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ONE-DIMENSIONAL NON-LINEAR WAVE PROPAGATION IN INCOMPRESSIBLE ELASTIC MATERIALS
( Department of Mathematics, University of Strathclyde )
We formulate the equations governing one-dimensional non-linear wave propagation in an incompressible heat-conducting elastic material which is transversely isotropic or isotropic in its undeformed state and obtain exact discontinuous solutions of these equations in which the states are uniform on either side of a discontinuity propagating through the material. To obtain further solutions we need to approximate the equations and this we do either by neglecting entropy changes or by assuming that the material does not conduct heat. In either approximation the governing system of equations is hyperbolic under certain restrictions on the internal energy density. We can therefore apply general results on solutions of hyperbolic systems with n dependent and two independent variables to investigate the simple waves, shocks, and other discontinuities which can propagate in the material. We find that in a transversely isotropic material a pair of fast and a pair of slow simple waves can propagate as can corresponding pairs of shocks. In an isotropic material a pair of linearly polarized transverse simple waves or shocks and a pair of circularly polarized waves or discontinuities can propagate. In either material we can also have a contact discontinuity which is, however, transient. We determine the conditions under which the various shocks and other discontinuities are evolutionary, that is, stable to small disturbances, and for an isotropic material investigate the relation between the requirements that the entropy increase across a transverse shock and that the shock be evolutionary. We obtain results on the ways in which these simple waves and discontinuities can propagate by considering the problem of an elastic half-space initially at rest, the surface of the half-space being given a uniform motion by the sudden application of shearing stresses to it.