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# Propagation of instability in dielectric elastomers

When an electric voltage is applied across the thickness of a thin layer of an dielectric elastomer, the layer reduces its thickness and expands its area. This electrically induced deformation can be rapid and large, and is potentially useful as soft actuators in diverse technologies. Recent experimental and theoretical studies have shown that, when the voltage exceeds some critical value, the homogenous deformation of the layer becomes unstable, and the layer deforms into a mixture of thin and thick regions. Subsequently, as more electric charge is applied, the thin regions enlarge at the expense of the thick regions. On the basis of a recently formulated nonlinear field theory, this paper develops a meshfree method to simulate numerically this instability.

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## Comments

## A few questions and comments

Jinxiong,

Very interesting paper. I particularly like the fact that you are able to calculate solutions on the

unstable equilibrium paths. But that leads to one of my questions.

The paper says that the Newton-Raphson solution algorithm fails to converge for some points on the unstable

path in the neo-Hookean model. I'm surprised that convergence is achieved for any points on the unstable

paths for Arruda-Boyce models also.

The paper doesn't mention any special techniques for following unstable paths.

I must admit that I'm puzzled by Figure 6-- the C-D section in particular. Near the beginning and end

of this section the separate loading/unloading paths look almost like some kind of hysteresis is present.

I don't understand the mechanism for this in an elastic material. But it certainly doesn't look like

a bifurcation path, either.

More fundamentally, I've been wondering how important it is to trace these unstable equilibrium

paths since the experiment is not going to follow them. One of the most interesting things in

the present paper is that when the path becomes stable again (section E), the deformation goes

back to uniform. So I'm still puzzled about the wrinkles in the experiments.

Bill

## Treatment of the unstable region

You raised a very good question, Bill.

Actually this point was addressed in early versions of the draft and later got deleted as this paper is not numerical oriented any more. No program would have survived if we used voltage increments, as the material model itself is unstable at that region, if the voltage is

controlled. Here we played a little trick, by controlling the increment of total charge

on the electrode, while maintaining the equi-potential boundary condition. Technically, we combined the DOFs corresponding to the nodal potentials, and added a charge

increment to this combined nodal variable. An analogy in mechanical load is adding a rigid plate to a surface (so same displacement), and loading the whole rigid plate by an increment in the total force.

The discrepency in the loading and unloading curve comes from the multivalued behavior of the material, i.e. under same voltage, we can have different charge densities on the electrode, all in equilibrium. The jump between them depends on the loading conditions and imperfections.

Let me know if this makes sense to you.

Wei

## Thank you for your insightful questions

Hi, Bill and Wei,

Thank you for your insightful questions and comments. Today I am busying myself writing and debugging another code so I missed your helpful discussions. So I will read carefully your questions and comments and try my best to answer your questions.

First, I will talk about something about tracking the unstable path. From figure 6 we can see that if we use voltage as loading increments, when the voltage reaches the peak value, the voltage must decrease. At this point, even you decrease the increments of voltage factitiously, the solution will go back and follow the previous solutions if no special treatment is introduced. Alternatively, we use charge as loading increments because charge is monotonically increasing. This technique works only for very small jump when the solution at the peak is near to the solution on the horizontal line. If the jump is big, i.e., the difference between the latest known homogeneous solution and the first inhomogeneous solution is big, the last homogeneous solution is not a good guess of the inhomogeous solution and in this case the New-Raphson iteration method can not converge. To be honest, this remains an issue for our programm and so far our programm can not fallow the unstable path when big jump occurs.

Actually, our problem is the same as the post-buckling problem in strucutral mechanics or the computation of birfurcation in nonlinear dynamics. As far as I know, the problem is solved and there are some approaches are available. One of the method I used many years ago is the parametric continuation method, e.g. the arc-length continuation method. The key point of the continuation method is to convert increments of a physical parameter, e.g., charge or potential in our method, to a geometric parameter-the arc length. The increments of physical parameters may increase or decrease during the whole solution diagram, while the arc length of the solution path is definitely increasing monotically. This point is manifested if you just use a one degree freedom system. So you have an additional variable and you need an additional equation added to the New-Raphson method. This additional equation is called arc-length equation. Therefore, even the original Jacobin matrix is singular, the matrix of the augmented system is no longer singular. In this way the method can follow any unstable path and can obtain the whole solution diagram. This method can be implemented at the cost of more complicated algorithm and programming. We have not implemented continuation method yet, but hopefully this can be done in the follow up works.

I also agree with you that for engineering applications only the first stable solution is important and in most cases the actuators will breakdown before the second stable state is observed. I am not clear whether this big jump is useful for some potential applications.

Hope that my reply can answer your first question. I will read your other questions and Wei's brilliant answers carefully and hopefully can involve into this hot discussion.

Jinxiong

## Equilibrium paths and arc length methods

Jinxiong,

Thanks very much for your detailed reply.

