Hybrid finite element implementation of two-potential constitutive modeling of dielectric elastomers

Abstract

Dielectric elastomers are increasingly studied for their potential in soft robotics, actuators, and haptic devices. Under time-dependent loading, they dissipate energy via viscous deformation and friction in electric polarization. However, most constitutive models and finite element (FE) implementations consider only mechanical dissipation because mechanical relaxation times are much larger than electric ones. Accounting for electric dissipation is crucial when dealing with alternating electric fields. Ghosh et al. (2021) proposed a fully coupled three-dimensional constitutive model for isotropic, incompressible dielectric elastomers. We critically investigate their numerical scheme for solving the initial boundary value problem (IBVP) describing the time-dependent behavior. We find that their fifth-order explicit Runge-Kutta time discretization may require excessively small or unphysical time steps for realistic simulations due to the stark contrast in mechanical and electric relaxation times. To address this, we present a stable implicit time-integration algorithm that overcomes these constraints. We implement this algorithm with a conforming FE discretization to solve the IBVP and present the mixed-FE formulation implemented in FEniCSx. We demonstrate that the scheme is robust, accurate, and capable of handling finite deformations, incompressibility, and general time-dependent loading. Finally, we validate our code against experimental data for VHB 4910 under complex time-dependent electromechanical loading, as studied by Hossain et al. (2015).

Figures (11)

• Figure 1: Rheological model of a dielectric elastomer. The elastic springs and the capacitor signify the elastomer's ability to store energy via mechanical deformation and via electric potential respectively. The dashpot and resistor on the other hand represent the elastomer's ability to dissipate energy via viscous dissipation and friction during the electric polarization process respectively. We refer the interested reader to ghosh2021two for a detailed discussion of the model.
• Figure 2: Applied Stretch $\lambda(t)$ and lagrangian electric field $E_1(t)$ considered for solving the system of nonlinear ODEs \ref{['eq:ODES_Cv_Ev']} in Sections (\ref{['sec:RK5_ODE']})-(\ref{['sec:BDF']}). The loading rates are chosen due to their practical relevance in experiments (see hossain2015comprehensive)
• Figure 3: The stress-stretch response of VHB 4910 subjected to electro-mechanical ($S^{(e)}_{un}$) and purely-mechanical ($S^{(m)}_{un}$) loadings for three different loading rates. In the electro-mechanical case, we consider both the electric and mechanical dissipation whereas in the purely-mechanical case we consider the effect of only mechanical dissipation
• Figure 4: The stress-stretch response of VHB 4910 subjected to electro-mechanical ($S^{(e)}_{un}$) with and without electric dissipation ($E^v$). We note that the stresses are practically indistinguishable from each other and hence for monotonic loadings explicit modeling of the evolution of $E^v$ may not be necessary.
• Figure 5: The stress-stretch response of VHB 4910 subject to the loadings \ref{['eq:appliedDeformations_elecF']} computed using the different time-integration algorithms summarized in Sections (\ref{['sec:RK5_ODE']})-(\ref{['sec:BDF']}). Note that the maximum stable time increment for the RK5 solver was $10^{-6}$ seconds, whereas for BE it was $10^{-2}$ seconds. The maximum stable time increment for BDF was in the range ($10^{-4}$, $10^{-3}$) seconds.

Theorems & Definitions (7)

• Remark 1: $5$th order accuracy
• Remark 2: Comparison between the different time-integration methods
• Remark 3: Stable time-increment ($\Delta t$) for the different time-integration methods
• Remark 4: Total time-to-solution (TTS) and cost
• Remark 5: Preserving the constraint $\det(\mathbf{C}^v) = 1$
• Remark 6: RK5 as a suitable choice to integrate \ref{['eq:ODES_Cv_Ev']}
• Remark 7: Backward-Euler as the time-integration algorithm