Nonlinear electroelastic deformations of dielectric elastomer composites: I—Ideal elastic dielectrics

This paper puts forth homogenization solutions for the macroscopic elastic dielectric response—under finite deformations and finite electric fields—of ideal elastic dielectric composites with two-phase isotropic particulate microstructures. Specifically, solutions are presented for three classes of microstructures: (i) an isotropic iterative microstructure wherein the particles are infinitely polydisperse in size, (ii) an isotropic distribution of polydisperse spherical particles of a finite number of different sizes, and (iii) an isotropic distribution of monodisperse spherical particles. The solution for the iterative microstructure, which corresponds to the viscosity solution of a Hamilton–Jacobi equation in five “space” variables, is constructed by means of a novel high-order WENO finite-difference scheme. On the other hand, the solutions for the microstructures with spherical particles are constructed by means of hybrid finite elements. Prompted by the functional features shared by all three obtained solutions, a simple closed-form approximation is proposed for the macroscopic elastic dielectric response of ideal elastic dielectric composites with any type of (non-percolative) isotropic particulate microstructure. As elaborated in a companion paper, the proposed approximate solution proves particularly useful as a fundamental building block to generate approximate solutions for the macroscopic elastic dielectric response of dielectric elastomer composites made up of non-Gaussian dielectric elastomers filled with nonlinear elastic dielectric particles.