Mathematical Analysis of the Armstrong - Frederick Model from the Theory of Inelastic Deformations of Metals. First Results and Open Problems

This paper tries to analyse, from the mathematical point of view, the system of equations proposed by P.J. Armstrong and C.O. Frederick and used in the theory of inelastic deformations of metals. We present global in time existence result for so called "weak-type admissible solutions" for the quasistatic Armstrong-Frederick model. This nonmonotone model is here written as a model of pre-monotone type with a nonassociated flow rule. Moreover, a monotone model is presented, which has a structure that is very similar to the Armstrong-Frederick model