Mohammed Hjiaj

A complete stress update algorithm for the non-associated Drucker-Prager model including treatment of the Apex

Abstract Numerical techniques based on convex analysis are applied to the non-associated Drucker–Prager model (without hardening) for which the plastic behavior is completely described by a unique function, called bi-potential. Among advantages of the present approach, motivated by mechanical considerations, a variational stress update algorithm along with coupled extremum principles can be derived. The time-integration algorithm is considered in detail and it is shown how the method can conveniently treat the singular point present in the Drucker–Prager model (apex). The existence of weak extremum principles allows using Mathematical Programming techniques and thereby obtains a robust algorithm even in the presence of large time increment and strong non-associativity. Numerical examples of incremental limit analysis for both the associated and the non-associated cases are presented.

A variational inequality-based formulation of the frictional contact law with a non-associated sliding rule

We present a variational formulation of a complex frictional contact law with anisotropic friction condition and a non-associated sliding rule. The distinguishing characteristic of the proposed formulation is that the interface law, as well as its inverse, derive from a single scalar-valued function called bi-potential. This function, which depends on both the velocities and the associated forces, is split into the sum of two dual pseudo-potentials for standard multivalued laws. The main advantages of the formalism are the compact form taken by the present complex law and convexity property of the bi-potential that can be exploited for numerical purposes.

A Posteriori Error Control of Finite Element Approximations for Coulomb's Frictional Contact

This paper is concerned with the frictional unilateral contact problem governed by Coulombs law. We define an a posteriori error estimator based on the concept of error in the constitutive relation to quantify the accuracy of a finite element approximation of the problem. We propose and study different mixed finite element approaches and discuss their properties in order to compute the estimator. The information given by the error estimates is then coupled with a mesh adaptivity technique which provides the user with the desired quality and minimizes the computation costs. The numerical implementation of the error estimator as well as optimized computations are performed.

Bifurcation analysis for shear localization in non-polar and micro-polar hypoplastic continua

In this paper, shear localization in granular materials is studied as a bifurcation problem based on a conventional (non-polar) and a micro-polar continuum description. General bifurcation conditions are formulated for a non-polar hypoplastic model and its micro-polar continuum extension. These conditions define stress, couple stress and density states at which weak discontinuity bifurcation may occur. The stress states for bifurcation are then compared with the peak stress states, which define a bounding surface for the accessible stress domain in the principal stress space. The results show that, in a micro-polar continuum description, the constitutive model may no longer be associated with weak discontinuity bifurcation.

Formulation of non-standard dissipative behavior of geomaterials

Abstract.In this paper, fundamental mathematical concepts for modeling the dissipative behavior of geomaterials are recalled. These concepts are illustrated on two basic models and applied to derive a new form of the evolution law of the modified Cam-clay model. The aim is to discuss the mathematical structure of the constitutive relationships and its consequences on the structural level. It is recalled that non-differentiable potentials provide an appropriate means of modeling rate-independent behavior. The Cam-clay model is revisited and a standard version is presented. It is seen that this standard version is non-dissipative, which at the same time explains why a non-standard version is needed. The partial normality is exploited and an implicit variational formulation of the modified Cam-clay model is derived. As a result, the solution of boundary-value problems can be replaced by seeking stationary points of a functional.

Generalized Legendre-Fenchel Transformation

The aim of the paper is to characterize transformations that preserve the potential structure of a relationship between dual variables. The first step consists in “geometrizing” the condition for the existence of a potential. Having at hand this formulation, it becomes clear that the canonical similitudes represents the class of transformations that preserves the potential form of a relationship. Next, we derive the conditions under which canonical similitudes preserve the convexity of the potential or change it into concavity. This new class of transformations provides a generalization of the Legendre-Fenchel transformation.

An Extension of Limit Analysis Theorems to Incompressible Material with a Non-Associated Flow Rule

The aim of the paper is to propose an extension of limit analysis theorems for a non-associated flow rule. The approach is to construct, as a first step, a variational formulation for boundary value problems involving rigid plastic materials with a non-associated flow rule. The method used here, proposed by the first author, is based on an extension of Auchmuty’s work on variational principles for non-potential operator equations. Having at our disposal the variational formulation, it is a simple matter to derive the bound theorems of limit analysis. It is found that for a non-associated flow rule the lower and the upper bound are coupled.

An a Posteriori Error Estimator for Finite Element Approximations of Coulomb’s Friction Problem

ABSTRACT This paper deals with an adaptative strategy for F.E. computations of the frictional unilateral contact problem governed by Coulomb’s law. This strategy uses an a posteriori error estimator which is based on constitutive relation residuals to quantify the accuracy of a finite element approximation of the problem. The numerical implementation of the error estimator and the procedure of adaptivity are carried out.