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

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.

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 𝜸_{𝐢𝐣𝐤𝐥} 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 𝜎_𝑖𝑘=𝜕𝑔₀𝜕𝜀_𝑖𝑘+𝜎^𝐿_𝑖𝑘,𝜎^𝐿_𝑖𝑘=122𝜖−𝑎₁𝐸_𝑘𝐸_𝑖−12𝜖+𝑎₂𝐸_𝑚𝐸_𝑚𝛿_𝑖𝑘, (1) where 𝜕𝑔₀/𝜕𝜀_𝑖𝑘 is the stress without the electromagnetic field in the medium. As shown in [7, 8] 𝜎^𝐿_𝑖𝑘 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, ClassicalEnergyconservation∶_𝑉𝑑𝑢𝑑𝑉−𝑑𝑤−𝑑𝑄=0,ClassicallinearPVP∶𝛿Π=_𝑉𝛿𝑢𝑑𝑉−𝛿𝑤−𝛿𝑄=0. (2)

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 enthalpy 𝑔=𝑈−𝐸_𝑖𝐷_𝑖, where 𝑈 is the internal energy, can be expanded in the series of 𝜺 and 𝐄: 1𝑔=2𝐶_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗𝜀_𝑘𝑙−12𝜖_𝑘𝑙𝐸_𝑘𝐸_𝑙−𝑒_𝑘𝑖𝑗𝐸_𝑘𝜀_𝑖𝑗−12𝑙_{𝑖𝑗𝑘𝑙}𝐸_𝑖𝐸_𝑗𝜀_𝑘𝑙,𝐶_{𝑖𝑗𝑘𝑙}=𝐶_{𝑗𝑖𝑘𝑙}=𝐶_{𝑖𝑗𝑙𝑘}=𝐶_{𝑘𝑙𝑖𝑗},𝑙_{𝑖𝑗𝑘𝑙}=𝑙_{𝑗𝑖𝑘𝑙}=𝑙_{𝑖𝑗𝑙𝑘}=𝑙_{𝑘𝑙𝑖𝑗},𝑒_𝑘𝑖𝑗=𝑒_𝑘𝑗𝑖,𝜖_𝑘𝑙=𝜖_𝑙𝑘. (3) The constitutive equations are 𝜎_𝑘𝑙=𝜕𝑔𝜕𝜀_𝑘𝑙=𝐶_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗−𝑒_𝑗𝑘𝑙𝐸_𝑗−12𝑙_{𝑖𝑗𝑘𝑙}𝐸_𝑖𝐸_𝑗,𝐷_𝑘=−𝜕𝑔𝜕𝐸_𝑘=𝜖_𝑘𝑙+𝑙_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗𝐸_𝑙+𝑒_𝑘𝑖𝑗𝜀_𝑖𝑗≈𝜖_𝑘𝑙𝐸_𝑙. (4) 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 1𝑔=2𝐶_{𝑖𝑗𝑘𝑙}𝜀_𝑗𝑖𝜀_𝑙𝑘+𝑔^𝑒,𝑔^𝑒1=−2𝐷_𝑘𝐸_𝑘+Δ_𝑘𝑙𝜀_𝑙𝑘,Δ_𝑘𝑙=𝑒_𝑚𝑘𝑙𝐸_𝑚𝑘, (5) where 𝑔^𝑒 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 𝑎^int 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 that 𝐮, 𝜑, 𝐮^env, 𝜑^env satisfy their boundary conditions on their own boundaries 𝑎_𝑢, 𝑎_𝜑, 𝑎^env_𝑢, 𝑎^env_𝜑, and the continuity conditions on the interface 𝑎^int. The physical variational principle with the electric Gibbs free energy is [7, 8, 13–17]:𝛿Π=𝛿Π₁+𝛿Π₂−𝛿𝑊^int=0,𝛿Π₁=_𝑉𝛿𝑔𝑑𝑉+_𝑉𝑔^𝑒𝛿𝑢_𝑖,𝑖𝑑𝑉−𝛿𝑊,𝛿Π₂=_𝑉env𝛿𝑔^env𝑑𝑉+_𝑉env𝑔^𝑒env𝛿𝑢^env_𝑖,𝑖𝑑𝑉−𝛿𝑊^env,𝛿𝑊=_𝑉𝑓_𝑘−𝜌𝑢̈_𝑘𝛿𝑢_𝑘𝑑𝑉−_𝑉𝜌_𝑒+𝛿𝜑𝑑𝑉_𝑎𝜎𝑇^∗_𝑘𝛿𝑢_𝑘𝑑𝑎−_𝑎𝐷𝜎^∗𝛿𝜑𝑑𝑎,𝛿𝑊^env=_𝑉env𝑓^env_𝑘−𝜌𝑢̈^env_𝑘𝛿𝑢^env_𝑘𝑑𝑉−_𝑉env𝜌^env_𝑒𝛿𝜑^env𝑑𝑉+_𝑎env𝜎𝑇^∗env_𝑘𝛿𝑢^env_𝑘𝑑𝑎−_𝑎env𝐷𝜎^∗env𝛿𝜑^env𝑑𝑎,𝛿𝑊^int=_𝑎int𝑇_𝑘^∗int𝛿𝑢_𝑘𝑑𝑎−_𝑎int𝜎^∗int𝛿𝜑𝑑𝑎, (6) where 𝐟, 𝐓^∗, 𝜎^∗, are given body force per volume, surface traction per area, and surface electric charge density, 𝐟^env, 𝐓^∗env, 𝜎^∗env, and 𝐓^∗int, 𝜎^∗int are also given values in the environment and on the interface, respectively. 𝐧=−𝐧^env is the outward normal on the interface of the body.

