The Two-Dimensional Structure of Dynamic Boundary Layers and Shear Bands in Thermoviscoplastic Solids

Abstract

A general boundary layer theory for thermoviscoplastic solids which accounts for inertia, rate sensitivity, hardening, thermal coupling, heat convection and conduction, and thermal softening is developed. In many applications of interest, the boundary layer equations can be considerably simplified by recourse to similarity methods, which facilitates the determination of steady-state and transient fully non-linear two-dimensional solutions. A simple analysis of the asymptotic behavior of the steady-state solutions leads to a classification of stable and unstable regimes. Under adiabatic conditions, the resulting material stability criterion coincides with that previously derived by Molinari and Clifton [(1987) Analytical characterization of shear localization in thermoviscoplastic solids. J. Appl, Mech. 54, 806–812] by a quasi-static, one-dimensional analysis. The transition from initially stable to unstable behavior can also be conveniently described by similarity methods. This provides a powerful semi-analytical tool for the interpretation of impact tests exhibiting dynamic shear bands, and for the characterization of the two-dimensional structure of such bands. It follows from the theory that, if the velocity of the impactor is held steady, the leading tip of the shear band propagates at a constant speed. This shear band tip speed follows readily from the theory as a function of the impact velocity and material parameters. The two-dimensional velocity, stress, temperature and plastic work fields attendant to the propagating shear band are also determined.