On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics

Summary

This investigation concerns equilibrium fields with discontinuous displacement gradients, but continuous displacements, in the theory of finite plane deformations of possibly anisotropic, compressible elastic solids. “Elastostatic shocks” of this kind, which resemble in many respects gas-dynamical shocks associated with steady flows, are shown to exist only if and when the governing field equations of equilibrium suffer a loss of ellipticity. The local structure of such shocks, near a point on the shockline, is studied with particular attention to weak shocks, and an example pertaining to a shock of finite strength is explored in detail. Also, necessary and sufficient conditions for the “dissipativity” of time-dependent equilibrium shocks are established. Finally, the relevance of the analysis carried out here to localized shear failures-such as those involved in the formation of Lüders bands-is discussed.

Since this paper was submitted for publication, Professor James R. Rice has pointed out to us that the dissipation inequality (6.28), which was in essence postulated in the present work, could have been deduced from the thermodynamic requirement of a positive rate of entropy production together with the energy identity (6.23). It is presumed in such a derivation that the underlying quasi-static time dependent equilibrium shock constitutes an isothermal process.