static electric force and measurement principle of material

Hindawi Publishing CorporationSmart Materials ResearchVolume 2012, Article ID 712103, 8 pagesdoi:10.1155/2012/712103

Research Article

Static Electric Force and Measurement Principle ofMaterial Constants in Electrostrictive Material

Zhen-Bang Kuang

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China

Correspondence should be addressed to Zhen-Bang Kuang, [email protected]

Received 30 November 2011; Accepted 23 December 2011

Academic Editor: Ma Jan

Copyright © 2012 Zhen-Bang Kuang. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Electrostrictive materials convert electrical energy into mechanical energy and vice versa. They are extensive applied as intelligentmaterials in the engineering structures. The governing equations in electrostrictive media under the quasistatic electric field arevery important for the measurement of material constants and the research on the strength and function. But some theoreticalproblems should be further clarified. In this paper, the electric force acting on the material is studied and the complete governingequations will be given. In this paper a possible method to measure electrostrictive coefficients is also discussed.

1. Introduction

The measurement method of material constants in an elec-trostrictive material is somewhat controversial between au-thors. Shkel and Klingenberg [1] considered that “The ulti-mate deformation depends on the elastic properties of thefixtures attached to the material (e.g., the electrodes). The(electrostrictive) coefficients γijkl are therefore not strictlymaterial parameters, but rather characteristics of the entiresystem.” Zhang et al. [2] pointed out that “In general ina nonpiezoelectric material such as the polyurethane elas-tomers investigated, the electric field induced strain can becaused by the electrostrictive effect and also by the Maxwellstress effect. The electrostrictive effect is the direct couplingbetween the polarization and mechanical response in thematerial. . . . On the other hand Maxwell stress, which isdue to the interaction between the free charges on theelectrodes (Coulomb interaction) and to electrostatic forcesthat arise from dielectric inhomogeneities.” Guillot et al. [3]considered that “strictly speaking, the Maxwell stress tensordoes not belong to the electrostrictive equations, but thatit should be taken into account in the measurements. . . .it is possible to factor out its contribution to the total res-ponse of the film and therefore to identify the isolated con-tribution due to the (electrostrictive) tensor only.” Thakurand Singh [4] considered that: “In most of the recent exper-iments concerning determination of electrostrictive param-

eters in elastic dielectrics, several researchers used incorrectequations without considering the contribution from theedge effect, the shear stress and suitable boundary condi-tions. This led to wrong predictions of experimental resultsparticularly for materials with high Poisson ratios. Errorsin the estimation of induced strains, varying from an un-derestimation of 202% to an overestimation of 168%, havebeen pointed out in the case of polycarbonate (PC).” Somematerial scientists felt puzzled about the objectivity of theelectrostrictive coefficients.

In the books of Stratton [5] and Landau and Lifshitz[6], the formula of the stress in an isotropic electrostrictivematerial is

σik = ∂g0

∂εik+ σLik,

σLik =(

12

)(2ε − a1)EkEi

−(

12

)(ε + a2)EmEmδik,

(1)

where ∂g0/∂εik is the stress without the electromagnetic fieldin the medium. As shown in [7, 8] σLik in this formula isjust the pseudo total stress [9, 10] which is the sum of theMaxwell stress and the Cauchy stress introduced by the con-stitutive equations. In Pao’s paper [11], he considered thatthe expression of the Maxwell stress is not unique and he gave

2 Smart Materials Research

some different expressions by different authors. McMeekingand Landis [12] considered that “Since there are no experi-ments that can separate the effects of the Cauchy andMaxwell stresses unambiguously, it is generally more prof-itable to consider their sum and not to try identify them sep-arately,” “we obviate the need to develop a constitutive theorythat is consistent with a pre-determined formulation of theMaxwell stress, as can often be found in the literature on elec-trostrictive materials. Instead, the constitutive model can besimplified to one that embraces simultaneously the Cauchy,Maxwell, electrostrictive and electrostatic stresses, which inany case cannot be separately identified from any experi-ment.”

