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

A variational functional for the unit cell for a doubly periodic in-plane problem is presented, based on the variational principle in elasticity in conjunction with the double quasi-periodicity of the displacement field and the double periodicity of the stress and strain fields. Then by combining with the eigenfunction expansions of the complex stress functions satisfying the traction-free conditions on the crack surfaces, an eigenfunction expansion-variational method for the unit cell model is developed. The general doubly periodic boundary conditions for a unit cell are considered, so the present method can be used to solve the general doubly periodic crack problems. The convergency analysis of the numerical results demonstrates the high efficiency and accuracy of the present method. Finally, for several general doubly periodic crack arrays, the influence of the stress intensity factors on the crack arrangement is examined.