# Static Electric Force and Measurement Principle of Material Constants in Electrostrictive Material

Authors: Zhen-Bang Kuang
Journal: Smart Materials Research
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

The measurement method of material constants in an electrostrictive material is somewhat controversial between authors. Shkel and Klingenberg [1] considered that “The ultimate deformation depends on the elastic properties of the fixtures attached to the material (e.g., the electrodes). The (electrostrictive) coefficients γijkl are therefore not strictly material parameters, but rather characteristics of the entire system.” Zhang et al. [2] pointed out that “In general in a nonpiezoelectric material such as the polyurethane elastomers investigated, the electric field induced strain can be caused by the electrostrictive effect and also by the Maxwell stress effect. The electrostrictive effect is the direct coupling between the polarization and mechanical response in the material. … On the other hand Maxwell stress, which is due to the interaction between the free charges on the electrodes (Coulomb interaction) and to electrostatic forces that arise from dielectric inhomogeneities.” Guillot et al. [3] considered that “strictly speaking, the Maxwell stress tensor does not belong to the electrostrictive equations, but that it should be taken into account in the measurements. … it is possible to factor out its contribution to the total response of the film and therefore to identify the isolated contribution due to the (electrostrictive) tensor only.” Thakur and Singh [4] considered that: “In most of the recent experiments concerning determination of electrostrictive parameters in elastic dielectrics, several researchers used incorrect equations without considering the contribution from the edge effect, the shear stress and suitable boundary conditions. This led to wrong predictions of experimental results particularly for materials with high Poisson ratios. Errors in the estimation of induced strains, varying from an underestimation of 202% to an overestimation of 168%, have been pointed out in the case of polycarbonate (PC).” Some material scientists felt puzzled about the objectivity of the electrostrictive coefficients.In the books of Stratton [5] and Landau and Lifshitz [6], the formula of the stress in an isotropic electrostrictive material is
(1)σik=∂g0∂εik+σikL,σikL=(12)(2ϵ-a1)EkEi-(12)(ϵ+a2)EmEmδik,
where ∂g0/∂εik is the stress without the electromagnetic field in the medium. As shown in [7, 8] σikL in this formula is just the pseudo total stress [9, 10] which is the sum of the Maxwell stress and the Cauchy stress introduced by the constitutive equations. In Pao’s paper [11], he considered that the expression of the Maxwell stress is not unique and he gave some different expressions by different authors. McMeeking and Landis [12] considered that “Since there are no experiments that can separate the effects of the Cauchy and Maxwell stresses unambiguously, it is generally more profitable to consider their sum and not to try identify them separately,” “we obviate the need to develop a constitutive theory that is consistent with a pre-determined formulation of the Maxwell stress, as can often be found in the literature on electrostrictive materials. Instead, the constitutive model can be simplified to one that embraces simultaneously the Cauchy, Maxwell, electrostrictive and electrostatic stresses, which in any case cannot be separately identified from any experiment.”From the above and other literatures we can find that different author has different understanding about the governing equations and the Maxwell stress for the electrostrictive materials. We offered the physical variational principle [7, 8, 13–17] based on the thermodynamics in the nonlinear electroelastic analysis to get the governing equations and get the Maxwell stress naturally. The strength problem in engineering is determined by the Cauchy stress, which is connected with the constitutive equation, and the Maxwell stress is an external effective static Coulomb electric force. In this paper we give a general method to determine the static electric force or the Maxwell stress acting on the material by using the migratory variation of φ in the energy principle. We are sorry that we do not have the ground to do experiments, so in this paper only a possible method to measure electrostrictive coefficients is discussed.


