Sixth GAMM Seminar on Microstructures - Abstract

Alber, Hans-Dieter

Two-scale convergence versus phase shift convergence in homogenization of the equations of viscoplasticity

In two recent articles the justification of the homogenization of models for viscoplastic material behavior was studied by Sergiy Nesenenko (for rate dependent and rate independent models) and by Alexander Mielke and Aida Timofte (for rate independent models). Though the proofs are based on very different methods, the investigations are closely related. Yet, the finally proved convergence results are different, and it is not obvious how they are connected. A. Mielke and A. Timofte proved a type of convergence, which they termed strong two scale convergence, whereas S. Nesenenko proved a type of convergence, which for the sake of this talk I call phase shift convergence. It is important to understand the relation between these two types of convergence. In this talk I discuss this question and show that they are very similar, though the question, whether they are completely equivalent, remains finally open.
[2ex] bf References
[1ex] [1] H.-D. Alber: Justification of homogenized models for viscoplastic bodies with microstructure. In: K. Hutter, H. Baaser (eds.), Deformation and failure in metallic materials. Abschlussbuch des Sonderforschungsbereichs 298. Lecture Notes in Applied and Computational Mechanics bf 10, Springer 2003, p. 295--319.
[.8ex] protect [2] A. Mielke, A. Timofte: Two scale homogenization for evolutionary variational inequalities via the energetic formulation. Manuscript 2006, submitted to SIAM J. Math. Anal.
[.8ex] protect [3] S. Nesenenko: Homogenization and regularity in viscoplasticity. Dissertation, Fachbereich Mathematik der Technischen Universität Darmstadt, 2006.
[.8ex] protect [4] S. Nesenenko: Homogenization in viscoplasticity. Manuscript 2006. Accepted for publication in SIAM J. Math. Anal.