Summary
This paper reconsiders the problem of determining the elastostatic field near the tip of a crack in an all-round infinite body deformed by a “Mode III” loading at infinity to a state of anti-plane shear. The problem is treated for a class of incompressible, homogeneous, isotropic elastic materials whose constitutive laws permit a loss of ellipticity in the governing displacement equation of equilibrium at sufficiently severe shearing strains. The analysis represents a generalization of that reported in an earlier study and, as before, is carried out for the “small-scale nonlinear crack problem”, in which a crack of finite length is replaced by a semi-infinite one, and the nonlinear field far from the crack-tip is matched to the near field predicted by the linearized theory. The methods employed in the present paper are necessarily largely qualitative, since they apply to all materials in the class considered. The principal feature of the resulting elastic field is the presence of two symmetrically located curves issuing from the crack-tip and bearing discontinuities in displacement gradient and stress.
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References
J. K., Knowles and Eli, Sternberg, Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example, Journal of Elasticity 10 (1980) 81–110.
J. K., Knowles, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, International Journal of Fracture, 13, (1977) 611.
J. K., Knowles and Eli, Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics, Journal of Elasticity 8 (1978) 329.
J. K., Knowles, On the dissipation associated with equilibrium shocks in finite elasticity. Journal of Elasticity 9 (1979) 131.
J. R., Rice, Stresses due to a sharp notch in a work-hardening elastic-plastic material loaded by longitudinal shear, Journal of Applied Mechanics 34 (1967) 287.
R., Courant and K. O., Friedrichs, Supersonic flow and shock waves, Interscience, New York, 1948.
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The results communicated in this paper were obtained in the course of an investigation supported in part by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.
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Knowles, J.K., Sternberg, E. Anti-plane shear fields with discontinuous deformation gradients near the tip of a crack in finite elastostatics. J Elasticity 11, 129–164 (1981). https://doi.org/10.1007/BF00043857
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DOI: https://doi.org/10.1007/BF00043857