René Chambon

Plastic continuum with microstructure, Local Second Gradient Theories for Geomaterials : Localization Studies

Many plastic second gradient models have been developed in the last 10 years. However some plastic second gradient models are nonlocal ones. This paper is an attempt to give a general framework to deal with local second gradient theories within theories with microstructure, keeping in mind future applications for geomaterials. It is advocated that particular elasto-plastic local models with microstructure, namely local second gradient and Cosserat second gradient models, which are the least developed in the literature have some advantages which are somewhat promising. One main objective of this paper is to present these two families of models. The first one (Cosserat second gradient model) is shown to be well adapted to granular materials. The second model family is likely to be a good model for cohesive geomaterials. Another aim of this work is to give a method to obtain basic solutions, which can be seen as localization analysis, in one and two dimensions cases. The key point of this method is the use of patch conditions between loading and unloading parts.

Localization criteria for non‐linear constitutive equations of geomaterials

This paper is devoted to two specific aspects of shear band analysis for geomaterials, which are basically, non-standard materials. Problems arising from incremental non-linearity (multilinear or thoroughly non-linear models) and consequences of discontinuities of the constitutive equations with respect to loading parameters, are especially investigated. Copyright

Shear band analysis and shear moduli calibration

Strain localization is a well known phenomenon, generally associated with plastic deformation and rupture in solids, especially in geomaterials. In this process, deformation is observed to concentrate in narrow zones called shear bands. This phenomenon has been studied extensively in the last 20 years by different researchers, experimentally, theoretically and numerically. A criterion for the onset of localization can be predicted solely on the basis of the constitutive law of the material, using the so-called shear band analysis. This criterion gives the critical orientation, and the critical stress state and strain for a given loading history. An important point, already stressed by Vardoulakis in 1980, is that in particular, out-of-axes shear moduli play a central role in the criterion. These are the moduli involved in the response to a deviatoric stress increment with principal axes oriented at 45° from total stress principal axes. Out-of-axes shear moduli are difficult parameters to calibrate; common tests, with fixed principal stress and strain directions, do not provide any information on these moduli, as long as they remain homogeneous. Still, real civil engineering and environmental problems are definitely not simple axisymmetric triaxial tests; practical modeling involves complex stress paths, and need complex parameters to be calibrated. Only special tests, like compression–torsion on hollow cylinder tests, or even more complex tests can be used for shear moduli calibration. However, shear band initiation in homogeneous, fixed-axes tests does activate out-of-axes shear. Hence, it is natural that shear band analysis makes shear moduli enter into the analysis. Then, a typical inverse analysis approach can be used here: experimental observation of strain localization in triaxial tests can be used together with a proper shear band analysis for the model considered, in order to determine out-of-axes shear moduli. This approach has been used for a stiff marl in the framework of a calibration study on a set of triaxial tests. The steps of the method are presented, and the bifurcation surface in the stress space is exhibited.

A review of two different approaches to hypoplasticity

Non-linearity and irreversibility are striking features of soil behavior, affecting the response of any geotechnical “structure”, be it, for example, a foundation, an excavation, an earth dam, or a natural slope. From a mathematical viewpoint (i.e., at the constitutive level), different strategies have been proposed to deal with such features of soil behavior, including:

Loss of uniqueness and bifurcation vs instability: some remarks

ABSTRACT The present work is an attempt to clarify the different notions of bifurcation and loss of stability. Although these notions have been already discussed by many authors, we believe that they still deserve some discussion. In fact, many different “definitions” of stability exist. This is not a problem for conservative systems, for which it turns out that all these definitions do coincide. Nor is this a problem if the system incorporates some viscous dissipation. However, for other cases, e.g. non conservative systems incorporating dry friction dissipation, which are of relevance in geomechanics, this is very important, because different definitions may yield different results. We recall first the definition of Lyapunov stability, which is related to the dynamic study of the influence of initial conditions on the solution of the mechanical equations. In this framework, we have a look at the linearization of the equations and its consequences. Then we go back to Hills work and try to give the basis of the so-called “Hill stability criterion”. Finally, we “define” an other stability criterion which has been criticized by Hill in some early paper. Through the study of a very simple mechanical system, we exhibit some differences between all these notions, showing that a system can be stable according to one criterion but unstable according to other criteria. It is necessary to mention which definition of stability is used. Any existing stability criterion has to be thought of as an heuristic method. On the contrary, bifurcation or loss of uniqueness is a more clear concept. Shear band localization, controlability, and invertibility can be actually seen as particular cases of bifurcation. However, bifurcation criteria have to be considered carefully, since they almost always incorporate linearization, which in most cases is not strictly justified. Some examples are given of general results which have been obtained without the use of any linearization procedure.

Micro-fracture instabilities in granular solids

We study the appearance of instable behaviors (like strain localization bands) in elastic solids, as a consequence of micro-fracture. A two-scale approach of computational homogenization is considered. The macroscopic behavior is obtained by unit cell finite element computations. On the micro-level, we consider a granular structure with each grain made of a large strain elastic material. Inter-granular boundaries are modeled with cohesive laws, friction and unilateral contact. We show that decohesion between grains give rise to macro-instabilities, indicated by the loss of ellipticity, typical for deformation localization bands. The relation between the microscopic softening on inter-granular boundaries and the onset of macro-instabilities is pointed out on some numerical examples. The influence of the cohesive law and friction parameters is analyzed.

Numerical post failure methods in multiphysical problems

La rupture dans les geomateriaux est souvent precedee par la formation de fines bandes de localisation des deformations. La formation de ces bandes de localisation est un processus non negligeable, etudie a la fois sur le plan experimental et sur le plan theorique. Cet article resume les principaux phenomenes observes sur les processus localises et propose quelques outils theoriques et numeriques necessaires a la caracterisation de ces processus de localisation. Afin de tenir compte des interactions entre les differentes phases des milieux poreux, une technique de regularisation basee sur des modeles de type second gradient est etendue aux couplages mutiphysiques.

A finite deformation second gradient theory of plasticity

Extending the previous work by Chambon et al. [2] to the finite deformation regime, a local second gradient theory of plasticity for isotropic materials with microstructure is developed based on the multiplicative decomposition of the deformation gradient, the additive decomposition of the second deformation gradient and the principle of maximum dissipation.

Local Second Gradient Models and Damage Mechanics: 1D Post-Localization Studies in Concrete Specimens

Continuum damage mechanics is often used as a framework for describing the variations of the elastic properties of due to micro-structural degradations. Experimentally, concrete specimens exhibit a network of microscopic cracks that nucleate sub-parallel to the axis of loading. Due to the presence of heterogeneities in the material (aggregates surrounded by a cement matrix), tensile transverse strains generate a self-equilibrated stress field orthogonal to the loading direction, a pure mode I (extension) is thus considered to describe the behaviour even in compression. This rupture mode must be reproduced numerically. This is the reason why the failure criterion of the chosen constitutive law is expressed in terms of the principal extensions and that a tension test is modelled at the end of this paper. The influence of micro-cracking due to the external loads is introduced via damage variables, ranging from 0 for the undamaged material to 1 for a completely damaged material.

A second gradient elastoplastic cohesive-frictional model for geomaterials

Abstract Starting from some experimental observations on shear strength of stiff clays [3,2], a isotropic, geometrically non-linear second gradient elastoplastic model is proposed for pressure dependent, brittle geomaterials. The development of the model follows the general theory presented in [1]. Due to the internal length scale provided by the microstructure, the model is ideally suited for the analysis of failure problems in which strain localization into shear band occurs, see, e.g., [4,5].