The variation of the virtual electric potential 𝜑 is divided into local variation 𝛿_𝜑𝜑 and migratory variation 𝛿_𝑢𝜑, and the similar divisions in 𝐄, so we have 𝛿𝜑=𝛿_𝜑𝜑+𝛿_𝑢𝜑,𝛿_𝑢𝜑=𝜑_,𝑝𝛿𝑢_𝑝=−𝐸_𝑝𝛿𝑢_𝑝,𝛿𝐸_𝑖=𝛿_𝜑𝐸_𝑖+𝛿_𝑢𝐸_𝑖,𝛿_𝜑𝐸_𝑖=−𝛿_𝜑𝜑_,𝑖,𝛿_𝑢𝐸_𝑖=𝐸_𝑖,𝑝𝛿𝑢_𝑝. (7) The derivation of 𝛿𝐄 is in Appendix A. Finishing the variational calculation finally we get the governing equations:𝑆_{𝑗𝑘,𝑗}+𝑓_𝑘=𝜌𝑢̈_𝑘,𝐷_𝑖,𝑖=𝜌_𝑒𝑆,in𝑉,_𝑗𝑘𝑛_𝑗=𝑇^∗_𝑘,on𝑎_𝜎,𝐷_𝑖𝑛_𝑖=−𝜎^∗,on𝑎_𝐷,𝜎^𝑀_𝑖𝑘=𝐷_𝑖𝐸_𝑘−12𝐷_𝑛𝐸_𝑛𝛿_𝑖𝑘,𝑆_𝑘𝑙=𝜎_𝑘𝑙+𝜎^𝑀_𝑘𝑙=𝐶_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗−𝑒_𝑗𝑘𝑙𝐸_𝑗−12𝑙_{𝑖𝑗𝑘𝑙}𝐸_𝑖𝐸_𝑗+𝐷_𝑖𝐸_𝑘−12𝐷_𝑛𝐸_𝑛𝛿_𝑖𝑘, (8) where 𝜎^𝑀_𝑖𝑘 is the Maxwell stress and 𝑆_𝑘𝑙 is the pseudo total stress [9, 10]. In the environment we have 𝑆^env_{𝑖𝑗,𝑖}+𝑓^env_𝑗=𝜌^env𝑢̈^env_𝑗,𝐷^env_𝑖,𝑖=𝜌^env_𝑒,in𝑉^env,𝑆^env_𝑖𝑗𝑛^env_𝑖=𝑇^∗env_𝑗,on𝑎^env_𝜎,𝐷^env_𝑖𝑛^env_𝑖=−𝜎^∗env,on𝑎^env_𝐷. (9) On the interface we have 𝑆_𝑖𝑗−𝑆^env_𝑖𝑗𝑛_𝑖=𝑇_𝑗^∗int,𝐷_𝑖−𝐷^env_𝑖𝑛_𝑖=−𝜎^∗int,on𝑎^int. (10)

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:𝑢_𝑖=𝑢^∗_𝑖,on𝑎_𝑢,𝜑=𝜑^∗,on𝑎_𝜑,𝑢^env_𝑖=𝑢^∗env_𝑖,on𝑎^env_𝑢,𝜑^env=𝜑^∗env,on𝑎^env_𝜑,𝑢_𝑖=𝑢^env_𝑖,𝜑=𝜑^env,on𝑎^int. (11) Equations (8)–(11) are the governing equations in the electroelastic analysis.

