Localization limiters in transient problems

In materials with strain softening, closed form solutions have shown that the energy dissipation associated with the localization and damage processes vanishes. For finite element solutions, the dissipation depends on the size of the element in the localized zone. To remedy this situation, non-local and imbricate formulations have previously been proposed. In this paper, a simple remedy based on introducing additional higher order terms in the governing equations in the strain-softening portions of the domain is proposed. It is shown that the resulting formulation provides solutions which are independent of element size with a finite energy dissipation. Example solutions are given for one-dimensional rod problems and one-dimensional spherically symmetric problems.