Boundary element analysis of non-planar three-dimensional cracks using complex variables

• A complex variables BEM scheme with second order approximations is developed.
• The method employs planar triangular boundary elements and analytical integration.
• The Cauchy-Pompeiu formula is used to reduce area integrals to contour ones.
• Numerical results for a penny-shaped crack and two non-planar cracks are presented.
• Some of the results are tabulated to serve as benchmarks for future investigations.

This paper reports new developments on the complex variables boundary element approach for solving three-dimensional problems of cracks in elastic media. These developments include implementation of higher order polynomial approximations for the boundary displacement discontinuities and more efficient analytical techniques for evaluation of integrals. The approach employs planar triangular boundary elements and is based on the integral representations written in a local coordinate system of an element. In-plane components of the fields involved in the representations are separated and arranged in certain complex combinations. The Cauchy–Pompeiu formula is used to reduce the integrals over the element to those over its contour and evaluate the latter integrals analytically. The system of linear algebraic equations to find the unknown boundary displacement discontinuities is set up via collocation. Several illustrative numerical examples involving a single (penny-shaped) crack and multiple (semi-cylindrical) cracks are presented.