If we introduce the body electric force 𝑓^𝑒_𝑘, 𝑓^𝑒env_𝑘, and surface electric force 𝑇^𝑒_𝑘, 𝑇^𝑒env_𝑘 in the materials as 𝑓^𝑒_𝑘=𝜎^𝑀_{𝑗𝑘,𝑗},𝑇^𝑒_𝑘=𝜎^𝑀_𝑗𝑘𝑛_𝑗;𝑓^𝑒env_𝑘=𝜎^𝑀env_{𝑗𝑘,𝑗},𝑇^𝑒env_𝑘=𝜎^𝑀env_𝑗𝑘𝑛^env_𝑗. (12) Then the variational principle equation (6) is reduced to 𝛿Π^′=𝛿Π^′₁+𝛿Π^′₂−𝛿𝑊^′int=0,𝛿Π^′₁=_𝑉𝛿𝑔𝑑𝑉−𝛿𝑊^′,𝛿Π^′₂=_𝑉env𝛿𝑔^env𝑑𝑉−𝛿𝑊^′env,𝛿𝑊^′=_𝑉𝑓_𝑘+𝑓^𝑒_𝑘−𝜌𝑢̈_𝑘𝛿𝑢_𝑘𝑑𝑉−_𝑉𝜌_𝑒+𝛿𝜑𝑑𝑉_𝑎𝜎𝑇^∗_𝑘−𝑇^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑎−_𝑎𝐷𝜎^∗𝛿𝜑𝑑𝑎,𝛿𝑊^′env=_𝑉env𝑓^env_𝑘+𝑓^𝑒env_𝑘−𝜌^env𝑢̈^env_𝑘𝛿𝑢^env_𝑘−𝑑𝑉_𝑉env𝜌^env_𝑒𝛿𝜑^env+𝑑𝑉_𝑎env𝜎𝑇^env∗_𝑘−𝑇^𝑒env_𝑘𝛿𝑢^env_𝑘−𝑑𝑎_𝑎env𝐷𝜎^env∗𝛿𝜑^env𝑑𝑎,𝛿𝑊^′int=_𝑎int𝑇_𝑘^int∗+𝑇^𝑒env_𝑘−𝑇^𝑒_𝑘𝛿𝑢_𝑘−𝑑𝑎_𝑎int𝜎^int∗𝛿𝜑𝑑𝑎. (13) In (13) the variations of 𝛿𝑢 and 𝛿𝜑 are completely independent, that is, it is not needed to consider the migratory variation 𝛿_𝑢𝜑. Equations (8)–(10) are reduced to 𝜎_{𝑖𝑗,𝑖}+𝑓_𝑗+𝑓^𝑒_𝑗=𝜌𝑢̈_𝑗,𝐷_𝑖,𝑖=𝜌_𝑒𝜎,in𝑉,_𝑖𝑗𝑛_𝑖=𝑇^∗_𝑗−𝑇_𝑗^𝑒∗,on𝑎_𝜎,𝐷_𝑖𝑛_𝑖=−𝜎^∗,on𝑎_𝐷;𝜎^env_{𝑖𝑗,𝑗}+𝑓^env_𝑖+𝑓^𝑒env_𝑖=𝜌𝑢̈^env_𝑖,𝐷^env_𝑖,𝑖=𝜌^env_𝑒,in𝑉^env𝜎^env_𝑖𝑗𝑛^env_𝑗=𝑇^∗env_𝑖−𝑇^∗𝑒env_𝑖,on𝑎^env_𝜎,𝐷^env_𝑖𝑛^env_𝑖=−𝜎^∗env,on𝑎^env_𝐷,𝜎_𝑘𝑙−𝜎^env_𝑘𝑙𝑛_𝑙=𝑇_𝑘^∗int−𝑇^𝑒_𝑖+𝑇^𝑒env_𝑖,𝐷_𝑘𝑛_𝑘−𝐷^env_𝑘𝑛_𝑘=−𝜎^∗int,on𝑎^int. (14)

