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# A method to analyze electromechanical stability of dielectric elastomer actuators

This letter describes a method to analyze electromechanical stability of dielectric elastomer actuators. We write the free energy of an actuator using stretches and nominal electric displacement as generalized coordinates, and pre-stresses and voltage as control parameters. When the Hessian of the free-energy function ceases to be positive-definite, the actuator thins down drastically, often resulting in electrical breakdown. Our calculation shows that stability of the actuator is markedly enhanced by pre-stresses.

Update 9 September 2007. This paper is now published as Method to analyze electromechanical stability of dielectric elastomers, Applied Physics Letters 91, 061921 (2007).

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## Comments

## "stress" in the dielectric

Xuanhe,

When I read this draft of your paper last May it started me thinking about what does it really mean that s33 is zero in the dielectric. The recent, lively discussion (http://imechanica.org/node/1001 ) caused me to think about this again. I look at this question of stress in the dielectric very fundamentally. There is a sheet of positive charge on one face

of the dielectric and negative charge on the opposite face. Since

they attract each other, some force has to be applied to keep them

apart. I accept that s33 = 0.0-- because that is the way it is

defined. But I am interested in a stress measure that tells me

something about the

internal forces in the dielectric that are keeping the free

charges apart. I don't think this is contrary to any statements I've

seen about this including Feynman's.

Bill

## Bill, good

Bill, good question.However, it's really difficult to answer what's the electric force (or internal force) inside a solid dielectric.

I would like to quote Feynman that "This is a very difficult problem which has not been solved, because it is,

in a sense, indeterminate. If you put charges inside (or on the surface in your

case) a dielectric solid, there are many kinds of pressures and strains. You

cannot deal with virtual work without including also the mechanical energy

required to compress the solid, and it is a difficult matter, generally speaking,

to make a unique distinction between the electrical forces and mechanical

forces due to solid material itself. Fortunately, no one ever really needs to

know the answer to the question proposed. He may sometimes want to know how

much strain there is going to be in a solid, and that can be worked out. But it

is much more complicated than the simple result we got for liquids.”

I think it's because of this difficulty a number of people were trying to formulate the field theories for deformable dielectrics avoiding the electric forces inside dielectrics. This is also one of the motivations for our current work on nonlinear field on deformable dielectrics.

This paper has been publised on applied physics letters. The fact that we didn't use electric forces or Maxwell stress in the method described in the paper makes it applicable to dielectric polymers with various dielectric properties.

## Dielectric Stress

Bill,

I have found it best (for me anyway) to think of the stress created by an electric field as a (phenominological) body force. The stress in the 'transverse' direction is (nearly) zero because the tractions on the surfaces are zero. I initially thought of the electrodes as applying a surface traction but I think this is the wrong way to consider it. As I mentioned in my other comment in this thread, think about the case where you hold the electrodes a fixed distance apart and slightly away from the dielectric. Thinking of the problem as electrodes squeezing the material would suggest the dielectric does not deform. But it would (I think).

Unfortunately, I have posed this question to a few people who should know (physicists and EEs) and have gotten different answers. That said, I am fairly convinced the 'electrodes' have nothing to do with the dielectric stress (and that EEs should not be trusted ). It's the electric field (however generated) that matters.

Eric

## stress in dielectric

Xuanhe,

The Feynman quote you included in your response is exactly what I was thinking of when I mentioned Feynman above. I'll bet I've read that quote at least 50 times trying to extract every bit of insight I can!

There seems to be fairly widespread agreement with Feynman's comment about trying separate mechanical and electrical forces inside the dielectric. However, when he says that "no one ever really needs to

know the answer to the question proposed", I

hopethe question he means is how to separate these forces; not how to calculate their sum. Because I think it is important to know this total force (or stress).At any rate, I hope we can agree that there is a force in the dielectric holding the free charges on the faces apart. And if there is a force, we can calculate a stress measure as well. In this other thread,

Re: Re: stress and strain in dielectrics, there was a misconception that the strain in the dielectric was similar to free thermal expansion of a body where there is strain but no stress. I just wanted to clear that up.

Now we can discuss whether that stress is important (I think it is) and how to calculate it.

Bill

## What keeps a positive charge and a negative charge apart?

Let us begin with a vacuum. In this vacuum, we have two point charges, one positive, and the other negative. They attract each other. We say they attract each other on the basis of an experimental fact: we need to apply an external force to keep them apart.

Are the above statements really correct? Not exactly. A hydrogen atom consists of a proton (a positive charge) and an electron (a negative charge). We know that the two charges do keep apart, in a way, without any external force. Someone had to invent quantum mechanics to account for the stability of the hydrogen atom. In this case, talking about a force that counteracts the electrostatic attraction does not get one very far.