From your paper and this discussion, it seems clear that there are two equilibrium paths. Both of these paths should be reasonably continuous. For the perfect structure, I'm assuming that the secondary path intersects the primary path at least once at a bifurcation point-- possibly on section B of the curve in your figure 6. When you introduce an imperfection, you convert this bifurcation problem into one that includes an equlibrium path with a limit point. This path is also continuous (i.e. no "jumps"). However, in the vicinity of the bifurcation point (singular point), this imperfect structure is so ill-conditioned numerically that it is extremely difficult to obtain a converged solution. I think that is why you are seeing what appears to be a jump; its just very difficult to obtain converged solutions for the intermediate points on the equilibrium path.

You may be able to obtain some additional insights by studying the behavior of the perfect structure. Specifically, if you trace the primary equilibrium path and observe sign changes in the determinant of the jacobian, it should be possible to estimate the location(s) of the bifurcation points.

I agree with your comments about arc-length methods. You'll notice I made a couple of comments about these in another reply on this thread. As I mentioned below, I have never tried these when the arc length contains two different types of degrees-of-freedom. I suspect there may be some sticky issues of how to scale the different DOF types in computing the arc length. One reason why I say this is that ABAQUS contains an arc length algorithm that is very effective in problems with only displacement DOFs. However, this algorithm is disabled when the problem contains different DOF types (e.g. displacement and voltage).

Bill

## Arc-length method for mixed DOFs?

Hi Bill,

Thanks for your valuable comments! You suggested a feasible way to obtain the critical load value, and I will try it later.

I know ABAQUS uses the arc-length method or you previously mentioned the Riks mehtod for postbuckling analysis. And the reported researches show that this method is effective for static postbuckling analysis. Of course, most of these reported researches involve only only type of DOFs, the displacements. What interests me is you mentioned that the method is disable for mixed DOFs. I have no experience of using ABAQUS to analyze postbuckling analysis and I am not clear what behind this invalidation of arc-length method.

Is there anybody who use Riks method in ABAQUS to analyze stability problems with different types of degree-of-freedoms (displacements plus whatever else)? Can anybody give us some insightful suggestions?

Jinxiong

## Back to voltage control if we use arc length method?

Hi Bill,

Another interesting question. Previously Wei mentioned that we must introduce some triky treatment of potential on the boundary where charge was applied and controled as load increments. I am still thinking that the voltage-charge curve should be continous even in the jump area. I mean following the jump line the voltage decreases continuously with charge almost fixed. If the jump is big, small variation of charge may correspond to big change of voltage. This is the reason why New-Raphson method fails to converge.

However, the votage can increase or decrease automatically if we use arc length method, because we actually use the length of arc as parameters. If that is the case, maybe we can back to potential loading and we can avoid any additional treatment of potential!

Can I have your comments on this point?

Jinxiong

## charge control

Hi Jinxiong,

I actually think that the charge control strategy described by Wei is quite a good approach to this problem. In fact, after reflecting on this more, I think my suggestions about the possible

needfor an arc-length algorithm may have been a bit hasty.I think it is more likely that the convergence difficulties are simply due to numerical ill-conditioning near the bifurcation point. Have you done any numerical experiments increasing the size of the imperfection to see to what extent that improves convergence?

Bill

## Unstable equilibrium paths

Wei,

Thanks for the clarifications.

Your charge control approach sounds like a very good strategy for solving this type of

problem. So, why was it not able to trace the complete path for the neo-Hookean material?

My other question concerns Figure 6. What is responsible for the

different loading and unloading paths? That wasn't clear to me from the paper. If you

don't add the imperfection to the model, do the numerical results follow the

analytical curve?

Thanks.

Bill

## "Ustable" equilibrium path

Another insightful question, Bill.

The analytical curve is only valid for an "homogeneous deformation state", when there is a little inhomogeneity, the unstable homogeneous state will separate into coexistent regions of two stable states, just like phase separation. The cost is to form a additional boundary or transition band between the two states. The imperfection is introduced to have enough inhomogeneity to enable such transition.

However, mathematically, there's always inhomogeneity, in the mesh grids, and even in the calculation error. It's just a question of how sensitive the system is to the inhomogeneity.

For the neo-Hookean material, the curve does not go up again, which means the coexistent states include one thin state of thickness 0. Or, simply there are no coexistent states. Once you have a separation, the system goes all the way to 100% thin state. Such a change is too significant for Newton-Raphson method to handle. So the calculation fails instead of the collapse of the film.

For Boyce-type of materials, if we don't introduce any imperfection, and use a square

block and a uniform mesh, the calculation can go into part of the unstable state. This does not mean our calculation preduces artificial results. It's just the curve is in the unstable regime for an "homogeneous deformation state". But if you look at the deformation pattern, it is not uniform any more. The block is more or less tilted. This means the cost of the transition region is so expensive that the system can not afford to have coexistent states. Instead, it has mostly a thick region and some little transition at the edge. Such a inhomogeneous deformation state is actually stable.

