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# A nonlinear field theory of deformable dielectrics

Zhigang Suo, Xuanhe Zhao, and William H. Greene

**Abstract**, Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both conservative and dissipative dielectrics. As applications of the theory, we discuss material laws for conservative dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.

** **

**Introduction**

All materials contain electrons and protons. In a dielectric, these charged particles form bonds, and move relative to one another by short distances in response to a voltage or a force. That is, all dielectrics are deformable. The notion of a rigid dielectric is as fictitious as that of a rigid body: they are idealizations useful for some purposes, but misleading for others.

Deformable dielectrics are central in diverse technologies (Newhham, 2005; Uchino, 1997; Sessler, 1987; Campbell, 1998). Our own interest is renewed by recent innovations in materials, principally organics capable of large deformation, including electrostrictive polymers (Zhang, 1998; Chu et al., 2006), cellular electrets (Graz, et al., 2006), liquid crystal elastomers (Warner and Terentjev, 2003), and elastomers capable of large deformation under electric field (e.g., Pelrine, et al, 2000). Also emerging are technologies to produce patterns of electric charge with small features (Jacobs and Whitesides, 2001;** **McCarty et al., 2006). Potential applications of these materials and technologies include transducers in large-area, flexible electronics (e.g., displays, artificial muscles, and sensitive skins), as well as in devices at small length scales. Phenomena of electric-field induced motion and instability have also been actively studied (e.g., Li and Aluru, 2002; Gao and Suo, 2003; Suo and Hong, 2004; Huang, 2005; Lu and Salac, 2006; Zhu et al., 2006).

Although the atomic origin of dielectric deformation has long been understood, how to formulate a field theory remains controversial. Many theories have been formulated (e.g., Becker, 1982; Landau and Lifshitz, 1984; Toupin, 1956; Eringen, 1963; Pao, 1978; Eringen and Maugin, 1989; Maugin, et al., 1992; Kuang, 2002), invoking different approximations and postulates. On these theories, Pao (1978) remarked, "That there are so many coexisting theories and results for a subject so fundamental in nature may sound very surprising to experimentalists, for theories can usually be sorted out, or proven to be fallacious by carefully designed experiments. The difficulty here is that the electromagnetic fields inside matter are expressed in terms of field variables which cannot be directly measured in laboratories." Recent critiques of these theories may be found in Rinaldi and Brenner (2002), and in McMeeking and Landis (2005).

Pao's remarks were directed to general theories of electromagnetism in matter, but we find that his remarks apt for theories of deformable dielectrics. To give some ideas of the controversies involved, we mention two difficulties.

One difficulty has to do with the notion of electric force. Consider, for example, a parallel-plate capacitor, consisting of an insulating medium and two electrodes, with a battery maintaining a positive charge on one electrode, and a negative charge on the other (Fig. 1). If the insulating medium is a vacuum or a fluid, we must apply a force (e.g., by using a weight) to maintain equilibrium. In this case, there is no ambiguity as to what the electric force is: the force between the two electrodes can be measured by the weight. Maxwell (1891) converted this force into a state of stress in the medium. When the insulating medium is a solid dielectric, however, the electric force cannot be measured. Indeed, for many common solid dielectrics subject to a voltage, the two electrodes appear to repel, rather than attract, each other (Newnham, 2005). The atomic origin of this phenomenon is clear. Influenced by the voltage between the electrodes, charges inside the dielectric tend to displace relative to one another, often accompanied by an elongation of the material in the direction of the electric field.

On the force between electric charges in a solid, Feynman (1964) remarked, "This is a very difficult problem which has not been solved, because it is, in a sense, indeterminate. If you put charges inside 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."

A second difficulty has to do with work conjugates. It is a simple matter to show that, when a dielectric solid deforms, the true electric field and the true electric displacement are *not* work conjugates. Although this fact does not preclude them from being used to formulate the field theory of deformable dielectrics, the non-conjugates do lead to complications, and their almost exclusive, and sometimes erroneous, use in the literature contributes to the controversies.

While the first difficulty was noted by nearly all authors on the subject, we have not found any explicit discussion on the second. The dubious status of electric force is unsettling as most textbooks start by defining electric field by the force acting on a test charge divided by the amount of charge. In a solid dielectric, the electric force is not a measurable quantity, so that this definition is not operational. A common approach is to forgo this definition, and regard the field equations of electrostatics as the starting point. But to discuss deformation one has to link the electric field to a force, and this connection is usually made by invoking work. In this connection, some authors (e.g., Landau and Lifshitz, 1984; Line and Glass, 1977) assumed that the true electric field and the true electric displacement are work conjugates. While this assumption does not lead to serious errors for infinitesimal deformation, it does for finite deformation.

In this paper, in the spirit of Feynman's remark, instead of dwelling on the meaningless notion of electric force, we ask questions concerning measurable quantities. In effect, we ask, given an applied voltage and applied force, how much does one electrode move relative to the other, and how much charge flows from one electrode to the other? Instead of leaving various fields undefined, we define them by operational procedures.

