Electroelastic Response of Isotropic Dielectric Elastomer Composites With Deformation-Dependent Apparent-Permittivity Matrix

Dielectric elastomers have been identified over the past twenty years as potential enablers of a variety of new technologies   ranging from artificial muscles for soft robotics to energy harvesters and Braille tactile displays. As a result, these stretchable dielectrics received a lot of experimental attention to determine their electromechanical response when undergoing finite deformations while being subjected to finite electric fields. Experimental evidence suggests that dielectric elastomers are so-called isotropic ideal dielectrics, that is, isotropic deformable dielectrics wherein the relation between the Eulerian electric displacement and the Eulerian electric field is linear, and that,  irrespectively of the deformations and electric fields the material is subjected to. This being true in particular when the material is undeformed and subjected to small electric fields, the constant second order-tensor - and by extension, the constant material coefficient - can be naturally identified with and referred to as the dielectric permittivity of the material.

Yet, another set of experimental data indicates that the dielectric constitutive relation between d and e in a dielectric elastomer may in fact strongly depend on the deformations the material undergoes. In contrast with the case of ideal dielectrics detailed above, these materials have been described as having a ``deformation-dependent permittivity''. Although intuitively descriptive, this terminology is slightly misleading since, strictly speaking and as recalled above, the dielectric permittivity is the constant second-order tensor that relates the electric displacement and the electric field when the material is not deformed and subjected only to small electric fields.

Aside from dielectric elastomers, dielectric elastomer composites are now being considered as promising pathways to enabling the breadth of technological applications envisioned for stretchable dielectrics thanks to their superior electromechanical properties. These materials are made out of a soft insulating elastomer matrix - in other words, a dielectric elastomer - embedding typically high-permittivity or (semi-)conducting particles. While current frameworks for the electromechanical response of dielectric elastomer composites treat the dielectric elastomer matrix as an ideal dielectric, this work is concerned with dielectric elastomer composites comprised of a dielectric elastomer matrix with a deformation-dependent apparent permittivity

This presentation puts forth an approximate yet accurate free energy for the elastic dielectric response - under finite deformations and finite electric fields - of non-percolative dielectric elastomer composites made out of a non-Gaussian dielectric elastomer matrix with deformation-dependent apparent permittivity isotropically filled with nonlinear elastic dielectric particles that may exhibit polarization saturation. While the proposed free energy applies in its most general form to arbitrary isotropic non-percolative microstructures, closed-form specializations are recorded for the practically relevant cases of rigid or liquid-like spherical particles. The proposed free energy is exact by construction in the asymptotic context of small deformations and moderate electric fields and is shown to remain accurate for arbitrary large deformations and electric fields via comparisons with full field finite-element simulations. The proposed constitutive model is deployed to probe the electrostriction response of these dielectric elastomer composites and corresponding results reveal that their elastic dielectric response strongly depends on the deformation-dependent apparent permittivity of the matrix they comprise.