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Analyzing wrinkling instabilities in constrained dielectric elastomers: A symplectic eigenvalue approach

DOI：10.7511/jslx20231110002

 作者 单位 E-mail 张腾 雪城大学 机械与航空系, 美国纽约, NY 13244 tzhang48@syr.edu

辛弹性力学已广泛应用于弹性学中各种边值问题的精确解、计算表面波模式以及预测多层超弹性薄膜中的表面褶皱。本文展示了辛分析框架还可应用于受约束介电弹性体中的表面褶皱。机械和电位移向量是两个基本变量来描述介电弹性体中机械变形与电场紧密耦合。褶皱的临界电压可以通过引入基本变量的对偶变量来从辛本征值问题中解决。本文采用扩展的W-W（Wittrick-Williams）算法和精确的积分方法，准确而高效地解决制定的辛本征值问题的本征值。通过将褶皱电压和波数与有无表面能的褶皱基准结果进行比较，验证了辛分析的有效性。辛分析框架简洁且适用于其他不稳定问题，如分层电介质弹性体、磁弹性不稳定性以及层压复合结构的微观和宏观不稳定性。

Symplectic elasticity has been widely used to find the exact solutions of various boundary value problems in elasticity, compute the surface wave modes, and predict surface wrinkles in multilayer hyper-elastic films.Here, we show that the symplectic analysis framework can also be applied to surface wrinkles in constrained dielectric elastomers, where the mechanical deformation is tightly coupled with the electric field.The critical voltage for wrinkles can be solved as a symplectic eigenvalue problem after introducing the dual variables to the primary variables of mechanical and electric displacement vectors.We employ the extended Wittrick-Williams (W-W) algorithm and precise integration method to solve the eigenvalues of the formulated symplectic eigenvalue problem accurately and efficiently.The symplectic analysis is validated by comparing the wrinkle voltage and wavenumber with benchmark results of wrinkles with and without surface energy.The symplectic framework is concise and applicable to other instability problems such as layered dielectric elastomers, magnetoelastic instabilities and the micro- and macro-instabilities of laminated composite structures.