Gérard Michaille

Multi-materials with strong interface: Variational modelings

We introduce a simplified model for a multi-material made up of two elastic bodies connected by a strong thin material layer whose stiffness grows as 1/e. The model is obtained by identifying the Γ-limit of the stored strain energy functionalof the physical problem when the thickness e of the intermediate layer tends to zero. The intermediate layer behaves as a stiffening elastic membrane. Furthermore, in the linear anisotropic case, we establish the strong convergence of the exactsolution toward the solution of the limit problem.

Variational limit of a one-dimensional discrete and statistically homogeneous system of material points

Abstract The energy of a discrete system of material points situated at random on a line and subjected to nearest-neighbor interactions, almost surely converges in a variational sense to a deterministic energy defined on spaces of functions with bounded variation.

A MATHEMATICAL MODEL FOR A PSEUDO-PLASTIC WELDING JOINT

An elementary situation in welding involves the perfect assembly of two adherents and a strong adhesive occupying a thin layer. The bulk energy density of the hyperelastic adherents grows superlinearly while that of the pseudo-plastic adhesive grows linearly with a stiffness of the order of the inverse of its thickness e. We propose a simplified but accurate model by studying the asymptotic behavior, when e goes to zero, through variational convergence methods: at the limit, the intermediate layer is replaced by a pseudo-plastic interface which allows cracks to appear.

Young measures generated by a class of integrands: a narrow epicontinuity and applications to Homogenization

Abstract Given a subset E of convex functions from R N+k into R which satisfy growth conditions of order p>1 and an open bounded subset O = O × O ⊥ of R N+k , we establish the continuity of a map μ↦Φμ from the set of all Young measures on O ×E equipped with the narrow topology into a set of suitable functionals defined in L p ( O ,W 1,p 0 ( O ⊥ )) and equipped with the topology of Γ-convergence. Some applications are given in the setting of periodic and stochastic homogenization.

Analysis of the jump and approximate gradient for solutions of Mumford-Shah type energy functionals

The main objective of this paper is to introduce and illustrate a new tool stemming from Young measure theory in order to capture concentrations, jump sign and gradient oscillations of sequences of SBV-functions. We show how this notion of measure can be applied for the analysis of approximating solutions of Mumford-Shah type energy functionals in the one dimensional case.

Variational approximation of a constraint signal by a Mumford-Shah type energy functional

In this paper, the main objective is to establish an existence result of a variational model in image segmentation constrained by a given vector field. In the one dimensional case, we give a discrete version converging in a variational way to the continuous model. We finally describe the numerical analysis of this model with application in image segmentation.