Modeling of smart materials with thermal effects: dynamic and quasi-static evolution

International audienceWe present a mathematical model for linear magneto-electro-thermo-elastic continua, as sensors and actuators can be thought of, and prove the well-posedness of the dynamic and quasi-static problems. The two proofs are accomplished, respectively, by means of the Hille-Yosida theory and of the Faedo-Galerkin method. A validation of the quasi-static hypothesis is provided by a nondimensionalization of the dynamic problem equations. We also hint at the study of the convergence of the solution to the dynamic problem to that to the quasi-static problem as a small parameter – the ratio of the largest propagation speed for an elastic wave in the body to the speed of light – tends to zero

An asymptotic strain gradient Reissner-Mindlin plate model

In this paper we derive a strain gradient plate model from the three-dimensional equations of strain gradient linearized elasticity. The deduction is based on the asymptotic analysis with respect of a small real parameter being the thickness of the elastic body we consider. The body is constituted by a second gradient isotropic linearly elastic material. The obtained model is recognized as a strain gradient Reissner-Mindlin plate model. We also provide a mathematical justification of the obtained plate model by means of a variational weak convergence result

A new duality approach to elasticity

International audienceThe displacement-traction problem of three-dimensional linearized elasticity can be posed as three different minimization problems, depending on whether the displacement vector field, or the stress tensor field, or the strain tensor field, is the unknown. The objective of this paper is to put these three different formulations of the same problem in a new perspective, by means of Legendre-Fenchel duality theory. More specifically, we show that both the displacement and strain formulations can be viewed as Legendre-Fenchel dual problems to the stress formulation. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new duality approach to elasticity

Numerical validation of an Homogenized Interface Model

International audienceThe aim of this paper is to numerically validate the effectiveness of a matched asymptotic expansion formal method introduced in a pioneering paper by Nguetseng and Sànchez Palencia [1] and extended in [2], [3]. Using this method a simplified model for the influence of small identical heterogeneities periodically distributed on an internal surface to the overall response of a linearly elastic body is derived. In order to validate this formal method a careful numerical study compares the solution obtained by a standard method on a fine mesh to the one obtained by asymptotic expansion. We compute both the zero and the first order terms in the expansion. To efficiently compute the first order term we introduce a suitable domain decomposition method

Asymptotic expansions and domain decomposition

International audienceWe apply the domain decomposition method to linear elasticity problems for multi-materials where the heterogeneities are concentrated in a thin internal layer. In the first case the heterogeneities are small, identical and periodically distributed on an internal surface and in the second one all the thin, curved internal layer is made of an elastic material much more strong than the surrounding one. In the first case the domain decomposition is used to efficiently solve the non-standard transmission problems obtained by the asymptotic expansion method. In the second case a non-standard membrane transmission problem originates from a surface shell like energy.La méthode de décomposition de domaine est appliquée a des problèmes d'élasticité linéaire avec des hétérogénéités. Dans un premier cas il s'agit d'une couche fine contenant des hétérogénéités réparties de façon périodique, dans un second cas d'une couche interne constitué d'un matériau avec un module de Young beaucoup plus grand que celui du matériau qui l'entoure. Dans le premier cas la décomposition de domaine est utilisée pour résoudre efficacement un problème avec des conditions de transmission non standard obtenu par développement asymptotique, Dans le deuxième cas, une condition de type Vencell apparai

Jonction forte entre deux solides : modèle simplifié et algorithme de résolution

Nous étudions un problème modèle non classique de transmission décrivant une multi-structure composée de deux solides reliés par une jonction forte. En utilisant une méthode de décomposition de domaines, le problème se ramène à une équation définie sur l'interface. Dans le cas thermique, cette équation est de la forme (I+G)g=F. On montre que les propriétés de G entraînent la convergence q-superlinéaire de l'algorithme GMRES

A geometrical deduction of the kinematics of some plate's models

Nous donnons une déduction purement géométrique des équations cinématiques de Kirchhoff-Love et de celles de Reissner-Mindlin dans le cas d'une plaque simplement connexe. Cette déduction est obtenue uniquement à partir des relations de compatibilité de Saint Venant et de la représentation intégrale de Cesàro-Volterra. Aucune information concernant le matériau constitutif de la plaque ou le chargement n'est utilisée