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Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example

Knowles, J. K. and Sternberg, Eli (1980) Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example. Journal of Elasticity, 10 (1). pp. 81-110. ISSN 0374-3535. doi:10.1007/BF00043136.

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This investigation aims at the elastostatic field near the edges (tips) of a plane crack of finite width in an all-round infinite body, which — at infinity — is subjected to a state of simple shear parallel to the crack edges. The analysis is carried out within the fully nonlinear equilibrium theory of homogeneous and isotropic, incompressible elastic solids. Further, the particular constitutive law employed here gives rise to a loss of ellipticity of the governing displacement equation of equilibrium in the presence of sufficiently severe anti-plane shear deformations. The study reported in this paper is asymptotic in the sense that the actual crack is replaced by a semi-infinite one, while the far field is required to match the elastostatic field predicted near the crack tips by the linearized theory for a crack of finite width. The ensuing global boundary-value problem thus characterizes the local state of affairs in the vicinity of a crack-tip, provided the amount of shear applied at infinity is suitably small. An explicit exact solution to this problem, which is deduced with the aid of the hodograph method, exhibits finite shear stresses at the tips of the crack, but involves two symmetrically located lines of displacement-gradient and stress discontinuity issuing from each crack-tip.

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Additional Information:© 1980 Sijthoff & Noordhoff International Publishers. Received February 16, 1979. The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.
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Office of Naval Research (ONR)N00014-75-C-0196
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ID Code:51214
Deposited By: Tony Diaz
Deposited On:04 Nov 2014 17:52
Last Modified:10 Nov 2021 19:07

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