From the above and other literatures we can findthat different author has different understanding about thegoverning equations and the Maxwell stress for the electro-strictive materials. We offered the physical variational princi-ple [7, 8, 13–17] based on the thermodynamics in the non-linear electroelastic analysis to get the governing equationsand get the Maxwell stress naturally. The strength problemin engineering is determined by the Cauchy stress, which isconnected with the constitutive equation, and the Maxwellstress is an external effective static Coulomb electric force.In this paper we give a general method to determine thestatic electric force or the Maxwell stress acting on the ma-terial by using the migratory variation of ϕ in the energyprinciple. We are sorry that we do not have the ground todo experiments, so in this paper only a possible method tomeasure electrostrictive coefficients is discussed.

2. The Physical Variational Principle

In literatures [7, 8, 13–17] we proposed that the first law ofthermodynamics includes two contents: energy conservationlaw and physical variational principle (PVP), that is,

Classical Energy conservation :∫VdudV − dw − dQ = 0,

Classical linear PVP : δΠ =∫VδudV − δw − δQ = 0.

(2)

We proposed the physical variational principle as a basicprinciple in the continuum mechanics. From this principlewe get the governing equations of the nonlinear electroelasticmaterials. For this principle with the electric Gibbs functionwe can simply illustrated as follows.

Under the small deformation the electric Gibbs functionor electric enthalpy g = U − EiDi, where U is the internalenergy, can be expanded in the series of ε and E:

g =(

12

)Cijklεi j εkl −

(12

)εklEkEl − eki jEkεi j

−(

12

)li jklEiEjεkl,

Cijkl = Cjikl = Cijlk = Ckli j ,

li jkl = l jikl = li jlk = lkli j ,

eki j = ek ji, εkl = εlk.(3)

The constitutive equations are

σkl = ∂g

∂εkl= Cijklεi j − ejklEj −

(12

)li jklEiEj ,

Dk = − ∂g

∂Ek=(εkl + li jklεi j

)El + eki jεi j ≈ εklEl.

(4)

In (3) and (4) σ , ε, D, E are the stress, strain, electricdisplacement, and electric field, respectively, C, e, ε, l are theelastic coefficient, piezoelectric coefficient, permittivity, andthe electrostrictive coefficient, respectively. Using (4), (3) isreduced to

g =(

12

)Cijklε jiεlk + ge,

ge = −(

12

)(DkEk + Δklεlk), Δkl = emklEmk,

(5)

where ge is the part of g related to the electric field. Thevalue Δ : ε in (5) can be neglected due to its very small.In the electroelastic analysis the dielectric, its environmentand their common boundary aint consociate a system andshould be considered together, because the electric field existsin every material except the ideal conductor. In this paperthe variables in the environment will be denoted by a rightsuperscript “env,” and the variables on the interface will bedenoted by a right superscript “int.” In the environment (3)–(5) are all held.

Under the assumption that u, ϕ, uenv, ϕenv satisfy theirboundary conditions on their own boundaries au, aϕ, aenv

u ,aenvϕ , and the continuity conditions on the interface aint. The

physical variational principle with the electric Gibbs freeenergy is [7, 8, 13–17]:

δΠ = δΠ1 + δΠ2 − δW int = 0,

δΠ1 =∫Vδg dV +

∫Vgeδui,idV − δW ,

δΠ2 =∫V env

δgenv dV +∫V env

ge envδuenvi,i dV − δW env,

δW =∫V

(fk − ρuk

)δuk dV −

∫VρeδϕdV

+∫aσT∗k δuk da−

∫aDσ∗δϕda,

δW env =∫V env

(f envk − ρuenv

k

)δuenv

k dV −∫V env

ρenve δϕenv dV

+∫aenvσ

T∗envk δuenv

k da−∫aenvD

σ∗envδϕenv da,

δW int =∫aint

T∗ intk δuk da−

∫aint

σ∗ intδϕda,

(6)

Smart Materials Research 3

where f , T∗, σ∗, are given body force per volume, surfacetraction per area, and surface electric charge density, fenv,T∗env, σ∗env, and T∗ int, σ∗ int are also given values in the en-vironment and on the interface, respectively. n = −nenv is theoutward normal on the interface of the body.

