RIEMANN PROBLEMS FOR 5X5 SYSTEMS OF FULLY NONLINEAR EQUATIONS RELATEDTO HYPOPLASTICITY

The equations of motion for two-dimensional deformations of an incompr essible elastoplastic material involve five equations, two equations e xpressing conservation of momentum, and three constitutive laws, which we take in the rate form, i.e. relating the stress rate to the strain rate. In hypoplasticity, the constitutive laws are homogeneous of deg ree one in the stress and strain rates. This property has the conseque nce that although the equations are not in conservation form, there is nonetheless a natural way to characterize planar shock waves. The Rie mann problem is the initial value problem for plane waves, in which th e initial data for stress and velocity consist of two constant vectors separated by a single discontinuity. The main result is that, under a ppropriate assumptions, the Riemann problem has a scale invariant piec ewise constant solution. The issue of uniqueness is left unresolved. I ndeed, we give an example satisfying the conditions for existence, for which there are many solutions. Using asymptotics, we show how soluti ons of the Riemann problem are approximated by smooth solutions of a s ystem regularized by the addition of viscous terms that preserve the p roperty of scale invariance.