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Applied Math Seminar
An Adaptive Finite Element Method for Computing Shear Band Formation
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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 outweight the tendency of the material to harden, thus creating a destabilizing mechanism which competes with internal dissipation. Experimental and numerical investigations, in cases where the degree of thermal softening is large, suggest that this competition results in instability in the form of shear bands. The objective is to elucidate numerically the interplay of thermal softening and strain-hardening in shearing deformations of strain-rate dependent materials. We consider a simple model problem, namely the unidirectional simple shear of an infinite slab. This model, in spite of its simplicity, incorporates the essential material behavior necessary for shear band modeling. In particular we use adaptive finite element method of any order for spatial discretization. Adaptivity is the spatial variable, is a necessity to correctly capture the singular phenomena (formation of shock and blow-up). Further implicit Runge-Kutta methods with strong stability properties and variable time-step are used as time-stepping mechanisms. The resulting numerical schemes are of implicit-explicit type, of any order in space and time and simple to implement. |