The variation of the virtual electric potential ϕ is dividedinto local variation δϕϕ and migratory variation δuϕ, and thesimilar divisions in E, so we have

δϕ = δϕϕ + δuϕ, δuϕ = ϕ,pδup = −Epδup,

δEi = δϕEi + δuEi, δϕEi = −δϕϕ,i, δuEi = Ei,pδup.(7)

The derivation of δE is in Appendix A. Finishing the vari-ational calculation finally we get the governing equations:

Sjk, j + fk = ρuk, Di,i = ρe, in V ,

Sjknj = T∗k , on aσ , Dini = −σ∗, on aD,

σMik = DiEk −(

12

)DnEnδik,

Skl = σkl + σMkl = Cijklεi j − ejklEj

−(

12

)li jklEiEj + DiEk −

(12

)DnEnδik,

(8)

where σMik is the Maxwell stress and Skl is the pseudo totalstress [9, 10]. In the environment we have

Senvi j,i + f env

j = ρenvuenvj , Denv

i,i = ρenve , in V env,

Senvi j nenv

i = T∗envj , on aenv

σ ,

Denvi nenv

i = −σ∗env, on aenvD .

(9)

On the interface we have(Si j − Senv

i j

)ni = T∗ int

j ,(Di −Denv

i

)ni = −σ∗ int, on aint.

(10)

The above variational principle requests prior that thedisplacements, the electric potential satisfy their own bound-ary conditions and the continuity conditions on the inter-face, so the following equations should also be added to go-verning equations:

ui = u∗i , on au, ϕ = ϕ∗, on aϕ,

uenvi = u∗env

i , on aenvu , ϕenv = ϕ∗env, on aenv

ϕ ,

ui = uenvi , ϕ = ϕenv, on aint.

(11)

Equations (8)–(11) are the governing equations in the elec-troelastic analysis.

If we introduce the body electric force f ek , f e envk , and

surface electric force Tek , Te env

k in the materials as

f ek = σMjk, j , Tek = σMjk nj ;

f e envk = σM env

jk, j ,

Teenvk = σM env

jk nenvj .

(12)

Then the variational principle equation (6) is reduced to

δΠ′ = δΠ′1 + δΠ′2 − δW ′int = 0,

δΠ′1 =∫Vδg dV − δW ′,

δΠ′2 =∫V

envδgenvdV − δW ′env,

δW ′ =∫V

(fk + f ek − ρuk

)δukdV −

∫VρeδϕdV

+∫aσ

(T∗k − Te

k

)δukda−

∫aDσ∗δϕda,

δW ′env =∫V env

(f envk + f e env

k − ρenvuenvk

)δuenv

k dV

−∫V env

ρenve δϕenvdV

+∫aenvσ

(Tenv∗k − Te env

k

)δuenv

k da

−∫aenvD

σenv∗δϕenvda,

δW ′int =∫aint

(T int ∗k + Te env

k − Tek

)δukda

−∫aint

σ int ∗δϕda.

(13)

In (13) the variations of δu and δϕ are completely inde-pendent, that is, it is not needed to consider the migratoryvariation δuϕ. Equations (8)–(10) are reduced to

σi j,i +(f j + f ej

)= ρu j , Di,i = ρe, in V ,

σi jni = T∗j − Te∗j , on aσ , Dini = −σ∗, on aD;

σenvi j, j +

(f envi + f e env

i

) = ρuenvi , Denv

i,i = ρenve , in V env

σenvi j nenv

j = (T∗envi − T∗e env

i

), on aenv

σ ,

Denvi nenv

i = −σ∗env, on aenvD ,

(σkl − σenv

kl

)nl = T∗ int

k − Tei + Te env

i ,

Dknk −Denvk nk = −σ∗ int, on aint.

