Implicit numerical integration for a kinematic hardening soil plasticity model

Soil models based on kinematic hardening together with elements of bounding surface plasticity, provide a means of introducing some memory of recent history and stiffness variation in the predicted response of soils. Such models provide an improvement on simple elasto‐plastic models in describing soil behaviour under non‐monotonic loading. Routine use of such models requires robust numerical integration schemes. Explicit integration of highly non‐linear models requires extremely small steps in order to guarantee convergence. Here, a fully implicit scheme is presented for a simple kinematic hardening extension of the Cam clay soil model. The algorithm is based on the operator split methodology and the implicit Euler backward integration scheme is proposed to integrate the rate form of the constitutive relations. This algorithm maintains a quadratic rate of asymptotic convergence when used with a Newton–Raphson iterative procedure. Various strain‐driven axisymmetric triaxial paths are simulated in order to demonstrate the efficiency and good performance of the proposed algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

基于广义塑性力学的土体次加载面循环塑性模型(Ⅱ):本构方程与验证

按广义塑性力学原理，导出了土体次加载面循环塑性模型的本构方程，建立了相应的加卸载准则以及模型参数的确定方法。通过多种应力路径下土的本构响应的模拟，表明次加载面循环塑性模型能较好地反映循环荷载作用下土体呈现的非线性、滞回性与变形的积累性三方面主要特征，初步验证了模型的有效性。     

Explicit integration of a porosity‐dependent hydro‐mechanical model for unsaturated soils

An adaptive substepping explicit integration scheme is developed for a porosity‐dependent hydro‐mechanical model for unsaturated soils. The model is referred to as the modified σ –Θ model in this paper, which features the employment of the subloading surface plasticity and the stress–saturation approach. On numerical aspects, convex/nonconvex subloading surfaces in the σ –Θ space may result in incorrect loading–unloading decisions during the integration. A new loading–unloading decision method is developed here to solve the problem and then embedded into the explicit integration scheme for the modified σ –Θ model. In addition, to enhance the accuracy of the explicit integration, local errors from both hydraulic and mechanical components are included in the error control for each substep. A drift correction method is also developed to ensure the state point lies on the subloading surface in the σ –Θ space within a set error level. The performance of the loading–unloading decision method for the modified σ –Θ model is discussed through comparing it with the conventional loading–unloading decision method. The importance of involving the hydraulic component in the error control is also demonstrated. The accuracy and efficiency of the proposed adaptive substepping explicit integration scheme for the modified p–Θ model are also studied via several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

A unified bounding surface plasticity model for unsaturated soils

A unified constitutive model for unsaturated soils is presented in a critical state framework using the concepts of effective stress and bounding surface plasticity theory. Consideration is given to the effects of unsaturation and particle crushing in the definition of the critical state. A simple isotropic elastic rule is adopted. A loading surface and a bounding surface of the same shape are defined using simple and versatile functions. The bounding surface and elastic rules lead to the existence of a limiting isotropic compression line, towards which the stress trajectories of all isotropic compression load paths approach. A non‐associated flow rule of the same general form is assumed for all soil types. Isotropic hardening/softening occurs due to changes in plastic volumetric strains as well as suction for some unsaturated soils, enabling the phenomenon of volumetric collapse upon wetting to be accounted for. The model is used to simulate the stress–strain behaviour observed in unsaturated speswhite kaolin subjected to three triaxial test load paths. The fit between simulation and experiment is improved compared to that of other constitutive models developed using conventional Cam‐Clay‐based plasticity theory and calibrated using the same set of data. Also, the model is used to simulate to a high degree of accuracy the stress–strain behaviour observed in unsaturated Kurnell sand subjected to two triaxial test load paths and the oedometric compression load path. For oedometric compression theoretical simulations indicate that the suction was not sufficiently large to cause samples to separate from the confining ring. Copyright © 2005 John Wiley & Sons, Ltd.     