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 permittivity 𝜖. 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 𝛿Π=𝛿Π₁+𝛿Π₂−𝛿𝑊^int=0,𝛿Π₁=𝛿_𝑉𝑔𝑑𝑉=ℎ𝜎₂₂𝛿𝑢_2,2−𝐷₂𝛿𝐸₂−12𝐸₂𝐷₂𝛿𝑢_2,2𝛿Π₂=0,𝛿𝑊^int=0., (15) 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 𝛿_𝜑𝐸₂=𝜑()−𝜑ℎ+𝛿ℎℎ=−𝜑𝛿ℎℎ²,−𝐷₂𝛿𝐸₂=𝐷₂𝐸₂𝛿ℎℎ. (16) Due to the electric potential 𝜑 on the electrode plate is constant, we have 𝛿_𝑢𝐸₂=𝐸_2,𝑝𝛿𝑢_𝑝=0, so 𝛿𝐸₂=𝛿_𝜑𝐸₂. Therefore, finally we get 𝛿Π=ℎ𝜎₂₂𝛿𝑢_2,2−𝐷₂𝛿𝐸₂−12𝐸₂𝐷₂𝛿𝑢_2,2=ℎ𝐷₂𝜎₂₂𝛿ℎℎ+12𝜖𝐸²₂𝛿ℎℎ=0⟹𝑇₂=𝜎₂₂=−𝜖𝐸²₂2. (17)

We can also use the Maxwell stress to derive the force acting on the dielectric directly. According to (10), we have (𝑆_𝑖𝑗−𝑆^env_𝑖𝑗)𝑛_𝑖=0 or 𝑇_𝑖=(𝜎_𝑖𝑗−𝜎^env_𝑖𝑗)𝑛_𝑖=−𝜎^𝑀_𝑖𝑗𝑛_𝑗, where 𝜎^𝑀env_𝑖𝑗=0 in the electrode is used. In the present case we have 𝑇₂=−𝜎^𝑀₂₂=−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 displacement 𝛿ℎ 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 𝐹⋅𝛿𝑢=𝜎₂₂1𝛿ℎ=𝜑𝛿𝑞−21𝜑𝛿𝑞=2=1𝜑𝛿𝑞2𝜑𝛿𝜖𝐸₂=𝜖2𝜑𝜑𝛿ℎ𝜖=−2𝜑ℎ²𝛿ℎ⟹𝜎₂₂𝜖=−2𝐸²₂, (18) 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 𝛿𝑊^𝑒=−𝛿_𝑢𝛿Π,_𝑢Π=_𝑉𝑔_,𝐸⋅𝛿_𝑢𝐸𝑑𝑉+_𝑉𝑔^𝑒𝛿𝑢_𝑘,𝑘𝑑𝑉+_𝑉𝜌_𝑒𝛿_𝑢+𝜑𝑑𝑉_𝑎𝐷𝜎^∗𝛿_𝑢𝜑𝑑𝑎+_𝑉env𝑔^env_,𝐸⋅𝛿_𝑢𝐸^env+𝑑𝑉_𝑉env𝑔^𝑒env𝛿𝑢^env_𝑖,𝑖𝑑𝑉+_𝑉env𝜌^env_𝑒𝛿_𝑢𝜑^env+𝑑𝑉_𝑎env𝐷𝜎^∗env𝛿_𝑢𝜑^env𝑑𝑎+_𝑎int𝜎^∗int𝛿_𝑢𝜑𝑑𝑎,𝛿𝑊^𝑒=_𝑉𝑓^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑉+_𝑎𝜎𝑇^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑎+_𝑉env𝑓^𝑒env_𝑘𝛿𝑢^env_𝑘+𝑑𝑉_𝑎env𝜎𝑇^𝑒env_𝑘𝛿𝑢^env_𝑘𝑑𝑎+_𝑎int𝑇^𝑒_𝑘−𝑇^𝑒env_𝑘𝛿𝑢_𝑘𝑑𝑎. (19) It is easy to prove that 𝑓^𝑒_𝑘, 𝑇^𝑒_𝑘, 𝑓^𝑒env_𝑘, 𝑇^𝑒env_𝑘 derived from (19) are identical with that in (12), see Appendix B. If the environment is neglected, then we get _𝑉𝑓^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑉+_𝑎𝜎𝑇^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑎=−_𝑉𝑔_,𝐸⋅𝛿_𝑢𝐸𝑑𝑉+_𝑉𝑔^𝑒𝛿𝑢_𝑘,𝑘+𝑑𝑉_𝑉𝜌_𝑒𝛿_𝑢𝜑𝑑𝑉+_𝑎𝐷𝜎^∗𝛿_𝑢𝜑𝑑𝑎. (20)

