-orientation of the sand deposition. In a word, the anisotropy will have a strong influence on the volumetric change during shearing.

It is admitted, however, that above discussion is still needed to be confirmed with experiment. It is clear from the figures that during cyclic shearing, overconsolidation gets higher and higher, in other words, the density is getting higher, resulting in the difficulty to further compression.

Influence of stress-induced anisotropy

As has been mentioned in the introduction, the stress-induced anisotropy has great influence on the mechanical behavior of sand subjected to consequential loadings. In order to investigate this influence, two sand samples [r] and [s], which have almost the same properties in the state variables but only different stress-induced anisotropy ζ₀, at the beginning of undrained triaxial cyclic loading tests, as shown in Table 10. The initial stress-induced anisotropy is acquired in a pre-loading test in which the sample in an isotropic stress condition is loaded in a half-cycle loading under drained and constant mean stress condition. The loading direction of the samples [r] and [s], however, is different, resulting a different stress-induced anisotropy, as shown in Fig. 17. Figure 18 shows the simulated results of the samples, the effective stress path and stress-strain relation in the triaxial cyclic loading tests. From the figure, it is known that behavior of the samples with different stress-induced anisotropy is quite different. Further discussion about the influence of the stress-induced anisotropy can be referred to the work by Ye et al. [43]

In order to investigate the influence of the stress-induced anisotropy on the mechanical behavior of rather dense sand, we prepared a set of sands whose material parameters are almost the same as those of Toyoura Sand listed in Table 2, except for the parameter b_r, which controls the developing rate of the stress-induced anisotropy. The initial values of the state variables are listed Table 11, from which it is known that these sands considered here are rather dense sands. The sand [i], is exactly the Toyoura sand that has a very similar property with the sand [7] listed in Table 7. Figure 19 shows the behavior of cyclic mobility for these sands subjected to cyclic loading under undrained conditions. It can be seen from the figure that the faster the development of the stress-induced anisotropy is, the easier the stress path will run into the cyclic mobility region, that is, the sand is much easier to be liquefied. The above discussion gives us a very simple and effective way to determine the value of the evolution parameter of anisotropy b_r.

Influence of structure

In order to identify the influence of the structure, the same set of sands listed in Table 7 is taken into consideration. Four of the sands listed in Table 7, namely, sands [2], [4], [7] and [8], are considered. As shown in Table 12, those sands with the structure, marked by [a], [b], [c] and [d], are totally the same as the sands, marked by [2], [4], [7] and [8] lasted in Table 7. While the sands without the structure, are marked by [e], [f], [g] and [h], are almost the same as the sands marked by [2], [4], [7] and [8], with the only difference that they have no structure, that is, R^* is always equal to 1.0. In this case, however, it should be pointed out that these four sands [e], [f], [g] and [h], are not originated from the unique loose sand listed in Figs. 5 and 7. Strictly speaking, they are different sands.

As to the sands with the structure, Fig. 20(a) shows the mechanical behavior of the sands [a], [b], [c] and [d], with different densities in undrained cyclic compression tests. It is clear from the figure that very loose sand will fail along the way directly towards the original, the zero effective stress state in the effective stress space, before the cyclic mobility has a chance to occur. For medium dense sand, however, cyclic mobility does occur. Dense sand will never show cyclic mobility. Figure 7 shows the mechanical behavior of the set of sands with different densities in undrained triaxial compression tests. We believe that these results are very familiar to our readers who are interested in soil mechanics. The above results mean that the mechanical behavior of sand, subjected to monotonic/cyclic loading under undrained conditions, can be uniquely and properly described by the proposed model no matter what density it may have.

As to the sands without the structure, Fig. 20(b) shows the behavior of the set of sands with different densities in undrained cyclic compression tests. Comparing the results with those in Fig. 20(a), in which the structure is considered, it is clear that loose sand liquefies with cyclic mobility, which is totally different from the simulated results in which the structure is considered. This implies that when the structure is not considered, it is impossible to describe the liquefaction behavior of loose sand properly. For dense sand, though the structure does not affect the behavior of the sand in the cyclic mobility region so much as the loose sand, it will also affect the shape of the effective stress path more or less, as shown in the figure.

Difference between clayey soils and sandy soils

In the works by Asaoka et al. [28], it is pointed out that the difference between clayey soils and sandy soils depends on two factors, namely, the rate of loss in overconsolidation and the rate of the collapse of the structure during static shearing. For sandy soils, the rate of loss in overconsolidation is very slow, while the rate of the collapse of the structure is very fast. On the contrary, for clayey soils, the rate of loss in overconsolidation is very fast, while the rate of the collapse of the structure is very slow. Under cyclic loading conditions, however, it is necessary to confirm whether this conclusion still remains valid. A set of soils with the same initial conditions but different values for a and m, which control the changing rates of overconsolidation and the collapse of the structure, are investigated for their behavior when subjected to cyclic loading under undrained conditions. Table 13 and 5 list the initial values of the state variables, and the changed material parameters a and m. The other material parameters are the same as those in Table 7. The parameters a and m of samples [l], [m], [n] and [o] are different, meaning that they are totally different soils. The soil [l], however, is Toyoura sand.

Figure 21 shows the different behavior of soils from sandy soil to clayey soil subjected to cyclic loading under undrained conditions. From this figure, it is very clear that by changing parameters a and m, the difference between sandy soils and clayey soils can be easily and uniquely identified. For instance, in the case of soil [l], m = 0.01 and a = 0.50, which means that the loss of overconsolidation is very slow while the collapse of the structure is very fast, a typical cyclic mobility behavior is observed. In the case of soil [o], m = 0.30 and a = 0.05, which means that the loss of overconsolidation is very fast while the collapse of the structure is very slow, however, a typical clayey soil behavior under cyclic loading is observed. The soils [m] and [n] just show the mechanical behavior of intermediate soil like silt. From the above discussion, it is very easy to explain why clay does not liquefy, namely, it is just the reason that the structure of clay collapses more slowly than sandy clay so that it is possible to resist the cyclic shearing. For sand, unfortunately, its structure will collapses so fast that it liquefies easily if the density is loose enough.