SANISAND-FN: An evolving fabric-based sand model accounting for stress principal axes rotation

SANISAND is the name of a family of bounding surface plasticity constitutive models for sand within the framework of critical state theory, which have been able to realistically simulate the sand behavior under conventional monotonic and cyclic loading paths. In order to incorporate the important role of evolving fabric anisotropy, one such model was modified within the framework of the new anisotropic critical state theory and named SANISAND-F model. Yet the response under continuous stress principal axes rotation requires further modification to account for the effect of ensuing noncoaxiality on the dilatancy and plastic modulus. This modification is simpler than what is often proposed in the literature, since it does not incorporate an additional plastic loading mechanism and/or multiple dilatancy and plastic modulus expressions. The new model named SANISAND-FN is presented herein and is validated against published data for loading that includes drained stress principal axes rotation on Toyoura sand.     

Modeling the behavior of sandstone based on generalized plasticity concept

With the concept of generalized plasticity, a constitutive model for describing the deformation behavior of sandstone is proposed in this paper. This proposed model is characterized by the following features: (i) nonlinear elasticity under hydrostatic and shear loading; (ii) associated flow rule for pre‐peak simulation; (iii) substantial plastic deformation during shear loading; and (iv) significant shear dilation and distortion prior to the failure state. This model requires 10 material parameters, including three for elasticity and seven for plasticity. All of the parameters can be determined, in a straightforward manner, by the suggested procedures. The proposed model has been validated by comparing the triaxial test results of the Mushan sandstone under different hydrostatic stress, different stress paths, and cyclic loading condition. It is also versatile in simulating the deformation behaviors of two other sandstones. Upon slight modification of the model, the post‐peak behavior of sandstone can be reasonably predicted using proposed model. Copyright © 2012 John Wiley & Sons, Ltd.     

Investigation of monotonic and cyclic behavior of sand using a bounding surface plasticity model

Developing the pore water pressures in loose to medium sands below the water table may lead to liquefaction during earthquakes. The simulation of liquefaction (cyclic mobility and flow liquefaction) in sandy soils is one of the major challenges in constitutive modeling of soils. This paper presents the simulation of sand behavior using a critical state bounding surface plasticity model (Dafalias and Manzari’s model, 2004) during monotonic and cyclic loading. The drained, undrained, and cyclic triaxial tests were simulated using Dafalias and Manzari’s model. The simulation results showed that the model predicts behavior of sand, reasonably well. Also, for CSR?<?0.2, number of cycles for liquefaction is significantly increased. The residual strength of Babolsar sand is produced when it is deformed to an axial strain of 20 to 25%.     

Constitutive model for cyclic behaviour of cohesionless sands

This paper presents a constitutive model for describing the stress-strain response of sands under cyclic loading. The model, formulated using the critical state theory within the bounding surface plasticity framework, is an upgraded version of an existing model developed for monotonic behaviour of cohesionless sands. With modification of the hardening law, plastic volumetric strain increment and unloading plastic modulus, the original model was modified to simulate cyclic loading. The proposed model was validated against triaxial cyclic loading tests for Fuji River sand, Toyoura sand and Nigata sand. Comparison between the measured and predicted results suggests that the proposed modified model can capture the main features of cohesionless sands under drained and undrained cyclic loading.     

A macro‐element for a circular foundation to simulate 3D soil–structure interaction

This paper presents a non‐linear interface element to compute soil–structure interaction (SSI) based on the macro‐element concept. The particularity of this approach lies in the fact that the foundation is supposed to be infinitely rigid and its movement is entirely described by a system of global variables (forces and displacements) defined in the foundation's centre. The non‐linear behaviour of the soil is reproduced using the classical theory of plasticity. Failure is described by the interaction diagram of the ultimate bearing capacity of the foundation under combined loads. The macro‐element is appropriate for modelling the cyclic or dynamic response of structures subjected to seismic action. More specifically, the element is able to simulate the behaviour of a circular rigid shallow foundation considering the plasticity of the soil under monotonic static or cyclic loading applied in three directions. It is implemented into FedeasLab, a finite element Matlab toolbox. Comparisons with experimental monotonic static and cyclic results show the good performance of the approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Mathematical modelling of monotonic and cyclic undrained clay behaviour

