Non-local theory solution for a plane rectangular crack in a 3D infinite transversely isotropic elastic material under a time-harmonic elastic P-wave

The non-local theory solution for a plane rectangular crack in a 3D infinite transversely isotropic elastic material is presented under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. In order to overcome the mathematical difficulties, a two-dimensional non-local kernel is used instead of a three-dimensional one for 3D dynamic problem to obtain the stress near the crack edges. With the help of the Fourier transform, the problem is formulated into three pairs of dual integral equations with the jumps of displacement across the crack surfaces as the unknown variables. To solve the dual integral equations, the jumps of displacement across the crack surface are directly expanded as a series of Jacobi polynomials. Numerical examples show how the stress is influenced by the geometric shape of the rectangular crack, the circular frequency of the incident waves and the lattice parameter of the material on the dynamic stress fields near the crack edges. Unlike the classical solutions, the present solution exhibits no stress singularity along the rectangular crack edges, i.e. the dynamic stress field near the rectangular crack edges is finite. Thus, this allows us to use the maximum stress as a fracture criterion.