The singular limit of a hyperbolic system and the incompressible limit of solutions with shocks and singularities in nonlinear elasticity

Discontinuous solutions with shocks for a family of almost incompressible hyperelastic materials are studied. An almost incompressible material is one whose deformations are not a priori constrained but whose stress response reacts strongly (of order ${\varepsilon ^{ - 1}}$) to deformations that change volume. The material class considered is isotropic and admits motions that are self-similar, exhibit cavitation, and are energy minimizing. For the initial-value problem when considering the entire material, the solutions converge (as $\varepsilon$ tends to zero) to an isochoric solution of the limit (incompressible) system with the corresponding arbitrary hydrostatic pressure being the singular limit of the pressures in the almost incompressible materials. The shocks, if they exist, disappear: their speed tends to infinity and their strength tends to zero.