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Brittle fracture in a periodic structure with internal potential energy. Quasi-static analysis

We consider a linearly elastic body consisting of two equal symmetrically arranged layers (or half-planes) connected by a structured interface as a prospective crack path. The interface is comprised by periodic discrete system of bonds. In the initial state with no external forces, the bonds are assumed to be stressed in such a way that tensile and compressive forces of the same value alternate. In the general considerations, the layers are assumed to be of a general, unspecified {\em periodic} structure, where such self-equilibrated residual stresses can also exist. A two-line chain and an anisotropic lattice are examined as illustrative examples. We consider the states of the body-with-a-crack under the residual stresses and under a combined action of the remote forces and residual stresses. Analytical solutions to the considered problems are presented. The solutions are based on a selective discrete transform introduced. In particular, it is found that a formula for local-to-global energy release ratio, written in terms of the Wiener-Hopf equation kernel, is very general. We show that the residual stresses can manifest themselves in a number of phenomena such as the crack bridging, `porous' in front of the crack and crack growth irregularities. We also demonstrate analytically and graphically that the residual stresses can result in increase as well as in decrease of the crack resistance depending on the internal energy level. The quasi-static considerations suggest different scenarios of the crack growth depending on the internal energy level.