Hong-en LI* , Yong-jun HE , Guang-ya FAN , Tong-chun LI, Manuel PASTOR

1. Dam Safety Management Department, Nanjing Hydraulic Research Institute, Nanjing 210029, P. R. China

2. Dam Safety Management Center of Ministry of Water Resources, Nanjing 210029, P. R. China

3. College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, P. R. China

4. Department of Applied Mathematics, Technical University of Madrid, Madrid 28040, Spain

1 Introduction

The soil constitutive model, which has attracted tremendous attention, is the basic equation for geotchenical analysis. In 1958 Roscoe defined the critical state of soil. Later, the famous Cambridge elastoplastic constitutive model for clay was proposed by introducing the classic elastoplastic theory into soil mechanics. Since then numerous elastoplastic models with different features have been put forward by researchers all over the world. In elastoplastic models, deformation is decomposed into a recoverable elastic part and unrecoverable plastic part, which are calculated according to the elastic theory and the plastic incremental theory,respectively. With the models, the dilatancy, overconsolidation, anisotropy, cyclic mobility,liquefaction, and other mechanical properties of soil can be simulated conveniently. Therefore,elastoplastic models have a great amount of potential to predict the deformation and failure mechanism of soil under complex loading conditions.

Of all the elastoplastic models, the generalized plasticity model (Pastor et al. 1990)has achieved considerable success in modeling the behavior of soil (Zienkiewicz et al. 1999). This kind of model is based on the boundary surface variant of the generalized plasticity theory.The yield or plastic potential surface and the hardening rule are not specified directly. Instead,the direction vectors are involved. According to the framework of the generalized plasticity theory (Zienkiewicz and Mróz 1984), many improvements have been made in order to analyze more complicated behaviors of soil that the original model could not deal with (Pastor et al.1993; Bolzon et al. 1996; Tamagnini and Pastor 2004; Ling and Yang 2006; Tonni et al. 2006;Manzanal et al. 2008; Manzanal 2008). This paper focuses on recently enhanced models based on the generalized plasticity theory and their applicability in soil with different properties.

2 Original generalized plasticity model and its limitations

The generalized plasticity theory, as well as a generalized constitutive model(Pastor-Zienkiewicz mark III model, PZ-III model for short), was first proposed by Zienkiewicz and Mróz (1984)and later extended by Pastor et al. (1990)for various types of soils. The main advantage of the model lies in its capability of simulating strain-stress response in different initial conditions under monotonic and cyclic loading without needing an explicit definition of yield or plastic potential surface. For the original model, there are 12 material parameters that require definitions. Generally, all the parameters are identified by monotonic and cyclic triaxial tests, except that in certain cases some parameters are adopted from previous experience if full test records are unavailable.

The original PZ-III model has been successfully applied in simulating distinct behaviors of soil and has shown appreciable adaptability and forecasting ability(Zienkiewicz et al. 1999). However, some limitations of the original model restrict its wide application. Therefore, based on the generalized plasticity theory, many researchers have proposed various enhanced models to improve the capability of the original model (Pastor et al. 1993; Bolzon et al. 1996; Bahda et al. 1997; Zhang et al. 2001a, 2001b;Fernández-Merodo et al. 2004; Tamagnini and Pastor 2004; Ling and Yang 2006; Manzanal and Fernández-Merodo 2006; Tonni et al. 2006; Manzanal et al. 2008).

It is well known that the behavior of sand depends on its density and confining pressure.For a determinate density, sand shows its loose behavior at sufficiently high confining pressures, and dense behavior at sufficiently low confining pressures. Therefore, the most serious limitation of the original model is that it regards sands of the same type with different initial densities (or initial void ratios)and confining pressures as different materials, which means that the model requires a number of parameters for a single type of sand with different densities under different confining pressures. Bahda et al. (1997)first introduced the state parameter to simulate the cyclic loading of sand. Then, Manzanal and Fernández-Merodo(2006), Tonni et al. (2006), and Ling and Yang (2006)proposed different generalized plasticity models for sand by introducing the concept of a state parameter to avoid the requirement of a number of parameters in the original model. The details of the state parameter-based model for saturated soil are analyzed in Section 3.

