Makrem Arfaoui

Further Development in Thermodynamic Approach for Thermoviscoelastic Materials

In this paper we derive linear and nonlinear theories ofthermoviscoelasticity for thermorheologically simple materials. Thisderivation is based on two theories: the Thermodynamic of IrreversibleTheory and the Generalized Viscoelastic Standard Theory. These twotheories, along with other assumptions, lead to three-dimensionalthermoviscoelastic models. The derivation we give here is an extensionof Schaperys models. We will soon state the relations expressing thebalance of energy, i.e., the specific heat capacity and the heatconduction equation.

Numerical convergence and stability of mixed formulation with X-fem cut-off

In this paper, we are concerned with the mathematical and numerical analysis of convergence and stability of the mixed formulation for incompressible elasticity in cracked domains. The objective is to extend the extended finite element method (X-FEM) cut-off analysis done in the case of compressible elasticity to the incompressible one. A mathematical proof of the inf-sup condition of the discrete mixed formulation with X-FEM is established for some enriched fields. We also give a mathematical result of quasi-optimal error estimate. Finally, we validate these results with numerical tests. On s’interresse dans ce papier à l’analyse mathématique et numérique de la conver-gence et de la stabilité de la formulation mixte d’un problème d’élasticité incompressible dans un domaine fissuré. L’objectif est d’étendre l’étude faite sur la variante X-FEM cut-off, dans le cas de l’élasticité compressible, au comportement incompressible. Une preuve mathématique de la condition inf-sup de la formulation mixte discrète avec X-FEM est établie pour certains champs enrichis. Nous donnons également un résultat mathématique de la quasi-optimalité de l’estimation d’erreur. Enfin, nous validons ces résultats avec des tests numériques.