Multiplicative electro-elasticity of electroactive polymers accounting for micromechanically-based network models

Electro-active polymers are materials which exhibit coupled electro-mechanical behavior at large strains. They respond by a deformation to an applied electrical field and are applied in advanced industrial environments as sensors and actuators, for example in robotics, biomimetics and smart structures.A predictive modeling of these materials must account for characteristic features of their microstructure, consisting of an amorphous matrix of cross-linked polymer chain matrix and possibly a fraction of crystalline particles. The polymer network has an important effect on the electro-mechanical overall response, in particular in the large strain regime. The key intention of this work is to outline a general modeling structure for electroactive polymers on the macroscopic continuum level, that incorporates existing micromechanically-based network models for cross-linked polymers in a modular format. In order to account for electric dipoles associated with crystalline particle aggregates embedded into the polymer network, we consider a link to the network model based on multiplicative decompositions of the deformation gradient into electrically-induced and stress-producing parts. This includes a separate constitutive modeling of an electrically-induced stretch driven by the particle dipoles. Here, we incorporate Lee- and Clifton-type right and left decompositions, where the latter seems more appropriate for the modeling of electroactive polymers due to its dependence on the true electric field. Furthermore, micromechanical structures for a deformation-dependent permittivity of the polymer chain matrix are taken into account. We develop details of a unified modeling structure and its numerical implementation for those alternative kinematic assumptions, and combine it with the so-called microsphere network model of rubber elasticity, that exploits a homogenization over a chain orientation space. This provides an advanced model problem for the application of the proposed constitutive framework for EAPs.