The equations of polyconvex elastodynamics can be embedded to an augmented symmetric hyperbolic system. This property provides a stability framework between solutions of the viscosity approximation of polyconvex elastodynamics and smooth solutions of polyconvex elastodynamics. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approxima...

The equations of polyconvex elastodynamics can be embedded to an augmented symmetric hyperbolic system. This property provides a stability framework between solutions of the viscosity approximation of polyconvex elastodynamics and smooth solutions of polyconvex elastodynamics. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approxima...

We consider a model of stress relaxation approximating the equations of elastodynamics. Necessary and sufficient conditions are derived for the model to be equipped with a global free energy and to have positive entropy production, and the resulting class allows for both convex and non-convex equilibrium potentials. For convex equilibrium potentials, we prove a strong dissipation estimate and t...

We consider a variational scheme developed by S. Demoulini, D. M. A. Stuart and A. E. Tzavaras [Arch. Rat. Mech. Anal. 157 (2001)] that approximates the equations of three dimensional elastodynamics with polyconvex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elastodynamics before shock formation. The proof...

In the current presentation, we suggest a very simple and effective post-processing procedure to increase the order of accuracy in time for numerical results obtained at time integration of linear elastodynamics problems by the trapezoidal rule. This technique is based on a new exact, closed-form, a-priori error estimator for time integration of linear elastodynamics equations by the trapezoida...

The embedding of the equations of polyconvex elastodynamics to an augmented symmetric hyperbolic system provides in conjunction with the relative entropy method a robust stability framework for approximate solutions [18]. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The mode...

In this paper we consider classical and variational of fracture dynamic problems with taking into account a contact interaction of the crack edges. An algorithm for the solution of the elastodynamic contact problem with unilateral constrain has been developed. It arises from the boundary variational principle of elastodynamics for bodies with cracks and unilateral constrains on their sides. Som...

Existence and convergence results are proved for a regularized model of dynamic brittle fracture based on the Ambrosio–Tortorelli approximation. We show that the sequence of solutions to the time-discrete elastodynamics, proposed by Bourdin, Larsen & Richardson as a semidiscrete numerical model for dynamic fracture, converges, as the time-step approaches zero, to a solution of the natural time-...

In this paper we extend the result from [9] (V. Jovanović, C. Rohde, Error estimates for finite volume approximations of classical solutions for nonlinear systems of balance laws, SIAM J. Numer. Anal., 43 (2006)), where, among other things, an h — error estimate in the L — norm for the elastodynamics system has been established. We first derive the general error estimate from [9, Theorem 4.4] i...

We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from the well-known Cauchy-Born rule.