(14)

3. The Static Electric Force Actingon a Dielectric

3.1. The Static Electric Force Acting on the Dielectric in aCapacitor. In any electromagnetic textbook we can find howto determine the static electric force acting on the dielectricin a capacitor consisted of two parallel electrode plates andfilled dielectric with permittivity ε. Assume the length andwidth of the plates are infinite, the distance h between twoplate electrodes is very small. The electric potential on theupper plate electrode is smaller than that on the lower plate.


Let the coordinate origin be located at its center, the planex1–x3 is parallel, and axis x2 is perpendicular to the electrodeplates. There is no external force on the plate and insidethe dielectric and the deformation is small. The electric fieldinside the dielectric of the capacitor is homogeneous andE = E2n, n = i2 due to that the plates are infinite, whereE2 = ϕ/h, ϕ is the difference of electric potentials betweentwo electrodes. The static electric force acting on electrodeplates can be obtained by the energy method.

In this simple case the static electric force can be directlyderived from the general equation of the physical variationalprinciple. According to (6) in this case we have

δΠ = δΠ1 + δΠ2 − δW int = 0,

δΠ1 = δ∫Vg dV = h

[σ22δu2,2 −D2δE2 −

(12

)E2D2δu2,2

]

δΠ2 = 0, δW int = 0.

,

(15)

Give a virtual displacement under the constant electric po-tential on the electrode plate. It is noted that though ϕ isconstant on the plate, but after virtual displacement ϕ ischanged inside the dielectric. For a fixed x the change of theelectric field due to changed ϕ is

δϕE2 = ϕ

(h + δh)− ϕ

h= −ϕδh

h2, −D2δE2 = D2E2

δh

h.

(16)

Due to the electric potential ϕ on the electrode plate isconstant, we have δuE2 = E2,pδup = 0, so δE2 = δϕE2.Therefore, finally we get

δΠ = h[σ22δu2,2 −D2δE2 −

(12

)E2D2δu2,2

]

= hD2

[σ22

(δh

h

)+(

12

)εE2

2

(δh

h

)]= 0

=⇒ T2 = σ22 = −εE22

2.

(17)

We can also use the Maxwell stress to derive the force ac-ting on the dielectric directly. According to (10), we have(Si j − Senv

i j )ni = 0 or Ti = (σi j − σenvi j )ni = −σMi j nj , where

σM envi j = 0 in the electrode is used. In the present case we

have T2 = −σM22 = −D22/2ε. If the force on the electrode plate

is zero, then the static electric force acting on the dielectric is−D2

2/2ε, which is identical with that in (17).In the electric textbooks the derivation is as follows:

given the upper plate a virtual displacement δh along x2, theelectric charge on the electrode plate increases δq due to fixedϕ, so the electric source supplies the energy ϕδq. The energyin the dielectric increases δ(ϕq/2) = ϕδq/2 and the remainedenergy ϕδq/2 is used to overcome the work produced by

the static electric force on the dielectric, that is, for virtualdisplacement δu we have

F · δu = σ22δh = ϕδq −(

12

)ϕδq =

(12

)ϕδq

=(

12

)ϕδ(εE2) =

(ε2

)ϕδ(ϕ

h

)

= −(ε2

)(ϕ

h

)2

δh =⇒ σ22 = −(ε2

)E2

2,

(18)

which is identical with that in (17). At least this result partlyproves the general theory is correct.

3.2. The Static Electric Force in General Case. Though thestatic electric force acting on dielectric of a capacitor wasderived in textbooks from the energy method as shown inabove section, but the energy method did not be used toderive the Maxwell stress in the general case.

