Instability and ill-posedness in the deformation of granular materials

A linear stability analysis of the partial differential equations of granular flow is performed. The constitutive relations include (i) elastic effects, (ii) non-associated flow rules and (iii) shear-strain hardening. Conditions for the equations to be well-posed and to be stable are derived. It turns out that there are two qualitatively different kinds of instability; which appears first depends on the parameters of the constitutive relations. If one kind appears first, then in a constitutive test a period of inhomogeneous deformation is predicted to occur before the formation of shear bands. If the other kind appears first, then shear bands are predicted to form in an approximately homogeneous sample. Some constitutive experiments are analysed from this viewpoint, and the analysis suggests some new experimental work.