Energy Criteria for Finite Hyperelasticity.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The equations of hyperelasticity have the special feature that their natural entropy is not a globally convex function. Strict convexity of the entropy function is essential in formulating a physically reasonable entropy criterion for shock waves. In this paper we show that the natural entropy of the equations of hyperelasticity is uniformly convex when restricted to the shock curves. This fact enables us to prove the equivalence of the entropy criterion and Laxs shock conditions for existence of weak shocks for problems that are genuinely nonlinear. Furthermore, for problems that are not necessarily genuinely nonlinear we study the generalized E-condition and show that it is indeed a generalization of the entropy condition. Finally, we consider the viscosity criterion which requires that a motion of a hyperelastic body is the limit of smooth motions of a family of viscoelastic materials. The relationship between the energy criterion, the E-condition, and the viscosity criterion is then discussed. Author