Local and Global Dynamics of a Functionally Graded Dielectric Elastomer Plate

TL;DR

This paper studies the complex nonlinear vibrations of a functionally graded dielectric elastomer plate under electromechanical loads, revealing how graded parameters influence stability, resonance, and chaos, and establishing a benchmark for smart material design.

Contribution

It introduces a comprehensive analysis of local and global dynamics in graded dielectric elastomer plates, including new methods for stability and chaos detection.

Findings

Functionally graded parameters significantly affect stability and natural frequencies.

Identification of safe operating regions using Hamiltonian energy methods.

Analysis of chaos and resonance phenomena in graded elastomer plates.

Abstract

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, Poincar\'e maps, largest Lyapunov exponent criteria, bifurcation diagram of Poincar\'e maps, and frequency-stretch curves. The results provide a benchmark for developing functionally graded soft smart materials.