Response to Comment on “On electromechanical stability of dielectric elastomers” [Appl. Phys. Lett. 94, 096101 (2009)] Free

Recently Liu et al.¹ commented the paper reported by Zhao and Suo² focused on the analysis of the electromechanical stability of neo-Hookean elastomers assuming that the elastomer behaves as Mooney–Rivlin material. In a further comment,³ they extend their analysis to our recent paper on the stability of neo-Hookean-type elastomers,⁴ that is, materials having a single-constant strain energy function. They conclude that the assumption that the elastic energy is a functional of two material constants also gives a good account of our results for neo-Hookean elastomers. We think that in their study they fail to consider several important issues.

First of all, the condition $s>0$ [next line after formula (5) of the Ref. 3] is not a stability condition as stated after formula (8) of the comment. In fact, stability conditions derived from Eq. (8) are

If $k$ is positive as is currently assumed, then Eq. (1a) is automatically fulfilled and the only stability requirement is given by Eq. (1b). The same is true for the result expressed as formula (11) and the comments subsequent to formula (12) of the Ref. 3. If one assumes $s>0$⁠, then the distinction between the character of the first equation in Eq. (9) (a mechanical assumption) and of the second equation in Eq. (9) (stability condition), as well as between the first and second equations in Eq. (13), should be made explicit.

Incidentally, the second formula in Eq. (9) is wrong, the correct one is

Maybe it is a typographical error.

Second, $C1$ as well as $C2$ are taken as material constants as corresponds to a Mooney–Rivlin material. However, the electrical field could affect these “material constants.” For instance, electrostriction produced by the electric field can destroy the isotropy of the system and consequently the material constants should be modified accordingly. We notice, at this respect, the example proposed by Dorfmann and Ogden⁵ (see the p. 177 of Ref. 5) where the parameter $C1$ is dependent on the electric field.

Third, it is important to notice that the condition $s1=s2$ does not necessarily imply that $λ1=λ2$ as stated in their comment.³ In fact, if one assumes constant values for $si(i=1,2)$ and put $s1=s2$ (case of equal dead loads), then by equalizing Eqs. (4) and (5) of Ref. 3 and subsequent factorization the following expression is obtained:

This implies that two types of solutions are possible (in contrast with the case of a neo-Hookean material considered in Ref. 2, where $k=0$⁠). The symmetric solution, $λ1=λ2$⁠, and the antisymmetric one, given by the squared bracket in Eq. (3). This fact gives rise to unstable behavior and to an interesting pitchfork bifurcation phenomenon as studied by Liu⁶ for the case where the electric field is absent.

In fact, more insights into the present problem could be obtained if, instead of making $λ1=λ2$ in the expression for the hessian [Eq. (7) of Ref. 3], the authors consider the stability conditions as obtained directly from it. Thus, for example, if the positive sign of the minor containing the terms 22, 23, and 33 of the Hessian matrix is considered, the following condition is obtained:

which reduces to

when $λ1=λ2$⁠.

Note that this expression is not equivalent to Eq. (1a) of the present paper. In fact, the condition (5) can be stronger than Eq. (1a) if the following inequality is fulfilled:

which can be expressed in terms of the electric field on account of the relationship⁴ $D̃=ελ12λ22Ẽ$ which reduces to $D̃=ελ4Ẽ$ if $λ1=λ2$ as follows:

a condition which can be fulfilled if high enough electric fields were applied to the sample. More stability conditions (obviously, not all equivalent) should be obtained from the Hessian matrix. For this reason a premature use of the condition $λ1=λ2$ could leave hidden some stability conditions, relevant to the problem.

Finally, and even more important, the assumption made by the authors that the Maxwell stress tensor is not relevant to the problem is rather problematic. Electric fields always induce stress, and in order to take into account all the effects on the sample, Maxwell stress tensor should be accounted for a more accurate analysis of the present problem. This can modify substantially the problem and the associated bifurcation phenomenon.