## 2. The Physical Variational Principle

In literatures [7, 8, 13–17] we proposed that the first law of thermodynamics includes two contents: energy conservation law and physical variational principle (PVP), that is,
(2)ClassicalEnergyconservation:∫VdudV-dw-dQ=0,ClassicallinearPVP:δΠ=∫VδudV-δw-δQ=0.We proposed the physical variational principle as a basic principle in the continuum mechanics. From this principle we get the governing equations of the nonlinear electroelastic materials. For this principle with the electric Gibbs function we can simply illustrated as follows.Under the small deformation the electric Gibbs function or electric enthalpyg=U-EiDi, where U is the internal energy, can be expanded in the series of ε and E:
(3)g=(12)Cijklεijεkl-(12)ϵklEkEl-ekijEkεij-(12)lijklEiEjεkl,Cijkl=Cjikl=Cijlk=Cklij,lijkl=ljikl=lijlk=lklij,ekij=ekji,ϵkl=ϵlk.
The constitutive equations are
(4)σkl=∂g∂εkl=Cijklεij-ejklEj-(12)lijklEiEj,Dk=-∂g∂Ek=(ϵkl+lijklεij)El+ekijεij≈ϵklEl.
In (3) and (4) σ, ε, D, E are the stress, strain, electric displacement, and electric field, respectively, C, e, ϵ, l are the elastic coefficient, piezoelectric coefficient, permittivity, and the electrostrictive coefficient, respectively. Using (4), (3) is reduced to
(5)g=(12)Cijklεjiεlk+ge,ge=-(12)(DkEk+Δklεlk),Δkl=emklEmk,
where ge is the part of g related to the electric field. The value Δ:ε in (5) can be neglected due to its very small. In the electroelastic analysis the dielectric, its environment and their common boundary aint consociate a system and should be considered together, because the electric field exists in every material except the ideal conductor. In this paper the variables in the environment will be denoted by a right superscript “env,” and the variables on the interface will be denoted by a right superscript “int.” In the environment (3)–(5) are all held.Under the assumption thatu, φ, uenv, φenv satisfy their boundary conditions on their own boundaries au, aφ, auenv, aφenv, and the continuity conditions on the interface aint. The physical variational principle with the electric Gibbs free energy is [7, 8, 13–17]:(6)δΠ=δΠ1+δΠ2-δWint⁡=0,δΠ1=∫VδgdV+∫Vgeδui,idV-δW,δΠ2=∫VenvδgenvdV+∫Venvgeenvδui,ienvdV-δWenv,δW=∫V(fk-ρük)δukdV-∫VρeδφdV+∫aσTk*δukda-∫aDσ*δφda,δWenv=∫Venv(fkenv-ρükenv)δukenvdV-∫VenvρeenvδφenvdV+∫aσenvTk*envδukenvda-∫aDenvσ*envδφenvda,δWint⁡=∫aint⁡Tk*int⁡δukda-∫aint⁡σ*int⁡δφda,
where f, T*, σ*, are given body force per volume, surface traction per area, and surface electric charge density, fenv, T*env, σ*env, and T*int, σ*int are also given values in the environment and on the interface, respectively. n=-nenv is the outward normal on the interface of the body.The variation of the virtual electric potentialφ is divided into local variation δφφ and migratory variation δuφ, and the similar divisions in E, so we have
(7)δφ=δφφ+δuφ,δuφ=φ,pδup=-Epδup,δEi=δφEi+δuEi,δφEi=-δφφ,i,δuEi=Ei,pδup.
The derivation of δE is in Appendix A. Finishing the variational calculation finally we get the governing equations:(8)Sjk,j+fk=ρük,Di,i=ρe,inV,Sjknj=Tk*,on⁡aσ,Dini=-σ*,on⁡aD,σikM=DiEk-(12)DnEnδik,Skl=σkl+σklM=Cijklεij-ejklEj-(12)lijklEiEj+DiEk-(12)DnEnδik,
where σikM is the Maxwell stress and Skl is the pseudo total stress [9, 10]. In the environment we have
(9)Sij,ienv+fjenv=ρenvüjenv,Di,ienv=ρeenv,inVenv,Sijenvnienv=Tj*env,on⁡aσenv,Dienvnienv=-σ*env,on⁡aDenv.
On the interface we have
(10)(Sij-Sijenv)ni=Tj*int⁡,(Di-Dienv)ni=-σ*int⁡,on⁡aint⁡.The above variational principle requests prior that the displacements, the electric potential satisfy their own boundary conditions and the continuity conditions on the interface, so the following equations should also be added to governing equations:(11)ui=ui*,on⁡au,φ=φ*,on⁡aφ,uienv=ui*env,on⁡auenv,φenv=φ*env,on⁡aφenv,ui=uienv,φ=φenv,on⁡aint⁡.
Equations (8)–(11) are the governing equations in the electroelastic analysis.If we introduce the body electric forcefke, fkeenv, and surface electric force Tke, Tkeenv in the materials as
(12)fke=σjk,jM,Tke=σjkMnj;fkeenv=σjk,jMenv,Tkeenv=σjkMenvnjenv.
Then the variational principle equation (6) is reduced to
(13)δΠ′=δΠ1′+δΠ2′-δW′int⁡=0,δΠ1′=∫V̅δgdV-δW′,δΠ2′=∫V̅envδgenvdV-δW′env,δW′=∫V(fk+fke-ρük)δukdV-∫VρeδφdV+∫aσ(Tk*-Tke)δukda-∫aDσ*δφda,δW′env=∫Venv(fkenv+fkeenv-ρenvükenv)δukenvdV-∫VenvρeenvδφenvdV+∫aσenv(Tkenv*-Tkeenv)δukenvda-∫aDenvσenv*δφenvda,δW′int⁡=∫aint⁡(Tkint⁡*+Tkeenv-Tke)δukda-∫aint⁡σint⁡*δφda.
In (13) the variations of δu and δφ are completely independent, that is, it is not needed to consider the migratory variation δuφ. Equations (8)–(10) are reduced to
(14)σij,i+(fj+fje)=ρüj,Di,i=ρe,inV,σijni=Tj*-Tje*,on⁡aσ,Dini=-σ*,on⁡aD;σij,jenv+(fienv+fieenv)=ρüienv,Di,ienv=ρeenv,inVenvσijenvnjenv=(Ti*env-Ti*eenv),on⁡aσenv,Dienvnienv=-σ*env,on⁡aDenv,(σkl-σklenv)nl=Tk*int⁡-Tie+Tieenv,Dknk-Dkenvnk=-σ*int⁡,on⁡aint⁡.


