This paper aims to investigate the local oscillations of a dielectric elastomer with emphasis on the effect of the second invariant of the deformation tensor in finite strain theory on such oscillations. Four hyperelastic constitutive models are utilized in this study, namely the classic Gent model, a modified Gent model, a modified version of the Pucci–Saccomandi model, and the Mooney–Rivlin model. We derive the governing equations for all the models through the use of the Euler–Lagrange formulation. We explore the local responses of the system in the framework of the nonlinear frequency responses by solving the equations of motion using the multiple time scales method. In addition, to provide an in-depth analysis, we analyze static and dynamic electromechanical instabilities by plotting the voltage-stretch diagram and the time signature. The numerical results for damped and undamped dielectric elastomers are simulated and compared. We analyze the system’s response for the aforementioned hyperelastic constitutive functions, and compare the differences and similarities of models. Based on the results, two saddle–node bifurcations occur in the system. Generally, increasing the second invariant parameters of the hyperelastic models, decreases the response amplitude. As contribution of the second invariant in the Mooney–Rivlin model is increased, for both damped and undamped systems, softening turns into hardening nonlinearity. We also show that the second invariant tunes the bifurcation points and instability of the system. Moreover, it can control the static and dynamic pull-in and snap-through voltages.

本文旨在研究介电弹性体的局部振荡，重点是有限应变理论中变形张量的第二不变量对这种振荡的影响。本研究使用了四种超弹性本构模型，即经典的 Gent 模型、修正的 Gent 模型、修正版的 Pucci-Saccomandi 模型和 Mooney-Rivlin 模型。我们通过使用欧拉-拉格朗日公式推导出所有模型的控制方程。我们通过使用多时间尺度方法求解运动方程，在非线性频率响应的框架内探索系统的局部响应。此外，为了提供深入分析，我们通过绘制电压拉伸图和时间特征来分析静态和动态机电不稳定性。对阻尼和无阻尼介电弹性体的数值结果进行了模拟和比较。我们分析了系统对上述超弹性本构函数的响应，并比较了模型的异同。根据结果​​，系统中出现两个鞍点分岔。通常，增加超弹性模型的第二个不变参数会降低响应幅度。随着 Mooney-Rivlin 模型中第二个不变量的贡献增加，对于有阻尼和无阻尼系统，软化都会变成硬化非线性。我们还表明，第二个不变量调整了系统的分岔点和不稳定性。此外，它还可以控制静态和动态的引入和快速通过电压。