We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character – considering either the electric field E , the electric displacement D or the electric polarization P . The first variation of the energy functional results in the set of Euler–Lagrange partial differential equations, which are the equilibrium equations, boundary conditions and certain constitutive equations for the electroelastic system. The partial differential equations for obtaining the bifurcation point have also been found using the bilinear functional based on the second variation. We show that the well-known Maxwell stress in a vacuum is a natural outcome of the derivation of equations from the variational principles and does not depend on the formulation used. As a result of careful analysis, it is found that there are certain terms in the bifurcation equation that appear to be difficult to obtain using ordinary perturbation-based analysis of the Euler–Lagrange equation. From a practical viewpoint, the formulations based on E and D result in simpler equations and are expected to be more suitable for analysing problems of stability as well as post-buckling behaviour.

中文翻译：

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使用非线性静电静力学的三种不同变分公式研究平衡方程及其扰动

我们使用三种不同的变分公式推导出非线性电弹力方程，其中包括变形函数和代表电特性的独立场变量——考虑电场 E、电位移 D 或电极化 P。能量泛函的第一个变体产生了一组欧拉-拉格朗日偏微分方程，它们是电弹性系统的平衡方程、边界条件和某些本构方程。还使用基于第二变体的双线性函数找到了用于获得分岔点的偏微分方程。我们表明真空中众所周知的麦克斯韦应力是从变分原理推导出方程的自然结果，不依赖于所使用的公式。仔细分析的结果是，发现分岔方程中有些项似乎难以使用欧拉-拉格朗日方程的普通微扰分析获得。从实用的角度来看，基于 E 和 D 的公式会产生更简单的方程，预计更适合分析稳定性和屈曲后行为问题。