## 3. The Static Electric Force Acting on a Dielectric

### 3.1. The Static Electric Force Acting on the Dielectric in a Capacitor

In any electromagnetic textbook we can find how to determine the static electric force acting on the dielectric in a capacitor consisted of two parallel electrode plates and filled dielectric with permittivityϵ. Assume the length and width of the plates are infinite, the distance h between two plate electrodes is very small. The electric potential on the upper plate electrode is smaller than that on the lower plate. Let the coordinate origin be located at its center, the plane x1–x3 is parallel, and axis x2 is perpendicular to the electrode plates. There is no external force on the plate and inside the dielectric and the deformation is small. The electric field inside the dielectric of the capacitor is homogeneous and E=E2n, n=i2 due to that the plates are infinite, where E2=φ/h, φ is the difference of electric potentials between two electrodes. The static electric force acting on electrode plates can be obtained by the energy method.In this simple case the static electric force can be directly derived from the general equation of the physical variational principle. According to (6) in this case we have
(15)δΠ=δΠ1+δΠ2-δWint⁡=0,δΠ1=δ∫VgdV=h[σ22δu2,2-D2δE2-(12)E2D2δu2,2]δΠ2=0,δWint⁡=0.,
Give a virtual displacement under the constant electric potential on the electrode plate. It is noted that though φ is constant on the plate, but after virtual displacement φ is changed inside the dielectric. For a fixed x the change of the electric field due to changed φ is
(16)δφE2=φ(h+δh)-φh=-φδhh2,-D2δE2=D2E2δhh.
Due to the electric potential φ on the electrode plate is constant, we have δuE2=E2,pδup=0, so δE2=δφE2. Therefore, finally we get
(17)δΠ=h[σ22δu2,2-D2δE2-(12)E2D2δu2,2]=hD2[σ22(δhh)+(12)ϵE22(δhh)]=0⟹T2=σ22=-ϵE222.We can also use the Maxwell stress to derive the force acting on the dielectric directly. According to (10), we have (Sij-Sijenv)ni=0 or Ti=(σij-σijenv)ni=-σijMnj, where σijMenv=0 in the electrode is used. In the present case we have T2=-σ22M=-D22/2ϵ. If the force on the electrode plate is zero, then the static electric force acting on the dielectric is -D22/2ϵ, which is identical with that in (17).In the electric textbooks the derivation is as follows: given the upper plate a virtual displacementδh along x2, the electric charge on the electrode plate increases δq due to fixed φ, so the electric source supplies the energy φδq. The energy in the dielectric increases δ(φq/2)=φδq/2 and the remained energy φδq/2 is used to overcome the work produced by the static electric force on the dielectric, that is, for virtual displacement δu we have
(18)F⋅δu=σ22δh=φδq-(12)φδq=(12)φδq=(12)φδ(ϵE2)=(ϵ2)φδ(φh)=-(ϵ2)(φh)2δh⟹σ22=-(ϵ2)E22,
which is identical with that in (17). At least this result partly proves the general theory is correct.


