Two interacting non-elliptical rigid harmonic inclusions loaded by couples

Xu Wang, Peter Schiavone

We use conformal mapping techniques to design two interacting non-elliptical rigid inclusions, each of which is loaded by a couple, which ensure the so-called `harmonic condition' in which the original mean stress in the matrix remains undisturbed after the introduction of the inclusions. We show that for prescribed Poisson's ratio and corresponding geometric parameters, several restrictions are necessary on the external loadings to ensure the harmonic condition. It is seen from our analysis that: (i) the interfacial and hoop stresses are uniformly distributed along each of the inclusion-matrix interfaces; (ii) the interfacial normal and hoop stresses along the two interfaces are completely determined by the Poisson's ratio and the constant mean stress in the matrix whilst the interfacial tangential stress along the two interfaces can be completely determined by the moments of the couples and the areas of the two inclusions; (iii) the existence of the applied couples will influence the non-elliptical shapes of the two rigid harmonic inclusions when the moment to area ratios for the two inclusions differ.