Jean-François Babadjian

Quasi-static Evolution in Nonassociative Plasticity: The Cap Model

Non-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollication of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.

3D–2D analysis of a thin film with periodic microstructure

The purpose of this article is to study the behavior of a heterogeneous thin film whose microstructure oscillates on a scale that is comparable to that of the thickness of the domain. The argument is based on a 3D-2D dimensional reduction through a

A variational approach to the local character of

\Gamma

G

-convergence analysis, techniques of two-scale convergence and a decoupling procedure between the oscillating variable and the in-plane variable.

-closure : the convex case

Abstract This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a G-closure problem. Under convexity and p-growth conditions ( p > 1 ), it is proved that all such possible effective energy densities obtained by a Γ-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.

Approximation of dynamic and quasi-static evolution problems in elasto-plasticity by cap models

This work is devoted to the analysis of elasto-plasticity models arising in soil mechanics. Contrary to the typical models mainly used for metals, it is required here to take into account plastic dilatancy due to the sensitivity of granular materials to hydrostatic pressure. The yield criterion thus depends on the mean stress and the elasticity domain is unbounded and not invariant in the direction of hydrostatic matrices. In the mechanical literature, so-called cap models have been introduced, where the elasticity domain is cut in the direction of hydrostatic stresses by means of a strain-hardening yield surface, called a cap. The purpose of this article is to study the well-posedness of plasticity models with unbounded elasticity sets in dynamical and quasi-static regimes. An asymptotic analysis as the cap is moved to in nity is also performed, which enables one to recover solutions to the uncapped model of perfect elasto-plasticity.

Enhancement of Electromagnetic Fields Caused by Interacting Subwavelength Cavities

This article is devoted to the asymptotic analysis of the electromagnetic elds scattered by a perfectly conducting plane containing two sub-wavelength rectangular cavities. The problem is formulated through an integral equation, and a spectral analysis of the integral operator is performed. Using the generalized Rouche theorem on operator valued functions, it is possible to localize two types of resonances, symmetric and anti-symmetric, in a neighborhood of each zero of some explicit function, associated to the limiting geometry. For the symmetric modes, the elds in the cavities interact in phase, and the system of two cavities essentially acts as a dipole. In the anti-symmetric case, the fields oscillate in anti-phase, and the system behaves like a quadripole. Asymptotic expansions of the resonances, the far-eld and the near-eld are given.

Hyperbolic structure for a simplified model of dynamical perfect plasticity

This paper is devoted to confronting two different approaches to the problem of dynamical perfect plasticity. Interpreting this model as a constrained boundary value Friedrichs’ system enables one to derive admissible hyperbolic boundary conditions. Using variational methods, we show the well-posedness of this problem in a suitable weak measure theoretical setting. Thanks to the property of finite speed propagation, we establish a new regularity result for the solution in short time. Finally, we prove that this variational solution is actually a solution of the hyperbolic formulation in a suitable dissipative/entropic sense, and that a partial converse statement holds under an additional time regularity assumption for the dissipative solutions.

Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination

This work is devoted so show the appearance of different cracking modes in linearly elastic thin film systems by means of an asymptotic analysis as the thickness tends to zero. By superposing two thin plates, and upon suitable scaling law assumptions on the elasticity and fracture parameters, it is proven that either debonding or transverse cracks can emerge in the limit. A model coupling debonding, transverse cracks and delamination is also discussed.

Stability of Quasi-Static Crack Evolution through Dimensional Reduction

This paper deals with quasi-static crack growth in thin films. We show that, when the thickness of the film tends to zero, any three-dimensional quasi-static crack evolution converges to a two-dimensional one, in a sense related to the Г - convergence of the associated total energy. We extend the prior analysis of [2] by adding conservative body and surface forces which allow us to remove the boundedness assumption on the deformation field.