We investigate the nonlinear vibrations of a functionally graded dielectric elastomer plate subjected to electromechanical loads. We focus on local and global dynamics in the system. We employ the Gent strain energy function to model the dielectric elastomer. The functionally graded parameters are the shear modulus, mass density, and permittivity of the elastomer, which are formulated by a common through-thickness power-law scheme. We derive the equation of motion using the Euler-Lagrange equations and solve it numerically with the Runge-Kutta method and a continuation-based method. We investigate the influence of the functionally graded parameters on equilibrium points, natural frequencies, and static/dynamic instability. We also establish a Hamiltonian energy method to detect safe regions of operating gradient parameters. Furthermore, we explore the effect of the functionally graded parameters on chaos and resonance by plotting several numerical diagrams, including time histories, phase portraits, Poincaré maps, largest Lyapunov exponent criteria, bifurcation diagram of Poincaré maps, and frequency-stretch curves. The results provide a benchmark for developing functionally graded soft smart materials.

我们研究了功能梯度介电弹性体板在机电负载下的非线性振动。我们关注系统中的本地和全球动态。我们采用 Gent 应变能函数来模拟介电弹性体。功能梯度参数是弹性体的剪切模量、质量密度和介电常数，它们是通过常见的全厚度幂律方案制定的。我们使用欧拉-拉格朗日方程导出运动方程，并使用龙格-库塔方法和基于连续的方法对其进行数值求解。我们研究了功能梯度参数对平衡点、固有频率和静态/动态不稳定性的影响。我们还建立了哈密顿能量方法来检测操作梯度参数的安全区域。此外，我们通过绘制几个数值图来探索功能梯度参数对混沌和共振的影响，包括时程、相图、庞加莱图、最大李雅普诺夫指数准则、庞加莱图分岔图和频率拉伸曲线。该结果为开发功能分级软智能材料提供了基准。