This paper develops the equilibrium equations describing the flexoelectric effect in soft dielectrics under large deformations. Previous works have developed related theories using a flexoelectric coupling tensor of mixed material–spatial character. Here, we formulate the model in terms of a flexoelectric tensor completely defined in the material frame, with the same symmetries of the small-strain flexocoupling tensor and leading naturally to objective flexoelectric polarization fields. The energy potential and equilibrium equations are first expressed in terms of deformation and polarization, and then rewritten in terms of deformation and electric potential, yielding an unconstrained system of fourth order partial differential equations (PDE). We further develop a theory of geometrically nonlinear extensible flexoelectric rods under open and closed circuit conditions, with which we examine analytically cantilever bending and buckling under mechanical and electrical actuation. Besides being a simple and explicit model pertinent to slender structures, this rod theory also allows us to test our general theory and its numerical implementation using B-splines. This numerical implementation is robust as it handles the electromechanical instabilities in soft flexoelectric materials.

本文提出了描述大变形下软电介质中的柔电效应的平衡方程。先前的工作已经使用混合材料的柔电耦合张量（空间特性）开发了相关理论。在这里，我们根据材料框架中完全定义的柔电张量来制定模型，具有与小应变柔耦合张量相同的对称性，并自然导致目标柔电极化场。势能方程和平衡方程首先用形变和极化表示，然后用形变和电势重写，得到不受约束的四阶偏微分方程组（PDE）。我们进一步发展了一种在开路和闭路条件下的几何非线性可扩展挠性电杆的理论，通过该理论，我们研究了在机械和电驱动下的悬臂弯曲和屈曲分析。除了是与细长结构相关的简单而明确的模型之外，这种杆理论还使我们能够使用B样条曲线测试通用理论及其数值实现。这种数值实现是稳健的，因为它可以处理软柔性电材料中的机电不稳定性。