As a mental aid in formulating the theory, imagine that each material particle in a dielectric is attached with a weight and a battery, and then prescribe a field of virtual displacement and a field of virtual voltage. We will use virtual work to define fields inside media, an approach well established in mechanics, but perhaps less so in electrostatics. Associated with the work done by the weights and inertia, we define the stress inside the dielectric as the conjugate to the gradient of displacement. Associated with the work done by the batteries, we define the electric displacement inside the dielectric as the conjugate to the gradient of electric potential. The approach requires no additional postulate beyond what we mean by work, displacement, charge and inertia. The approach does not start with field equations, but produces them as consequences. The theory is applicable to finite deformation, and to both conservative and dissipative dielectrics.

We write the body of the paper with minimal digression, hoping that a reader with basic knowledge of electrostatics and mechanics can appreciate the theory. Section 2 reviews elementary facts of work, energy and electromechanical coupling, using a generic transducer. Section 3 uses a homogenous field to illustrate a procedure to define quantities per unit length, area, and volume, a procedure that we generalize in Section 4 to inhomogeneous fields in three dimensions. Section 5 sketches the material laws for conservative media. Section 6 applies the theory to fluid dielectrics, and recovers the Maxwell stress. Section 7 discusses solid dielectrics. Section 8 applies the theory to infinitesimal fields superimposed upon a given field.

Our approach allows us to choose, among many alternatives, measures of stress, strain, electric field, and electric displacement. The body of the paper will focus on nominal quantities using material coordinates. Various Appendices describe alternative formulations and link to the existing literature. We show that our theory recovers the results of McMeeking and Landis (2005), who formulated a theory of deformable dielectrics by using spatial coordinates and the true electric field and true electric displacement. These authors started with an electric force, but concluded that this force cannot be measured in solid dielectrics. A parallel reading of that paper and the present one should provide a fuller understanding of both approaches.

**Update on 4 June 2007.** I attach the slides to be presented on McMat 2007 in Austin, on 6 June 2007.

**Update on 27 June 2007**. This paper is published in the Journal of the Mechanics and Physics of Solids: http://dx.doi.org/10.1016/j.jmps.2007.05.021

Attachment | Size |
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deformable dielectrics 1 Jan 2007.pdf | 230.72 KB |

deformable dielectrics revision 2007 05 09.pdf | 140.82 KB |

2007.06.06 dielectric elastomer slides.pdf | 470.71 KB |

2007.06.06 dielectric elastomer slides.ppt | 2.6 MB |

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## Comments

## A nice and important piece of work

Hi, Zhigang,

The paper "a nonlinear field theory of deformable dielectrics" is a very nice and important piece of work in the field of deformable dielectrics theory. In general, electric quantities and the mechanical quantities are coupled, and the work conjugate quantities can be obtained from the potential function (e.g., the Helmholtz free energy). I believe this new theory formulated by Suo et at al will be very useful in solving many related problems.

## some previous work on dielectric crystals

Dear Zhigang:

An interesting work! And thank you for citing my paper.

After reading the abstract and introduction, especially the comment by Pao YH, I was reminded some previous work done by my PhD advisor, PCY Lee, and his advisor RD Mindlin (who was also Pao's advisor). I am sure your work is very different because of the nonlinearity just by the title (I have not read through the entire paper yet). As far as I know, both Lee and Mindlin's work are on linear theoreis. Nevertheless, it might be of interest to note the previous works. Below is the citation and link to one paper by Lee, in which he cited an earlier work by Mindlin.

PCY Lee, A VARIATIONAL PRINCIPLE FOR THE EQUATIONS OF PIEZOELECTROMAGNETISM IN ELASTIC DIELECTRIC CRYSTALS. J. Appl. Phy. 69, 7470-7473 (1991).

RH

## earlier works on dielectrics by Lee

Here are a couple of earlier works on dielectrics by Lee and Cakmak of Princeton University.

Physical Review B,1(7), 1970, pp. 3525-3537.Int. J. Solids Structures,Vol.7, 1971, pp.523-537.Phys. Rev. B,9(12), 1974, pp. 5291-5299.RH

## Zhigang, Nice presentation

Zhigang,

Nice presentation as usual! See also important publications by Dorfmann and Ogden (Acta Mechanica 174, 167-183 (2005), for instance) where significant intersections with your work can be found.

Regards,

Kosta

## On papers by Dorfmann and Ogden

Dear Kosta:

Thank you so much for pointing out the paper by Dorfmann and Ogden to us. With this lead, we've found a number of other papers written by them in the last few years. These papers have, as you've pointed out, significant intersections with our work. For example, they have also adopted material description in their formulation, and the similarity does not stop here.

