To facilitate the understanding of the mechanisms underlying the electric breakdown of dielectric elastomers, we derive a one-dimensional (1d) model for axisymmetric necking in a dielectric membrane subjected to equibiaxial stretching and an electric field, starting from the three-dimensional (3d) nonlinear electroelasticity theory. Our reduction is built on the variational asymptotic method, so that the resulting 1d model is asymptotically self-consistent. The 1d model offers an easier and more efficient way to analyze axisymmetric necking in a dielectric membrane in the linear, weakly nonlinear and fully nonlinear regimes. It delivers results identical to the 3d theory in the linear and weakly nonlinear regimes, and near-identical results in the fully nonlinear regime due to its asymptotic self-consistency. We demonstrate the straightforward implementation of this 1d model by solving it using the Rayleigh--Ritz method and validate it by comparison with finite-element simulations. The 1d model enables a precise calculation of the minimum thickness that a dielectric membrane can reach when necking instability occurs and quantitative assessment of the effects of imperfections so that the integrity of a dielectric elastomer actuator can be evaluated with respect to electric breakdown. The developed methodology is not problem-specific and can also be applied to analyze similar phenomena in other soft materials subjected to any fields (e.g., the axisymmetric necking of stretched plastic membranes).
First integrals of the axisymmetric shape equation of lipid membranes
脂膜的形状方程[Zhong-can and Helfrich（1987）PRL 59 2486]是一个四阶偏微分方程。在轴对称条件下，Zheng和Liu [Zheng and Liu（1993）PRE 48 2856]将该方程转化为二阶常微分方程。在这里，我们试图进一步减少这个二阶常微分方程的一阶常微分方程寻求第一积分根据诺特定理。当二阶常微分方程在构象变换下不变时，我们确实找到了第一积分。利用薄膜应力张量得到了第一积分的力学解释。
Simplified discrete model for axisymmetric dielectric elastomer membranes with robotic applications
Hohai University（河海大学）；University of Birmingham（伯明翰大学）；Nanyang Technological University（南洋理工大学）；University of Michigan（密歇根大学）；Newcastle University（纽卡斯尔大学）
Biomembranes undergo complex, non-axisymmetric deformations governed by Kirchhoff-Love kinematics and revealed by a three dimensional computational framework
