Some Aspects of a Flat Elliptical Crack Under Shear Stress

Discussion: “Optimum Thin-Walled Pressure Vessels of Anisotropic Materials” (Gerard, G., and Lakshmikantham, C., 1966, ASME J. Appl. Mech., 33, pp. 623–628)

“…In their introduction, the authors stated that their analysis is pertinent to pressure vessels designed according to ultimate as well as yield. The writer questions the validity of the analysis in regard to design for ultimate strength of ductile metals (including 1 the HCP alloys, typified by titanium alloys, which exhibit the texture hardening cited by Backofen). For properly manufactured pressure vessels of these materials, failure is most likely to be ductile rather than brittle.…”

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In light of the foregoing analogy, the stress-intensity factors ki, kt for an isotropic and transversely isotropic solid differ only in that v is replaced by 1 -fi/a.Since k% and kt are independent of the shear modulus ju, there is no need to redefine the constants A and B as it was done in equations (1) and (2) of the discussion. Furthermore, it is the opinion of the authors that the discusser did not provide the pertinent information in his work [1] for finding the expressions of ki and kz-Incidentally, the derivation of h? and kz from the crack-border stress field, based on the limiting forms of the elliptical coordinates (£, T], f) to the polar coordinates (r, 6), is by no means an easy task.

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“…Another point that requires clarification is concerned with the method of solution used by the discusser [1] to solve the crack problem. From the existing knowledge of the isotropic solution, he conjectures that the boundary conditions of constant shear stresses applied to the crack surfaces can be satisfied by taking u ~ Z x '\ for 2 = 0, x 2 /a 2 + y 2 /b 2 ^ 1 (10)…”

“…A detailed proof of this theorem for the transversely isotropic case follows readily from the work of Kassir and Sih [5] for isotropic elasticity. In addition, the mixed boundary-value problem, described by the four conditions in [1], can be solved directly by application of the method developed in the paper and the potential solutions in [2,3], The problem of specifying shear stresses on the crack surfaces as a linear function of x and y can also be solved explicitly. However, the results for general loading conditions, say, applied stresses given by polynomials in x", y" where n ^ 2, appear to be restrictive.…”

Discussion: “Three-Dimensional Stress Distribution Around an Elliptical Crack Under Arbitrary Loadings” (Kassir, M. K., and Sih, G. C., 1966, ASME J. Appl. Mech., 33, pp. 601–611)

“…As an example, consider the problem of an elliptical crack in a transversely isotropic medium as treated in [1], The boundary conditions on the surfaces of the crack are = q; r"2 = 0, for 2 = 0, x 2 /a 2 + y 2 /b 2 $ 1…”