Implementation algorithm for a single hardening constitutive model for frictional materials

An advanced elasto‐plastic constitutive model for frictional materials, whose incremental version is presented in a companion paper (Int. J. Numer. Anal. Meth. Geomech., 2002; 26 :647), is implemented in a user‐defined material module. The general calculation strategy inside this module is presented and discussed, including the initial intersection of the yield surface and the techniques for updating of stresses and hardening modulus. Several integration schemes are implemented in the module and their capabilities in relation to the advanced, three‐dimensional constitutive model are evaluated. The forward Euler, modified Euler, and Runge–Kutta–Dormand–Prince integration schemes are explained in detail, compared, and evaluated in view of error tolerances and computational efficiency. Copyright © 2002 John Wiley & Sons, Ltd.     

A constitutive model for granular materials with grain crushing and its application to a pyroclastic soil

A constitutive model for granular materials is developed within the framework of strain–hardening elastoplasticity, aiming at describing some of the macroscopic effects of the degradation processes associated with grain crushing. The central assumption of the paper is that, upon loading, the frictional properties of the material are modified as a consequence of the changes in grain size distribution. The effects of these irreversible microscopic processes are described macroscopically as accumulated plastic strain. Plastic strain drives the evolution of internal variables which model phenomenologically the changes of mechanical properties induced by grain crushing by controlling the geometry of the yield locus and the direction of plastic flow. An application of the model to Pozzolana Nera is presented. The stress–dilatancy relationship observed for this material is used as a guidance for the formulation of hardening laws. One of the salient features of the proposed model is its capability of reproducing the stress–dilatancy behaviour observed in Pozzolana Nera, for which the minimum value of dilatancy always follows the maximum stress ratio experienced by the material. Copyright © 2002 John Wiley & Sons, Ltd.     

Explicit solutions for the instantaneous undrained contraction of hollow cylinders and spheres in porous elastoplastic medium

In this article we present closed‐form solutions for the undrained variations in stress, pore pressure, deformation and displacement inside hollow cylinders and hollow spheres subjected to uniform mechanical pressure instantaneously applied to their external and internal boundary surfaces. The material is assumed to be a saturated porous medium obeying a Mohr–Coulomb model failure criterion, exhibiting dilatant plastic deformation according to a non‐associated flow rule which accounts for isotropically strain hardening or softening. The instantaneous response of a porous medium submitted to an instantaneous loading is undrained, i.e. without any fluid mass exchange. The short‐term equilibrium problem to be solved is now formally identical to a problem of elastoplasticity where the constitutive equations involve the undrained elastic moduli and particular equivalent plastic parameters. The response of the model is presented (i) for extension and compression undrained triaxial tests, and (ii) for unloading problems of hollow cylinders and spheres through the use of appropriately developed closed‐form solutions. Numerical results are presented for a plastic clay stone with strain hardening and an argilite with strain softening. The effects of plastic dilation, of the strain softening law and also of geometry of the cavity on the behaviour of the porous medium have been underlined. Analytical solutions provide valuable benchmarks enabling various numerical methods in undrained conditions with a finite boundary to be verified. Copyright © 2002 John Wiley & Sons, Ltd.     

A note on formulas for localized failure of frictional materials in compression and biaxial loading modes

The paper investigates aspects of the localization analysis of frictional materials. We derive closed formulas and diagrams for the inclination angle of critical discontinuity surfaces which develop in homogeneous compression and biaxial loading tests. The localization analysis is based on a Drucker–Prager‐type elastoplastic hardening model for non‐associated plastic flow at small strains, which we represent in spectral form. For this type of constitutive model, general analytical formulas for the so‐called critical hardening modulus and the inclination angle of critical discontinuity surfaces are derived for the plane strain case. The subsequent treatment then specializes these formulas for the analysis of compression and biaxial loading modes. The key contribution here is a detailed analysis of plane strain deformation modes where the localized failure occurs after subsequent plastic flow. The derived formulas and diagrams can be applied to the checking of an accompanying localization analysis of frictional materials in finite‐element computations. Copyright © 2001 John Wiley & Sons, Ltd.     

