Uniqueness, loss of ellipticity and localization for the time-discretized, rate-dependent boundary value problem with softening

The discretization in time of the initial boundary value problem for rate-dependent (elastic-viscoplastic) solid materials in presence of softening is investigated in this paper. The emphasis is put on uniqueness, loss of ellipticity and localization. It is found that the time-discretized problem resembles the incremental problem for rate-independent materials and softening may lead to ill-posedness (loss of ellipticity) if the time step is greater than a critical value. It is well established that the implication of loss of ellipticity for the numerical simulations after spatial discretization is the pathological mesh dependency of the computed results. We furnish a method to compute the critical time step and demonstrate its use for a simple example problem. Copyright (C) 2010 John Wiley & Sons, Ltd.