Catastrophic Thinning of Dielectric Elastomers

We provide an energetic insight into the catastrophic nature of thinning instability in soft electroactive elastomers. This phenomenon is a major obstacle to the development of giant actuators, yet it is neither completely understood nor modeled accurately. In excellent agreement with experiments, we give a simple formula to predict the critical voltages for instability patterns; we model their shape and show that reversible (elastic) equilibrium is impossible beyond their onset. Our derivation is fully analytical, does not require finite element simulations, and can be extended to include prestretch and various material models.

Figure 1

(Right panels a,b) First recorded experimental evidence of catastrophic localization of thinning deformations (pull-in instability). Reprinted from [3], with the permission of AIP Publishing. (Left panel) Sketch of the experimental setting and qualitative description of the localizations.

Figure 2

(Left panel) Experimental creasing instability for nonprestretched one-side constrained films. Reprinted from [30], with the permission of AIP Publishing. (Right panel) Predicted failure precursor.

Figure 3

(a) Alignment of creases in the direction of higher prestretch as voltage increases for one-side constrained films. Reprinted from [8], with the permission of John Wiley & Sons, Inc. (b) Failure precursors with ${\lambda }_1=2$, ${\lambda }_2=1$.

Figure 4

Creasing instability after uniaxial prestretch. Comparison of experiments (the dashed line) [8] and theory (the solid curves, plotted for different values of $n$, the number of links in the Arruda-Boyce model). Here, ${\overline{E}}_c=E_c/\sqrt{2/3}$; see the Supplemental Material [15] for model calibration. A thick solid curve shows the previous qualitative trend based on dimensional or FEM analysis [8].

Figure 5

Pull-in instability in equibiaxially strained films prestretched by dead loads. A comparison of experimental (dots) [7] and theoretical (solid curves) critical voltages (in kilovolts). Curves $a$, $b$, $c$, and $d$ correspond to dead loads weighing 20, 25.5, 31, and 35.6 g, respectively: they describe homogeneous paths until failure and are modeled by the solid curves. Their intersection with the $V_c$ curve [given by Eq. (2)] corresponds to catastrophic thinning (see the Supplemental Material [15] for details on the calibration). The Hessian approach can predict only one failure, denoted by $\mathsf{x}$.

Figure 6

Electroelastic free energy $\psi /\mu$ for a one-side constrained film without prestretch (${\lambda }_1={\lambda }_2=1)$. The film is a slightly compressible neo-Hookean material of thickness $h=0.01L$, where $L$ is a typical lateral length scale and $\kappa /\mu =1000$, with $\kappa$ being the initial bulk modulus [see Eqs. (7) and (8) in the Supplemental Material 15]. For $E<E_c=0.816$, the homogeneous solution with ${\lambda }_3=1$ is the unique energy minimizer; see the upper surface. Above $E_c$, the energy is still convex in ${\lambda }_3$, but it becomes nonconvex in $\nabla {\lambda }_3$; see the lower surface. Failure precursors are then energetically favored, but no energy minimizers exist. In the shaded Hessian plane (at $\parallel \nabla {\lambda }_3\parallel =0$), Hessian stability takes place; there, the energy is convex in ${\lambda }_3$ and the failure threshold cannot be captured.