A discrete thermodynamic approach for anisotropic plastic–damage modeling of cohesive‐frictional geomaterials

A discrete plastic–damage model is developed for cohesive‐frictional geomaterials subjected to compression‐dominated stresses. Macroscopic plastic strains of material are physically generated by frictional sliding along weakness planes. The evolution of damage is related to the evolution of weakness planes physically in connection with the propagation of microcracks. A discrete approach is used to account for anisotropic plastic flow and damage evolution, by introducing two stress invariants and one plastic hardening variable for each family of sliding weakness planes. Plastic flow in each family is coupled with damage evolution. The proposed model is applied to typical geomaterials and comparisons between numerical predictions and experimental data are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

A hierarchical approach for constitutive modelling of geologic materials

A hierarchical concept is proposed for the development of constitutive models to account for various factors that influence behaviour of (geologic) materials. It permits evolution of models of progressively higher grades from the basic model representing isotropic hardening with associative behaviour. Factors such as non-associativeness and induced anisotropy due to friction and cyclic loading, and softening are introduced as corrections or perturbations to the basic model. The influence of these factors is captured through non-associativeness manifested by deviation from normality of the plastic strain increments to the yield surface, F. Details of four models: isotropic hardening with associative behaviour, isotropic hardening with non-associative behavioural anisotropic hardening and strain-softening with a damage variable are presented. They are verified with respect to laboratory multiaxial test data under various paths of loading, unloading and reloading for typical soils, rock and concrete. The proposed concept is general, yet sufficiently simplified in terms of physical understanding, number of constants and their physical meanings, determination of the constants and implementation.     

A theoretical framework for constructing elastic/plastic constitutive models of triaxial tests

Modern ideas of thermomechanics are used to develop families of models describing the elastic/plastic behaviour of cohesionless soils deforming under triaxial conditions. Once the form of the free energy and dissipation potential functions have been specified, the corresponding yield loci, flow rules, isotropic and kinematic hardening rules as well as the elasticity law are deduced in a systematic manner. The families contain the classical linear frictional (Coulomb type) models and the classical critical state models as special cases. The generalized models discussed here include non‐associated flow rules, shear as well as volumetric hardening, anisotropic responses and rotational yield loci. The various parameters needed to describe the models can be interpreted in terms of ratio of the plastic work, which is dissipated, to that which is stored. Non‐associated behaviour is found to occur whenever this division between dissipated and stored work is not equal. Micro‐level interpretations of stored plastic work are discussed. The models automatically satisfy the laws of thermodynamics, and there is no need to invoke any stability postulates. Some classical forms of the peak‐strength/dilatancy relationship are established theoretically. Some representative drained and undrained paths are computed. Copyright © 2002 John Wiley & Sons, Ltd.     

Integration and calibration of a plasticity model for granular materials

A new macroscopic constitutive model for non‐cohesive granular materials, with the focus on coarse‐sized materials (railway ballast), is presented. The model is based on the concepts of rate‐independent isotropic plasticity. The Backward Euler rule is used for integrating the pertinent evolution equations. The resulting incremental relations are solved in the strain space that is extended with the internal (hardening) variables. The model is calibrated using data from Conventional Triaxial Compression (CTC) tests, carried out at the University of Colorado at Boulder. A function evaluation method is used for the optimization, whereby a ‘multi‐vector’ strategy for choosing the appropriate start vector is proposed. Copyright © 2002 John Wiley & Sons, Ltd.     

Stress‐dependent hardening and failure surfaces of dry sand

During several triaxial compression experiments on plastic hardening, softening, and failure properties of dense sand specimens, it was found on various stress paths that the size of the failure surface was not constant. Instead, it changed depending on the current state of hydrostatic pressure. This finding is in contrast to the standard opinion consisting of the fact that the failure surface remains constant, once it has been reached during an experiment or in situ. In general, the behaviour of cohesionless granular‐material‐like sand is somehow characterised in between fluid and solid, where the solid behaviour results from the angle of internal friction and the confining pressure. Although the friction angle is an intrinsic material property, the confining pressure varies with the boundary conditions, thus defining different solid properties like plastic hardening, softening, and also failure. Based on our findings, it was the goal of the present contribution to introduce an improved setting for the plastic strain hardening and softening behaviour including the newly found yield properties at the limit state. For the identification of the material parameters, a complete triaxial experimental analysis of the tested sand is given. The overall elasto‐plasticity concept is validated by numerical computations of several laboratory foundation‐ and slope‐failure experiments. The performance of the proposed approach is compared with the standard concept of a constant failure surface, where the corresponding yield surfaces are understood as contours of equivalent plastic work or plastic strain. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite deformation analysis of geomaterials

