iMechanica - electromechanical instability - Comments

Dear Prof. Norris Thank

Xuanhe Zhao — Mon, 17 Sep 2007 11:18:18 -0400

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.

critical electric field in thin elastomers

Andrew Norris — Sun, 16 Sep 2007 22:03:14 -0400

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!

Re: Thin film electromechanical instability

Zhigang Suo — Sun, 16 Sep 2007 10:47:24 -0400

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.

Thin film electromechanical instability

Andrew Norris — Sun, 16 Sep 2007 10:34:51 -0400

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

Yes, it has been considered

Wei Hong — Mon, 10 Sep 2007 23:40:37 -0400

Hi Hua,

Yes, it can be understood as such, that this difference drives the swelling of the gel.

(The difference will drive solvent molecules diffuse into the gel. To be more exact, it should be the difference in chemical potential, the partial derivative of the free energy with respect to the concentration.)

If there were no elastic force of the network, the gel will keep swelling forever, as the concentration is always lower than the exterior.

However, as for the elastic force, there will be a equilibrium state, in which the driving force from the difference in concentraion balances the elastic force.

Wei

contribution of this difference into free energy?

Hua Li — Mon, 10 Sep 2007 23:03:32 -0400

Hi Wei,

Sounds great. Do you think if this difference also makes its own contributions into the free energy of your gel system?

Yes, there is difference

Wei Hong — Mon, 10 Sep 2007 09:16:24 -0400

Hi Hua,

There is difference. Inside the gel, the solvent is mixed with network (We define concentration as number of molecules per unit volume, not just the concentration of ions with respect to liquid only.) , while ourside, it is either pure solvent or the mixture of solvent with something else.

Wei

Hi, Wei, thank you for your

Hua Li — Mon, 10 Sep 2007 06:27:20 -0400

Hi, Wei, thank you for your response, but i still want you to clarify that "3) The contribution from C is just the energy of mixing (14)." In your system, is there difference of concentrations between the interior gel and exterior solution?

An elaborated version of this paper

Xuanhe Zhao — Fri, 07 Sep 2007 21:30:56 -0400

An elaborated version of this paper has been accepted by Physical Review B. Please take a look at the second attachment.

Dear Prof

Xuanhe Zhao — Fri, 07 Sep 2007 16:47:47 -0400

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.

dT = 0

Wei Hong — Fri, 07 Sep 2007 16:20:53 -0400

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

It might just be the title

Wei Hong — Fri, 07 Sep 2007 16:20:08 -0400

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.

Hi Han. Thanks a lot for

Xuanhe Zhao — Fri, 07 Sep 2007 16:07:06 -0400

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 of Nonlinear 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 of Nonlinear 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.

Dielectric Stress

ericmock — Fri, 07 Sep 2007 15:41:59 -0400

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

Dielectric breakdown determined by other material properties?

ericmock — Fri, 07 Sep 2007 15:26:44 -0400

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)?