However, if you keep on loading, the transition region will grow and graduate eat up the whole block. That's when the curve will deviate from the analytical solution.

For the difference in loading and unloading, I guess it is because the imperfection we introduced does not have a same contribution to the transitions. (One being local thinning and the other being local popup) It is not a big difference and the imperfection we introduced is artificial, I don't think that's a physical phenomenon that worth to be studied.

Wei

## Numerical solutions for alternate equilibrium paths

Wei,

Thanks for the

detailed and thoughtful reply. Do you mind a couple more questions and

comments? One is based on a comment from your last reply that I thought I had

understood from the paper.

I've included parts of your reply below (italics font) so I

can refer to them directly.

For the neo-Hookean material, the curve does not go upagain, which means the coexistent states include one thin state of thickness 0.

Or, simply there are no coexistent states. Once you have a separation,

the system goes all the way to 100% thin state. Such a change is too

significant for Newton-Raphson method to handle. So the calculation fails

instead of the collapse of the film.

In my previous reply, my

first thought was that your charge-control algorithm would follow the

equilibrium path almost until the block reached zero-thickness. However, after

reflecting on this more, I think it may be necessary to constrain also the

displacementincrements in addition to the charge. This would be a Riks (or Crisfield) type

algorithm that you may be familiar with. I must admit, though, that I have no

experience with these algorithms for mixed degree-of-freedom types (e.g.

displacement and potential) in the solution vector.

For Boyce-type of materials, if we don't introduce any imperfection,and use a square

block and a uniform mesh, the calculation can go into part of the unstable

state. This does not mean our calculation preduces artificial results.

It's just the curve is in the unstable regime for an "homogeneous

deformation state". But if you look at the deformation pattern, it

is not uniform any more. The block is more or less tilted. This means the

cost of the transition region is so expensive that the system can not afford to

have coexistent states. Instead, it has mostly a thick region and some

little transition at the edge. Such a inhomogeneous deformation state is

actually stable.

However, if you keep on loading, the transition regionwill grow and graduate eat up the whole block. That's when the curve will

deviate from the analytical solution.

I thought this is exactly the problem you show results for

in Figure 4 where the numerical solutions lie right on the analytical curves? I

had assumed these numerical solutions had uniform displacements because they

agree so well with the analytical solutions. What am I misunderstanding?

For the difference in loading and unloading, Iguess it is because the imperfection we introduced does not have a same

contribution to the transitions. (One being local thinning and the other being

local popup) It is not a big difference and the imperfection we introduced is

artificial, I don't think that's a physical phenomenon that worth to be

studied.

One of the main reasons for my questions is to try and

understand the little C-D segment of the equilibrium path in your Figure 6. I'm

guessing that it actually intersects the B and E sections of the primary

equilibrium path but that it is numerically very difficult to get converged

solutions closer than you have to the primary path? Does this speculation agree

with your numerical experiences? Here is another guess about the "loading" and

"unloading" behavior. I'm guessing that the Newton-Raphson algorithm is just

(somewhat) arbitrarily converging to one equilibrium path or the other based on

whether you are "loading" or "unloading" in your solution algorithm. If so, this

"choice" doesn't really imply anything about how the solid actually behaves.

Thanks again.

Bill

## numerical and analytical

Yes, you are right, Bill.

To obtain the unstable result of homogeneous deformation, one also needs to contrain the deformation so that it is uniformly compressed. We have given it a little bit thought before making the plot, such as combine the DOFs of the vertical displacement of the top-nodes. However, different from the charge control, the deformation constraint is artificial, and the only result it can get is to have the calculation follow the analytical curve all the way, which is sort of borring anyway.

For the plots in Fig. 4, actually there is a little discrepency. The block does show some inhomogeneous deformation on the corner. But as the region is small, it contributes little to the overal measurements, say the voltage and total charge.

For the interesting transition points near C or D, although thinking as an ideal thermodynamic system, the preferable line would be horizontal line CD intersects with the primary pathes. However, to have the coexistent states, as the system does in between C and D, it has to pay extra energy to form an transition band between the two phases. Which can be seen on fig 7 C and D, the transition band is having a width of about the thickness of the film. The extra cost in forming the band prevent the film to jump from the primary curve to CD directly. Instead, it overshoots a little before making the transition. The amount of the overshoot depends on the formation energy of the transition band.

You can also think in this way: The CD segment corresponds to the coexistent states of the two thickness, so you do need both stable states. However, the transition band from thick to thin states has finite width. If the necking region is even smaller than that width (e.g. the state a little bit prior to Fig. 7C), the coexistent of states simply can not happen. Comparing to the state of having a partially necking in the middle, the system would rather prefer a uniform thick state. That's why we always have an overshoot before making the turn.

You guessing might also be right that where to make the turn also depends on the algorithm itself, as the discrepency between loading and unloading. There is no proof that N-R method would prefer an energetically more favorable state, and the convergence region of N-R method is initialvalue dependent and somewhat arbitrary or fractal.

Wei