Nonlinear oscillation of a dielectric elastomer balloon

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Jian Zhu, Shengqiang Cai, and Zhigang Suo

Polymer International 59 (2010) p. 378–383

wiki entry by Emily Russell, Fall 2010

The article can be found here.

Overview

This paper gives a theoretical treatment of a balloon made of a dielectric elastomer (elastic polymer), where the radius of the balloon is affected not only by internal pressure, but also by the voltage applied across the membrane. Under appropriate conditions, such a balloon will oscillate around a stable equilibrium radius; these oscillations may be excited not only by a driving voltage at the resonant frequency, but super- and sub-harmonically by driving voltages at fractional and integer multiples of the resonant frequency.

Model

The balloon is considered as a spherical, elastic membrane which is incompressible (constant total volume), and has a fixed dielectric constant. The system is then described in terms of the stretch of the balloon, $\lambda = r / R$, where r is the deformed radius of the balloon and R its undeformed radius, and the electric displacement, $D = Q / 4 \pi r^2$, with Q the magnitude of the surface charges induced by the applied voltage.

The free-energy function of the balloon is modeled as having a dielectric contribution $D^2 / 2\epsilon$, and a neo-Hookian elastic contribution, $\mu / 2 * (2\lambda^2 + \lambda^{-4} - 3)$. The variation in free energy with $\lambda$ and D is then equated to the virtual work done by the applied voltage, the internal pressure, and the inertial force as the radius changes, to give a nonlinear differential equation for $\lambda$, $\frac{d^2 \lambda}{d T^2} + 2\lambda - 2\lambda^{-5} - (\frac{pR}{\mu H})\lambda^2 - 2(\frac{\epsilon \Phi^2}{\mu H^2}) \lambda^3 = 0$.

The balloon is first treated as being under constant pressure and voltage; up to some critical pressure, the balloon can reach an equilibrium stretch, and the natural frequency of oscillations around this equilibrium is calculated; both $\lambda_{eq}$ and $\omega_0$ are calculated algebraically. An AC voltage component is then added to the model, and the response of the balloon to this driving force is computed numerically.

An interesting note is that the final equations are all presented in terms of dimensionless quantities; the stretch itself is dimensionless, and the time, pressure, and voltage are all scaled by the appropriate combinations of the density, undeformed radius and thickness, shear modulus, and dielectric constant of the material. Dimensionless quantities - as discussed at some length in the reading for this topic - allow the direct comparison of systems which have different parameters but the same intrinsic timescale, or determination of the result of the application of a given pressure or voltage to one system by comparison to the effect of the same scaled pressure or voltage to a very different system.

Results

Figure 1. Subject to a static pressure and voltage, the balloon may reach a state of equilibrium. The pressure is plotted as a function of the equilibrium stretch at several values of the voltage.

For a given applied voltage, there is some critical pressure above which there is no equilibrium; the balloon will expand indefinitely (until in a real system it pops). This critical pressure is lowered by increasing the applied voltage, which thins the membrane; at a given sub-critical pressure, the equilibrium stretch therefore also increases with applied voltage (see Fig. 1). Conversely, at a given pressure, there is a critical voltage, above which the balloon is unstable. For pressures and voltages which allow a stable equilibrium, a small perturbation from equilibrium will oscillate with some characteristic frequency.

Figure 2. Excited by a sinusoidal voltage, the balloon resonates at several values of the frequency of excitation $\Omega$. The oscillating amplitude of the balloon is plotted as a function of the frequency of excitation for $p R / \mu H = 0.1$ and $\epsilon \Phi_{dc}^2 / \mu H^2 = 0.1$, at selected values of $\Phi_{ac} / \Phi_{dc}$.

The more interesting results are the response to a driving force, here applied through an oscillating component of the voltage. The balloon resonates strongly when driven at its natural frequency, but also when driven at twice the natural frequency, or at a fraction of the natural frequency (Fig. 2); these resonances are results of the nonlinearity of the system. When driven at these frequencies, the balloon still oscillates at its natural frequency (in contrast to a simple harmonic oscillator, which always oscillates at the drive frequency), hence the terms sub- and super-harmonic. The authors note briefly that harmonic and superharmonic resonances have been observed experimentally, but that subharmonic resonance has not yet been seen.

The authors also mention that a hysteresis is observed in the system; if the excitation frequency is varied in time around the natural frequency, the amplitude of the response exhibits a discontinuous jump, and this jump occurs at a different frequency depending on whether the frequency is increasing or decreasing with time. The result is presented graphically, but any intuitive understanding of the hysteresis is obscured by some rather ponderous mathematics, with most of the steps left as exercises to the reader and only the final results given.

I am curious to what degree the effects seen in this model are determined by the details of the model, such as the inclusion of the $\lambda^{-4}$ term in the elastic contribution to the free energy, and to what degree they are more general. The paper presents the interesting situation of a relatively simple model giving rise to unexpected resonances and hysteresis; however, the model is not quite so simple as it might be, which detracts somewhat from the elegance of the results, and the extra complication is neither fully explained nor fully justified to the reader who is not familiar with elastomeric properties and behaviors.