The problem of microdefects in the matrix and debondings along a circular inclusion: An integral equation solution (Q1812860)

The most usual kind of damage in composite materials is the occurence and the propagation of cracks. Generally they initiate from debondings between matrix and inclusion. The interaction of these debondings with cracks and/or holes either pre-existing or generated during the service provokes the propagation either of the debondings or the cracks. In order to predict such a damage the knowledge of the distribution of stresses is of course necessary. Obviously, the shape of the inclusion has an important influence on the distribution of stresses. In order to simplify the problem we have considered a circular inclusion having \(n\) debondings along the interface and interacting with \(m\) cracks and holes of arbitrary shape. The problem of the debondings has already been considered [ \textit{N. I. Ioakimidis} and \textit{P. S. Theocaris}, Rev. Roum. Sci. Techn., Ser. Mec. Appl. 23, 563-575 (1978; Zbl 0399.73087)]. The proposed problem could be solved using the method of previous papers [\textit{G. Tsamasphyros} and \textit{P. S. Theocaris}, ibid. 25, 839-856 (1980; Zbl 0464.65012)]. This would result in a system of \((n+m)\) singular integral equations (S.IEs) along the debondings and the cracks and/or holes. But the numerical solution of these S.IEs presents serious numerical difficulties and may diverge, since the unknown distribution of dislocation along the debondings has complex singularities. In order to avoid these difficulties we must eliminite the \(n\) S.IEs along the debondings. In this paper we describe a method of such elimination. That means that the proposed method leads to a system of \(m\) S.IEs only along the cracks and/or holes of the matrix.