The difference between their approach and ours is the starting point. They started with the Maxwell-Faraday theory of electrostatics, and invoked quantities like electric body force, Maxwell stress, and polarization. We started with the idea of work done by a weight and a battery, and derived Maxwell-Faraday theory of electrostatics.

The two approaches lead to the same results. For example, our equation (5.14) is the same as their equations (35) and (36).

That the two approaches lead to the same testable results means, to us, that the additional postulate of electric body force is unnecessary.

As we noted in Section 6, for deformable dielectrics, any attempt to separate stress and call part of the stress the Maxwell stress must be arbitrary, and will lead to no observable results. McMeeking and Landis (2005) have reached the same conclusion. The remarkable results of Dorfmann and Ogden also lend support to this view.

We have adopted the following attitude: If a practice is difficult to understand and has no observable consequence , then we should not build a theory upon it.

How do you think?

## On concepts

Dear Zhigang,

I think that the electric body force (as well as the gravitational body force) is a physically appealing concept. It is not a kind of internal variable though one cannot measure it directly. I also find the concept of the Maxwell stress useful. (No stress is measurable, by the way!) May be I have to give more thought to both concepts to get disappointed with them. I did not accomplish that yet.

Whether to favor a variational formulation of the theory or not is a matter of taste. Landau & Lifshitz, for example, are crazy about variational formulations.

-Kosta

## Please choose a definition of electric body force you like

Dear Kosta: In a homogeneous state, stress is measurable in the sense that it is the applied force (say due to a weight) divided by the area. In the paper, we simply generalized this simple operation to inhomogeneous field.

Incidentally, as you probably have realized it, Landau and Lifshitz did not invoke the notion of electric body force. Nor did Feynman. Nor did Becker. Have you found any definition of electric body force (in a deformable dielectric) that you are comfortable with? If you point to a source, we can have a more focused discussion.

## You cannot resolve the uncertainty by ignoring it

Zhigang, this is the point – you can only measure the resultant force not the force per unit area. The latter can be done if you assume that the force is distributed equally over the area. Such an assumption is evident for the homogeneous state and not for the inhomogeneous one.

I call the electric body force physically appealing because it is a resultant of microscopic forces acting on the charged particles (see, for example, Continuum Mechanics of Electromagnetic Solids by GA Maugin). It is true that we do not know exactly how the resultant should be computed. The latter is, probably, the reason why Landau did not want to include the forces in his book. Should we reject the concept of the electric body forces if we do not understand how to specify them unmistakably?

I do not think that you can resolve the uncertainty by ignoring it.

## Once more on electric body force in a dielectric

Kosta: To see if a quantity is meaningful, we can watch for two things:

I submit that the notion of electric body force in a dielectric fails on both. If you disagree with this assertion, would you please point to a definition and describe a phenomenon in which we need the electric body force?

## example

Zhigang, the cited Dorfmann-Ogden paper gives a partial answer:

1. The quantity is defined in Eq.(7).

2. Together with other quantities defined in the paper it is possible to formulate a boundary value problem. The solved examples could interpret the corrsponding phenomena if observed.

Ogden and Dorfmann do not provide experimental data to test the BVP solution. Imagine that the experiments are in a good correspondence with the theory. In the latter case your requirements are met.

## Same governing equations, with or without electric body force

Thank you, Kosta. Now let's be specific.

1. Dorfmann and Ogden did not derive equation (7) in their paper. They gave the formula and then cited the review article by Pao (1978). Pao reviewed several inequivalent expressions for electric body force, and had this to say: "Whether we can now consider it as the body force in the mechanical momentum equation is far from certain." See more of his reservations on p. 232. Other authors have raised similar concerns, e.g., Becker.

2. The final set of governing equations reached by Dorfmann and Ogden are

These same equations are obtained in our paper (summarized at the beginning of Section 8), without invoking any electric body force. So far as this set of equations are concerned, the electric body force will play no role: you can get the same prediction whether you assume it or not.

3. Regarding your previous remark "whether to favor a variational formulation of the theory or not is a matter of taste". I agree with you, in the sense that formulating any theory is a matter of taste. The nature will operate regardless we have a theory or how ugly our theory is. To me, introducing the notion of electric body force for dielectric is yet another postulate, especially if different people can have different formulas. I'm happy to show that you don't need it to arrive your governing equations.

You might like this quote of Einstein, "I believe that we should adhere to the strict validity of the energy principle until we shall have found important reasons for renouncing this guiding star."

In our paper, we have showed, by adhering to what work is, all equations can be derived without ever talking about the electric body force.

## You are right

Dear Zhigang,

You are right that the electric body force can be included in the constitutive equation. This is possible if the force is a divergence of some quantity. The latter is not always the case and Landau notices that the external charges, if exist, can define a separate electric body force.

