Inclusions with Uniform Stress in a Bounded Elastic Domain

A single elliptical or ellipsoidal inclusion with an arbitrary uniform eigenstrain is known to achieve a constant stress field when embedded in an elastic medium provided the edge of the medium is sufficiently far from the inclusion (i.e. the interaction between the inclusion and the edge of the medium is negligible). In this paper, we aim to answer the question as to whether there exists an inclusion of certain configuration (with a uniform eigenstrain) that remains to possess a constant stress when embedded in a bounded medium whose edge interacts significantly with it. Specifically, we consider the anti-plane shear case of an inclusion with a uniform eigenstrain in a circular medium with a traction-free edge. We derive a sufficient and necessary condition ensuring the uniformity of the stress within the inclusion, which further leads to a nonlinear system of equations with respect to an infinite group of parameters characterizing the shape of the inclusion. We obtain convergent solutions for the truncated version of the nonlinear system using numerical techniques, and illustrate the corresponding shape of the inclusion in a few numerical examples. Our results for the case corresponding to small inclusion size and small edge-inclusion distance (relative to the radius of the medium) are well-consistent with the existing results for an inclusion with uniform stress in a semi-infinite medium with a traction-free surface, while those for centrally placed inclusions achieving uniform stress capture the classical case of centric circular inclusion accurately. The results presented in this paper provide a strong evidence for the existence of inclusions possessing uniform stress in an elastic bounded domain subjected to common external boundary conditions under anti-plane shear deformation.

Automated summary

An elliptical or ellipsoidal inclusion with a uniform eigenstrain can generate a constant stress field in an elastic medium, assuming that the edge of the medium does not significantly interact with the inclusion. This paper investigates the possibility of achieving uniform stress in an inclusion with a uniform eigenstrain placed in a bounded medium with a traction-free edge. The study focuses on the anti-plane shear case of an inclusion in a circular medium and establishes a condition for the uniformity of stress within the inclusion. Numerical techniques are employed to find convergent solutions for a truncated version of the nonlinear system of equations, and the shape of the inclusion is illustrated through numerical examples. The results provide evidence for the existence of inclusions with uniform stress in elastic bounded domains subjected to common external boundary conditions under anti-plane shear deformation.