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ORAL PRESENTATION
Session: 6 Coarse-graining and mesoscopic dynamics - some mathematical aspects
(scheduled: Wednesday, 14:50 )
Stress relaxation theories in the approximation of polyconvex elastodynamics by viscoelasticity
T. Tzavaras
Department of Mathematics, University of Maryland, USA
We consider a model of stress relaxation approximating the equations of elastodynamics in Lagrangian coordinates. Necessary and sufficient conditions are derived for the model to be equipped with a global free energy and to have positive entropy production. For convex equilibrium potentials, we prove a relative energy estimate that provides convergence to the equations of elastodynamics when the solutions are smooth. In the case of polyconvex elastodynamics the situation is delicate because it is known from general theory of relaxation processes that the approximating system cannot have a convex energy as well. On the other hand, the equations of polyconvex elastodynamics can be embedded to an augmented symmetric hyperbolic system. We devise a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The model is endowed with an entropy function which is not convex but rather of polyconvex type. Using relative entropy ideas we prove a stability estimate and convergence of the stress relaxation model to polyconvex elastodynamics in the smooth regime. © IWNET 2006
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and Kleanthi for IWNET 2006