A distributional approach to the geometry of dislocations at the mesoscale

We develop a theory to represent dislocated single crystals at the mesoscopic scale by considering concentrated effects, governed by the distribution theory combined with multiple-valued kinematic fields. Our approach gives a new understanding of the continuum theory of defects as developed by Kroener (1980) and other authors. Fundamental 2D identities relating the incompatibility tensor to the Frank and Burgers vectors are proved under global strain assumptions relying on the geometric measure theory, thereby giving rise to rigorous homogenisation from mesoscopic to macroscopic scale.Comment: article soumi

Damage-driven fracture with low-order potentials: asymptotic behavior, existence and applications

We study the $\Gamma$-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinematical constraints such as crack non-interpenetration. Existence results are also addressedComment: 41 pages, 4 Figure

Topological sensitivity analysis for elliptic differential operators of order 2m

Abstract The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m ⩾ 1 . The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders

A Temperature-Dependent Damage Model for Ductile Crack Initiation and Propagation with Finite Strains

Temperature-dependent ductile fracture of solids and shell

Damage and fracture evolution in brittle materials by shape optimization methods

International audienceThis paper is devoted to a numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation , including kinking and branching. Several numerical examples in 2d and 3d are discussed

Constrained ALE-Based Discrete Fracture in Shells with Quasi-Brittle and Ductile Materials

Shell fracture using Arbitrary Lagrangian-Eulerian method