### 3.2. The Static Electric Force in General Case

Though the static electric force acting on dielectric of a capacitor was derived in textbooks from the energy method as shown in above section, but the energy method did not be used to derive the Maxwell stress in the general case.In the general case the Maxwell stress is introduced by the migratory variations of electric field. So the static electric force from (6) and (13) can be written as
(19)δWe=-δuΠ,δuΠ=∫Vg,E⋅δuEdV+∫Vgeδuk,kdV+∫VρeδuφdV+∫aDσ*δuφda+∫Venvg,Eenv⋅δuEenvdV+∫Venvgeenvδui,ienvdV+∫VenvρeenvδuφenvdV+∫aDenvσ*envδuφenvda+∫aint⁡σ*int⁡δuφda,δWe=∫VfkeδukdV+∫aσTkeδukda+∫VenvfkeenvδukenvdV+∫aσenvTkeenvδukenvda+∫aint⁡(Tke-Tkeenv)δukda.
It is easy to prove that fke, Tke, fkeenv, Tkeenv derived from (19) are identical with that in (12), see Appendix B. If the environment is neglected, then we get
(20)∫VfkeδukdV+∫aσTkeδukda=-(∫Vg,E⋅δuEdV+∫Vgeδuk,kdV+∫VρeδuφdV+∫aDσ*δuφda).UsingD=Dnn+Dtt and E=Enn+Ett, and the continuous condition (10) and (11) of the D, E on the interface, the Maxwell stress can also be rewritten as(21)n⋅(σ-σenv)=T̃*int⁡,T̃*int⁡=T*int⁡+n⋅(σMenv-σM),n⋅(σMenv-σM),=[n⋅(Denv⊗Eenv)-(12)(Denv⋅Eenv)n]-[n⋅(D⊗E)-(12)(D⋅E)n]=(12)[Dn(Enenv-En)-(Dtenv-Dt)Et]n,
where n is the unit normal, subscripts n and t mean the normal and tangential direction respectively; there is no sum on n and t. Equation (21) points out that in the case of small strain the boundary surface traction corresponding to the Maxwell stress is along the normal direction.


