Finite Plane and Anti-Plane Elastostatic Fields with Discontinuous Deformation Gradients Near the Tip of a Crack

In this paper the fully nonlinear theory of finite deformations of an elastic solid is used to study the elastostatic field near the tip of a crack. The special elastic materials considered are such that the differential equations governing the equilibrium fields may lose ellipticity in the presence of sufficiently severe strains. The first problem considered involves finite anti-plane shear (Mode III) deformations of a cracked incompressible solid. The analysis is based on a direct asymptotic method, in contrast to earlier approaches which have depended on hodograph procedures. The second problem treated is that of plane strain of a compressible solid containing a crack under tensile (Mode I) loading conditions. The material is characterized by the so-called Blatz-Ko elastic potential. Again, the analysis involves only direct local considerations. For both the Mode III and Mode I problems, the loss of equilibrium ellipticity results in the appearance of curves ("elastostatic shocks") issuing from the crack-tip across which displacement gradients and stresses are discontinuous.