Rigid inclusion in an elastic matrix revisited

This paper deals with the plane deformation concerning a rigid inclusion embedded in an infinite elastic matrix under a uniform remote stress. The rigid inclusion is assumed to be perfectly bonded to the matrix and its geometry is described by a polynomial mapping function with an arbitrary number of terms. The potential functions representing the stress field in the matrix are obtained in an explicit form using the Cauchy integral techniques other than the pure series expansion techniques which are not that wieldy in expanding the derivation. In addition to the explicit solution, specific examination of the stress distribution around some polygonal rigid inclusions is conducted by increasing the number of terms in the mapping function (a strictly polygonal shape of the inclusion corresponds to an infinite number of terms, and since inclusions in engineering are often chamfered, it is reasonable and of reference significance to take finite terms in the mapping function). We find that for a quasi-polygonal inclusion defined by a finite-term mapping function, increasing number of terms in the mapping function leads to only a slight change in the stress distribution at a large majority of the interface, and particularly not all the stress components in the vicinity of the corners of the inclusion increase as the corners of the inclusion become sharper with increasing number of terms in the mapping function. Specially, for the stress components that are intensified unboundedly at the corners of the inclusion with increasing number of terms in the mapping function, we establish and numerically justify a simple exponential relation between them and the curvature of the interface at the corners.

Automated summary

This paper investigates the plane deformation of a rigid inclusion embedded in an infinite elastic matrix under a uniform remote stress. The stress field in the matrix is represented by potential functions obtained using Cauchy integral techniques. The examination of stress distribution around polygonal rigid inclusions by increasing the number of terms in the mapping function reveals a simple exponential relationship between the stress components and the curvature of the interface at the corners.