## 3.1. The Static Electric Force Acting on the Dielectric in a Capacitor

In any electromagnetic textbook we can find how to determine the static electric force acting on the dielectric in a capacitor consisted of two parallel electrode plates and filled dielectric with permittivityϵ. Assume the length and width of the plates are infinite, the distance h between two plate electrodes is very small. The electric potential on the upper plate electrode is smaller than that on the lower plate. Let the coordinate origin be located at its center, the plane x1–x3 is parallel, and axis x2 is perpendicular to the electrode plates. There is no external force on the plate and inside the dielectric and the deformation is small. The electric field inside the dielectric of the capacitor is homogeneous and E=E2n, n=i2 due to that the plates are infinite, where E2=φ/h, φ is the difference of electric potentials between two electrodes. The static electric force acting on electrode plates can be obtained by the energy method.In this simple case the static electric force can be directly derived from the general equation of the physical variational principle. According to (6) in this case we have
(15)δΠ=δΠ1+δΠ2-δWint⁡=0,δΠ1=δ∫VgdV=h[σ22δu2,2-D2δE2-(12)E2D2δu2,2]δΠ2=0,δWint⁡=0.,
Give a virtual displacement under the constant electric potential on the electrode plate. It is noted that though φ is constant on the plate, but after virtual displacement φ is changed inside the dielectric. For a fixed x the change of the electric field due to changed φ is
(16)δφE2=φ(h+δh)-φh=-φδhh2,-D2δE2=D2E2δhh.
Due to the electric potential φ on the electrode plate is constant, we have δuE2=E2,pδup=0, so δE2=δφE2. Therefore, finally we get
(17)δΠ=h[σ22δu2,2-D2δE2-(12)E2D2δu2,2]=hD2[σ22(δhh)+(12)ϵE22(δhh)]=0⟹T2=σ22=-ϵE222.We can also use the Maxwell stress to derive the force acting on the dielectric directly. According to (10), we have (Sij-Sijenv)ni=0 or Ti=(σij-σijenv)ni=-σijMnj, where σijMenv=0 in the electrode is used. In the present case we have T2=-σ22M=-D22/2ϵ. If the force on the electrode plate is zero, then the static electric force acting on the dielectric is -D22/2ϵ, which is identical with that in (17).In the electric textbooks the derivation is as follows: given the upper plate a virtual displacementδh along x2, the electric charge on the electrode plate increases δq due to fixed φ, so the electric source supplies the energy φδq. The energy in the dielectric increases δ(φq/2)=φδq/2 and the remained energy φδq/2 is used to overcome the work produced by the static electric force on the dielectric, that is, for virtual displacement δu we have
(18)F⋅δu=σ22δh=φδq-(12)φδq=(12)φδq=(12)φδ(ϵE2)=(ϵ2)φδ(φh)=-(ϵ2)(φh)2δh⟹σ22=-(ϵ2)E22,
which is identical with that in (17). At least this result partly proves the general theory is correct.


## 3.2. The Static Electric Force in General Case

Though the static electric force acting on dielectric of a capacitor was derived in textbooks from the energy method as shown in above section, but the energy method did not be used to derive the Maxwell stress in the general case.In the general case the Maxwell stress is introduced by the migratory variations of electric field. So the static electric force from (6) and (13) can be written as
(19)δWe=-δuΠ,δuΠ=∫Vg,E⋅δuEdV+∫Vgeδuk,kdV+∫VρeδuφdV+∫aDσ*δuφda+∫Venvg,Eenv⋅δuEenvdV+∫Venvgeenvδui,ienvdV+∫VenvρeenvδuφenvdV+∫aDenvσ*envδuφenvda+∫aint⁡σ*int⁡δuφda,δWe=∫VfkeδukdV+∫aσTkeδukda+∫VenvfkeenvδukenvdV+∫aσenvTkeenvδukenvda+∫aint⁡(Tke-Tkeenv)δukda.
It is easy to prove that fke, Tke, fkeenv, Tkeenv derived from (19) are identical with that in (12), see Appendix B. If the environment is neglected, then we get
(20)∫VfkeδukdV+∫aσTkeδukda=-(∫Vg,E⋅δuEdV+∫Vgeδuk,kdV+∫VρeδuφdV+∫aDσ*δuφda).UsingD=Dnn+Dtt and E=Enn+Ett, and the continuous condition (10) and (11) of the D, E on the interface, the Maxwell stress can also be rewritten as(21)n⋅(σ-σenv)=T̃*int⁡,T̃*int⁡=T*int⁡+n⋅(σMenv-σM),n⋅(σMenv-σM),=[n⋅(Denv⊗Eenv)-(12)(Denv⋅Eenv)n]-[n⋅(D⊗E)-(12)(D⋅E)n]=(12)[Dn(Enenv-En)-(Dtenv-Dt)Et]n,
where n is the unit normal, subscripts n and t mean the normal and tangential direction respectively; there is no sum on n and t. Equation (21) points out that in the case of small strain the boundary surface traction corresponding to the Maxwell stress is along the normal direction.