The mathematical structure and numerical analysis of classical small deformation elasto–plasticity is generally well established. However, development of large deformation elastic–plastic numerical formulation for dilatant, pressure sensitive material models is still a research area. In this paper we present development of the finite element formulation and implementation for large deformation, elastic–plastic analysis of geomaterials. Our developments are based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A consistent linearization of the right deformation tensor together with the Newton method at the constitutive and global levels leads toward an efficient and robust numerical algorithm. The presented numerical formulation is capable of accurately modelling dilatant, pressure sensitive isotropic and anisotropic geomaterials subjected to large deformations. In particular, the formulation is capable of simulating the behaviour of geomaterials in which eigentriads of stress and strain do not coincide during the loading process. The algorithm is tested in conjunction with the novel hyperelasto–plastic model termed the B material model, which is a single surface (single yield surface, affine single ultimate surface and affine single potential surface) model for dilatant, pressure sensitive, hardening and softening geomaterials. It is specifically developed to model large deformation hyperelasto–plastic problems in geomechanics. We present an application of this formulation to numerical analysis of low confinement tests on cohesionless granular soil specimens recently performed in a SPACEHAB module aboard the Space Shuttle during the STS‐89 mission. We compare numerical modelling with test results and show the significance of added confinement by the thin hyperelastic latex membrane undergoing large stretching. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical simulations of simple shear with non‐coaxial soil models

This paper investigates the effects of a non‐coaxial model on simulated stress–strain behaviour of granular materials subject to simple shearing under various initial conditions. In most cases, a significant difference of predictions between coaxial and non‐coaxial modelling is found during the early stage in shearing. With the increase in shearing, non‐coaxial simulations approach and tend to coincide with coaxial simulations. It is also found that the roles of non‐coaxial modelling in simulating simple shear behaviour are considerably influenced by hardening rules, flow rules, initial static lateral pressure coefficients. In some cases, the non‐coaxial modelling gives a similar simulation as the coaxial modelling. In other cases, the non‐coaxial modelling decreases the hardening response or softening response of materials, compared with the coaxial modelling. Under certain conditions, the predicted peak strength of materials with non‐coaxial modelling is larger than that for coaxial modelling. Some of these observations can be attributed to the amount of principal stress rotation in various cases analysed. Others can be attributed to the difference between the directions of the non‐coaxial plastic flow and those for coaxial plastic flow. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress–dilatancy based modelling of granular materials and extensions to soils with crushable grains

Stress–dilatancy relations have played a crucial role in the understanding of the mechanical behaviour of soils and in the development of realistic constitutive models for their response. Recent investigations on the mechanical behaviour of materials with crushable grains have called into question the validity of classical relations such as those used in critical state soil mechanics. In this paper, a method to construct thermodynamically consistent (isotropic, three‐invariant) elasto‐plastic models based on a given stress–dilatancy relation is discussed. Extensions to cover the case of granular materials with crushable grains are also presented, based on the interpretation of some classical model parameters (e.g. the stress ratio at critical state) as internal variables that evolve according to suitable hardening laws. Copyright © 2004 John Wiley & Sons, Ltd.     

Approximate model for blunt objects indenting cohesive‐frictional materials

An approximate two‐dimensional model for indentation of blunt objects into various types of rigid‐perfectly plastic cohesive‐frictional material is derived. Particular emphasis is placed on considering indentation as a process involving evolution of the boundary of material displaced by the indenter. Force–penetration relationships are obtained by an incremental approach utilizing key kinematic and static information from indentation of a flat punch. Albeit approximate, the proposed model applies to arbitrary indenter geometry and weightless or ponderable cohesive‐frictional materials exhibiting associated or non‐associated plastic flow. Two specific indenter geometries, the cylinder and blunt wedge, are explored in detail. Favorable agreement is found between the analytic results and those obtained using the finite element method (FEM). For both the wedge and cylinder, it is further shown that accurate analytic expressions relating indentation force explicitly to penetration can be derived. In the case of the wedge and weightless material, predictions of indentation force obtained from the derived expressions are very close to those computed from implicit equations available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Implicit integration of a mixed isotropic–kinematic hardening plasticity model for structured clays

