Explicit time-discretisation of elastodynamics with some inelastic processes at small strains

The 2-step staggered (also called leap-frog) time discretisation of linear 2nd-order Hamiltonian systems (typically linear elastodynamics in a stress-velocity form) is extended for a 3-step staggered discretisation applicable for systems involving some internal variables subjected to a dissipative evolution. After spatial discretisation, a-priori estimates and convergence is proved under the usual CFL-condition. Applications to specific problems in continuum mechanics of solids at small stains are considered, in particular linearized plasticity, diffusion in poroelastic media, damage, or adhesive contact. Numerical implementation and some computational 2-dimensional simulation of waves emitted by a rupture (delamination) of an adhesive contact illustrate the abstract theory and efficiency of the explicit method.