This work concerns the simulation of viscoelastic behavior of polymer at different states. Viscoelastic modeling of polymer was performed from the solid state to the liquid state via a multiphase approach which is largely used to deal with the fluid structure interaction. To ensure the appreciation of the FSI, viscoelasticity is considered in two parts: an elastic one and viscous other where the main idea is to use a mixed formulation in three fields (u, v, p) (displacement, velocity, pressure), with u and v, represented the primary variables of a strain and a strain rate formulation. We are led to the Navier-Stokes compressible problem with extra-stress, which is solved by using the Mixed Finite Element. The present work contributes some stabilization elements to the numerical simulation of multiphase problem by the monolithic approach.Comparison between the literature and experiments was performed through the validation of an elastic case and the viscoelastic Kelvin-Voigt model in the context of Lagrangian framework as well as Eulerian framework. The extension of the methodology to a visco-hyper-elastic is given through the modeling and validation on material point on the finite elements library CimLib®. Finally, a stabilization scheme of the EVSS type is adopted for viscoelastic Kelvin-Voigt model, hyper-elastic Neo-Hookean model, and also visco-hyper-elastic model which proposed an open door in computational modeling, not only with viscoelasticity but also complex dynamic application.

Ce travail a été effectué dans le cadre du projet Rem3D® dans lequel participent plusieurs entreprises avec l'objectif de développer un logiciel d'injection en 3D par éléments finis. L'objectif est de développer une méthode numérique pour modéliser le comportement viscoélastique des polymères de l'état solide à l'état liquide à travers une approche multiphasique qui est largement utilisé pour traiter le problème de l'interaction fluide-structure (IFS). La philosophie est d'utiliser une formulation mixte de trois champs (u, v, p) (déplacement, vitesse, pression), où u et v représentent les principales variables de déformation et de vitesse de déformation. Nous sommes amenés au problème de Navier-Stokes compressibles avec l'extra-contrainte, qui est résolu en utilisant la méthode des éléments finis mixte. Le présent travail contribue aussi certains éléments de stabilisation pour la simulation numérique des problèmes multiphasiques par l'approche monolithique.Comparaison entre la littérature et l'expérience est accompli par la validation du cas élastique et cas modèle viscoélastique de Kelvin-Voigt dans le lagrangien approche ainsi qu'eulérien approche. L'extension de la méthodologie au modèle visco-hyper-élastique est débuté par la modélisation et la validation au point matériel, puis l'implémentation dans la bibliothèque des éléments finis CimLib®. Enfin, un schéma stabilisation de résolution du type EVSS est adopté pour le modèle viscoélastique de Kelvin-Voigt, le modèle visco-hyper-élastique de Néo-Hookean, et aussi le modèle visco-hyper-élastique qui propose une prometteuse porte ouverte dans la simulation et modélisation, non seulement pour la viscoélasticité, mais aussi pour les applications dynamique complexes. / This work concerns the simulation of viscoelastic behavior of polymer at different states. Viscoelastic modeling of polymer was performed from the solid state to the liquid state via a multiphase approach which is largely used to deal with the fluid structure interaction. To ensure the appreciation of the FSI, viscoelasticity is considered in two parts: an elastic one and viscous other where the main idea is to use a mixed formulation in three fields (u, v, p) (displacement, velocity, pressure), with u and v, represented the primary variables of a strain and a strain rate formulation. We are led to the Navier-Stokes compressible problem with extra-stress, which is solved by using the Mixed Finite Element. The present work contributes some stabilization elements to the numerical simulation of multiphase problem by the monolithic approach.Comparison between the literature and experiments was performed through the validation of an elastic case and the viscoelastic Kelvin-Voigt model in the context of Lagrangian framework as well as Eulerian framework. The extension of the methodology to a visco-hyper-elastic is given through the modeling and validation on material point on the finite elements library CimLib®. Finally, a stabilization scheme of the EVSS type is adopted for viscoelastic Kelvin-Voigt model, hyper-elastic Neo-Hookean model, and also visco-hyper-elastic model which proposed an open door in computational modeling, not only with viscoelasticity but also complex dynamic application.

Interaction of fluid with elastic tube is complicated issue studied by many scientific departments around the world. Object of this thesis is to analyze simplified one-dimensional model. At the beginning, used balance equations and basics of hyper-elasticity are presented. Then we review three most common materials used for the description of blood vessels and other soft tissues. For these materials we introduce a method which we use to derive a relation between tube deformation and transmural pressure (i.e. difference between inner and outer pressure). In mathematical section we give brief review of theory of nonlinear hyperbolic equations and some relatively new results in the field of existence and uniqueness of a solution of one-dimensional hyperbolic system. The "building stone" of these results is a solution of the so-called Riemann problem. We use a method for finding exact solutions to the Riemann problem to analyze studied model of fluid-tube interaction and study dependence of the qualitative behavior of the solution on the material properties of the tube wall.