  Softening material models are known to trigger spurious localizations.This
may be shown theoretically by the existence of solutions with zero dissipation
when localization occurs and numerically with spurious mesh dependency and
localization in a single layer of elements. We introduce in this paper a new
way to avoid spurious localization. The idea is to enforce a Lipschitz
regularity on the internal variables responsible for the material softening.
The regularity constraint introduces the needed length scale in the material
formulation. Moreover, we prove bounds on the domain affected by this
constraint. A first one-dimensional finite element implementation is proposed
for softening elasticity and softening plasticity.