In the general case the Maxwell stress is introduced bythe migratory variations of electric field. So the static electricforce from (6) and (13) can be written as

δWe = −δuΠ,

δuΠ =∫Vg,E · δuE dV +

∫Vgeδuk,kdV +

∫Vρeδuϕ dV

+∫aDσ∗δuϕda +

∫V env

genv,E · δuEenvdV

+∫V env

ge envδuenvi,i dV +

∫V env

ρenve δuϕ

env dV

+∫aenvD

σ∗envδuϕenv da +

∫aint

σ∗ intδuϕda,

δWe =∫Vf ek δuk dV +

∫aσTekδuk da +

∫V env

f e envk δuenv

k dV

+∫aenvσ

Te envk δuenv

k da +∫aint

(Tek − Te env

k

)δuk da.

(19)

It is easy to prove that f ek , Tek , f e env

k , Te envk derived from

(19) are identical with that in (12), see Appendix B. If theenvironment is neglected, then we get

∫Vf ek δuk dV +

∫aσTekδuk da

= −(∫

Vg,E · δuE dV +

∫Vgeδuk,k dV

+∫Vρeδuϕ dV +

∫aDσ∗δuϕda

).

(20)


Using D = Dnn + Dtt and E = Enn + Ett, and the con-tinuous condition (10) and (11) of the D, E on the interface,the Maxwell stress can also be rewritten as

n · (σ − σenv) = T∗ int,

T∗ int = T∗ int + n ·(σM env − σM

),

n · (σM env − σM),

=[

n · (Denv ⊗ Eenv)−(

12

)(Denv · Eenv)n

]

−[

n · (D⊗ E)−(

12

)(D · E)n

]

=(

12

)[Dn(Eenvn − En

)− (Denvt −Dt)Et

]n,

(21)

where n is the unit normal, subscripts n and t mean thenormal and tangential direction respectively; there is no sumon n and t. Equation (21) points out that in the case of smallstrain the boundary surface traction corresponding to theMaxwell stress is along the normal direction.

4. Measurement of Material Constants inIsotropic Electrostrictive Materials

For isotropic materials we have

Cijkl = λδi jδkl + G(δikδjl + δilδ jk

),

li jkl = a2δi jδkl +(a1

2

)(δikδjl + δilδ jk

),

εi j = εδi j , eki j = 0.

(22)

So (3) and (4) are reduced to

g =(

12

)λεiiεkk + Gεi jεi j −

(12

)εEkEk

−(

12

)(a2EiEiεkk + a1EiEjεi j

),

(23)

σkl = λεiiδkl + 2Gεkl −(

12

)(a2EiEiδkl + a1EkEl),

Dk = εEk + (a2Ekεmm + a1Elεkl) ≈ εEk,

Skl = σkl + σMkl = λεiiδkl + 2Gεkl −(

12

)(a2 + ε)EmEmδkl

+(

12

)(2ε − a1)EkEl.

(24)

The first formula in (24) is just the usual form of theconstitutive equation, where a1 and a2 are known as elec-trostrictive coefficients. From (8)–(11), it is known thatsolving S is easier than solving σ , so in experiments the mea-sured variables usually are (S, ε, E). So that in usual experi-ments the measured material coefficients are 2ε − a1 anda2 + ε. Therefore, when we do experiments we should clearlyprovide what variables are used. In isotropic materials or

materials without the electromagnetic body couple variablesS, σ , and σM are all symmetric.

As an example, we discuss the electroelastic field of anisotropic rectangular dielectric with material constants ε, a1,a2. The length, width and height of the dielectric are l, b,and h, respectively. Let the coordinate origin be located atits center, the axes are parallel to its edges, the axis x2 isperpendicular to its middle plane. Assume the electric field indielectric is homogeneous E = E2n, n = i2 and the dielectricis free from the external force (f , T∗ int = 0).

