Regularizations and relaxations of time-continiuous problems in plasticity

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During the last decade, the theory of finite-strain elastoplasticity has been developed quite rapidly. The major impulses were the discovery that time-incremental problems can be formulated as minimization problems and the recent developments in the field of microstructures in infimizing sequences of functionals. While most of the mathematical theory treats the formation of microstructure via global minimization in static problems, it is desirable to derive models for the evolution of microstructure under slowly varying loads. The theory for laminate evolution in discrete time steps will be extended to the continuous time-level.This project is devoted to the study of temporal evolution models for plasticity and for systems with microstructures in general. Using spatial regularization via higher gradients and temporal regularization via viscosity we derive mathematical models that allow for an existence theory for solutions without microstructure. The temporal regularization will lead to viscous, time-continuous solutions, which converge to so-called BV solutions in the vanishing-viscosity limit. While this limiting procedure is understood in finite- dimensional systems and simple semilinear partial differential equations, the application to finite-strain plasticity is topic of the present research.The relaxation of a sequence of incremental minimization problems will be attacked using weighted energy-dissipation functionals and a development by !-convergence. Finally, it is planned to derive suitable scalings for discrete dislocation densities and pinning sites, such that the dynamic problem leads to a !-limit describing macroscopic dislocation models in the line-tension limit.