If we place a positive charge and a negative charge inside a solid dielectric, what keeps them apart? We may say that all these proptons and electrons and their quantum mechanical interactions keep our two charges apart. Does this statement help us? Not now, because we cannot do this complicated calculation.

Thus, we go back to a phenomenological approach, and formulate a theory using quantities that we can measure, as done in this APL.

Now, Bill, if you could pose an observable phenomenon that could not be addressed by the theory, we would have to extend the theory.

## about the free-energy function

Xuanhe, your talk yesterday interested me and I just went through your paper this morning. I followed most part of it, but got stuck at the free-energy function(seems Mike also asked about this). I do not quite see how you justify that the "free-energy function" W, thus G, that you defined, should be the one to miminize for equilibium judgement. Is the entropy contribution implicitly included in the form of "W"? Maybe you already answered this in other papers though.

Thanks,

Li Han

## There is no entropy contribution in this model

Li Han,

The entropy contribution (due to heat transfer), TdS has been neglected in the model. Or you can think in this way, the process is assumed to be reversible, or isentropic, so that dS=0.

Wei

## Hi Han. Thanks a lot for

Hi Han. Thanks a lot for the interests in our work. Even though the free energy function is really the focus of our another paper , you asked a very good question.

The material we study here is elastomer, which follows the rule of rubber elasticity. We need to go through one basic concept of rubber elasticity to answer the question. One basic assumption of rubber elasticity is

internal energy of rubber does not change with deformation at all (P310 ofNonlinear Solid Mechanics).For elastic rubber (without electric field):

Let denote U as the internal energy of the dielectric elastomer. The Helmholtz free energy of rubber by definition is

W=U-ST

Considering isothermal condition (dT=0) and the basic assumption for rubber elasticity (dU=0), we have

dW=-TdS

As a consequnce of the Gaussian statistical theory of a molecular network, we can get

W=1/2(lambda1^2+lambda2^2+lambda3^2-3)

This is the first term in Eq 5 of this paper. This is also the "entropy contribution" in your question. For a detailed deriviation, you may refer to

(P310-319 ofNonlinear Solid Mechanics).For dielectric elastomer (with electric field):

The internal energy is a function of dielectric displacement D, i.e. U(D). For ideal dielectric elastomer:

U=1/2*D^2/epsilon

This is the second term in Eq 5 of this paper. There is no entropy contribution from this term.

Hope this helps.

XH

## dT=0 or dS=0?

Xuanhe and Wei, thanks for the prompt reply. But I seemed given two different assumptions, which are not necessarily compatible. I personally feel more comfortable with dT=0 though, since I do not think the reversible process assumption is really safe considering the collaps of the dielectric at high voltage.

Li Han

## dT = 0

Li Han,

Sorry about the previous mistake.

Here the "free energy" is indeed the Helmholtz free energy.

For rubber, under the assumption of constant T, the change in the free energy is all from entropy (of configuration).

dW = -TdS

But the configuration entropy S is a function of the deformation.

Almost all material laws of rubber are derived from this assumption, directly or indirectly.

So in that sense, the model has already considered the entropic effect of the rubber chains.

Wei

## Dielectric breakdown determined by other material properties?

This is an interesting formulation and similar to something I toyed around with a while back. I was interested in what I had heard called the pull-in instability, which I think is the same phenomena you're studying. I did not take the analysis as far as you have. I was trying to see how much non-linear stiffening would be necessary to prevent this instability.

It is quite amazing that you can see significant effects of pre-stretching the material with such a simple model. Again, people have understood that pre-stretch improves performance but I had really never heard a good explanation for why. And I had not anticipated that such a simple (don't take 'simple' the wrong way, simple is good) analysis would yield any insight.

This leads to one thing that troubles me a bit about your predictions. As the title implies, I think dielectric breakdown strength is an independent material property. However, you suggest it is determined from other material properties. Ultimately I think there is confusion between the pull-in instability and true dielectric breakdown. You frequently read engineers developing devices with dielectric elastomers say the pre-stretch increases the breakdown strength. This is often explained as a result of polymer chains being stretch (which for some reason increases the breakdown strength). Your result may suggest that it is not really breakdown strength that is being altered by pre-stretch but the pull-in failure mode. This would actually be a much better explanation in my mind.

Re breakdown being an independent property, it seems analogous to modulus and elastic strength. These are obviously two very different material properties. Likewise, it would seem like dielectric strength should be independent of elastic modulus and dielectric constant. Consider an experiment in which you take two rigid electrodes and hold them a _fixed_ distance apart with the dielectric in between. What would happen (assuming you have a matching dielectric fluid surrounding everything)?

