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Maxwell stress of a dielectric elastomer subjected to electric field

Lianhua Ma's picture

 

Hello everyone,

As reported in some references, Maxwell stress of a dielectric  elastomer subjected to electric field can be determined by the following formula

P=e*E^2    (1) -------e=e0*e1, e0 is the dielectric permittivity of vacuum, e1 is the relative dielectric permittivity of the elastomer, and E is the applied electric field.

However I found another expression from  “Electrodynamics of Continuous Media”. (Landau and E.M. Lifshitz.,Course of Theoretical Physics. Vol. 8. 2nd Edition. Butterworth-Heinemann. Oxford. 1984. L.D. ), which is  described as follow,

(sigma)ik=(e/4/pi)(EiEk-1/2E^2 deltaik)     -(2)   pi=3.14

For a flat elatomer, the Maxwell stress in the direction of thickness can be written as,  (sigma)=(e/8/pi)*E^2  -----(3)


Eq(3) is definetely different from (1) ( in absence of 8*pi), I am confused with the expressions. 


Could you tell me which one is correct?

Thanks
L.H

Comments

Zhigang Suo's picture

Landau and Lifshitz used the centimeter gram second (CGS) system of units.  The conversion of this system to the meter kologram second (MKS) system is described in this Wikipedia article.

In particular, the factor 4pi is due to the difference of the two systems of units.  The two sets of equations have identical physical content. 

Prashant Saxena's picture

Second equation is correct (or more general I should say)... you can deduce the first one out of it using certain simplifications.

A good reference reading would be "On electric body forces and Maxwell stresses in nonlinearly electroelastic solids" by Bustamante, Dorfmann, Ogden.

Lianhua Ma's picture

Thanks, Zhigang and Psaxena,

Just as you mentioned, the difference of factor between Eq.(1) and (3) is due to the conversion of CGS system to MKS system.

I have still a little question for you.

According to the "Various extensions of the CGS system to electromagnetism" described in Wikipedia article, Eq(3) can be rewritten as the following form in SI (International System of Units) system,

 (sigma)=(e/8/pi)*E^2 (CGS)= (e/2)*E^2  (SI)-----(4)    the main difference of the two systems of units lies in the factor 4pi. (point charges)

But now Eq(4) in SI system,  (sigma)= (e/2)*E^2  (SI), is still different from Eq(1) P=e*E^2 (SI), with the difference of a factor 1/2.

Can you help me out with this?

Many thanks,
L.H

 

Zhigang Suo's picture

This difference was explained in my first lecture (slide 16) at a winter school on dielectric transducers.  Because the elastomer is incompressible, the state of deformation induced by a voltage can be represented in three ways.  The three states of stress differ by superposing hydrostatic stress.  All three states of stress give the same state of deformation:  the elastomer reduces the thickness and expands the area.

A detailed derivation of the Maxwell stress for elastomers is given in the notes for a part of a course on advanced elasticity. 

Lianhua Ma's picture

Thank you, Zhigang.

I can understand the three states of stress in your lecture posted here.

I think I have a further understanding of Maxwell stress by studing your notes.

Before I made a mistake in presuming the Maxwell stress only exist in the direction of Electric field.


According to your lectures and  the theory of "Electrodynamics of Continuous Media",  Maxwell stresses  exist in every direciton under
uniaxial electric field E.

For example, considering a flat dielectric elastomer subjected to  electric field E in thickness (z direction), the Maxwell stress in z direction is 1/2*e*E^2, Meanwhile, Maxwell stresses are also induced in both x and y direction with the magnitude  -1/2*e*E^2, despite the elastomer is free in x and y direction.

Is that right? 

It seems not to be easily understood by our "feeling", because of the free edge in x and y direciton. 

Here the in-plane stresses can be regarded as body foces in x and y direciton.

In some papers, for  a dielectric elastomer subjected to electric field E in thickness direction,  Maxwell stress induced by E is assumed to exist in thickness direciton only(e*E^2), and the stresses in other direcitons are treated as zero.   That is not a reasonable restriction? despite the two states of stress give the same state of deformation.

If we want to know the "actual" stress, that is to say, we measure the stress (in thickness direction)  by expriment test for the elastomer subjected to E,  is the compression stress (in thickness direction)  e*E^2 or 1/2*e*E^2 ? 

Thanks
L.H.

 

Zhigang Suo's picture

Dear Lianhua:  I need to get ready to go to airport, so this reply will be short. 

Thank you for your questions.  We struggled with the same questions when we started.  We have tried to resolve these questions in the following papers:

These papers may take some time to read.  I have tried to condense the essential ideas in the lecture notes

Now back to your specific questions.

  1. How to interpret the Maxwell stresses in the directions transeverse to the direction of the electric field?  I have not found a more intuitive interpretation than going through the derivation in the lecture notes.  But some experimental demonstrations might help.  On slide 8 of the first lecture for the winter school on dielectric transducers, I described a classical experiment.
  2. What is the actual stress in an elastomer subject to voltage?  Zero.  The voltage-induced deformation can be thought in the same way as thermal strain.  When a freestanding material is subject to a change in temperature, the material changes its size.  This change is described as thermal expansion.  The stress in the material is zero.  The stress in the context of voltage-induced deformation is explained on p. 4 of the lecture notes.  
Lianhua Ma's picture

I appreciate your reply very much, Zhigang, and I learned a lot from you notes and lectures.
I am reading your two interesting papers posted here, I think both of them are very helpful to me.

Can we understand the Maxwell stress as follow?

Despite Maxwell stress can be treated as a body force, and we can determine analytically and numerically the magnitude of Maxwell stress (not zero in FE simulation),  In fact, the

The introduction of "body force" is just used to analyze the deformation induced by voltage, and the obtained "deformation"(displacement) of elastomer is reliable and effective, whereas the calculated "stress" is not actual stress in nature.

Zhigang Suo's picture

As discussed in our papers and the lecture notes, the Maxwell stress only applies only when the dielectric behavior is liquid-like, unaffected by deformation.  In general, the voltage-generated deformation can be treated as part of a material model.  The idea of body force can be confusing, and is not needed.

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