A constitutive law for dielectric elastomers subject to high levels of stretch during combined electrostatic and mechanical loading: Elastomer stiffening and deformation dependent dielectric permittivity

Abstract

We construct a constitutive law for the response of dielectric elastomers subject to high levels of stretch during combined electrostatic and mechanical loading. The constitutive law is based on a statistical mechanics analysis of a freely jointed chain, due to Kuhn and Grün [1–3], that relates the force of extension and polarizability anisotropy of a polymer chain to its fractional extension, r/nl, through the inverse Langevin function. We utilize a Padé [4] approximant that accurately represents the inverse Langevin function through the entire range of fractional extensions. Thereafter, we cast this machinery into the 8-chain lattice [5,6] and model an elastomer as a heavily interpenetrated network of 8-chain lattices. We assume that the motions of each lattice are affine with the overall deformation of the elastomer. In this fashion, the fractional extension of each chain, r/nl, is linked to the stretch ratios. With such an approach, we obtain a materially objective free energy density and an expression for the dielectric permittivity of the elastomer that depends on the current state of deformation and the overall stretch level. The elastic free energy density depends on two parameters, the small deformation shear modulus and the chain extensibility limit. We observe that the present model and the well established Arruda and Boyce [5], Gent [7], and neoHookean models are all special cases of the eight chain model of the elastic free energy density presented in this work. The isotropic part of the dielectric permittivity and the electrostrictive coefficient depend on the dilatation. The dielectric permittivity remains isotropic under a pure dilatation, but otherwise becomes anisotropic during deformation. The form of the permittivity resembles that of the deformation dependent permittivity presented by Jiménez and McMeeking [8]. However, in the model presented in this work, the electrostrictive coefficient is not only affected by dilatation but also becomes a function of the current level of deformation through the first invariant of the left Green-Cauchy tensor. We utilize the free energy density of the dielectric elastomer to compute the response of a thin film actuator subject to electrostatic and mechanical loading. In this model, the actuator is allowed to have different levels of in-plane limit stretch, and the through thickness permittivity is allowed to increase or decrease with in-plane extension of the actuator. We establish a parameter space map, extensibility limit versus electrostrictive coefficient of the elastomer, for which our constitutive law is relevant to the behavior of dielectric elastomers. With this approach, we study the actuation, electric charge storage, and stability characteristics of the actuator. From the results of our calculations we clearly identify two types of actuator behavior: actuators that exhibit electromechanical instability (type A), and actuators that do not exhibit this instability (type B). We establish that type A actuators develop hysteresis loops in a similar manner to those identified by Zhao, Hong and Suo [9] and Jiménez and McMeeking [8], for dielectric elastomers with constant isotropic permittivity that stiffen during straining, in the case of the former, and for dielectric elastomers that do not stiffen but exhibit a through thickness permittivity that increases/decreases with straining, in the case of the latter. Finally, we show that, while pre-stretch reduces the electric potential and electric charge levels required to operate the actuator, and simultaneously enhances the sensitivity of the actuator to electric potential, it has a detrimental effect on the sensitivity to electric charge.