The proposed general analytical model describes the anisotropic, elasto-plastic, path-dependent, stress-strain-strength properties of inviscid saturated clays under undrained loading conditions. The model combines properties of isotropic and kinematic plasticity by introducing the concept of a field of plastic moduli which is defined in stress space by the relative configuration of yield surfaces. For any loading (or unloading) history, the instantaneous configuration is determined by calculating the translation and contraction (or expansion) of each yield surface. The stress-strain behaviour of clays can thus be determined for complex loading paths and in particular for cyclic loadings. The stress-strain relationships are provided for use in finite element analyses. The model parameters required to characterize the behaviour of any given clay can be derived entirely from conventional triaxial or simple shear soil test results. The model's extreme versatility is demonstrated by using it to formulate the behaviour of the Drammen clay under both monotonic and cyclic loading conditions. The parameters are determined by using solely the results from monotonic and cyclic strain-controlled simple shear experimental tests, and the model's accuracy is evaluated by applying it to predict the results of other tests such as (1) cyclic stress-controlled simple shear tests, (2) monotonic triaxial loading compression and unloading extension tests, and (3) cyclic stress- and strain-controlled triaxial tests on, this same clay. The theoretical predictions are found to agree extremely well with the experimental test results.     

On hypoelasticity and plasticity approaches to constitutive modelling of inelastic behaviour of soils

A comparative discussion of constitutive rate equations generated by using the hypoelasticity and plasticity theories is presented. If is shown that these two approaches are equivalent for the loading process but differ essentially in defining the unloading (or reverse loading) paths. The multisurface plasticity formulation proves to be more convenient in incorporating memory rules of particular loading events, whereas loading-unloading criteria used in hypoelasticity may result in an unrealistic prediction of material response for some loading paths, and in particular, for cyclic loading. The violation of the continuity condition between loading–unloading domains in proposed constitutive relations^3,4 limits their applicability to some particular classes of loading paths.     

Non‐isothermal plasticity model for cyclic behaviour of soils

On the one hand, it has been observed that liquefaction‐induced shear deformation of soils accumulates in a cycle‐by‐cycle pattern. On the other hand, it is known that heating could induce plastic hardening. This study deals with the constitutive modelling of the effect that heat may have on the cyclic mechanical properties of cohesive soils, a relatively new area of interest in soil mechanics. In this paper, after a presentation of the thermo‐mechanical framework, a non‐isothermal plasticity cyclic model formulation is presented and discussed. The model calibration is described based on data from laboratory sample tests. It includes numerical simulations of triaxial shear tests at various constant temperatures. Then, the model predictions are compared with experimental results and discussed in the final section. Both drained and undrained loading conditions are considered. The proposed constitutive model shows good ability to capture the characteristic features of behaviour. Copyright © 2007 John Wiley & Sons, Ltd.     

A two-surface plasticity model for cyclic behavior of saturated clay

This paper presents a two-surface plasticity model for describing some important features of saturated clay under cyclic loading conditions, such as closed hysteresis loops, cyclic shakedown and degradation, and different stress–strain relations for two-way loading. The model, namely ACC-2-C, is based on the elastoplastic model ACC-2 (an adapted Modified Cam Clay model with two yield surfaces) developed by Hong et al. (Acta Geotech 11(4):871–885, 2015). The small-strain nonlinearity concept is adopted to achieve the nonlinear characteristics of clay during unloading–loading stage. The new hardening law related to accumulated deviatoric plastic strain is proposed for the inner surface to describe the cyclic shakedown and degradation. Following the advantages of the ACC-2 model, the constitutive equations are simply formulated based on the consistency condition for the inner yield surface. The model is conveniently implemented in a finite element code using a stress integration scheme similar to the Modified Cam Clay model. The simulation results are highly consistent with experimental data from drained and undrained isotropic cyclic triaxial tests in normally consolidated saturated clay under both one-way and two-way loadings.

