QUARTERLY OF APPLIED MATHEMATICS VOLUME LV, NUMBER 3 SEPTEMBER 1997, PAGES 485-504 THE SINGULAR LIMIT OF A HYPERBOLIC SYSTEM AND THE INCOMPRESSIBLE LIMIT OF SOLUTIONS WITH SHOCKS AND SINGULARITIES IN NONLINEAR ELASTICITY By RUSTUM CHOKSI Courant Institute, New York University, New York, NY Abstract. Discontinuous solutions with shocks for a family of almost incompressible hyperelastic materials are studied. An almost incompressible material is one whose defor- mations are not a priori constrained but whose stress response reacts strongly (of order e-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 e 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. 1. Introduction to the problem. In this article we give support for the following conjecture in mechanics: An incompressible nonlinear elastic material can be regarded as the limit of a family of almost incompressible materials; materials whose deformations are not a priori constrained but whose stress response reacts strongly to deformations that change volume. This family will consist of compressible materials all sharing a basic constitutive relation for the stress modulo an extra pressure term of order 1/e. The arbitrary hydrostatic pressure resulting in the incompressible case is actually a singular limit of the almost incompressible pressures that depend exclusively on the motion. Such almost incompressible materials were originally discussed by Spencer [11] and such a limiting relationship was noted in TVuesdell and Noll [12, p. 122], The idea of an incompressible limit has been well-studied for fluids using relevant solutions in smooth (Sobolev) spaces: Ebin [5] and Klainerman and Majda [7], [8]. Hence, very general results can be obtained using a priori estimates. For elastic solids, the arguments for fluids have been extended by Schochet [10] again working with solutions (motions) with Lp derivatives; the idea being that the equations of motion can be written Received October 3, 1994. 1991 Mathematics Subject Classification. Primary 35L67, 73C50. ©1997 Brown University 485