You try to confine the discussion to the electric fields in dielectrics only. If, however, you take a wider view of magnetic/electromagnetic interactions in both dielectrics and conductors then the necessity of the concept of the electromagnetic body force seems to be evident and accepted by all authors, including Landau. If you agree with the latter then why should we sacrifice the poor body forces in a very special case of the electromagnetic deformation?

## Electrodynamics in deformable media

Dear Kosta:

You have raised an interesting challenge. I have not spent much time with electrodynamics, so I cannot address this challenge properly now. But I have been doing some reading. Here are a list of things I've noticed.

1. There are many versions of theories and, according to Pao (1978), none is universally accepted. His review was written almost 30 years ago. I'm curious if anything of substance has happened lately. If you and others know, please let me know.

2. I quickly read the paper by Peter Lee (as pointed out by Rui Huang above). Peter invoked no body force in his formulation. As Rui pointed out, Peter's formulation was for infinetesimal deformation. I think I have managed to translate Peter's formulation to one for finite deformation. But I need to go through the steps a few more times before I post anything, as I'm not familiar with electrodyanmics.

3. If we are just talking about static magnetics, I believe the situation is similar to electrostatics. No body force is needed.

4. I've been working on electromigration for many years. One may regard the electron wind force as a body force, but it seems to be a very different concept from the body force that we are talking about. In a review article written in 1996, I did talk about a virtual work formulation of the electron wind force. It might be interesting to see how these ideas relate.

## continuum electrodynamics and more

Dear Zhigang:

I found more works on nonlinear theories of deformable dielectrics (ferromagnetics, electroelasticity, etc.). You may find some from the attached article about H.F. Tiersten (one of Mindlin's students). Apparently, Tiersten had done quite a bit on dielectrics in general and electrodynamics in particular. However, his work is not well known for some reasons.

RH

## Thank you, Rui, for pointing out the work of Tiersten

Thank you, Rui, for pointing out the work of Tiersten. More than 10 years ago, when I first started working on cracks in piezoelectric ceramics, I looked at his book on linear piezoelectric plates. I'm unaware of his other works. Have you finished reading our paper posted here? If you do, perhaps you can comment on if we have missed anything from Tiersten's work. I'll also begin to read his papers. Thank you again.

## A question concerning Maxwell stress in dielectric fluids

Kosta:

Equation (6.7) in our paper gives an expression of the stress in compressible dielectric fluids. This expression is not new, and is attributed to Helmhotz in textbooks. We just show how this expression comes out from our approach. To us, this expression, together with (6.4), is the material law for compressible dielectric fluids.

I have two questions for you:

My answers to the questions are

I'm curious if you will reach different answers.

Note added on 18 January 2007. Here is the entry in Wikipedia on the Maxwell stress. This entry only defines the Maxwell stress in a vacuum.

## Zhigang, By Maxwell stress I

Zhigang,

By Maxwell stress I define the stress appearing under the electric potential in the absence of mechanical loads. The non-Maxwellian stress appears under the mechanical loading in the absence of the electric field.

I agree that it can be difficult to formalize this notion in the nonlinear case...

## Dorfmann is an iMechanician now!

Kosta: Thank you very much for introducing this paper to me. I followed it up, not only located the paper and read it, and also located one of its authors: Luis Dorfmann. He has joined the faculty of Tufts University, a few miles from Harvard. Today we had a lunch together and spent some time comparing our approaches. He has now posted a more recent version of his work with Ogden. Amazing. It has just been a few days. I posted our paper on line on 1 January 2007, Monday. Today is Friday.

Thank you for bringing us together.

## This is what iMechanica for!

This is what iMechanica for!

## Thank you for encouragements and suggestions.

The paper was submitted to a journal yesterday after it was posted on iMechanica. We'll follow up on the suggestions posted here in revising the paper. In the journal paper we'll also cite this thread of discussion, so that readers of the journal paper will see your contributions. Happy new year!

## Elegant Work

Dear Zhigang:

This is very fundamental work with an elegant and insightful approach. I read the whole paper from the beginning to the end. This paper has clarified some important concepts that have confused us for a long time, such as how to treat the stress in a dielectric media. The paper showed that it is not necessary (or possible) to separate the stress by the "mechanical part" and the "dielectric part". Such a separation would be non-unique and does not correspond to measurable physical quantities. In addition to deformation, the results would also be very useful to calculate particle interactions. For instance, consider the interaction of two particles in an incompressible dielectric fluid. We can enclose one particle with a surface just outside the particle, and use the integration of the Maxwell stress to calculate the total force and torque this particle feels. Similar approach can be applied to consider the interaction of particles in deformable media and functionized particles. I am excited to see that with manuscript posting iMechanica has become media that allows real "up to the minute" communication of research advances. This is how Internet reshape the approach of scientific communication.

