Simplified discrete model for axisymmetric dielectric elastomer membranes with robotic applications

TL;DR

This work introduces a one-dimensional discrete differential geometry (DDG) model for axisymmetric dielectric elastomer membranes to study nonlinear electroelastic dynamics under inflation and voltage. The framework integrates elastic kinematics, hyperelastic constitutive laws, pneumatic and electrical actuation, and contact within an implicit time-integration scheme, enabling efficient, near real-time simulations. Thorough validation against hyperelastic benchmarks and electroelastic cases shows accurate predictions of pressure- and voltage-induced responses and snap-through phenomena. The authors illustrate three soft-robotic applications—a soft circular pump, a toroidal gripper, and a spherical actuator—highlighting how electrical loading amplifies functional ranges via snap-through. Overall, the DDG approach offers a fast, versatile tool for design, control, and optimization of electroelastic actuators in soft robotics.

Abstract

Soft robots utilizing inflatable dielectric membranes can realize intricate functionalities through the application of non-mechanical fields. However, given the current limitations in simulations, including low computational efficiency and difficulty in dealing with complex external interactions, the design and control of such soft robots often require trial and error. Thus, a novel one-dimensional (1D) discrete differential geometry (DDG)-based numerical model is developed for analyzing the highly nonlinear mechanics in axisymmetric inflatable dielectric membranes. The model captures the intricate dynamics of these membranes under both inflationary pressure and electrical stimulation. Comprehensive validations using hyperelastic benchmarks demonstrate the model's accuracy and reliability. Additionally, the focus on the electro-mechanical coupling elucidates critical insights into the membrane's behavior under varying internal pressures and electrical loads. The research further translates these findings into innovative soft robotic applications, including a spherical soft actuator, a soft circular fluid pump, and a soft toroidal gripper, where the snap-through of electroelastic membrane plays a crucial role. Our analyses reveal that the functional ranges of soft robots are amplified by the snap-through of an electroelastic membrane upon electrical stimuli. This study underscores the potential of DDG-based simulations to advance the understanding of the nonlinear mechanics of electroelastic membranes and guide the design of electroelastic actuators in soft robotics applications.

Figures (8)

• Figure 1: The three dimensional axisymmetric electroelastic membrane (a) can be generated by employing an order-reduced curve (b) and rotating it with respect to the $Z$ axis. (c) The membrane possesses an undeformed thickness $\bar{h}$ and a deformed thickness $h = \lambda \bar{h}$ when subjected to loads, including mechanical forces and electric potential difference between its top and bottom surfaces.
• Figure 2: Validations of three example hyperelastic structures: (a) The inflation of a circular plate with radius $\bar{R} = 7.5$ m and reference results are presented in liu2023computational. (b) A spherical balloon with radius $\bar{R}=10$ m and thickness $\bar{h}=0.1$ m and analytical solution is shown in holzapfel2002nonlinear. (c) The inflation of a toroidal membrane with outer radius $\bar{R}_{b} = 10$ m, inner radius $\bar{R}_{s} = 2$ m, and thickness $\bar{h}= 0.1$ m, with reference solutions generated using the approach outlined in liu2020coupled. The normalised pressure $\hat{p}$ is plotted against (d) the maximum normalised displacement during the pressurisation of the circular plate, (e) the stretch during the inflation of the spherical balloon, and (f) maximum normalised displacement during the inflation of the toroidal membrane.
• Figure 3: An electroelastic spherical balloon with radius $\bar{R} = 10$ m, thickness $\bar{h} = 0.1$ m, and $\hat{c}_1/\hat{c}_2 = 7$. (a) Variation of normalized pressure $\hat{p}$ over the stretch $\lambda$. The electrical load $\hat{E}$ varies within a set of values ${0}$, ${0.1538}$, ${0.2308}$, and ${0.3077}$. (b) Variation of electrical load over the stretch $\lambda$. The normalized pressure $\hat{p}$ varies within a set of values $\{9.5, 10.7, 11.8, 13.0, 14.2\} \times 10^{-3}$. Here, the symbols represent the discrete simulation, and the lines are from the analytical solution.
• Figure 4: An electroelastic toroidal membrane with outer radius $\bar{R}_b = 10$ m, inner radius $\bar{R}_s = 2$ m, thickness $\bar{h}=0.1$ m, and $\bar{c}_1/\bar{c}_2 = 7$, is examined. (a) Variation of normalised pressure $\hat{p}$ over the maximum normalised displacement $u_m/\bar{R}_b$. The electrical load $\hat{E}$ varies within a set of values $\{ 0, 0.1538, 0.2308, 0.3077 \}$. (b) Variation of electrical load over the maximum normalised displacement $u_m/\bar{R}_b$. The normalised pressure $\hat{p}$ varies within $\{ 0.0260, 0.0284, 0.0308, 0.0331, 0.0355 \}$. (c) Variation of the normalised pressure at limit point $\hat{p}_s$ over the aspect ratio $\bar{R}_s/\bar{R}_b$, while the electrical load $\hat{E}$ varies within $\{0, 0.1538, 0.3077, 0.4615 \}$. (d) Variation of electrical load at limit point over the aspect ratio $\bar{R}_s/\bar{R}_b$, while the normalised pressure $\hat{p}$ varies within $\{0.0260, 0.0284, 0.0710, 0.0947 \}$. Here, the symbols are from 3D FEM analysis and the lines are from the discrete simulation.
• Figure 5: A soft fluid pump utilizing an electroelastic circular membrane, with radius $\bar{R} = 0.1$ m and thickness $\bar{h}=10^{-4}$ m. (a) Illustrations are provided of the unactuated and actuated states of the fluid pump. (b) The relationship between the enclosed volume and the fluid pressure inside the membrane is shown. (c) Temporal evolution of the enclosed volume. (d) The temporal evolution of the inside fluid pressure and the electric potential applied to the membrane.