## 4. Measurement of Material Constants in Isotropic Electrostrictive Materials

For isotropic materials we have(22)Cijkl=λδijδkl+G(δikδjl+δilδjk),lijkl=a2δijδkl+(a12)(δikδjl+δilδjk),ϵij=ϵδij,ekij=0.
So (3) and (4) are reduced to
(23)g=(12)λεiiεkk+Gεijεij-(12)ϵEkEk-(12)(a2EiEiεkk+a1EiEjεij),(24)σkl=λεiiδkl+2Gεkl-(12)(a2EiEiδkl+a1EkEl),Dk=ϵEk+(a2Ekεmm+a1Elεkl)≈ϵEk,Skl=σkl+σklM=λεiiδkl+2Gεkl-(12)(a2+ϵ)EmEmδkl+(12)(2ϵ-a1)EkEl.The first formula in (24) is just the usual form of the constitutive equation, where a1 and a2 are known as electrostrictive coefficients. From (8)–(11), it is known that solving S is easier than solving σ, so in experiments the measured variables usually are (S,ε,E). So that in usual experiments the measured material coefficients are 2ϵ-a1 and a2+ϵ. Therefore, when we do experiments we should clearly provide what variables are used. In isotropic materials or materials without the electromagnetic body couple variables S, σ, and σM are all symmetric.As an example, we discuss the electroelastic field of an isotropic rectangular dielectric with material constantsϵ, a1, a2. The length, width and height of the dielectric are l, b, and h, respectively. Let the coordinate origin be located at its center, the axes are parallel to its edges, the axis x2 is perpendicular to its middle plane. Assume the electric field in dielectric is homogeneous E=E2n, n=i2 and the dielectric is free from the external force (f,T*int=0).


### 4.1. The Surrounding of the Dielectric Is Air

In the air there is no mechanical stress. On the interface we have(25)Etair=Et,Dnair=Dnon⁡interface;orE1air=E1=0,E3air=E3=0,D2air=D2(ϵ0E2air=ϵE2),x2=±h2,E1air=E1=0,E3air=E3=0,E2air=E2;x1=±l2,orx3=±b2,
where n and t mean the normal and tangential directions, respectively. In this case from (8)–(10) we get
(26)Sij,j=0,inV,(Sij-Sijair)ni=0,on⁡aint⁡⟹σij=(σijMair-σijM)int⁡,σ22M=(12)D2E2,σ11M=σ33M=-(12)D2E2,σijM=0,wheni≠j;on⁡aint⁡,σ22Mair=(12)D2airE2air,σ11Mair=σ33Mair=-(12)D2airE2air;σijMair=0,wheni≠j,on⁡aint⁡⟹σ11=σ33=-(12)[(ϵ2ϵ0)-ϵ]E22,σ22=(12)[(ϵ2ϵ0)-ϵ]E22.
Using the constitutive equation (24) we get
(27)-(12)[(ϵ2ϵ0)-ϵ]E22=(3λθ+2Gε11)-(12)a2E22,(12)[(ϵ2ϵ0)-ϵ]E22=(3λθ+2Gε22)-(12)(a2+a1)E22,-(12)[(ϵ2ϵ0)-ϵ]E22=(3λθ+2Gε33)-(12)a2E22.


### 4.2. The Dielectric and Air Are All between Two Parallel Infinite Rigid Electrodes

In electrodesE=0, so there is no Maxwell stress. In this case from (8)–(10) we get(28)Sij,j=0,inV,(Sij-Sijenv)ni=0,on⁡aint⁡⟹σij=[σijMenv-σijM]int⁡,σ22M=(12)ϵE22,σ11M=σ33M=-(12)ϵE22,σijM=0,wheni≠j,on⁡aint⁡,σ22Mplate=0,σ11Mair=σ33Mair=-(12)ϵ0E22,σijMenv=0,wheni≠j,on⁡aint⁡⟹σ11=σ33=(12)(ϵ-ϵ0)E22,σ22=-(12)ϵE22.
Using the constitutive equation (24) we get
(29)(12)(ϵ-ϵ0)E22=(3λθ+2Gε11)-(12)a2E22,-(12)ϵE22=(3λθ+2Gε22)-(12)(a2+a1)E22,(12)(ϵ-ϵ0)E22=(3λθ+2Gε33)-(12)a2E22.
In (27) and (29) 3θ=εii. In the discussed cases the third equation in (27) and (29) can be omitted. In experiments we can measure strains from the given electric field. From (27) or (29) or other improved methods we can get electrostrictive coefficients. It is clear that the environment has obviously effect.The above example is very simple and ideal. The real experimental set is more complex, but from above discussions, we can see that in order to get correct electrostrictive constants in experiments we need consider the entire system including the dielectric medium, its environment and their common boundary.


