Effects of couple stresses on anti-plane problems of piezoelectric media with inhomogeneities

The examination of the effect of couple stresses on anti-plane electro-mechanical behaviour of piezoelectric media is of interest. The constitutive equations of piezoelectricity for a transversely isotropic piezoelectric medium of crystal class C6v=6 mm are derived in the context of couple stress elasticity. In this framework, a characteristic length appears in the formulation of anti-plane problems, by which examination of the size effect is possible. Also stemming from this approach is a new elasticity constant defined as the ratio of couple stress to the curvature, which based on the assumption of positive definiteness of the internal energy density, must be positive. For demonstration, this paper considers a piezoelectric-inhomogeneity system consisting of a circular piezoelectric inclusion or void inside an infinite body made of piezoelectric or solely elastic material. A special case pertinent to the circular piezoelectric inclusion embedded in a piezoelectric matrix, which was previously examined within the limitations of the classical piezoelectricity, is re-examined by the proposed theory. It is shown that the mechanical and electrical fields drastically depend on the relative size of the inhomogeneity with respect to the characteristic length of the material. The effects of couple stresses on the stress and electric concentrations are also addressed.