Wei

## a few observations on mechanical and electric field quantities

While reading this interesting paper (not finished yet), something occurs to me as we consider mechanical and electric quantities in parallel. A few years ago when I was studying piezoelectricity, I noticed that there are several alternatives in describing the electromechanical coupling. One is relating stress to strain and electric displacement, and the other relates stress to strain and electric field. While the two are essentially equivalent (at least for linear problems), they lead to different definitions of piezoelectric coefficients that sometimes cause confusion. I chose to use the second approach by a parallel reasoning. As familiar to us mechanicians, stress can be considered area intensity of force, and strain is gradient of displacement. Similarly, electric displacement (a confusing name) may be considered as area intensity of charge and electric field intensity as gradient of electric potential. Here we see the parallelism: stress//electric displacement, strain//electric field intensity, force//electric charge, displacement//electric potential. I found such thinking helpful in doing electromechanical researches.

Now back to the paper we are discussing here. I have a few questions as below.

First, when considering a simple system in Section 2, Eq. (2.1) defines the free energy (U) as a function of mechanical displacement (l) and electric charge (Q). However, in Section 5 for inhomogeneous fields, Eq. (5.3) defines a free energy in terms of deformation gradient (F) and electric displacement (D). The two definitions are not parallel. This non-parallelism may have started in Section 3 when considering a virtual voltage. My parallel reasoning following Eq. (2.1) would use a virtual charge. The consequence is then, in Eq. (3.6), we have both a virtual voltage and a variation in charge, while in (3.3) the force remains constant with a virtual displacement. My question is then directed to Eq. (3.6): is this a correct definition of virtual work done by the battery in consistency with Eq. (2.1)?

The above thinking leads to a probably deeper question: is the battery a good model to begin with? What about the electrochemical energy inside the battery? Should it also be included from an energy consideration for a closed system? Alternatively, one may consider an isolated system with no grounded batteries. Taking the parallel capacitor as an example, the charge is conserved on the electrodes, while the electric potential varies. This would naturally lead to the use of virtual voltage.

I will stop here for now and ask other questions later. Thanks.

RH

## a few comments on (virtual) potential

Rui,

Perhaps I can clear up a couple of these issues. At any rate, I'll give it a try.

First, in equation 2.1, del_U is a real change in energy due to moving a real increment of charge (del_Q) through the potential, phi. Again, in equation 3.6, del_Q is a real increment of charge acted on by virtual potential. So only the potential is being varied.

I think the idea of a "battery" as a device for doing electrical work is analogous to a "weight" for doing mechanical work. The weight produces mechanical work with constant force and the battery produces electrical work with constant potential. You'll notice in equation 5.12 we define an electrical Gibbs free energy so we can treat potential as the independent variable. P. Lee and others have used the same approach.

Bill

## battery and weight

Bill:

Thanks for your answer. However, I don't think battery works analogously to the weight. For example, why did not you change the weight (P) to calculate the virtual work done by the weight?

I noticed Eq. (5.13), which is my favorite form following Mindlin and Peter Lee, with the strain and electric field as free variables in the material laws. Now I need to understand the difference between (5.3) and (5.13). Or, in other words, what is the difference between Helmholtz free energy and Gibbs free energy? Physically speaking. When should we use which?

RH

## Re: battery and weight

Rui,

Is your fundamental concern over the electrical work term in eq. 2.1? Do you have a different expression in mind?

The purpose of W^ is just so we can use E~ as the independent variable rather than D~. Both W and W^ are equally valid energy functions. But by using W~ in eq. 8.8, we end up with the familiar definitions for the dielectric permitivity and piezoelectric coefficients.

Bill

## Helmholtz vs. Gibbs, Battery vs. Weight, Virtual vs. actual

1. Helmholtz vs. Gibbs free energyThey are connected simply by a well known relation, i.e., equation (5.12) in our paper. You can use either one, depending on whether you would like to use E or D as the independent variable in your material law. As mentioned in the paragraph above (5.12), you wish to use E as the independent variable because it matches well with the structure of the field equations. However, if you'd like to study ferroelectric phase transition, for example, you will like to use D as independent variable. Because for a given E, you may get multiple values of D.

Physically, E is a force-like quantity, D is a deformation-like quantity (measuring the displacement of charges in dielectrics). It is interesting that the structure of the electrical field equations looks opposite from the mechanical field equations. Well, such is life.

2. Battery vs. WeightA wight is just as complex as a battery, if you watch all the coming and going of molecules inside the weight. However, both are very simple if all you care is how much work they do. For the weight, the work is force x displacement. For the battery, the work is voltage x charge. So long as we agree what is work, displacement and charge, the above statements define force and voltage.

An empirical fact is this: there is an external mechanism that can do work through displacement, and there is an external mechanism that can do work through flow of change. If you don't like the word weight and battery, you can call them whatever names you wish.

3. Virtual vs. actualVirtual work is not actual work. We use (4.1) to define stress, and (4.5) to define electric displacement. Since we endow these quantities no more properties than what are implied by their definitions, we can choose whatever we please to vary. In the way we use it, the virtual displacement has no relation to actual displacement, and need not even have the unit of displacement. Nor does the virtual displacement have to be small. It is merely a test function to define stress. Similar remarks go for the virtual voltage.

