A Mixed Finite Element Analysis For An Evolution Problem Modeling Shear Band Deformation

The mixed finite element method has been developed during 1960s and 1970s, which contains two or more approximation spaces mostly. This method shows a better numerical results than the general way in many fields, like elasticity, fluid dynamics. Babuska (1974) and Brezzi (1974) found the basic theory of the mixed method.Shear bands are thin regions in a thermoplastic material where the strain rate is very high due to the applied stresses. The phenomena has received increasing attention in recent years and is believed to be a crucial part of the deformation process in metal forming, ballistic impact and penetration. In this paper we are going to discuss a semi-discrete mixed finite element method for a nonlinear problem which modeling the antiplane shear deformations of a thermoplastic material. The one dimensional version of this model was considered in [1] by Madani and Maddocks, French and Garcia discussed a semi-discrete finite element approximation in [2]. Under some regularity assumptions for the exact and discrete solutions, they obtained an error estimate in L2 norm. Xie xiao-ping, Feng min-fu and Lu jin-shu developed a fully discrete finite element method for this model in [3], an alternating scheme was presented which lead to two decoupled linear algebraic subsystems, and an error estimate was derived. In this paper ,our purpose is to introduce a new variable which...