In recent years, more and more attention has been paid to unsaturated geomaterials, as the mechanical behaviors of unsaturated soil are quite different from those of saturated soil. Some specific characteristics of unsaturated geomaterials are as follows (Gens and Balkema 1996):(1)preconsolidation stress increases significantly with suction; (2)the soil state after wetting-induced collapse lies on the saturated consolidation line; (3)soil exhibits stress-path independent behavior in wetting tests; (4)strain reversal occurs in some wetting tests; and(5)shear strength increases with suction, and a critical state line (CSL)exists at constant suction values. In order to make the original model suitable for unsaturated soil analysis,various methods have been used to enlarge its application scope. In Section 4, the unsaturated generalized plasticity models are discussed in detail.

Finally, based on the modification of PZ-III models, a general modeling process based on generalized plasticity is obtained, and generalized plasticity models with special abilities can be easily derived.

3 Generalized plasticity models based on state parameters for saturated soil

Many researchers have illustrated the influence of density and confining pressure on the sand behavior (Ishihara 1993). The state parameter concept provides a powerful tool to deal with the density and pressure dependency of sand.

Uriel (1973)first proposed a definition of state parameters with a proper form of density at a critical state, and then Pastor and Uriel (1983)extended this definition to overconsolidated compacted clay. Been and Jefferies (1985)defined the state parameter ψ as the difference between the actual void ratio (e)and the void ratio at the critical state (ec)under the same confining pressure with the form of ψ = e - ec, which is widely accepted today.

Some researchers introduced state parameters into the PZ-III model with similar approaches (Bahda et al. 1997; Ling and Liu 2003; Tonni et al. 2003; Ling and Yang 2006;Manzanal and Fernández-Merodo 2006). The model proposed by Manzanal and Fernández-Merodo (2006), Manzanal (2008), and Manzanal et al. (2008, 2010a)is regarded as the most comprehensive one.

Manzanal (2008)presented the unified model based on state parameters considering the following aspects: (1)the dependency of dilatancy expression on density and confining pressure, (2)the relation between the maximum mobilized friction angle (or the maximum stress ratios)and softening behavior and its dependency on the initial state of soil, (3)the yield surface associated with state parameters existing in an explicit or implicit form, and (4)the dependence of isotropic plastic modulus on the variation of densities. Concerning these issues,the state parameter has been imbedded in both flow rules and formulations of plastic modulus.

3.1 Modified flow rule

According to a new expression for dilatancy proposed by Li and Daflias (2000), the dilatancy equation in the original model is modified as

where dgis dilatancy; η is the stress ratio, η= q p′; q is the deviatoric stress; p′ is the mean effective confining pressure; d0is the initial tangent slope of the volume strain-shear strain curve derived from drained triaxial compression tests; ηPTSis the stress ratio at the point of phase transformation, which depends on the state parameter ψ,ηPTS=Mgexp(m ψ); m is a model parameter, and its value is determined by the phase transformation line when dilatancy is zero (Ishihara et al. 1975); and Mgis the slope of CSL in a q-p′ plane. The dilatancy laws of dense and loose sands for the modified model and for the original one are depicted in Fig. 1.

Fig. 1 Dilatancy laws for modified and original models

To improve the original model, the non-associated flow rule is adopted in the modified model. The plastic direction vector in the original model, which controls the plastic flow direction, is achieved by substituting the new expression of dilatancy into Eq. (1)(Pastor et al.1990).

Unlike the method mentioned by Zienkiewicz et al. (1999), where the relation between Mf(a model parameter controlling the loading direction in the original model)and Mgis somehow related to the relative density Dr, a new formulation concerning the variation of void ratio is proposed:

where h1and h2are model parameters; ψqis a state parameter of an alternative form,= (e e)β; and βis a constant.0c

3.2 Plastic behavior

The isotropic plastic modulus in the original model is modified to incorporate the influences of the initial density and confining pressure. The formulation is as follows:

When the stress path includes the deviatoric stress q, the plastic modulus is modified with the state parameter as follows:

where

In Eq. (5), Hfstands for the degree of failure and Hsstands for material degradation under accumulated deviatoric plastic strain, which maintains the same expressions as the original model. In the original model, the function f only depends on the stress ratio η. In the modified model, a new form of the volumetric component Hvis proposed, which also depends on the state parameters,

3.3 Elastic behavior

In order to reflect the dependency of a shear elastic modulus on the void ratio and confining pressure, the expressions which have been applied to various models (Gajo and Wood 1999; Li and Dafalias 2000)are derived as follows:

where Gesis the shear elastic modulus, Kevis the bulk modulus, Gesois the model parameter reflecting the initial shear strength, and ν is the Poisson ratio.

