Stability by homogenization of thermoviscoplastic problems

In this paper we study the homogenization of the system of partial differential equations describing the quasistatic shearing of heterogeneous thermoviscoplastic materials. We first prove the existence and uniqueness of the solution of the system for the general model. We then define “stable by homogenization” models as the models where the equations in both the heterogeneous problems and the homogenized one are of the same form. Finally we show that three types of models, all three with non oscillating strain-rate sensitivity, are stable by homogenization: the viscoplastic model, the thermoviscous model and the thermoviscoplastic model under steady boundary shearing and body force. In these three models, the homogenized (effective) coefficients depend on the initial conditions, and, in the case of the thermoviscoplastic model, also on the boundary shearing and body force. Those theoretical results are illustrated by some numerical examples. Nicolas Charalambakis, Department of Civil Engineering, Aristotle University, GR 54124 Thessaloniki, Greece ; e-mail : [email protected] Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Bôıte courrier 187, 75252 Paris Cedex 05, France ; e-mail : [email protected]