圆形电极覆盖介质膜的轴对称颈缩
Axisymmetric necking of a circular electrodes-coated dielectric membrane
摘要：
研究了圆形电极覆盖介质膜在电场和面内张力共同作用下的稳定性。已知这种膜易受极限点不稳定性（也称为拉入不稳定性）的影响，这被广泛认为是电击穿的前兆。然而，有实验证据表明，极限点的不稳定性可能不一定是负责快速减薄和电击穿。我们探讨的可能性，后者是由于一个新的不稳定机制，即局部轴对称颈缩。首先推导了轴对称颈缩的分叉条件，并指出这种不稳定性可能发生在一类自由能函数的Treloar-Kearsley不稳定性或极限点不稳定性之前。弱非线性分析表明，近临界行为是由一个四阶非线性常微分方程的变系数。使用有限差分法求解此振幅方程，并证明了局部化的解决方案确实从齐次解分叉。基于这一分析和已知的纯机械情况下，我们可以推断，颈缩的演变遵循相同的三个阶段的启动，增长和传播的其他类似的本地化问题。目前的研究提供的见解，预计将在评估介电弹性体致动器的完整性相关。
英文摘要：
We investigate the stability of a circular electrodes-coated dielectric membrane under the combined action of an electric field and all-round in-plane tension. It is known that such a membrane is susceptible to the limiting point instability (also known as pull-in instability) which is widely believed to be a precursor to electric breakdown. However, there is experimental evidence showing that the limiting point instability may not necessarily be responsible for rapid thinning and electric breakdown. We explore the possibility that the latter is due to a new instability mechanism, namely localised axisymmetric necking. The bifurcation condition for axisymmetric necking is first derived and used to show that this instability may occur before the Treloar-Kearsley instability or the limiting point instability for a class of free energy functions. A weakly nonlinear analysis is then conducted and it is shown that the near-critical behavior is described by a fourth order nonlinear ODE with variable coefficients. This amplitude equation is solved using the finite difference method and it is demonstrated that a localised solution does indeed bifurcate from the homogeneous solution. Based on this analysis and what is already known for the purely mechanical case, we may deduce that the necking evolution follows the same three stages of initiation, growth and propagation as other similar localisation problems. The insight provided by the current study is expected to be relevant in assessing the integrity of dielectric elastomer actuators.
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