Figs. 2 and 3 show the performances of the generalized plasticity model based on state parameters for saturated soil, when applied to Toyoura sand under undrained and drained conditions (Verdugo and Ishihara 1996).

Fig. 2 Experiments and model predictions under undrained condition for Toyoura sand(p0is initial mean effective confining pressure)

Fig. 3 Experiments and model predictions under drained condition for Toyoura sand at p′=100 kPa(e0is initial void ratio)

In Fig. 2, a sand sample was tested at four different initial confining pressures, 100, 1 000,2 000, and 3 000 kPa. As shown in Fig. 3, the confining pressure of 100 kPa was selected and sand samples of the same kind with three different densities were tested. The same set of model parameters were adopted for predictions in all the cases, and the results show that the essential features of the sand behavior obtained are similar with those observed in the experiments.

4 Enhanced generalized plasticity models for unsaturated soil

4.1 Review of effective stress expressions and constitutive approaches

Unlike the constitutive model for the saturated soil, the choice of appropriate stress variables has often been an important foundation and an intensely debated issue in the establishment of a constitutive model for unsaturated soil. Different constitutive approaches for unsaturated soil could be obtained according to the type of stress variables adopted in their formulation. In this section, effective stress expressions and constitutive approaches are reviewed briefly in order to provide a clear perspective of existing constitutive models for unsaturated soil.

4.1.1 Effective stress expressions

Bishop (1959)proposed an expression of effective stress by modifying Terzaghi’s form as follows:

where χ is a positive scalar function depending on saturation degree Sr, σ is the total stress, δ is the Kronecker function, σ′ is the effective stress tensor, and uaand uware the pore air pressure and pore water pressure, respectively. Traditionally, the difference between these two variables is defined as suction s.

Although the effective stress expression proposed by Bishop has been successively used to model the strength of unsaturated soils, there is consensus on the limitations of this expression (Gens et al. 2006): (1)the expression is unable to explain the collapse during wetting, (2)the discontinuity problem occurs at the transition between saturated and unsaturated states, and (3)the material behavior is embodied in both the constitutive relation and the stress space as the parameter χ usually depends on material states (saturation degree),which has been argued often.

Many researchers have continued to work on the effective stress tensor approach. Lewis and Schrefler (1987)derived the effective stress tensor by averaging the contributions of various components:

4.1.2 Constitutive approaches for unsaturated soil

Sheng et al. (2008b)showed that the variations of volumetric behavior, strength behavior,and hydraulic behavior associated with saturation degree or the suction of unsaturated soil should be considered in generating a constitutive approach. Based on these issues, various constitutive models have been proposed to simulate the behaviors of unsaturated soil, which for the most part can be sorted into two categories: the bi-tensorial approach, and the extended critical state approach.

4.1.2.1 Bi-tensorial approach

Because of the limitations of the Bishop’s stress mentioned in Section 4.1.1, Bishop and Blight (1963)demonstrated that the principle of effective stress can be applied to saturated soil only when the effective stress path is taken into account, and that in the case of partly saturated soil, both the effective stress path and the path of suction should be considered. From then on, the so-called bi-tensorial framework based on the net stress and the suction stress tensors has been widely used for modeling the behaviors of unsaturated soil.

The experiments of Fredlund and Morgenstern (1977)provided the theoretical basis and justified the bi-tensorial approach, in which the constitutive equations were derived based on two independent components and the rate of suction was introduced as an external stress variable. In their tests, the suction, mean net stress, and mean effective stress were maintained as constants while the mean total stress, pore air pressure, and pore water pressure were variables. They concluded that any couples of the stress tensors mentioned above are suitable for the definition of constitutive equations. The most convenient coupling is the form of the effective stress tensor and suction (Fredlund and Rahardjo 1993).

Until the work of Houlsby (1997), the bi-tensorial approach was widely used in unsaturated soil areas. In Houlsby’s research, he analyzed the work input to an unsaturated granular material and obtained its power rate:

where W is the input work,ε is the strain tensor, n is the porosity, and I is the unit matrix.

He also confirmed the thermodynamic continuity of the effective stress tensor (Eq. (9))proposed by Shrefler (1984). The expression above could be rewritten as

Eq. (11)shows that the conjugated variables were properly chosen in the bi-tensorial approach.