Using 𝐃=𝐷_𝑛𝐧+𝐷_𝑡𝐭 and 𝐄=𝐸_𝑛𝐧+𝐸_𝑡𝐭, and the continuous condition (10) and (11) of the 𝐃, 𝐄 on the interface, the Maxwell stress can also be rewritten as𝐧⋅𝝈−𝝈^env=𝐓^∗int,𝐓^∗int=𝐓^∗int+𝐧⋅𝝈^𝑀env−𝝈^𝑀,𝐧⋅𝜎^𝑀env−𝜎^𝑀,=𝐧⋅𝐃^env⊗𝐄^env−12𝐃^env⋅𝐄^env𝐧−()−1𝐧⋅𝐃⊗𝐄2()𝐧=1𝐃⋅𝐄2𝐷_𝑛𝐸^env_𝑛−𝐸_𝑛−𝐷^env_𝑡−𝐷_𝑡𝐸_𝑡𝐧, (21) 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 have𝐶_{𝑖𝑗𝑘𝑙}=𝜆𝛿_𝑖𝑗𝛿_𝑘𝑙+𝐺𝛿_𝑖𝑘𝛿_𝑗𝑙+𝛿_𝑖𝑙𝛿_𝑗𝑘,𝑙_{𝑖𝑗𝑘𝑙}=𝑎₂𝛿_𝑖𝑗𝛿_𝑘𝑙+𝑎₁2𝛿_𝑖𝑘𝛿_𝑗𝑙+𝛿_𝑖𝑙𝛿_𝑗𝑘,𝜖_𝑖𝑗=𝜖𝛿_𝑖𝑗,𝑒_𝑘𝑖𝑗=0. (22) So (3) and (4) are reduced to 1𝑔=2𝜆𝜀_𝑖𝑖𝜀_𝑘𝑘+𝐺𝜀_𝑖𝑗𝜀_𝑖𝑗−12𝜖𝐸_𝑘𝐸_𝑘−12𝑎₂𝐸_𝑖𝐸_𝑖𝜀_𝑘𝑘+𝑎₁𝐸_𝑖𝐸_𝑗𝜀_𝑖𝑗,𝜎_𝑘𝑙=𝜆𝜀_𝑖𝑖𝛿_𝑘𝑙+2𝐺𝜀_𝑘𝑙−12𝑎₂𝐸_𝑖𝐸_𝑖𝛿_𝑘𝑙+𝑎₁𝐸_𝑘𝐸_𝑙,𝐷_𝑘=𝜖𝐸_𝑘+𝑎₂𝐸_𝑘𝜀_𝑚𝑚+𝑎₁𝐸_𝑙𝜀_𝑘𝑙≈𝜖𝐸_𝑘,𝑆_𝑘𝑙=𝜎_𝑘𝑙+𝜎^𝑀_𝑘𝑙=𝜆𝜀_𝑖𝑖𝛿_𝑘𝑙+2𝐺𝜀_𝑘𝑙−12𝑎₂+𝜖𝐸_𝑚𝐸_𝑚𝛿_𝑘𝑙+122𝜖−𝑎₁𝐸_𝑘𝐸_𝑙. (23)(24)

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 𝝈^𝑀 are all symmetric.

As an example, we discuss the electroelastic field of an isotropic 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 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 𝐸^air_𝑡=𝐸_𝑡,𝐷^air_𝑛=𝐷_𝑛𝐸oninterface;or_air1=𝐸₁=0,𝐸^air₃=𝐸₃𝐷=0,_air2=𝐷₂𝜖₀𝐸^air₂=𝜖𝐸₂,𝑥₂ℎ=±2,𝐸^air₁=𝐸₁=0,𝐸^air₃=𝐸₃𝐸=0,_air2=𝐸₂;𝑥₁𝑙=±2,or𝑥₃𝑏=±2, (25) where 𝑛 and 𝑡 mean the normal and tangential directions, respectively. In this case from (8)–(10) we get 𝑆_{𝑖𝑗,𝑗}𝑆=0,in𝑉,_𝑖𝑗−𝑆^air_𝑖𝑗𝑛_𝑖=0,on𝑎^int⟹𝜎_𝑖𝑗=𝜎^𝑀air_𝑖𝑗−𝜎^𝑀_𝑖𝑗^int,𝜎^𝑀₂₂=12𝐷₂𝐸₂,𝜎^𝑀₁₁=𝜎^𝑀₃₃1=−2𝐷₂𝐸₂,𝜎^𝑀_𝑖𝑗=0,when𝑖≠𝑗;on𝑎^int,𝜎^𝑀air₂₂=12𝐷^air₂𝐸^air₂,𝜎^𝑀air₁₁=𝜎^𝑀air₃₃1=−2𝐷^air₂𝐸^air₂;𝜎^𝑀air_𝑖𝑗=0,when𝑖≠𝑗,on𝑎^int⟹𝜎₁₁=𝜎₃₃1=−2𝜖²𝜖₀−𝜖𝐸²₂,𝜎₂₂=12𝜖²𝜖₀−𝜖𝐸²₂. (26) Using the constitutive equation (24) we get −12𝜖²𝜖₀−𝝐𝐸²₂=3𝜆𝜃+2𝐺𝜀₁₁−12𝑎₂𝐸²₂,12𝜖²𝜖₀−𝝐𝐸²₂=3𝜆𝜃+2𝐺𝜀₂₂−12𝑎₂+𝑎₁𝐸²₂,−12𝜖²𝜖₀−𝝐𝐸²₂=3𝜆𝜃+2𝐺𝜀₃₃−12𝑎₂𝐸²₂. (27)

