We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and bottom of the tank, and an electric field is generated by a potential difference between the electrodes. We solve the associated incremental boundary value problem of superimposed, inhomogeneous small-amplitude wrinkles, signalling the onset of instability. We decouple the resulting bifurcation equation into symmetric and antisymmetric modes. For a neo-Hookean dielectric plate, we show that a potential difference between the electrodes can induce a thinning of the plate and thus an increase of its planar area, similar to the scenarios encountered when there is no silicone oil. However, we also find that, depending on the material and geometric parameters, an increasing applied voltage can also lead to a thickening of the plate, and thus a shrinking of its area. In that scenario, Hessian instability and wrinkling bifurcation may then occur spontaneously once some critical voltages are reached.

中文翻译：

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外电场对介电板不稳定性的影响

我们研究了浸入充满硅油的罐中的电压激活介电板的理论非线性响应、Hessian 稳定性和可能的​​起皱行为。罐体顶部和底部放置固定刚性电极，通过电极间的电位差产生电场。我们解决了叠加的、不均匀的小幅度皱纹的相关增量边界值问题，这标志着不稳定的开始。我们将得到的分岔方程解耦为对称和反对称模式。对于新胡克介电板，我们表明电极之间的电位差会导致板变薄，从而增加其平面面积，类似于没有硅油时遇到的情况。然而，我们也发现，根据材料和几何参数，增加施加的电压也会导致板变厚，从而导致其面积缩小。在这种情况下，一旦达到某些临界电压，Hessian 不稳定性和起皱分叉可能会自发发生。