According to Houlsby’s conclusions and Eq. (9), a new generation of models for unsaturated soil was proposed based on the effective stress and suction with the extended critical state framework, where the hardening law depends on the saturation degree of soil, and the rate of suction is regarded as an internal variable. Jommi (2000)presented a general framework for critical state models of unsaturated soil. Then, isotropic compression tests were modeled (Gallipoli and Gens 2003; Wheeler et al. 2003; Sheng et al. 2004), and the shearing behavior of unsaturated soil was modeled (Tamagnini 2004; Tamagnini and Pastor 2004;Santagiuliana and Schrefler 2006). This kind of model is able to reproduce collapse tests during wetting and could avoid the discontinuity at the transition between saturated and unsaturated states.

Gens and Balkema (1996)showed that the bi-tensorial approach and extended critical state approach have the same form of the total strain rate:

4.2 Extended critical state approach-based generalized plasticity model for unsaturated clay

Following the associated flow rule, Tamagnini and Pastor (2004)modified the PZ-III model for clay based on the extended critical state approach, which was defined in the classic plane of two variables (the mean effective stress and deviatoric stress)with suction as an internal variable. The total strain rate was introduced in a decomposed form as Eq. (12). The new constitutive relation was written in a vector form as

where Deis the elastic matrix, ngLis the plastic flow direction vector, n is the unit direction vector, and Hsis the plastic modulus concerning suction. The third term in Eq. (13)is introduced as a new component that takes suction into account.

Consequently, all the variables in Eq. (13), including the plastic modulus HL, the direction vector n, and the plastic flow direction vector ngL, should be redefined according to the definition of the mean effective stress (Eq. (9)). HDMwas modified to accord with the new hardening mechanism induced by suction:

where ζ is the mobilized stress function, and γDMis a model parameter. J( s)is a function reflecting the additional contribution of hardening in the unsaturated state and it has the form of

where c is a constitutive parameter; and Sris related to the suction s, which forms the water retention curve (WRC). In the simple case, WRC was assumed to obey the simple function:

where m and n′are constants.

Because of hydraulic hysteresis occurring in inflow and outflow, WRC is generally not a unique-value curve. Therefore, the following equations proposed by Romero and Vaunat (2000)were used to model the irreversibility response in wetting-drying cyclic tests:

The main drying and wetting curves of WRC could be obtained by assuming different values for the constitutive parameters α and β′, with the parameters m, n′, and water ratio intercept ewmremaining invariable. The scanning curves were assumed to be linear:

where ksis a constant.

The plastic modulus Hsin Eq. (13)reflects the plastic strain due to the changes of suction,and it is defined as

where b is a new constitutive parameter that controls the shrinkage of the yield surface produced by the change of suction. The multiplying factors Hfand HDMform Hsto model the stress path considering the deviatoric stress and to take the overconsolidation into account.

It should be pointed out that the model is just suitable for clay and the associated flow rule has been used. However, the model provided a clear process for modifying the original model to produce an unsaturated one.

4.3 State parameter-based generalized plasticity model for unsaturated soil

Some limitations of the unsaturated model in Section 4.2 should be pointed out. First,although the associated flow rule has been adopted in the enhanced model, there is a common sense that the non-associated flow rule should be used when simulating the sand behavior.Second, the influences of density and confining pressure on the behavior of sand or clay are important for modeling unsaturated materials. Thus, the anterior unsaturated model requires a number of parameters if the initial conditions of materials have been changed.

Manzanal (2008)and Manzanal et al. (2010b)proposed an unsaturated PZ-III model from the unified point of view. Two pairs of conjugated variables suggested by Houlsby (1997)were used, where the stress variables were the effective stress tensor and suction, and the strain variables were the traditional strain and saturation degree.

4.3.1 Modification of effective stress tensor and CSL

A modified expression of the effective stress tensor has been applied in the model because of the existence of the residual degree of saturation. A definition of the relative degree of saturation is as follows:

where Sr0is the residual degree of saturation. Then, Eq. (9)is modified by incorporating Sre,and the new form of the effective stress tensor is proposed:

Some comparisons were performed to prove the better simulation ability of the new form of the effective stress tensor (Manzanal 2008). Fig. 4 shows that there is a small deviation at high pressures, but the improvement is obvious.

Because of the suction dependency of the normal consolidation line for unsaturated soil, a cementation variable ξ was introduced to normalize the normal consolidation lines (Gallipoli and Gens 2003), as follows:

where the function f( s), proposed by Haines (Fisher 1926), reflects the increment of the stable hydrostatic stress from zero suction to a given suction.

