Modeling electromechanical properties of layered electrets: Application   of the finite-element method
  We present calculations on the deformation of two- and three-layer electret systems. The electrical field is coupled with the stress-strain equations by means of the Maxwell stress tensor. In the simulations, two-phase systems are considered, and intrinsic relative dielectric permittivity and Young's modulus of the phases are altered. The numerically calculated electro-mechanical activity is compared to an analytical expression. Simulations are performed on two- and three-layer systems. Various parameters in the model are systematically varied and their influence on the resulting piezoelectricity is estimated. In three-layer systems with bipolar charge, the piezoelectric coefficients exhibit a strong dependence on the elastic moduli of the phases. However, with mono-polar charge, there is no significant piezoelectric effect. A two-dimensional simulation illustrated that higher piezoelectricity coefficients can be obtained for non-uniform surface charges and low Poisson's ratio of phases. Irregular structures considered exhibit low piezoelectric activity compared to two-layer structures. 
Enis Tuncer, Michael Wegener and Reimund Gerhard-Multhaupt