A Finite Element Method for computing Shear Band formation Th. Baxevanis, Th. Katsaounis, and A. Tzavaras Abstract. The objective of this work is to provide an in depth numerical study of the interplay between thermal softening and strain-hardening in shearing deformations of strain-rate dependent materials. We consider the unidirectional simple shearing of an infinite slab. This model, despite its simplicity, incorporates the essential features of shear band modeling. We employ an adaptive finite element method of any order for the spatial discretization. Adaptivity in the spatial variable, is a necessity to correctly capture these singular phenomena. Further the implicit Euler method with variable time-step is used for the time discretization. The resulting numerical scheme is of implicit-explicit type, of any order in space and simple to implement. 1. Introduction Dissipative mechanisms, such as viscosity or thermal diffusion, tend to stabilize the thermomechanical processes opposing the destabilizing influence of the nonlinearity of the material response. The competition is especially delicate when the strength of the dissipative mechanisms weakens in the course of the motion. At high strain rates, thermal softening can eventually outweigh the tendency of the material to harden, thus creating a destabilizing mechanism which competes with internal dis- sipation. Experimental and numerical investigations indicate that when the degree of thermal softening is large this competition results to instability and formation of shear bands. Shear bands are narrow regions of concentrated shearing deformation. Once the band is fully formed, the two sides of the region are displaced relatively to each other, however the material still retains full physical continuity from one side to the other. We refer to [19] for an excellent survey of the mechanical issues. The intent of this work is to provide an in-depth numerical study of the interplay of thermal softening and strain-hardening in shearing deformations of strain-rate dependent materials, using modern ideas of adaptive finite element methods so as to fully resolve the bands. We consider a simple model, the unidirectional simple