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

The paper of Zhao and Suo¹ describes a fully nonlinear electromechanical model for the phenomenon of electrical breakdown in thin elastomers. The purpose of this comment is to point out some analytical simplifications which provide further insight into their model and to provide explicit formulas useful for elastomer design.

The results here stem from the observation that the determinant of the Hessian $H$ of Eq. (4) in Ref. 1 may be factored, leading to semiexplicit formulas for the critical values of the electrical and mechanical parameters. It may be checked that the determinant reduces to a quadratic in $z$⁠,

where the nondimensional parameter $z$ is

and all other notation follows.¹ The roots of the quadratic are real and of opposite sign, so there is a unique positive value of $z$ at which the Hessian is no longer positive definite. It turns out that the same structure of the Hessian is retained for free energy of the form

where $U(λ1,λ2)=ψ(λ1,λ2,λ1−1λ2−1)$⁠. An equation similar to Eq. (1) is obtained, and taking the single positive root shows that the critical value of the electric field satisfies

where $Uij=∂2U∕∂λi∂λj$⁠. If the stretches $λ1$ and $λ2$ are prescribed, then Eq. (4) is sufficient to estimate the critical field strength. Otherwise, if the nominal stresses $s1$ and $s2$ are prescribed then the stretches are determined from

Under equibiaxial strain $λ1=λ2=λ$⁠, Eq. (4) becomes

where the critical value of the stress $s1=s2=s$ is

Consider the Ogden model for rubber² 

for which the critical electrical field strength is

If the stress is prescribed then $λ$ is given by

These parameterize the critical electrical and mechanical fields in terms of $λ$⁠.

Values of the critical breakdown voltage for the elastomer VHB $4905∕4910$ have been reported by Plante and Dubowsky³ and by Kofod et al.⁴ Assuming the Ogden model with $N=2$⁠, Plante and Dubowsky³ measured values of $α1=1.445(1.450)$⁠, $α2=4.248(8.360)$⁠, $μ1=43560(112,200)Pa$⁠, $μ2=117.4(0.1045)Pa$ for elastomer films of initial thickness $L3=1.5mm$ at low (high) stretch rates. Using these values, the critical breakdown voltage $Vc=L3Ẽc$ predicted by Eq. (9) is compared with the data of Refs. 3 and 4 in Fig. 1. The material dielectric constant was chosen as $ϵd=12$ to fit the curves with the data, where $ϵ=ϵdϵ0$ and $ϵ0=8.85×10−12F∕m$ is the free space permittivity. The agreement is reasonable, given that the experiments were not performed in a state of pure equibiaxial stress. Unlike previous theories, e.g., Ref. 5 the Zhao and Suo model provides an estimate of the breakdown strength that takes into account nonlinear electromechanical effects.

Some useful explicit results can be determined for the one term Ogden model $(N=1,α1,μ1,→α,μ)$⁠. Under equibiaxial stress the critical stretch satisfies $λ⩾λc$ where $λc=[(4+2α)∕(4−α)]1∕(3α)$ is the $s=0$ value. This obviously requires that $α<4$⁠. The critical field $Ec$ has a unique minimum at $λ0=[2(2α+1)∕(α−1)]1∕(3α)$ if $α>1$⁠. Zhao and Suo¹ considered $α=2$⁠, for which $λc≈1.26$⁠, $λ0≈1.47$ and the minimum value of $(ϵ∕μ)Ec$ is 1.038.

Finally, we note that the neo-Hookean constitutive model of Zhao and Suo¹ is apparently unique among the $N=1$ Ogden models in that it yields a simple formula for uniaxial stress. Thus, Eq. (5) with $N=1$⁠, $α=2$ for $j=2$ and $s2=0$ yields the relation $λ12=3λ22∕(λ26−1)$ between the stretches. Hence, we can parameterize the critical values in terms of $1<λ2⩽λc≈1.26$⁠:

In summary, the model of Zhao and Suo readily generalizes to arbitrary elastic strain energy. The explicit results reported here, such as Eq. (4), can be used to compare different elastic constitutive models, and should be helpful in the design of elastomeric actuators.