Adaptive Finite Element Methods and the Numerical Solution of Shear Band Problems

摘要

Shear bands are localized regions of very high shear strain which arise as a result of high rates of loading. They occur in metal forming and cutting processes and in impact and penetration problems. This reprint describes a mathematical model for the formation of shear bands in simple shear that involves the description of irreversible mechanical shear and the resulting heat release. The location of a shear band is unknown in advance, and the evolution results in large gradients of displacement, velocity, and temperature. Shear band formation, therefore, offers an interesting and physically important application of a code able to reslove small-scale transient structures. The authors use an adaptive finite element code to solve several problems involving shear band formation. The code automatically locates regions with large gradients and adaptively concentrates finite elements there in order to minimize approximately the discretization error per time step. Results show the developement of shear bands under many circumstances and indicate some possible mechanisms for their formation.