A fast and accurate analysis of the interacting cracks in linear elastic solids

Abstract

This paper presents a fast and accurate solution for crack interaction problems in infinite- and half- plane solids. The new solution is based on the method of complex potentials developed by Muskhelishvili for the analysis of plane linear elasticity, and it is formulated through three steps. First, the problem is decomposed into a set of basic problems, and for each sub-problem, there is only one crack in the solid. Next, after a crack-dependent conformal mapping, the modified complex potentials associated with the sub-problems are expanded into Laurent’s series with unknown coefficients, which in turn provides a mechanism to exactly implement in the form of Fourier series the boundary condition in each sub-problem. Finally, taking into account the crack interaction via a perturbation approach, an iterative algorithm based on fast Fourier transforms (FFT) is developed to solve the unknown Fourier coefficients, and the solution of the whole problem is readily obtained with the superposition of the complex potentials in each sub-problem. The performance of the proposed method is fully investigated by comparing with benchmark results in the literatures, and superb accuracy and efficiency is observed in all situations including patterns where cracks are closely spaced. Also, the new method is able to cope with interactions among a large number of cracks, and this capability is demonstrated by a calculation of effective moduli of an elastic solid with thousands of randomly-spaced cracks.