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

In electrodes 𝐄=𝟎, so there is no Maxwell stress. In this case from (8)–(10) we get𝑆_{𝑖𝑗,𝑗}𝑆=0,in𝑉,_𝑖𝑗−𝑆^env_𝑖𝑗𝑛_𝑖=0,on𝑎^int⟹𝜎_𝑖𝑗=𝜎^𝑀env_𝑖𝑗−𝜎^𝑀_𝑖𝑗^int,𝜎^𝑀₂₂=12𝜖𝐸²₂,𝜎^𝑀₁₁=𝜎^𝑀₃₃1=−2𝜖𝐸²₂,𝜎^𝑀_𝑖𝑗=0,when𝑖≠𝑗,on𝑎^int,𝜎^𝑀plate₂₂=0,𝜎^𝑀air₁₁=𝜎^𝑀air₃₃1=−2𝜖₀𝐸²₂,𝜎^𝑀env_𝑖𝑗=0,when𝑖≠𝑗,on𝑎^int⟹𝜎₁₁=𝜎₃₃=12𝜖−𝜖₀𝐸²₂,𝜎₂₂1=−2𝜖𝐸²₂. (28) Using the constitutive equation (24) we get 12𝜖−𝜖₀𝐸²₂=3𝜆𝜃+2𝐺𝜀₁₁−12𝑎₂𝐸²₂,−12𝜖𝐸²₂=3𝜆𝜃+2𝐺𝜀₂₂−12𝑎₂+𝑎₁𝐸²₂,12𝜖−𝜖₀𝐸²₂=3𝜆𝜃+2𝐺𝜀₃₃−12𝑎₂𝐸²₂. (29) In (27) and (29) 3𝜃=𝜀_𝑖𝑖. 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.

