Quasistatic problems in viscoplasticity theory: Existence of solutions and homogenization

We discuss quasistatic initial-boundary value problem with internal variables which model the deformation behavior of viscoplastic bodies at small strains. In the first part of the talk we introduce the class of constitutive equations of monotone type, discuss the relevance of this class to engineering applications and study the existence theory. In the second part of the talk these results are used to justify the formal homogenization of such initial-boundary value problems with periodic microstructure, called the microscopic problems. To this end it is first shown that the formally derived homogenized initial-boundary value problem has a solution. From this solution an asymptotic solution of the microscopic problem is constructed, and it is shown that the difference of the exact solution and the asymptotic solution converges to zero if the lengthscale of the microstructure tends to zero. This convergence proof is based on the notion of operator difference introduced by Vladimirov.