A harmonic rigid inclusion loaded by a couple in finite plane elasticity

We consider a rigid inclusion embedded in a matrix from a particular class of compressible hyperelastic materials (so-called harmonic materials) subjected to uniform remote stresses. The inclusion is loaded by a couple, and the inclusion-matrix system undergoes finite plane deformations. We examine whether it is possible to design the shape of the inclusion to achieve the well-known 'harmonic field condition' in which the sum of the normal stresses in the uncut matrix remains undisturbed after the introduction of the inclusion. Indeed, we find that the only such rigid inclusion is elliptical. Moreover, we show that when the remote loading is symmetric, the harmonic field condition is satisfied only when two further conditions on prescribed geometric and material parameters are met: One condition requires a relatively simple linear relationship between the moment of the couple and the remote shear stress; the second condition involves a rather complex nonlinear relationship between the two remote normal stresses and the moment of the couple. Once these conditions are satisfied, the interfacial normal and tangential stresses as well as the hoop stress are uniformly distributed along the boundary of the rigid inclusion. Furthermore, the interfacial normal and hoop stresses are dependent only on the remote normal stresses and the aspect ratio of the elliptical inclusion, whereas the interfacial tangential stress depends only on the moment of the couple and the area of the inclusion.