Due to the inherent viscosity of polymer, piezoelectric response in the thickness direction $\left(d_{33}\right)$ of cellular ferroelectret films usually depends on the time of measurement. In this letter, the micromechanical theory of viscoelastic composite was extended to predict the time dependence of the overall piezoelectric $d_{33}$ coefficient of voided charged polymer foam. Experiments were carried out to find the time spectra of piezoelectric $d_{33}$ coefficient of voided charged polypropylene film. Theoretical simulation agrees well with experiment data.

Cellular space-charge foams, which have very sensitive transducer properties and are called cellular ferroelectrets, emerged as a kind of novel electromechanical transducer materials. In recent years, the understanding of charging is advanced significantly. However, some fundamental theoretical aspects, as to the relation between the electromechanical behavior and their mesostructures, are still far from clear. As is well known, the transducer coefficients in the thickness direction strongly depend on the relative density of foam, which is essentially relevant to the mesostructure, such as void shape, void fraction, etc. In this article, a theoretical model based on micromechanics is proposed to estimate the overall electromechanical moduli of cellular ferroelectrets. From the viewpoint of composites, in this model, the voids with surplus charge are deemed as heterogeneous inclusions, which show piezoelectriclike effect under deformation. The matrix is the nonpolar polymer. Within the micromechanical framework for piezoelectric composites, the generalized Eshelby tensor is obtained for both isotropic and anisotropic matrix. The equivalent inclusion method is employed to treat inhomogeneities. Similar to the classical differential scheme for pure elasticity, the differential equation for the effective electromechanical moduli is derived, from which the overall behavior of cellular space-charge foam can be numerically solved. Detailed analysis is presented with respect to material parameters and geometrical variables of void. Finally, this model is used to simulate the inflation experiments of cellular ferroelectret film published in literature. Results show this model is capable of predicting the extremum of the effective electromechanical moduli. In the meantime, quantitative comparison indicates that theoretical simulation is in good consistency with experiment data.