Appendices

A. The Derivation of 𝛿E in (7)

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

Assume the variational functional is 𝐼=_𝑉𝐿𝑥_𝑖,𝑢_𝛼,𝑢_𝛼,𝑗𝑑𝑉, (A.1) where 𝐱, 𝐮, 𝐮_,𝑗 are the independent variable, dependent variable, and the derivative of the dependent variable, respectively. Assume an infinitesimal transformation 𝑥_𝑖⟶𝑥^′_𝑖=𝑥_𝑖+𝛿𝑢_𝑖𝑥_𝑗,𝜑𝑥,𝜑_𝑖⟶𝜑^′𝑥^′_𝑖=𝜑𝑥_𝑖+𝛿𝜑𝑥_𝑖,𝜑,𝛿𝜑=𝜑^′𝑥^′_𝑖−𝜑𝑥_𝑖=𝜑𝑥_𝑖+𝛿𝑢_𝑖+𝛿_𝜑𝜑𝑥_𝑖−𝜑𝑥_𝑖=𝛿_𝜑𝜑+𝛿_𝑢𝜑=𝛿_𝜑𝜑+𝜑_,𝑖𝛿𝑢_𝑖. (A.2) Noting 𝜕𝑥^′_𝑖𝜕𝑥_𝑗≈𝛿_𝑖𝑗+𝜕𝛿𝑢_𝑖𝜕𝑥_𝑗,𝜕𝑥_𝑖𝜕𝑥^′_𝑗≈𝛿_𝑖𝑗−𝜕𝛿𝑢_𝑖𝜕𝑥_𝑗,𝑗=𝜕𝑥^′_𝑖𝜕𝑥_𝑗≈1+𝜕𝛿𝑢_𝑖𝜕𝑥_𝑖, (A.3) from (A.2) we get 𝛿𝜑_,𝑗=𝜕𝜑^′𝑥^′_𝑖𝜕𝑥^′_𝑗−𝜕𝜑𝑥_𝑖𝜕𝑥_𝑗=𝜕𝜑𝑥_𝑖+𝛿𝜑𝑥_𝑖,𝜑𝜕𝑥_𝑘𝜕𝑥_𝑘𝜕𝑥^′_𝑗−𝜕𝜑𝑥_𝑖𝜕𝑥_𝑗=𝜕𝛿𝜑𝑥_𝑖,𝜑𝜕𝑥_𝑗−𝜕𝜑𝑥_𝑖𝜕𝑥_𝑘𝜕𝛿𝑢_𝑘𝜕𝑥_𝑗=𝜕𝛿_𝜑𝜑+𝜑_,𝑖𝛿𝑢_𝑖𝜕𝑥_𝑗−𝜕𝜑𝑥_𝑖𝜕𝑥_𝑘𝜕𝛿𝑢_𝑘𝜕𝑥_𝑗=𝜕𝛿_𝜑𝜑𝜕𝑥_𝑗+𝜕𝜑_,𝑖𝛿𝑢_𝑖𝜕𝑥_𝑗−𝜕𝜑𝑥_𝑖𝜕𝑥_𝑘𝜕𝛿𝑢_𝑘𝜕𝑥_𝑗=𝜕𝛿_𝜑𝜑𝜕𝑥_𝑗+𝜕𝜑_,𝑖𝜕𝑥_𝑗𝛿𝑢_𝑖. (A.4) Equation (A.4) is identical with 𝛿𝐄 in (7) in text.