In recent years, a number of constitutive models have been proposed to describe mathematically the mechanical response of natural clays. Some of these models are characterized by complex formulations, often leading to non‐trivial problems in their numerical integration in finite elements codes. The paper describes a fully implicit stress‐point algorithm for the numerical integration of a single‐surface mixed isotropic–kinematic hardening plasticity model for structured clays. The formulation of the model stems from a compromise between its capability of reproducing the larger number of features characterizing the behaviour of structured clays and the possibility of developing a robust integration algorithm for its implementation in a finite elements code. The model is characterized by an ellipsoid‐shaped yield function, inside which a stress‐dependent reversible stiffness is accounted for by a non‐linear hyperelastic formulation. The isotropic part of the hardening law extends the standard Cam‐Clay one to include plastic strain‐driven softening due to bond degradation, while the kinematic hardening part controls the evolution of the position of the yield surface in the stress space. The proposed algorithm allows the consistent linearization of the constitutive equations guaranteeing the quadratic rate of asymptotic convergence in the global‐level Newton–Raphson iterative procedure. The accuracy and the convergence properties of the proposed algorithm are evaluated with reference to the numerical simulations of single element tests and the analysis of a typical geotechnical boundary value problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Creep of saturated materials as a chemically enhanced rate‐dependent damage process

Material behaviour that exhibits characteristics of creep induced by a spontaneous mineral dissolution enhanced by material damage is studied. It is believed that the characteristic rates of the chemical processes involved determine the time‐rate dependence of the resulting strain. A basic model of a combined chemo‐plastic softening and chemically enhanced deviatoric strain hardening for saturated geomaterials is presented. Chemical softening is postulated to occur as a consequence of the net mass removal resulting from dissolution and precipitation of specific minerals occurring at the damage‐generated inter‐phase interfaces. Closed and open systems are discussed. In the former case, deformation at constant stress results entirely from a local compensation mechanism between the chemical softening and strain hardening. The classical three stages of creep are interpreted in terms of mechanisms of dissolution and precipitation, as well as the variation in the reaction surface areas involved in the mass exchange. In an open system, the above local mechanism is enhanced by the removal of mass via diffusion of species affecting the mass balance. Such a system is addressed via a boundary value problem as shown in an example. Copyright © 2007 John Wiley & Sons, Ltd.     

Kinematic hardening plasticity formulation of small strain behaviour of soils

An objective of this paper is to demonstrate that the small strain model developed by the authors can be incorporated into the conventional kinematic hardening plasticity framework to predict pre‐failure defor mations. The constitutive model described in this paper is constituted by three elliptical yield surfaces in triaxial stress space. Two inner surfaces are rotated ellipses of the same shape, representing the boundaries of the linear elastic and small strain regions, while the third surface is the modified Cam clay large‐scale yield surface. Within the linear elastic region, the soil behaviour is elastic with cross‐coupling between the shear and volumetric stress–strain components. Within the small strain region, the soil behaviour is elasto‐plastic, described by the kinematic hardening rule with an infinite number of loading surfaces defined by the incremental energy criterion. Within the large‐scale yield surface, the soil behaviour is elasto‐plastic, described by kinematic and isotropic hardening of the small strain region boundary. Since the yield surfaces have different shapes, the uniqueness of the plastic loading condition imposes a restriction on the ratio between their semi‐diameters. The model requires 12 parameters, which can be determined from a single consolidated undrained triaxial compression test. Copyright © 2000 John Wiley & Sons, Ltd.     

Elasto‐plastic model for cement‐treated sand

This paper presents an elasto‐plastic model for non‐linear analyses of cement‐treated sand. Various laboratory tests were systematically carried out to investigate the pre‐peak and post‐peak behaviours of a cement‐treated sand. On the basis of these experimental results, the new model was built within the framework of a relatively simple elasto‐plastic theory. Two failure criteria are employed to express tensile and shear failure characteristics observed in the experimental results of the cement‐treated sand. The proposed model can describe strain‐hardening and strain‐softening responses under both failure modes. In the strain‐softening rules, the smeared crack concept is used, and a characteristic length is considered to avoid the issue of mesh‐size dependency. Since the failure criterion and strain‐hardening/softening rules are based on the experimental evidences, the model is relatively easy to understand and the parameters used in the model have clear physical meaning. The proposed model was applied to simulate the behaviour of cement‐treated sand in various laboratory tests, allowing for a reasonable comprehensive evaluation. It was demonstrated that the proposed model is suitable for describing both the tensile and shear failure behaviours of cement‐treated sand. Copyright © 2006 John Wiley & Sons, Ltd.