Staggered explicit-implicit time-discretization for elastodynamics with dissipative internal variables

¹ Mathematical Institute, Math.-Phys. Faculty, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic
² Institute of Thermomechanics of the Czech Acad. Sci., Dolejškova 5, CZ-182 08 Praha 8, Czech Republic
³ Department of Applied Mathematics, University of California, Merced, 5200 North Lake road, Merced, CA 95343, USA
⁴ Institute of Applied and Computational Mathematics, Foundation for Research and Technology – Hellas, Nikolaou Plastira 100, Vassilika Vouton, GR-700 13 Heraklion, Crete, Greece

^* Corresponding author: tomas.roubicek@mff.cuni.cz; roubicek@karlin.mff.cuni.cz

Received: 16 October 2019
Accepted: 5 June 2020

Abstract

An extension of the two-step staggered time discretization of linear elastodynamics in stress-velocity form to systems involving internal variables subjected to a possibly non-linear dissipative evolution is proposed. The original scheme is thus enhanced by another step for the internal variables which, in general, is implicit, although even this step might be explicit if no spatial gradients of the internal variables are involved. Using an abstract Banach-space formulation, a priori estimates and convergence are proved under a CFL condition. The developed three-step scheme finds applications in various problems of continuum mechanics at small strain. Here, we consider in particular plasticity, viscoelasticity (creep), diffusion in poroelastic media, and damage.