B. The Derivation of (19)

From (19) we have 𝛿_𝑢Π=_𝑉𝑔_,𝐸⋅𝛿_𝑢𝐸𝑑𝑉+_𝑉𝑔^𝑒𝛿𝑢_𝑘,𝑘𝑑𝑉+_𝑉𝜌_𝑒𝛿_𝑢+𝜑𝑑𝑉_𝑎𝐷𝜎^∗𝛿_𝑢𝜑𝑑𝑎+_𝑉env𝑔^env_,𝐸⋅𝛿_𝑢𝐸^env+𝑑𝑉_𝑉env𝑔^𝑒env𝛿𝑢^env_𝑖,𝑖𝑑𝑉+_𝑉env𝜌^env_𝑒𝛿_𝑢𝜑^env+𝑑𝑉_𝑎env𝐷𝜎^∗env𝛿_𝑢𝜑^env𝑑𝑎+_𝑎int𝜎^∗int𝛿_𝑢𝜑𝑑𝑎=−_𝑉𝐷_𝑖𝐸_𝑝,𝑖𝛿𝑢_𝑝𝑑𝑉−_𝑉12𝐷_𝑘𝐸_𝑘𝛿𝑢_𝑗,𝑗+𝑑𝑉_𝑉𝜌_𝑒𝛿_𝑢𝜑𝑑𝑉+_𝑎𝐷𝜎^∗𝛿_𝑢𝜑𝑑𝑎−_𝑉env𝐷^env_𝑖𝛿_𝑢𝐸^env_𝑖−𝑑𝑉_𝑉env12𝐷^env_𝑘𝐸^env_𝑘𝛿𝑢^env_𝑗,𝑗𝑑𝑉+_𝑉env𝜌^env_𝑒𝛿_𝑢𝜑^env𝑑𝑉+_𝑎env𝐷𝜎^∗env𝛿_𝑢𝜑^env𝑑𝑎+_𝑎int𝜎^∗int𝛿_𝑢𝜑𝑑𝑎=−_𝑉𝐷_𝑖𝐸_𝑝,𝑖𝛿𝑢_𝑝𝑑𝑉+_𝑉𝜌_𝑒−𝐷_𝑖,𝑖𝛿_𝑢−1𝜑𝑑𝑉2_𝑎𝐷_𝑘𝐸_𝑘𝑛_𝑗𝛿𝑢_𝑗1𝑑𝑎+2_𝑉𝐷_𝑘𝐸_𝑘,𝑗𝛿𝑢_𝑗+𝑑𝑉_𝑎𝐷𝜎^∗𝛿_𝑢𝜑𝑑𝑎−_𝑉env𝐷^env_𝑖𝐸^env_𝑝,𝑖𝛿𝑢^env_𝑝+𝑑𝑉_𝑉env𝜌^env_𝑒−𝐷^env_𝑖,𝑖𝛿_𝑢𝜑^env−1𝑑𝑉2_𝑉^env𝐷^env_𝑘𝐸^env_𝑘𝑛^env_𝑗𝛿𝑢^env_𝑗+1𝑑𝑉2_𝑉^env𝐷^env_𝑘𝐸^env_𝑘,𝑗𝛿𝑢^env_𝑗𝑑𝑉+_𝑎env𝐷𝜎^∗env𝛿_𝑢𝜑^env𝑑𝑎+_𝑎int𝜎^∗int𝛿_𝑢𝜑𝑑𝑎. (B.1) Using (8)–(11), that is, 𝐷_𝑖𝑛_𝑖+𝜎^∗=0,on𝑎_𝐷,𝐷_𝑖,𝑖−𝜌_𝑒𝐷=0,in𝑉,_𝑖−𝐷^env_𝑖𝑛_𝑖=−𝜎^∗int,on𝑎^int (B.2) and adding terms ∫_𝑎𝐷_𝑖𝑛_𝑖(𝐸_𝑝𝛿𝑢_𝑝+𝛿_𝑢𝜑)𝑑𝑎=0 and ∫_𝑎env𝐷^env_𝑖𝑛^env_𝑖(𝐸^env_𝑝𝛿𝑢^env_𝑝+𝛿_𝑢𝜑^env)𝑑𝑎=0 to (B.1), then (B.1) is reduced to 𝛿_𝑢Π=−_𝑉𝐷_𝑖𝐸_𝑝,𝑖𝛿𝑢_𝑝𝑑𝑉−_𝑉12𝐷_𝑘𝐸_𝑘𝑛_𝑗𝛿𝑢_𝑗+𝑑𝑉_𝑉12𝐷_𝑘𝐸_𝑘,𝑗𝛿𝑢_𝑗𝑑𝑉+_𝑎𝐷_𝑖𝐸_𝑝𝑛_𝑖𝛿𝑢_𝑝−𝑑𝑎_𝑉env𝐷^env_𝑖𝐸^env_𝑝,𝑖𝛿𝑢^env_𝑝−𝑑𝑉_𝑉env12𝐷^env_𝑘𝐸^env_𝑘𝑛^env_𝑗𝛿𝑢^env_𝑗+𝑑𝑉_𝑉env12𝐷^env_𝑘𝐸^env_𝑘,𝑗𝛿𝑢^env_𝑗+𝑑𝑉_𝑎env𝐷^env_𝑖𝐸^env_𝑝𝑛^env_𝑖𝛿𝑢^env_𝑝=𝑑𝑎_𝑎𝜎𝜎^𝑀_𝑖𝑗𝑛_𝑖𝛿𝑢_𝑗𝑑𝑎−_𝑉𝜎^𝑀_{𝑖𝑗,𝑖}𝛿𝑢_𝑗+𝑑𝑉_𝑎env𝜎𝜎^𝑀env_𝑖𝑗𝑛^env_𝑖𝛿𝑢^env_𝑗𝑑𝑎−_𝑉𝜎^𝑀env_{𝑖𝑗,𝑖}𝛿𝑢^env_𝑗+𝑑𝑉_𝑎int𝜎^𝑀_𝑖𝑗𝑛_𝑖𝛿𝑢_𝑗𝑑𝑎+_𝑎int𝜎^𝑀env_𝑖𝑗𝑛^env_𝑖𝛿𝑢^env_𝑗𝑑𝑎. (B.3) From (19), we know that he work done by the equivalent static electric force is 𝛿𝑊^𝑒=_𝑉𝑓^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑉+_𝑎𝜎𝑇^𝑒_𝑘𝛿𝑢_𝑘𝑑𝑎+_𝑉env𝑓^𝑒env_𝑘𝛿𝑢^env_𝑘+𝑑𝑉_𝑎env𝜎𝑇^𝑒env_𝑘𝛿𝑢^env_𝑘𝑑𝑎+_𝑎int𝑇^𝑒_𝑘−𝑇^𝑒env_𝑘𝛿𝑢_𝑘𝑑𝑎. (B.4) From (B.3) and (B.4), we immediately get (19).