Finite plane and anti-plane elastostatic fields with discontinuous deformation gradients near the tip of a crack

Abstract

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 materials 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 elliptieity results in the appearance of curves ("elastostatic shocks") issuing from the crack-tip across which displacement gradients and stresses are discontinuous.

Two general integrals of singular crack tip deformation fields

“…These solutions, for stationary cracks, involve proportional plastic straining and hence can also be interpreted as valid nonlinear elastic solutions. Similar features of crack tip fields have been found for nonlinear materials which lose ellipticity, and strain soften, when deformed beyond a certain threshold (Knowles and Sternberg, 1981;Fowler, 1984). It seems that the results which follow apply to such cases, with the understanding that certain components of F~,(O) may have a Dirac singular pulse across the ray of discontinuity.…”

“…Here for simplicity it is assumed that the discontinuity remains on the ray 0 = 0 ° for some finite r, as in the anisotropic single crystal ideally plastic solutions (e.g., Rice and Nikolic, 1985), although such does not fully describe other cases of interest (e.g., Fowler, 1984) with post-elliptic softening. Thus letting e -~ 0 and observing that Tj must be continuous, eq.…”

Analysis of deformations near a crack tip in a compressible nonlinear elastic material

“…Knowles and Stemberg [l] have used the hodograph transformation method to study stresses near a crack tip in an elastic body made of a neo-Hookean material and deformed in antiplane shear. Subsequently, Fowler [2] used an asymptotic representation of the displacement field to study the same problem and obtained results in general agreement with those of Knowles and Stemberg [l]. These investigators found that for shearing in the z-direction, the component T, of the Cauchy stress tensor T becomes unbounded at the crack tip.…”

Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium

“…In particular, we observe from (3.15) that as p ~ 0, t~ -~ 1 and as p -", p .... t o --, 0. For 0 < p < Pmax (and so 0 < t~ < 1), the resulting stress fields are given by (3.10), (3.11), (3,13) as…”

“…In particular, the deformation gradient and stress fields could fail to be continuous across one or more surfaces -equilibrium shocks -in the body (see, e.g. [5,6,[10][11][12][13]). In this section it is shown that axisymmetric solutions with such discontinuities do not exist in the present problem.…”