M-integral for finite anti-plane shear of a nonlinear elastic matrix with rigid inclusions

The path-independent M-integral plays an important role in analysis of solids with inhomogeneities. However, the available applications are almost limited to linear-elastic or physically non-linear power law type materials under the assumption of infinitesimal strains. In this paper we formulate the M-integral for a class of hyperelastic solids undergoing finite anti-plane shear deformation. As an application we consider the problem of rigid inclusions embedded in a Mooney–Rivlin matrix material. With the derived M-integral we compute weighted averages of the shear stress acting on the inclusion surface. Furthermore, we prove that a system of rigid inclusions can be replaced by one effective inclusion.