Discontinuous Deformation Gradients in Plane Finite Elastostatics of Incompressible Materials. (I) General Considerations. (II) An Example

Citation

Abeyaratne, Rohan Chandra (1979) Discontinuous Deformation Gradients in Plane Finite Elastostatics of Incompressible Materials. (I) General Considerations. (II) An Example. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/seym-cd95. https://resolver.caltech.edu/CaltechTHESIS:03012018-170438409

Abstract

This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related is sue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of "elastostatic shocks" -- elastostatic fields with continuous displacements and discontinuous deformation gradients -- is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipticity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material. In order to illustrate the occurrence of elastostatic shocks in a physical problem, a specific displacement boundary value problem is studied. Here, a particular class of isotropic, incompressible, elastic materials which allow for a loss of ellipticity is considered. It is shown that no solution which is smooth in the classical sense exists to this problem for certain ranges of the applied loading. Next, we admit solutions involving elastostatic shocks into the discussion and find that the problem may then be solved completely. When this is done, however, there results a lack of uniqueness of solutions to the boundary value problem. In order to resolve this non-uniqueness, dissipativity and stability are investigated.

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