Theory of cracks in prestressed elastic bodies. Shear cracks and limiting cases

Abstract

Thus, using the methods and formulation of [6], the following new results have been derived in the present article. 1. Using the complex potentials introduced previously by the author, an exact solution is given in closed form for a plane transverse shear crack with an arbitrary loading of the edges of the crack, based on the solution of the Riemann-Hilbert problem for unequal roots, and on the Keldysh-Sedov formula for equal roots. The exact solution for a longitudinal shear crack was derived also. These solutions are given for prestressed bodies in general form for compressible and incompressible models and an arbitrary structure of the elastic potential. 2. The asymptotic distrioution of stresses and displacements near the tip of a crack was constructed. For a transverse shear crack it was shown that in the general case of unequal roots the order of the singularity differs from that given by classical elasticity theory. In the special case of unequal roots and in the general case of equal roots the order of the singularity agrees with the classical elasticity theory, but the coefficients near the singularity are different. Along the line of a crack the coefficients for stresses agree also. 3. It was shown that for an initially unstressed body the results obtained go over into the solution of classical elasticity theory described in [12, 13, 16].