An efficient numerical method for the solution of the second boundary value problem of elasticity for 3D-bodies with cracks

The second boundary value problem of elasticity for 3D-bodies containing cracks is considered. Presentation of the solution in the form of the double layer potential reduces the problem to a system of 2D-integral equations which kernels are similar for the body boundary and crack surfaces. For discretization of these equations, Caussian approximation functions centered at a set of nodes homogeneously distributed on the body and crack surfaces are used. For such functions, calculation of the elements of the matrix of the discretized problem is reduced to five standard 1D-integrals that can be tabulated. For planar cracks, these integrals are calculated in closed analytical forms. The method is mesh free, and for its performing, only node coordinates and surface orientations at the nodes should be defined. Calculation of stress intensity factors at the crack edges in the framework of the method is discussed. Examples of an elliptical crack, a lens-shaped crack, and a spherical body subjected to concentrated and distributed surface forces are considered. Numerical results are compared with the solutions of other authors presented in the literature. Convergence of the method with respect to the node grid steps is analyzed. An efficient algorithm of the node grid generation is proposed.