Analysis of doubly periodic in-plane cracks using the eigenfunction expansion-variational method

A variational functional for the unit cell for a doublyperiodic in-plane problem is presented, based on the variational principlein elasticity in conjunction with the double quasi-periodicity of thedisplacement field and the double periodicity of the stress and strainfields. Then by combining with the eigenfunction expansions of the complexstress functions satisfying the traction-free conditions on the cracksurfaces, an eigenfunction expansion-variational method for the unit cellmodel is developed. The general doubly periodic boundary conditions for aunit cell are considered, so the present method can be used to solve thegeneral doubly periodic crack problems. The convergency analysis of thenumerical results demonstrates the high efficiency and accuracy of thepresent method. Finally, for several general doubly periodic crack arrays,the influence of the stress intensity factors on the crack arrangement isexamined.