A note on ill-posedness for rate-dependent problems and its relation to the rate-independent case

The issue of ill-posedness for rate-dependent solids is investigated. We show that under some circumstances, the associated finite step problem obtained by discretization in time of the initial-value problem may be ill-posed. A critical time step may exist beyond which well-posedness is lost. For a sufficiently small time step however, well-posedness is guaranteed in general although situations may exist where this time step is too much a small. This failure of well-posedness occurs in general for implicit algorithms and for softening and/or non-associative flow. The results are illustrated through the simple example of J (2) associative viscoplasticity.