As a consequence of how we write (4.1), when the virtual displacement is replaced by the increment of actual displacement, the virtual mechanical work becomes actual mechanical work. As a consequence of how we write (4.5), when the virtual voltage is replaced by the actual voltage, the virtual electrical work becomes actual electrical work. These observations lead to (5.3) for conservative materials.

Perhaps the teaching of mechanics has added too much mystery to the idea of virtual displacement. In my class I have begun to call it fake displacement. Perhaps test function is a more respectable name.

## Rui,

Rui,

I had exactly the same concern as you have.

In the variational methods we see in traditional mechanics books, the variables with the delta are called "virtual" and a virtual displacement (potential) is associated with the virtual work, while a virtual force (charge) should result in the "virtual complimentary work".

On a surface on which the displacement (potential, e.g. the ground or an ideal battery) is given, we can not have change in the displacement (potential), and only change in the force (virtual force) is allowed, so that we can only write the complementary virtual work on it. (The virtual work on a displacement/potential boundary is simply 0.) The result would be a displacement (potential) boundary condition. On the other hand, when the traction (charge) is given, we can only have change in the displacement (potential). This will result in a force (charge) boundary condition.

But all these are just a way of expressing the idea. We can simply place the delta infront of the force/potential, and change the virtual work into complementary virtual work, and we will get the same result without any problem. (Although the "battery" might not be a good example, as people usually don't give the charge boundary condition on a battery, at least in textbooks.)

## A nonlinear field theory of deformable dielectrics

Hi Zhigang, Xuanhe and William,

This paper provides physical insight into electro-mechanical interactions and derives the governing equations in an elegant manner.

A number of different (equivalent) equilibrium formulations for electro-sensitive materials are available in the literature. These involve, in general, different definitions of stress tensor and electric body force and may provide potential for confusion and misinterpretation.

The use of the ‘total stress tensor’ in equation (5.6) allows the equilibrium equation to be written in a very simple form. In this formulation the influence of the electric field on the deforming continuum is incorporated through the stress tensor and not through body force terms.

Congratulations on this nicely written paper.

## On the traction force of the opposite charges on two electrodes

Dear Prof. Suo,

I am reading your nonlinear field theory of deformable dielectric. I am not very familiar with the nonlinear continuum theory. I have a question on the traction force of the opposite charges on the two electrodes. You illustrate a parallel-plate capacitor, loaded by both the voltage and the weight. The two electrodes are separated by a vacuum. The traction force of the opposite charges on the two electrodes is given by Eq.(2.7), which is balanced by the weight. In general, the interaction force of charges depends on the distance between them, i.e. the force is inversely proportional to the square of the distance if the charges are constant. However, Eq.(2.7) shows that the traction force is independent of the distance between the two electrodes. I don't understand the difference between your traction force and the interaction force of two separated charged particles. Maybe I have no deep understanding to your theory.

Jie

## Re: traction force of the opposite charges on two electrodes

Dear Jie: Thank you very much for reading our paper. (2.7) is a result in many textbooks (e.g., The Feynman Lectures on Physics, Vol. II, p.8-2). We used this result in our paper to illustrate the Maxwell stress in vacuum. Because the charge is distributed on two plates, rather than concentrated at two points, the expression of the force looks different. Like Feynman and many others, we reached this result by calculating work done by the voltage and the weight. I'm sure you can obtain the same result by integrating forces due to point charges. Hope that this note helps.

## Dear Prof. Suo, Thank you

Dear Prof. Suo,

Thank you very much for your reply. I will read the Feynman Lectures on Physics you mentioned for details.

Jie

## Thesis by Van de Ven

There is a nice book on the subject by Hutter and Van de Ven that was published in the late 1970's and seems to have very recently been revised and is back in print. Information about the second edition can be found here.

The original book was essentially a reproduction of Van de Ven's thesis (poor typesetting and all) that can be found at here.

Have not yet seen the second edition (I discovered its existence while doing some background research for this post) but am excited to take a look soon.

## Van de Ven's book

I finished my master's thesis under Fons Van de Ven roughly half a year ago. I remember seeing him with the just published second edition of his book. If you are interested, I will talk to him to find out how his book is being distributed.

"I am always ready to learn, although I do not always like to be taught." Winston Churchill

## Accepted: A nonlinear field theory of deformable dielectrics

I'm delighted that this paper is now accepted for publication in JMPS. The following is the comments of the reviewers and our responses. Both shed addition light on the topic. I have also attached the revised manuscript.

