Circular inclusion near a circular void: determination of elastic antiplane shear fields and configurational forces

The stress and displacement fields are determined inside and outside a circular inclusion located in the vicinity of a circular void in an infinite elastic solid, within a circular cylinder, or near the free surface of a half-space, in the case when the inclusion is characterized by a uniform eigenstrain of the antiplane shear type. The fields are obtained as the sum of their infinite-medium stress fields and the calculated auxiliary fields. It is shown that the fields outside the inclusion follow directly from the extended Milne-Thomson circle theorem, but not the fields inside the inclusion. The overall fields are interpreted as the superposition of the infinite-medium fields from the actual and the image inclusion of the appropriate location, radius, and eigenstrains. The stress amplification is evaluated for the inclusion approaching the boundary of the void, cylinder, or half-space. The configurational forces are then evaluated, associated with a relative translation of the void and inclusion, or the expansion of the void or inclusion. The J and M integrals along the boundary of the void are evaluated without using the solution of the entire boundary-value problem, but only the stress field for an inclusion in an unvoided infinite medium. This is accomplished by incorporating the result that the circumferential shear stress along the boundary of a traction-free circular void in an infinite isotropic solid under antiplane shear is twice the circumferential shear stress along the corresponding circle in an infinite solid without a void, under the same loading conditions. The energy release rate associated with a self-similar expansion of the inclusion is calculated from the determined elastic field around the inclusion and from the evaluated total strain energy of the system. The configurational forces on the inclusion in a circular cylinder and near the free surface of a half-space are also determined and discussed.