Conclusions

In this paper, the overall behavior of Toyoura Sand, a typical clean sand, is described in a unified way by the model proposed by the authors. For a given sand, the value of the material parameters are fixed and can be easily determined by conventional drained triaxial compression tests and undrained triaxial cyclic loading tests. By using the uniquely determined material parameters, the mechanical behavior of the sand under different loadings and drainage conditions can be simulated by the constitutive model. The capability of the model to give a unified description of the overall behaviors with fixed values of parameters is verified and the following conclusions can be given.

1) Eight parameters are needed to describe the behavior of Toyoura Sand, among which five parameters, M, N, λ, κ, and ν are the same as the ones of Cam-clay model. Other three parameters, a: the parameter controlling the collapse rate of structure, m: the parameter controlling the losing rate of overconsolidation and b_r: the parameter controlling the developing rate of stress-induced anisotropy, have clear physical meanings and can be easily determined by undrained triaxial cyclic loading tests and drained triaxial compression tests. In order to give a unified description of Toyoura Sand, the eight material parameters are kept the same value for all the tests under different loadings and drainage conditions.

2) The critical state line, as the threshold between plastic compression and plastic expansion, proposed in the present model, is fixed, no matter what kind of effective stress path there may be, which makes the model look much more efficient, more realistic, and easier to handle.

3) Simulations on a set of sands with different densities originally compacted from the same loose sand are conducted to verify the density-dependent behavior of the sand. The results reveal the fact that the mechanical behaviors of the sand subjected to cyclic loading under drained/undrained conditions can be uniquely and properly described by the constitutive model no matter what density it may have. It is confirmed theoretically that for loose sand, liquefaction happens without transition from contractive to dilative state; for medium dense sand, cyclic mobility occurs while for dense sand, liquefaction will not occur. It is not necessary to assign, in advance, which sand will liquefy or not. It is simply dependent on the state, namely, the overconsolidation ratio (density), the stress-induced anisotropy, and the structure. Furthermore, it is confirmed that if the sand subjected to cyclic loading under drained condition, then the consolidation of sand will occur; while under undrained condition, the liquefaction might happen, depending on its density.

4) Under undrained triaxial monotonic compression test, loose sand exhibits a peak strength in small strain level and then collapses and flows rapidly toward the origin of of the stress space, showing a typical strain-hardening/softening and contractive behaviors. For medium dense sand, stiffness of the sand decreases abruptly in certain strain level where a typical transition from contractive to dilative state occurs. Dense sand, however, only shows strain hardening. In undrained tests, all sand samples finally move towards CSL, while in drained tests, all sands approach to the same point in e-p space at critical state, irrespective of different initial densities at the beginning of shearing.

5) Confining-stress dependency of sand, a typical behavior of sand with the same density but under different confining stresses when subjected to undrained monotonic triaxial compression, can also be simulated properly.

6) Dense sand subjected to rather large cyclic loading under drained and constant-mean-effective-stress conditions is also simulated. The overall characteristics of the sand is predicted well by the model, for instance, the changes in dilatancy and stress-strain relations are quantitatively the same as the test results with a slight over-estimation of volume strain. Particular attention is paid to the volumetric contraction during cyclic loading, which shows in the test a small dilatancy at the very beginning and then turns to compression until it reaches a steady state where the compression stops. The simulation also describes the same behavior automatically without changing the values of any parameters

7) The difference between a sand and clay is just the difference in the collapse rate of the structure and the losing rate of the overconsolidation that happened to the soils when they were subjected to shearing. This statement is firstly given in the work by Asaoka et al. [34] for the soils under static loading. Present study, however, shows that this statement is not only valid for static loading, but also for the soils subjected to cyclic loading under undrained conditions. That is, for clayey soils, the structure collapses very slowly while the overconsolidation declines easily. For sands, however, the structure collapses very quickly while the overconsolidation declines very slowly. Under some circumstances, the overconsolidation may even increase during plastic shearing, which is different from the assumption adopted in the model proposed by Asaoka et al. [34] whereby the accumulation of overconsolidation only happens during the elastic unloading process. In a word, the mechanical behavior of soils subjected to cyclic loading under undrained conditions, can be uniquely described by the proposed model, no matter what kind of soil it may be.

8) By introducing the TS concept, the influence of intermediate principal stress on mechanical behavior of sand can be properly taken into consideration, without changing any value of the material parameters if compared with those adopted in conventional triaxial stress condition.

9) One of the most important features of the model is that the changing rate of overconsolidation, or density, is assumed to be controlled by two factors, one is the plastic component of stretching and the other is the increment of stress-induced anisotropy. The overconsolidation can also accumulate during a plastic loading process, which is totally different from any other constitutive models that always assumed that the overconsolidation is only accumulated in elastic unloading process. In a word, the stress-induced anisotropy also give a large influence on the change of density. It is also known from the simulation of the sand under undrained cyclic triaxial loading that the developing rate of the stress-induced anisotropy affects the degree of the cyclic mobility of sand greatly. The faster the rate is, the more likely the cyclic mobility of soils may occur.

It cannot say that the model can perfectly describe the various behaviors of Toyoura Sand, but that the model can give a unified description of Toyoura Sand with quite satisfactory accuracy only by using eight material parameters with fixed value. On the other hand, in some very specific aspects, such as the non-coaxial property, which have been pursued for years by many researchers, should be done in future to verify the applicability of the model.

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