Comment on “On electromechanical stability of dielectric elastomers” [Appl. Phys. Lett. 93, 101902 (2008)] Free

The dielectric elastomer film will encounter electrical breaking-down frequently in its working state due to the coupling effect of electric field and mechanical force field. Zhao and co-workers^1,2 proposed that the free-energy function of any form can be used to analyze the electromechanical stability of dielectric elastomer.

Further research was done on the stability of neo-Hookean silicone based elastomer by Díaz-Calleja et al.³ The stable domain and unstable domain of dielectric elastomers under two special conditions were determined. These results can help us understand the stability performance of neo-Hookean silicone more thoroughly.

Elastic strain energy function with two material constants is used to analyze the stability of Mooney–Rivlin-type silicone under two special loading conditions in purpose of further research the stable domain of this silicone. In this research, the dielectric elastomer is taken to be incompressible.

The free energy with two material constants can be written as the following:⁴ 

where $λ1,λ2$ denote the in-plane principle stretching ratios. $C1,C2$ are material constants, which can be determined by experiments, $D∼$ is the nominal electric displacement, and $ε$ denotes the permittivity of dielectric elastomer. According to the theory of Zhao and co-workers,^1,2$s1=∂W/∂λ1$⁠, $s2=∂W/∂λ2$⁠, $E∼=∂W/∂D∼$ are obtained, where $E∼$ presents the nominal electric field.

Introducing a dimensionless parameter $k$⁠, which depends on the material and the activated shape, let $C2=kC1$⁠. When $C1$ is a constant, as $k=0$⁠, $C2=0$⁠. The system’s free-energy function is changed to the form of Zhao and co-workers^1,2 and Díaz-Calleja at al.³ Let $s1=s2=0$⁠, $λ1=λ2=λ$⁠, Here, nominal electric field and nominal electric displacement are exactly expressed as

According to Eq. (3), the stretching ratio and the maximal nominal electric field $Emax∼$ can be obtained for different dielectric electromers by varying the value of $k$⁠. For example, in the case of $k=1/2$⁠, $Emax∼=0.9362C2/ε$⁠, and the critical stretch can be evaluated to be $λC=1.37$⁠; if $k=1/4$⁠, then $Emax∼=0.8174C2/ε$⁠, $λC=1.32$⁠; if $k=1/5$⁠, then $Emax∼=0.7925C2/ε$⁠, and $λC=1.30$⁠.

In the first case, we suppose that the in-plane stretch of dielectric elastomer is equal biaxial $λ1=λ2=λ$⁠. Let $C2=kC1$⁠, $C1>0$⁠, the free-energy function of electromechanical coupling system can be simplified as follows:

When $C1$ is a constant, as $k=0$⁠, $C2=0$⁠, The system’s free-energy function under the first special loading conditions is changed to the form of Díaz-Calleja et al.⁴ 

The corresponding nominal stress of dielectric elastomer is

Prestretch is applied to the dielectric elastomer $s>0$⁠; from Eq. (5), the range to keep the dielectric elastomer stable can be obtained,

The corresponding Hessian matrix is

Due to the steady electromechanical coupling system, Hessian Matrix is positive definite; hence,

Considering Eqs. (6) and (8), the governing equations of the dielectric elastomer’s stability are

where $y=D∼/εC1$⁠.

Figure 1 shows that for different values of $k$⁠, the steady domain of dielectric elastomer is only below both the two curves (the red and the blue lines, excluding these two curves); simultaneously, all other regions are instable. It can be seen from Fig. 1, when the material constant rate $k$ decreases, that the difference between red and blue curves increases.

In the second case, we take the stretching perpendicular to the plate or layer of polymer as $λ3=λ$⁠, $λ1=λ2=λ−1/2$ and $C2=kC1$ $(C1>0)$⁠; then the free-energy function of electromechanical coupling system can be simplified as follows:

The system’s free-energy function under the other special loading conditions is changed to the form of Díaz-Calleja et al.⁴ 

The nominal stress of the dielectric elastomer is

The corresponding Hessian matrix is

In Eq. (11) $s>0$ and the Hessian matrix [Eq. (12)] is positive definite; the governing equations of the dielectric elastomer’s stability in this case are

where $y=D∼/εC1$⁠.

Figure 2 shows that for different values of $k$⁠, the steady domain of dielectric elastomer is only between the red and the blue lines (excluding these two curves); simultaneously, all other regions are instable.

In this paper, the free-energy function with two material constants is introduced to analyze the electromechanical stability domain of dielectric elastomers. The stability governing equations and the stability domain of different dielectric elastomers are obtained in the case of different values of $k$⁠; simultaneously the nominal critical electric field corresponding to the different values of $k$ is calculated. These results provide useful guidelines for the design and fabrication of actuators based on dielectric elastomer.