Effect of the second invariant of the Cauchy-Green deformation tensor on the local dynamics of dielectric elastomers

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.