## It might just be the title

Good point, Eric. And I think it is exactly the point that this paper is trying to express.

It might just be the title that make you think there is confusion. The authors made this point in the paper (though may be less explicit). The problem this paper tried to solve is to explain the pull-in instability, instead of "derive" the dielectric breakdown voltage.

You made a very good analogy. I would say that the instability here is more like the critical load for buckling (in a complex structure) which depends on the modulus and the geometry of the structure.

However, they are also related in many applications. As the paper predicted, as the true breadown voltage is relatively high, in many cases what really happens is that the pull-in instability happens, and then the material breaks down because of the dramatic increase in true electric field. Just like in mechanical structures, many structural elements lose stability (buckle) before the yielding takes place, and the buckling is really the reason for later yielding. In other words, the pull-in instability is often more dangerous than the breakdown.

## Dear Prof

Dear Prof Mockensturm:

Thanks a lot for your interests in our work. You asked a number of good questions. Let me try to answer them one by one:

1. "I was interested in what I had heard called the pull-in instability,

which I think is the same phenomena you're studying. I did not take

the analysis as far as you have. "

The instability we analyze here is actually the "pull-in instability". However, we don't feel comfortable about this name, because we found that the reason for this instability is really the non-convex natrue of the free energy function of dielectric elastomer. Therefore, we would rather call it "electromechanical instability".

2. "I was trying to see how much non-linear stiffening would be necessary to prevent this instability."

This is a really good point. I would like to refer you to our another paper "Electromechanical coexistent states and hysteresis in dielectric elastomers." We found that one may prevent the electromechanical instability by increasing the crosslink density of polymer to a very high value. For example, the crosslink density is denoted by "1/N" in Arruda-Boyce's law for rubber elasticity. We found one need a value of N<2.6 to avoid the instability.

3. "As the title implies, I think dielectric breakdown strength is an

independent material property. However, you suggest it is determined

from other material properties. Ultimately I think there is confusion

between the pull-in instability and true dielectric breakdown."

We agree that "electromechanical instability" and "true dielectric breakdown" refer to different phenomena. We beleive that, for most of the case in dielectric elastomer, the "electromechanical instability" happens first and leads to the "true dielectric breakdown". Current experiment results also seems to support our assumption.

For example, Plante, J.S. and Dubowsky, S. "Large-Scale Failure Modes of Dielectric Elastomer Actuators ."

International Journal of Solids and Structures, Vol. 43, No. 25, pp. 7727-7751, December 2006.4. Prof Mockensturm, we notice a number of very good papers from your group on dielectric elastomers. Hope you can share some of them with us on Imechanica.

XH

## Thin film electromechanical instability

Xuanhe and Zhigang,

Your APL paper gives a really nice explanation of the loss of stability. It is the first proper description - using

finite electromechanical theory with minimal assumptions - that I am aware of.

On reading through the paper I realized that there are some algebraic

simplifications possible. The determinant of Hreduces to a quadratic in D^2. The quadratic always

has one positive and one negative root, so the nonzero critical value

of D can be found in fairly nice form as an explicit function of the

stretches. Using this, its possible to get simple expressions for the critical values under uniaxial and equal biaxial stress.

I wrote this up in a 1-page file that I sent off to APl as

a "comment". The above link is to a copy of the file on iMechanica.

Andy

## Re: Thin film electromechanical instability

Dear Andy:

Quickly looked through your note. Very

interesting! To upload the note to iMechanica, you can start a new

blog entry, and upload the note. You can then link your comment to

your blog entry.

## critical electric field in thin elastomers

Hi Zhigang,

Its probably pretty obvious that the same structure for the Hessian is maintained for very general forms of the free energy. The key is the decoupled vacuum electric energy. Using your APL paper, its easy to see that a free energy of the form

has critical electric field given by

where

Games could be played by asking for "optimal" forms of U.

My interest in all this is in understanding the electrostrictive effect in elastomers. This will require explicit coupling between the E and mechanical fields in the energy. That is a whole new ballgame - with lots of interesting things to be found I expect!

## Dear Prof. Norris Thank

Dear Prof. Norris

Thank you for your interests in our work.

One of the "optimal" forms of U, the strain energy function of an elastomer, has been indicated in our recent paper "Electromechanical coexistent states and hysteresis in dielectric elastomers." Physically, it means that an elastomer described by the Gaussian statistics will always have electromechanical instability. Only when the elastomer is so stiffened that its chains are near the extension limit, the instability may be avoided.

I totally agree that the effect of electrostriction is another interesting topic.

XH