     

A two-surface plasticity model for stiff clay

This paper presents a constitutive model for describing some important features of the behavior of natural stiff clay evidenced experimentally such as the limited elastic zone, the presence of strain hardening and softening, and the smooth transition from elastic behavior to a plastic one. The model, namely ACC-2, is an adapted Modified Cam Clay model with two yield surfaces: similarly to bounding surface plasticity theory, an additional yield surface—namely Inner yield surface—was adopted to account for the plastic behavior inside the conventional yield surface. A progressive plastic hardening mechanism was introduced with a combined volumetric-deviatoric hardening law associated with the Inner yield surface, enabling the plastic modulus to vary smoothly during loading paths. The main feature of the proposed model is that its constitutive equations can be simply formulated based on the consistency condition for the Inner yield surface, so that it can be efficiently implemented in a finite element code using a stress integration scheme similar to that of the Modified Cam Clay model. Furthermore, it is proved to be an appropriate model for natural stiff clay: the simulations of a set of tests along different mechanical loading paths on natural Boom Clay show good agreement with the experimental results.     

Development in modeling cyclic loading of sands based on kinematic hardening

In this paper, there is presented an elastoplastic constitutive model to predict sandy soils behavior under monotonic and cyclic loadings. This model is based on an existing model (Cambou‐Jafari‐Sidoroff) that takes into account deviatoric and isotropic mechanisms of plasticity. The flow rule used in the deviatoric mechanism is non‐associated and a mixed hardening law controls the evolution of the yield surface. In this research the critical state surface and history surface, which separates the virgin and cyclic states in stress space, are defined. Kinematic hardening modulus and stress–dilatancy law for monotonic and cyclic loadings are effectively modified. With taking hardening modulus as a function of deviatoric and volumetric plastic strain and with defining the history surface and stress reversal, the model has the ability to predict the sandy soils' behavior. All of the model parameters have clear physical meanings and can be determined from usual laboratory tests. In order to validate the model, the results of homogeneous tests on Hostun and Toyoura sands are used. The results of validation show a good capability of the proposed model. Copyright © 2009 John Wiley & Sons, Ltd.     

An elastoplastic model with combined isotropic–kinematic hardening to predict the cyclic behavior of stiff clays

This paper presents a kinematic hardening model for describing some important features of natural stiff clays under cyclic loading conditions, such as closed hysteretic loops, smooth transition from the elastic behavior to the elastoplastic one and changes of the compression slope with loading/unloading loops. The model includes two yield surfaces, an inner surface and a bounding surface. A non-associated flow rule and a kinematic hardening law are proposed for the inner surface. The adopted hardening law enables the plastic modulus to vary smoothly when the kinematic yield surface approaches the bounding surface and ensures at the same time the non-intersection of the two yield surfaces. Furthermore, the first loading, unloading, and reloading stages are treated differently by applying distinct hardening parameters. The main feature of the model is that its constitutive equations can be simply formulated based on the consistency condition for the inner yield surface based on the proposed kinematic hardening law; thereby, this model can be easily implemented in a finite element code using a classic stress integration scheme as for the modified Cam Clay model. The simulation results on the Boom Clay, natural stiff clay, have revealed the relevance of the model: a good agreement has been obtained between simulations and the experimental results from the tests with different stress paths under cyclic loading conditions. In particular, the model can satisfactorily describe the complex case of oedometric conditions where the deviator stress is positive upon loading (compression) but can become negative upon unloading (extension).     

Constitutive modeling of dense gravelly soils subjected to cyclic loading

A constitutive model for dense gravelly soils was developed to reproduce their responses under cyclic loadings. Its application aims at nonlinear dynamic analyses of earth structures involving gravelly soils, such as rockfill dams and railroad ballasts. The framework of generalized plasticity was modified to incorporate the concept of stress distance for better simulation of unloading and reloading responses. It was then combined with the theory of critical state soil mechanics to develop the constitutive model. The model has the following important features: unified simulation of particle breakage through translating critical state line, smooth transition from unloading to reloading in the stress space, and proper modeling of cyclic hysteresis, cyclic densification, and cyclic hardening of dense gravelly soils. Most of the model parameters can be obtained through simple calculation using conventional triaxial test results, and their calibration process was discussed. The model was used to simulate the cyclic responses of three gravelly soils with satisfactory accuracy. Copyright © 2014 John Wiley & Sons, Ltd.