On the Failure of Ellipticity and the Emergence of Discontinuous Deformation Gradients in Plane Finite Elastostatics.

摘要

This investigation concerns equilibrium fields with discontinuous displacements 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 shock-line,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 Lueders bands -is discussed. (Author)