A Circular Inclusion With Imperfect Interface: Eshelby’s Tensor and Related Problems

Eshelby’s tensor for an ellipsoidal inclusion with perfect bonding at interface has proven to have a far-reaching influence on the subsequent development of micromechanics of solids. However, the condition of perfect interface is often inadequate in describing the physical nature of the interface for many materials in various loading situations. In this paper, Airy stress functions are used to derive Eshelby’s tensor for a circular inclusion with imperfect interface. The interface is modeled as a spring layer with vanishing thickness. The normal and tangential displacement discontinuities at the interface are proportional to the normal and shear stresses at the interface. Unlike the case of the perfectly bonded inclusion, the Eshelby’s tensor is, in general, not constant for an inclusion with the spring layer interface. The normal stresses are dependent on the shear eigenstrain. A closed-form solution for a circular inclusion with imperfect interface under general two-dimensional eigenstrain and uniform tension is obtained. The possible normal displacement overlapping at the interface is discussed. The conditions for nonoverlapping are established.