Fig. 4 CSL of silty soil inq-p′plane obtained from critical states under different suctions(Maatouk et al. 1995)

The model assumes the uniqueness of the CSL of unsaturated soil by using the effective stress tensor proposed in Eq. (21), so the CSL maintains the linear form.

Fig. 5 shows the strong dependence of the critical state on suction and good performance of the proposed approach. The generalized form of the CSL for unsaturated soil can realize the smooth transition between the saturated state and the unsaturated state of the soil and avoid the sudden change of the CSL in the bi-tensiorial model. This form is especially beneficial to programming the FEM code because there is no need to determine whether the soil is saturated or unsaturated before confirming the CSL.

4.3.2 Hydraulic hysteresis of WRC

WRC describes the relation between suction and degree of saturation (Fig. 6). WRC in unsaturated soil mechanics has been researched for a long time. In the unsaturated PZ-III model proposed by Tamagnini and Pastor (2004), Eqs. (16)and (17)were used to define WRC.However, the dependency of WRC on void ratio could not be simulated properly. Therefore, a modified formulation of WRC was derived as follows (Fredlund and Xing 1994):

Fig. 5 Dependence of critical state on suction and performance of proposed approach in prediction of Speswhite kaolin soil

where s*, with the expression of s*=e?s, is the normalized suction used to evaluate the dependency of WRC on void ratio (Gallipoli and Gens 2003), and awand ? are model parameters.

Eq. (24)provides the boundaries, which are mainly the wetting and drying curves in Fig. 6, and the actual hydraulic state could be determined through Eq. (25)by linking the boundaries and the scanning curve characterized in the Sr-s*plane:

where kswas chosen to be constant in Eq. (18)in Tamagnini and Pastor’s model (Tamagnini and Pastor 2004), while a nonlinear interpolation rule concerning the hysteresis of the scanning curve was adopted here. kswas suggested by Li (2005)as

Fig. 6 Sketch of WRC

4.3.3 State parameter-based unsaturated PZ-III model

An expression of the total strain, similar to Eq. (13), is

where Hbis the plastic modulus, which is defined as

where w(ξ)is a function to estimate the effect of the cementation variable defined in Eq. (22)during the collapse test,

The model elastic behavior can be expressed with Eq. (7). As we have discussed in Section 4.2 regarding Tamagnini and Pastor’s model, the memory or overconsolidation factor HDMshould record the hardening mechanism induced by suction and degree of saturation. A new expression,J( s), is introduced, which concerns the cementation effects between the particles due to capillary water, as follows:

After coupling the state dependant WRC shown in Section 4.3.2, the proposed model has the capability of simulating the hydraulic effects.

So far, the enhanced state parameter-based unsaturated PZ-III model has been fully stated.In addition to the fact that only one group of parameters is sufficient to realize almost all the predictions of the unsaturated soil under different initial conditions, the irreversible response in wetting-drying paths and the mechanical effect on the hydraulic behavior can be simulated accurately by the proposed model.

5 General formation for modeling with generalized plasticity

After a detailed discussion of the enhanced generalized plasticity models, we could arrive at a general formulation for modeling the total strain rate:

where X can be substituted with suction, temperature, or other factors to meet various needs in its practical application. The plastic modulus HXcontrols the shrinkage of the yield surface caused by the changes of X.

At the same time, the memory factor HDMshould be modified to account for the hardening mechanism induced by both the plastic strain and internal state variable X,

The function J( X )should satisfy the condition J( X )=1 for X=0. γ is a parameter that needs to be calibrated. HXunder isotropic compression conditions for other different stress paths can be obtained as follows:

where αXis a constitutive parameter corresponding to p′.

Therefore, the generalized plasticity model for specific practical use could be easily realized by combining Eqs. (31), (32), and (33)based on the generalized plasticity theory.

6 Conclusions

This paper reviews the fundamental theories of generalized plasticity and the recent development of generalized plasticity models. The group of generalized plasticity models covers a fairly wide range of the characteristics of saturated and unsaturated geomaterials under different conditions. It is worth mentioning that the state parameter-based generalized plasticity models for saturated and unsaturated soils reveal the complicated mechanical properties of soil and sand.

After the summary of different enhanced models, general formation of generalized plasticity models was discussed. The effective and simple models within the generalized plasticity framework are obtained by introducing certain factors such as suction and temperature, which can meet different needs in practical application.

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