4.1. The Surrounding of the Dielectric Is Air. In the air thereis no mechanical stress. On the interface we have

Eairt = Et, Dair

n = Dn on interface; or

Eair1 = E1 = 0, Eair

3 = E3 = 0,

Dair2 = D2

(ε0E

air2 = εE2

), x2 = ±h

2,

Eair1 = E1 = 0, Eair

3 = E3 = 0,

Eair2 = E2; x1 = ± l

2, or x3 = ±b

2,

(25)

where n and t mean the normal and tangential directions,respectively. In this case from (8)–(10) we get

Si j, j = 0, in V ,

(Si j − Sair

i j

)ni = 0, on aint =⇒ σi j =

(σM airi j − σMi j

)int,

σM22 =(

12

)D2E2, σM11 = σM33 = −

(12

)D2E2,

σMi j = 0, when i /= j; on aint,

σM air22 =

(12

)Dair

2 Eair2 , σM air

11 = σM air33 = −

(12

)Dair

2 Eair2 ;

σM airi j = 0, when i /= j, on aint =⇒

σ11 = σ33 = −(

12

)[(ε2

ε0

)− ε]E2

2,

σ22 =(

12

)[(ε2

ε0

)− ε]E2

2.

(26)

Using the constitutive equation (24) we get

−(

12

)[(ε2

ε0

)− ε

]E2

2 = (3λθ + 2Gε11)−(

12

)a2E

22,

(12

)[(ε2

ε0

)− ε

]E2

2 = (3λθ + 2Gε22)−(

12

)(a2 + a1)E2

2,

−(

12

)[(ε2

ε0

)− ε

]E2

2 = (3λθ + 2Gε33)−(

12

)a2E

22.

(27)


4.2. The Dielectric and Air Are All between Two Parallel InfiniteRigid Electrodes. In electrodes E = 0, so there is no Maxwellstress. In this case from (8)–(10) we get

Si j, j = 0, in V ,

(Si j − Senv

i j

)ni = 0, on aint =⇒ σi j =

[σM envi j − σMi j

]int,

σM22 =(

12

)εE2

2, σM11 = σM33 = −(

12

)εE2

2,

σMi j = 0, when i /= j, on aint,

σM plate22 = 0, σM air

11 = σM air33 = −

(12

)ε0E

22,

σM envi j = 0, when i /= j, on aint =⇒

σ11 = σ33 =(

12

)(ε − ε0)E2

2,

σ22 = −(

12

)εE2

2.

(28)

Using the constitutive equation (24) we get

(12

)(ε − ε0)E2

2 = (3λθ + 2Gε11)−(

12

)a2E

22,

−(

12

)εE2

2 = (3λθ + 2Gε22)−(

12

)(a2 + a1)E2

2,

(12

)(ε − ε0)E2

2 = (3λθ + 2Gε33)−(

12

)a2E

22.

(29)

In (27) and (29) 3θ = εii. In the discussed cases the thirdequation in (27) and (29) can be omitted. In experiments wecan measure strains from the given electric field. From (27)or (29) or other improved methods we can get electrostrictivecoefficients. It is clear that the environment has obviouslyeffect.

The above example is very simple and ideal. The realexperimental set is more complex, but from above discus-sions, we can see that in order to get correct electrostrictiveconstants in experiments we need consider the entire systemincluding the dielectric medium, its environment and theircommon boundary.

5. Conclusions

In this paper, we discuss the nonlinear electroelastic analysisin the electrostrictive materials by the physical variationalprinciple. Given a general expression of the electric force.Using the governing equation proposed in this paper we givea possible method to correctly measure the electrostrictivecoefficients. It is shown that in order to get correct materialconstants in experiments we need take the correct governingequations and consider the entire system including thedielectric medium, its environment and their commonboundary.

Appendices

A. The Derivation of δE in (7)

In our previous papers δE in (7) was directly given, here weshall give a simple derivation.

Assume the variational functional is

I =∫VL(xi,uα,uα, j

)dV , (A.1)

where x, u, u, j are the independent variable, dependentvariable, and the derivative of the dependent variable,respectively. Assume an infinitesimal transformation

xi −→ x′i = xi + δui(xj ,ϕ

),

ϕ(xi) −→ ϕ′(x′i) = ϕ(xi) + δϕ

(xi,ϕ

),

δϕ = ϕ′(x′i)− ϕ(xi)

=[ϕ(xi + δui) + δϕϕ(xi)

]− ϕ(xi)

= δϕϕ + δuϕ = δϕϕ + ϕ,iδui.

