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).

翻译：介电弹性体因其在软体机器人、致动器和触觉设备中的潜在应用而日益受到研究。在时变载荷作用下，它们通过粘性变形和电极化过程中的摩擦耗散能量。然而，大多数本构模型和有限元（FE）实现仅考虑机械耗散，因为机械弛豫时间远大于电弛豫时间。在处理交变电场时，考虑电耗散至关重要。Ghosh等人（2021）提出了针对各向同性、不可压缩介电弹性体的全耦合三维本构模型。我们对其用于求解描述时变行为的初始边界值问题（IBVP）的数值方案进行了批判性研究。我们发现，由于机械和电弛豫时间的显著差异，其五阶显式Runge-Kutta时间离散化在真实模拟中可能需要过小或非物理的时间步长。为解决此问题，我们提出了一种稳定的隐式时间积分算法，以克服这些限制。我们采用协调有限元离散化实现该算法来求解IBVP，并给出了在FEniCSx中实现的混合有限元公式。我们证明该方案稳健、准确，能够处理有限变形、不可压缩性和一般的时变载荷。最后，我们针对Hossain等人（2015）研究的VHB 4910在复杂时变机电载荷下的实验数据，验证了我们的代码。