Effective elastic, dielectric and piezoelectric coefficients are crucial for modeling the constitutive behavior of piezo-and ferroelectric materials. A homogenization process provides the average properties of a representative volume element and allows for the mathematical treatment of boundary value problems within the context of classical continuum mechanics of homogeneous media. Cracks, in particular, have an impact on all these effective coefficients. A sophisticated model has to account for crack interaction as well as appropriate boundary conditions regarding electric charges and induced stresses on the crack faces. Results are obtained applying analytical homogenization approaches leading to closed-form solutions in the limiting case of non-interacting cracks.

In this paper we present two subjects of our actual research in the field. The first deals with the boundary conditions at the crack faces. The well known model by Hao and Shen gives opportunity to take the finite dielectric permeability of the crack into account, without having to solve the two-or three-dimensional coupled boundary value problem of solid material and crack medium. This approach, however, is based on the assumption of the electric field being perpendicular to the crack faces. We investigate this problem for arbitrary poling and field directions based on a combined analytical-numerical approach. The second focus of the paper is on the effective properties of piezoelectrics with cracks. Here, homogenization procedures are applied and extended towards coupled field problems including e.g. Maxwell stresses at internal boundaries and interfaces. Effective elastic, dielectric and piezoelectric constants exhibit interesting effects.

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The coupling of electric, magnetic and mechanical phenomena may have various reasons. The famous Maxwell equations of electrodynamics describe the interaction of transient magnetic and electric fields. On the constitutive level of dielectric materials, coupling mechanisms are manyfold comprising piezoelectric, magnetostrictive or magnetoelectric effects. Electromagnetically induced specific forces acting at the boundary and within the domain of a dielectric body are, within a continuum mechanics framework, commonly denoted as Maxwell stresses. In transient electromagnetic fields, the Poynting vector gives another contribution to mechanical stresses. First, a system of transient partial differential equations is presented. Introducing scalar and vector potentials for the electromagnetic fields and representing the mechanical strain by displacement fields, seven coupled differential equations govern the boundary value problem, accounting for linear constitutive equations of magnetoelectroelasticity. To reduce the effort of numerical solution, the system of equations is partly decoupled applying generalized forms of Coulomb and Lorenz gauge transformations [1,2]. A weak formulation is given to establish a basis for a finite element solution. The influence of constitutive magnetoelectric coupling on electromagnetic wave propagation is finally demonstrated with a simple one-dimensional example. Balance and constitutive equationsOn the one hand the coupling of electric, magnetic and elastic fields is described by the following constitutive equations:On the other hand these fields are also coupled through the Maxwell equations (without electric current) and the momentum balance. In the momentum balance the Maxwell stress T ij and the Poynting vector S i have to be considered. To reduce the number of field equations, scalar φ and vector A i potentials, trivially satisfying two of the four Maxwell equations, are introduced. So with the governing field equationsthe two potentials and displacement u i related to the magnetic induction B i , electric field E i and strain ij as