(A.2)

Noting

∂x′i∂xj

≈ δi j +∂δui∂xj

,

∂xi∂x′j

≈ δi j − ∂δui∂xj

,

j =∣∣∣∣∣∂x′i∂xj

∣∣∣∣∣ ≈ 1 +∂δui∂xi

, (A.3)

from (A.2) we get

δϕ, j = ∂ϕ′(x′i)

∂x′j− ∂ϕ(xi)

∂xj

= ∂[ϕ(xi) + δϕ

(xi,ϕ

)]∂xk

∂xk∂x′j

− ∂ϕ(xi)∂xj

= ∂δϕ(xi,ϕ

)∂xj

− ∂ϕ(xi)∂xk

∂δuk∂xj

=∂(δϕϕ + ϕ,iδui

)

∂xj− ∂ϕ(xi)

∂xk

∂δuk∂xj

=∂(δϕϕ

)

∂xj+∂(ϕ,iδui

)∂xj

− ∂ϕ(xi)∂xk

∂δuk∂xj

=∂(δϕϕ

)

∂xj+∂(ϕ,i)

∂xjδui.

(A.4)

Equation (A.4) is identical with δE in (7) in text.


B. The Derivation of (19)

From (19) we have

δuΠ =∫Vg,E · δuE dV +

∫Vgeδuk,k dV +

∫Vρeδuϕ dV

+∫aDσ∗δuϕda +

∫V env

genv,E · δuEenv dV

+∫V env

ge envδuenvi,i dV +

∫V env

ρenve δuϕ

env dV

+∫aenvD


∫aint

σ∗ intδuϕda

= −∫VDiEp,iδup dV −

∫V

(12

)DkEkδuj, j dV

+∫Vρeδuϕ dV +

∫aDσ∗δuϕda−

∫V env

Denvi δuE

envi dV

−∫V env

(12

)Denv

k Eenvk δuenv

j, j dV +∫V env

ρenve δuϕ

env dV

+∫aenvD


∫aint

σ∗ intδuϕda

= −∫V

(DiEp

),iδup dV +

∫V

(ρe −Di,i

)δuϕdV

−(

12

)∫aDkEknjδuj da +

(12

)∫V

(DkEk), jδuj dV

+∫aDσ∗δuϕda−

∫V env

(Denv

i Eenvp

),iδuenv

p dV

+∫V env

(ρenve −Denv

i,i

)δuϕ

env dV

−(

12

)∫V env

Denvk Eenv

k nenvj δuenv

j dV

+(

12

)∫V env

(Denv

k Eenvk

), jδuenv

j dV

+∫aenvD


∫aint

σ∗ intδuϕda.

(B.1)

Using (8)–(11), that is,

Dini + σ∗ = 0, on aD,

Di,i − ρe = 0, in V ,

(Di −Denv

i

)ni = −σ∗ int, on aint

(B.2)

and adding terms∫a Dini(Epδup + δuϕ)da = 0 and∫

aenv Denvi nenv

i (Eenvp δuenv

p + δuϕenv)da = 0 to (B.1), then (B.1)is reduced to

δuΠ = −∫V

(DiEp

),iδup dV −

∫V

(12

)DkEknjδuj dV

+∫V

(12

)(DkEk), jδuj dV +

∫aDiEpniδup da

−∫V env

(Denv

i Eenvp

),iδuenv

p dV

−∫V env

(12

)Denv

k Eenvk nenv

j δuenvj dV

+∫V env

(12

)(Denv

k Eenvk

), jδuenv

j dV

+∫aenv

Denvi Eenv

p nenvi δuenv

p da

=∫aσσMi j niδuj da−

∫VσMi j,iδuj dV

+∫aenvσ

σM envi j nenv

i δuenvj da−

∫VσM envi j,i δuenv

j dV

+∫aint

σMi j niδuj da +∫aint

σM envi j nenv

i δuenvj da.

