Asymptotic solution of spatial problems of elasticity theory about extended plane shear cracks (Q581100)

An asymptotic solution is obtained for spatial problems of elasticity theory concerning shear cracks occupying a plane domain extended along a certain curve. Terms of the expansion of the solution in a small parameter characterizing the extension of the crack are constructed on the basis of a system of integrodifferential equations in the displacement component of points of the crack surfaces. On the basis of the asymptotic formulas obtained, a dependence of the displacement of the surfaces, the stress intensity factors, and the specific increment of the total potential energy on the crack shape and the load is described, and the relationships of the type of congruence theorems are set up. Values of the stress intensity factors and the displacements of the surfaces that are computed for specific kinds of cracks by means of the asymptotic formulas are in agreement with the known exact values and with values obtained by a numerical method. The asymptotic solution for an analogous simpler problem about separation cracks that reduces to the solution of one integrodifferential equation is presented by \textit{R. V. Gol'dshtejn}, \textit{A. V. Kaptsov} and the author, ibid. 48, 854-863 (1984; Zbl 0585.73168).