Ms. Ref. No.: JMPS-D-07-00003

Title: A nonlinear field theory of deformable dielectrics

Journal of the Mechanics and Physics of Solids

Dear Huajian:

Thank you very much your message dated 6 May 2007, and for finding very perceptive reviewers. As you know, the problem we study has a long history, and the existing literature is complex and difficult. We have worked hard to simplify our presentation, and are delighted that both reviewers found our paper transparent.

Reviewer #1 stated that "this is a very interesting paper," but is not sure if the paper is worthy of publication in JMPS. Reviewer #2 stated that "The paper clarifies a number of concepts that are currently not clearly understood in the literature", and recommended the paper be published in JMPS.

We believe that this paper makes the following contributions:

1. A new formation of the mechanics of deformable dielectrics that removes superfluous postulates, which one can find many in the literature. Section 4 looks extremely simple and familiar. Its significance should be appreciated by comparing it to the classic papers of Toupin, Eringen, Maugin, and Tiersten, as well as with more recent papers by McMeeking and Landis, Rinaldi and Brenner, and Dorfmann and Ogden.

2. Central to our formulation are the weak statements (4.2) and (4.6). We have not found these weak statements in the literature. In a follow-up work (Zhou et al., 2007), we have used the weak statements to implement a finite element method to solve problems of large deformation.

3. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. However, in general, we show that the stress depends on electric field and deformation in a coupled way, so that Maxwell stress cannot be identified. Our theory does not rely on the notion of the Maxwell stress.

4. A formulation of problems involving small perturbations around a given state. The given state can be fully nonlinear. Problems include bifurcation and localization, which are known to be highly nonlinear problem. In terms of the nominal quantities, all field equations are linear; see Section 4. Nonlinearity comes in through the free-energy function. In a follow-up work (Zhao et al. available online), we have used the new theory to give the first theoretical interpretation of an experimentally observed instability reported in 2006.

In what follows, we will respond to the

comments of the reviews in detail. In

the revised manuscript, we have added the modifications described below,

corrected typos, updated references, and reworked paragraphs to improve

clarity.

Best wishes,

Zhigang

Reviewer #1Comments.This is a very interesting paper. In the absence of the large volume of prior work in this area I would recommend that this manuscript be accepted without significant revision. However, upon reading this I get the feeling that I am already aware of what is being presented, this has already been done. In addition to the general formulations by Toupin, Eringen, Maugin and others, the additional idea that the Maxwell stress cannot be separated from the total stress is a point that has also been emphasized by other authors.Response. As explained in the Introduction of the general formulations by Toupin, Eringen, Maugin invoked various many additional postulates and quantities. Given the broad applications of deformable dielectrics, and recent interest in dielectrics capable of large deformation, it seems to us that this paper makes a significant contribution by placing the subject on a secure theoretical foundation. Although qualms about the Maxwell stress in deformable solids have been expressed by several authors, notably by McMeeking and Landis (2005), all existing formulations start with some notion of Maxwell stress. By contrast, we show that one can formulate a theory without invoking the notion of the Maxwell stress. We have added the following to the manuscript.p.7 "We will first show that a procedure to define nominal stress in continuum mechanics is still applicable for deformable dielectrics. This procedure makes additional postulates in the literature of deformable dielectrics superfluous. We will then apply the same procedure to define the nominal electric displacement."

Comments. The significant contributions of the present manuscript are its presentation of these ideas in a very elegant and convincing manner and in its formulation in the reference configuration instead of the current configuration (although I think that this has been done as well). Given that the primary premise of the paper has appeared before, I would have liked to have seen the authors make some critical evaluations of the two approaches, reference versus current. Granted this may be a matter of taste, but I would imagine that there are some situations where one approach might be preferred over the other. For example, the mapping of a closed crack onto and open crack is singular and solving such a problem in the reference configuration might not be straightforward (I think this may not even be possible, but I could be wrong).Response. We are delighted that the reviewer thanks that our formulation is a significant contribution. However, formulation in reference configuration is not new, as pointed out by the reviewer. Our paper regards both current and reference states as well understood settings to formulate continuum theories. What is new in this paper, we believe, is our removal all superfluous postulates. This removal is done in both current and reference states. We have reworked a paragraph as follows:p. 6. "There is considerable flexibility in choosing measures of stress, strain, electric field, and electric displacement. So long as a theory relates measurable quantities, all alternative definitions are equally valid, and are related by transformations. However, a given boundary value problem may be easier to solve in terms of one set of variables than in terms of another. To avoid confusion, we will develop one set of measures (the nominal quantities) in the body of the paper and discuss alternatives in Appendix A and B."

Comments. The examples that are given, on small deformation and small electric displacement, and infinitesimal perturbations are interesting but don't really exercise the need for a finite deformation electrostatics theory.Response. In section 8, we described the approach to a class of problems involving small perturbations from a given state. The given state is arbitrarily nonlinear. Among examples given are bifurcation and localization. These problems can only be formulated with a fully nonlinear theory, so that one can ask whether the tensor of tangent modulus is positive-definite or not.Comment.Another issue in using one formulation versus another is the measurement of the material properties. If one just assumes a certain constitutive relationship then it is likely that there will be differences in the derivation of the stresses of the order of E*D which are in effect the Maxwell stresses. So while the separate identification of Maxwell stresses is notmeasurable, the subtle effects of these forces must still be measured.

