Elastic solutions for periodically layered strip with perfect bonding or with an interface crack

A non-homogenization approach to the analysis of a strip consisting of a finite number of isotropic elastic layers arranged periodically is presented. The result is obtained by the use of the representative cell method based on the discrete Fourier transform. Two types of problems are addressed. In the problems of the first type the bonding between the layers is perfect. The solution for this case is found in a closed form in terms of the Laplace integrals. The numerical results for different boundary conditions are given and a comparison with plate theory is carried out. In the problems of the second type there is a flaw (crack) at one of the interfaces. The solution in this case hinges on the analytic expression for the Green function corresponding to a single interface dislocation in the uncracked strip. By using this expression the crack problem is reduced to a singular integral equation. The influence of the elastic mismatch and other problem parameters on the fracture characteristics is examined. It is found that, similar to the case of the periodic plane, when the thinner layeres in the strip are stiffer, further increase of their stiffness may lead to the increase or to the decrease of the absolute value of the stress intensity factor depending upon the elastic mismatch between the composite constituents.