(B.3)

From (19), we know that he work done by the equivalentstatic electric force is

δWe =∫Vf ek δuk dV +

∫aσTekδuk da +

∫V env

f e envk δuenv

k dV

+∫aenvσ

Te envk δuenv

k da +∫aint

(Tek − Te env

k

)δuk da.

(B.4)

From (B.3) and (B.4), we immediately get (19).

References

[1] Y. M. Shkel and D. J. Klingenberg, “Material parameters forelectrostriction,” Journal of Applied Physics, vol. 80, no. 8, pp.4566–4572, 1996.

[2] Q. M. Zhang, J. Su, C. H. Kim, R. Ting, and R. Capps, “Anexperimental investigation of electromechanical responses ina polyurethane elastomer,” Journal of Applied Physics, vol. 81,no. 6, pp. 2770–2776, 1997.

[3] F. M. Guillot, J. Jarzynski, and E. Balizer, “Measurement ofelectrostrictive coefficients of polymer films,” Journal of theAcoustical Society of America, vol. 110, no. 6, pp. 2980–2990,2001.

[4] O. P. Thakur and A. K. Singh, “Electrostriction and elec-tromechanical coupling in elastic dielectrics at nanometricinterfaces,” Materials Science- Poland, vol. 27, no. 3, pp. 839–850, 2009.

[5] J. A. Stratton, Electromagnetic Theory, Mcgraw-Hill, New York,NY, USA, 1941.

[6] L. D. Landau and E. M. Lifshitz, Electrodynamics of ContinuumMedia, vol. 8, Pergamon Press, Oxford, UK, 1960.


[7] Z. B. Kuang, “Some problems in electrostrictive and magne-tostrictive materials,” Acta Mechanica Solida Sinica, vol. 20, no.3, pp. 219–227, 2007.

[8] Z. B. Kuang, Theory of Electroelasticity, Shanghai JiaotongUniversity Press, Shanghai, China, 2011.

[9] Q. Jiang and Z. B. Kuang, “Stress analysis in two dimensionalelectrostrictive material with an elliptic rigid conductor,”European Journal of Mechanics A, vol. 23, no. 6, pp. 945–956,2004.

[10] Q. Jiang and Z. B. Kuang, “Stress analysis in a cracked infiniteelectrostrictive plate with local saturation electric fields,”International Journal of Engineering Science, vol. 44, no. 20, pp.1583–1600, 2006.

[11] Y. H. Pao, “Electromagnetic force in deformable continua,”in Mechanics Today, S. Nemat-Nasser, Ed., Pergamon Press,Oxford, UK, 1978.

[12] R. M. McMeeking and C. M. Landis, “Electrostatic forces andstored energy for deformable dielectric materials,” Journal ofApplied Mechanics, vol. 72, no. 4, pp. 581–590, 2005.

[13] Z. B. Kuang, “Some variational principles in elastic dielectricand elastic magnetic materials,” European Journal of MechanicsA, vol. 27, no. 3, pp. 504–514, 2008.

[14] Z. B. Kuang, “Some variational principles in electroelasticmedia under finite deformation,” Science in China, Series G,vol. 51, no. 9, pp. 1390–1402, 2008.

[15] Z. B. Kuang, “Internal energy variational principles andgoverning equations in electroelastic analysis,” InternationalJournal of Solids and Structures, vol. 46, no. 3-4, pp. 902–911,2009.

[16] Z. B. Kuang, “Physical variational principle and thin plate the-ory in electro-magneto-elastic analysis,” International Journalof Solids and Structures, vol. 48, no. 2, pp. 317–325, 2011.

[17] Z. B. Kuang, “Some thermodynamic problems in continuummechanics,” in Thermodynamics—Kinetics of Dynamic Sys-tems, J. C. M. Pirajan, Ed., InTech Open Access Publisher,Rijeka, Croatia, 2011.

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