Deformation dependent dielectric permittivity and its effect on actuator performance and stability

We utilize a model for birefringence/permittivity based on the statistical mechanics of a Gaussian polymer chain to construct a relationship for the dependence of the dielectric permittivity of an elastomer on a general 3-dimensional state of deformation. The model, due to Kuhn and Grün (1942 [1]), expresses the birefringence/permittivity of a Gaussian polymer chain elastomer as a function of the end-to-end distance of the chains, and assumes that the motions of the chains are affine to the overall deformation. The outcome is an expression for the permittivity tensor of the elastomer as a function of its stretch ratios. The permittivity is isotropic in the undeformed state and under pure dilatation, but otherwise becomes anisotropic during deformation. With this model, we use the free energy of the elastomer to compute the response of a neo-Hookean thin film in an actuator configuration subject to electric and mechanical loading for conditions where the permittivity in the through thickness direction is allowed to increase or decrease with the in-plane extension of the thin film. With such an approach, we study the deformation characteristics of the actuator and its stability under through thickness electric fields. Our calculations show that the deformation dependent permittivity can hasten or postpone an electromechanical instability that can cause a sudden thinning of the dielectric, accompanied by in-plane stretching, when the through thickness electric field is raised above a critical magnitude. Specifically, we consider the case of an actuator exhibiting a through thickness permittivity that decreases with in-plane extension. We observe that in such an actuator the instability is delayed to a higher electric field than would be the case if the dielectric permittivity were independent of strain. Furthermore, we establish that upon removal of the electric field the system follows a different path in terms of potential versus charge, and so develops a hysteresis loop, similar to that identified by Zhao et al. (2007 [2]) for dielectric elastomers with constant isotropic permittivity, but that stiffen during straining.