## 4.1. The Surrounding of the Dielectric Is Air

In the air there is no mechanical stress. On the interface we have(25)Etair=Et,Dnair=Dnon⁡interface;orE1air=E1=0,E3air=E3=0,D2air=D2(ϵ0E2air=ϵE2),x2=±h2,E1air=E1=0,E3air=E3=0,E2air=E2;x1=±l2,orx3=±b2,
where n and t mean the normal and tangential directions, respectively. In this case from (8)–(10) we get
(26)Sij,j=0,inV,(Sij-Sijair)ni=0,on⁡aint⁡⟹σij=(σijMair-σijM)int⁡,σ22M=(12)D2E2,σ11M=σ33M=-(12)D2E2,σijM=0,wheni≠j;on⁡aint⁡,σ22Mair=(12)D2airE2air,σ11Mair=σ33Mair=-(12)D2airE2air;σijMair=0,wheni≠j,on⁡aint⁡⟹σ11=σ33=-(12)[(ϵ2ϵ0)-ϵ]E22,σ22=(12)[(ϵ2ϵ0)-ϵ]E22.
Using the constitutive equation (24) we get
(27)-(12)[(ϵ2ϵ0)-ϵ]E22=(3λθ+2Gε11)-(12)a2E22,(12)[(ϵ2ϵ0)-ϵ]E22=(3λθ+2Gε22)-(12)(a2+a1)E22,-(12)[(ϵ2ϵ0)-ϵ]E22=(3λθ+2Gε33)-(12)a2E22.


## 4.2. The Dielectric and Air Are All between Two Parallel Infinite Rigid Electrodes

In electrodesE=0, so there is no Maxwell stress. In this case from (8)–(10) we get(28)Sij,j=0,inV,(Sij-Sijenv)ni=0,on⁡aint⁡⟹σij=[σijMenv-σijM]int⁡,σ22M=(12)ϵE22,σ11M=σ33M=-(12)ϵE22,σijM=0,wheni≠j,on⁡aint⁡,σ22Mplate=0,σ11Mair=σ33Mair=-(12)ϵ0E22,σijMenv=0,wheni≠j,on⁡aint⁡⟹σ11=σ33=(12)(ϵ-ϵ0)E22,σ22=-(12)ϵE22.
Using the constitutive equation (24) we get
(29)(12)(ϵ-ϵ0)E22=(3λθ+2Gε11)-(12)a2E22,-(12)ϵE22=(3λθ+2Gε22)-(12)(a2+a1)E22,(12)(ϵ-ϵ0)E22=(3λθ+2Gε33)-(12)a2E22.
In (27) and (29) 3θ=εii. In the discussed cases the third equation in (27) and (29) can be omitted. In experiments we can measure strains from the given electric field. From (27) or (29) or other improved methods we can get electrostrictive coefficients. It is clear that the environment has obviously effect.The above example is very simple and ideal. The real experimental set is more complex, but from above discussions, we can see that in order to get correct electrostrictive constants in experiments we need consider the entire system including the dielectric medium, its environment and their common boundary.


## 5. Conclusions

In this paper, we discuss the nonlinear electroelastic analysis in the electrostrictive materials by the physical variational principle. Given a general expression of the electric force. Using the governing equation proposed in this paper we give a possible method to correctly measure the electrostrictive coefficients. It is shown that in order to get correct material constants in experiments we need take the correct governing equations and consider the entire system including the dielectric medium, its environment and their common boundary.



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*Source: 712103-2012-03-22.xml*