Deriving new evolution equations for microstructures via relaxation of variational incremental problems Original Research Article

We study material models for rate-independent inelasticity in situations where no internal length scale is given and formation of microstructure for the deformation gradients and the internal variables may occur. We develop a rational procedure for deriving consistent macroscopic models which allow for the computation of nontrivial effective quantities without resolving the fine scales. The method involves the relaxation of variational incremental problems which are derived from an elastic and a plastic potential. We use Young measures to describe the microscopic distribution of the internal variables as well as the quasiconvexification of the elastic stored-energy density with respect to the deformation gradient. The resulting model provides a new rate-independent model in terms of the deformation and the Young measure. The approach is based on a derivative free, energetic formulation using one functional for elastic energy storage and one for the dissipation distance. The latter is derived from a dissipation potential defining the dissipation distance between internal states and hence the Wasserstein distance between Young measures. This approach is strongly linked to an associated time-incremental problem which is a minimization problem of the type used for several years now in the engineering literature. The update algorithm for the incremental problem is discussed in detail, and two simple examples are given.