On the more philosophical issue of Maxwell stresses, I certainly agree that experimentally it is not possible to separate the Maxwell stress from the total stress, but can this be done in a model? Certainly, when we model a fluid as incompressible (or even compressible with zero bulk modulus) then an unambiguous formula for the Maxwell stress results. What if we model a dielectric solid as a simple cubic array of charges (or dipoles if you prefer). At each lattice point there are equal positive and negative charges. The electrostatic interaction between these charges is ignored (to avoid the infinity) but there is a linear "spring" connecting them such that the separation of these charges requires energy (or the dipole is initially zero and requires energy to be created). This model accounts for linear dielectric behavior. Next, link each of the pairs of charges to their nearest neighbors with a linear spring in order to account for elasticity (perhaps we need diagonal springs as well to stabilize the lattice). Now, we can apply electric fields to this lattice and compute the electrical forces on the dipoles and the forces due to the elasticity springs. Aren't the electrical forces related to the Maxwell stress? In such a model is it fair to claim that the identification of the Maxwell stress is "without merit"? The problem with measuring these different contributions in a real system is that the elastic springs are due to electromagnetic forces as well and ultimately an experimental distinction is impossible.

Response. We believe that the reviewer has addressed his/her concerns in the last sentence above. We wish to point to the following in our original manuscript:p.12 "By contrast, for compressible fluids, and for solid dielectrics discussed below, the stress (6.7) depends on deformation and electric displacement in a coupled way. Any attempt to separate them and call part of the stress the Maxwell stress must be arbitrary. The practice may provide temporary mental comfort, but on close examination is without merit."

Comments. Ultimately, the derivation in here is correct and likely to be worthy of publication. My only concern is that I am not sure that it is worthy of publication in JMPS. I tend to think that papers in JMPS should present new theories, which this does not, or they should significantly enhance the state of the art, which I don't think this does in its present form. I am suggesting a major revision so that the authors can perhaps address a problem where finite deformation is important. Another course of action is for the authors to withdraw the manuscript from JMPS and submit to another relevant journal.Response. We believe that this paper makes the following contributions:1. A new formation of the mechanics of deformable dielectrics that removes superfluous postulates in the literature. Section 4 looks extremely simple and familiar. Its significance should be appreciated by comparing it to the classic papers of Toupin, Eringen, Maugin, and Tiersten, as well as with more recent papers by McMeeking and Landis, Rinaldi and Brenner, and Dorfmann and Ogden.

2. Central to our formulation are the weak statements (4.2) and (4.6). In a followup work (Zhou et al., 2007), we have used the weak statements to implement a finite element method to solve problems of large deformation.

3. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. However, in general, we show that the stress depends on electric field and deformation in a coupled way, so that Maxwell stress cannot be identified. Our theory does not rely on the notion of the Maxwell stress.

4. A formulation of problems involving small perturbations around arbitrary given state. Problems include bifurcation and localization, which are known to be highly nonlinear problem. Note that in terms of the nominal quantities, all field equations are linear; see Section 4. Nonlinearity comes in through the free-energy function. In a followup work (Zhao et al. available online), we have used the new theory to give the first theoretical interpretation of an experimentally observed instability reported in 2006.

Reviewer #2:

Comments.This is a nice paper. It is very well written and easy to read. The paper clarifies a number of concepts that are currently not clearly understood in the literature. The paper can be published in JMPS after the following comments are addressed:1) The authors should compare the results from the theory with experimental data. By selecting one or two problems where there is experimental data the authors should compare theory with experiments. This will convince the reader that the theory accounts for all the physical mechanisms that are present in deformable dielectrics.

2) Different dielectrics behave differently under electrical potentials e.g. silicon dioxide and silicon nitride behave differently under electric field. It is not clear how the theory accounts for the differences in various dielectrics.

Response. This paper focuses on fundamental theory. Comparison with experimental observations is planed in follow up work. Our initial attempts are documented in Zhao et al. (available online), and Zhou et al. (available upon request). Different materials are differentiated by the free energy function, as described in Sections 5,6,7## Accepted: A nonlinear field theory of deformable dielectrics

Dr. Suo,

Reading the manuscript, I couldnt curb my curiosity..can I have copies of the papers Zhao et al. and Zhou et al.?

Many thanks

Aman

## Papers on nonlinear deformation of dielectric elastomers

Dear Aman: In addition to this paper, members of my group have posted the following papers on iMechanica:

These papers have shown how the basic formulation may be used to analyze some of the intriguing experimental observations reported recently. Please do let us know if you have comments.

## New items on deformable dielectrics