Shear band spacing in gradient-dependent thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in simple shear. The strain gradients are taken as independent kinematic variables and the corresponding higher order stresses are included in the balance laws, and the equation for the yield surface. Three different functional relationships, the power law, and those proposed by Wright and Batra, and Johnson and Cook are used to relate the effective strain rate to the effective stress and temperature. Effects of strain hardening of the material and elastic deformations are neglected. The homogeneous solution of the problem is perturbed and the stability of the problem linear in the perturbation variables is studied. Following Wright and Ockendon's postulate that the wavelength whose initial growth rate is maximum determines the minimum spacing between adjacent shear bands, the shear band spacing is computed. It is found that the minimum shear band spacing is very sensitive to the thermal softening coefficient/exponent, the material characteristic length and the nominal strain-rate. Approximate analytical expressions for the critical wave length for heat conducting nonpolar materials and locally adiabatic deformations of gradient dependent materials are also derived.