A Hybrid finite element implementation of two-potential constitutive model of dielectric elastomers

Abstract

There has been an increasing interest in the constitutive modeling of dielectric elastomers due to their potential in enabling new technologies such as soft robotics, actuators and haptic devices. Under realistic time-dependent loadings, dielectric elastomers are inherently dissipative. They dissipate energy both through viscous deformation and through friction in their electric polarization process. However, a majority of constitutive models and their corresponding Finite Element (FE) implementations consider only mechanical dissipation. The main reason for this bias is that the mechanical relaxation time of dielectric elastomers is much larger than their electric relaxation time. However, accounting for electric dissipation, in addition to mechanical dissipation, is crucial when dealing with applied alternating electric fields. A fully coupled 3-D constitutive model for isotropic and incompressible dielectric elastomers was proposed by Ghosh and Lopez-Pamies (2021). In this paper, we critically investigate the numerical scheme proposed in this paper to solve the initial boundary value problem (IBVP) that describes the time-dependent behavior of dielectric elastomers. We find that the scheme in Ghosh and Lopez-Pamies (2021), employing a fifth-order explicit Runge–Kutta time discretization, may lead to excessively small or nonphysical time steps for IBVPs simulating the behavior of real-world elastomers. This is because of the stark contrast in the relaxation times of mechanical dissipation and electric polarization. To this end, we first present a stable implicit time-integration algorithm that overcomes the unrealistic time-step constraints imposed by the fifth-order explicit Runge–Kutta algorithm in Ghosh and Lopez-Pamies (2021). We then deploy the algorithm along with a conforming FE discretization to solve the IBVP. We present implementations of the mixed-FE formulation of the governing equations for dielectric elastomers in FEniCSx. We also show that the numerical scheme is robust, accurate, capable of handling finite deformations, the incompressibility constraint of the rubber, and general time-dependent loading conditions. In the last part, the FE code is deployed to validate the experimental findings describing the electromechanical behavior of VHB 4910 (from 3M) under a complex time-dependent electromechanical load as studied in Hossain et al. (2015).