Perturbation model for composite materials applied to analysis of cracks

The behaviour of the stress intensity factor is investigated for long plane cracks with one tip interacting with a region of graded material properties. The material outside the region is considered to be homogeneous elastic. The analysis is based on assumed small differences in stiffness in the entire body. The linear extent of the body is assumed to be large compared with that of the graded region. The crack tip, including the graded region, is assumed embedded in a square-root singular stress field. The stress intensity factor for an arbitrarily shaped region is given by a singular integral. Solutions are presented for rectangular regions with elastic gradient parallel to the crack plane. The limiting case of infinite strip is solved analytically, leading to a very simple expression. Further, a fundamental case of material properties variation is considered, allowing the solution for an arbitrary variation to be represented by Fourier’s series expansion. An interesting feature of the solution is that the stress intensity factor remains finite and does neither vanish nor become unbounded as for the cases where modulus of elasticity posses jumps. A numerical study of the fundamental strip case, with finite variations of material properties, performed using the finite element method is communicated in brief. The analytical solution is compared with the numerical results and is shown to have a surprisingly large range of validity. If an error of 5% is tolerated, modulus of elasticity in the strip may drop by around 40% or increase with around 60%.