Comment on “Method to analyze electromechanical stability of dielectric elastomers”[Appl. Phys. Lett. 91, 061921 (2007)] Free

The elastic strain energy functional with two material constants can be written as follows:

where $λ1,λ2$ denote the in-plane principle stretch ratios and $C1$ and $C2$ are two material constants. According to Suo’s theory,¹ the free energy $W(λ1,λ2,D∼)$ is given by

where $D∼$ is the nominal electric displacement. Taking derivatives, according to Suo’s theory,^1–3 we obtain

where $E∼$ presents the nominal electric field and $ε$ denotes the electric constant of dielectric elastomer. Note the formulas that appeared in Suo’s theory¹ can be recovered if $C2$ in Eqs. (4) and (5) is taken to be zero.

The Hessian matrix is invoked herein,

The dielectric elastomer is uniformly prestretched such that the in-plane prestretch ratios equal each other, i.e., $s1=s2=s$⁠,$λ1=λ2=λ$⁠. Introducing a dimensionless quantity $k$⁠, which depends on the material and the activated shape, $C2=kC1$ when $C1$ is a constant, as $k=0$⁠, $C2=0$⁠. The system free energy function is changed to Suo’s form.¹ One can nondimensionalize $D∼$ and $E∼$ using Eqs. (4)–(6),

Figure 1 shows the maximum value of $E∼/C2/kε$ at nominal electric displacement $D∼/εC2/k$ with different ratios of $s/C2$ for $k=1,1/2,1/4,1/5$⁠. For $k=1/2$⁠,⁴ it is given as $Emax∼=0.9362C2/ε$⁠. Taking $C2=0.25×106Pa$⁠, $ε=4×10−11F/m$⁠, one can further obtain that $Emax∼≈1.04×108V/m$⁠, which is approximately the same magnitude as the reported breakdown field value. Also for $k=12$⁠, the critical stretch can be evaluated to be $λC=1.37$ and the corresponding strain in the thickness direction as 0.46. To compare, for the same $C1$ and $ε$⁠, the above quantities for the case of $k=15$ can be shown to be $Emax∼=0.7925C2/ε≈1.40×108V/m$⁠, $λC=1.30$⁠, and the strain in the thickness direction as 0.41. This indicates that for dielectric elastomer material or dielectric elastomer actuator of different structures, smaller $k$ leads to larger $Emax∼$⁠, smaller strain in thickness direction, and smaller critical elongation. Based on this observation, the method exposed above is capable of analyzing the stability of dielectric materials having different configurations and $k$’s. Specifically, one may conclude that smaller $k$ enhances markedly the stability.

Figure 2 reveals the relation between the two dimensionless quantities $λ1/λ1P$ and $E∼/C2/kε$⁠, where $λ1P$ is the prestretch ratio of dielectric elastomer and $λ1$ is the stretch ratio produced by the electric field. As shown in the figure, when the nominal stress increases, $E∼/C2/kε$ decreases. The curve matches well with both the experimental observations and Suo’s results.¹ 

To summarize, the elastic strain energy functional with two material constants is capable of analyzing the stability of dielectric elastomers. For dielectric material with smaller dimensionless constant $k$⁠, the corresponding nominal electric breakdown voltage becomes higher, which may be applied in the design of dielectric elastomer actuators.