Abstract
In this chapter, the physical variational principle is used to derive the governing equations of the nonlinear and linear electroelastic analyses in piezoelectric and pyroelectric materials. Some kinds of the physical variational principle are given. Applying the migratory variation of electric potential, the general expression of the static electric force is given. It is shown that for the physical nonlinear problem, such as the electrostrictive materials, in order to get correct governing equations and material constants in experiments, we need to consider the entire system including the dielectric medium, its environment, and their common boundary. When the temperature varies with time, the inertial entropy and generalized inertial entropy theories are used to derive the temperature wave equation and the mass diffusion equation. This theory is consistent with current known thermodynamic theory.
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Keywords
- Physical variational principle
- Governing equation
- Inertial entropy theory
- Temperature wave equation
- Mass diffusion equation
2.1 Electric Gibbs Free Energy Variational Principle in Piezoelectric Materials
2.1.1 Electric Gibbs Free Energy and Constitutive Equations
In Sect. 1.6, the physical variational principle (PVP) was proposed as a basic principle in the continuum mechanics for an electrically static state. In this chapter, we shall discuss its applications. At first, the electric Gibbs free energy variational principle in piezoelectric materials under the isothermal case is discussed. According to Eq. (1.69) in Sect. 1.5.4, the electric Gibbs free energy \( {g} \) is
Under the small deformation, \( {g} \) can be expanded in the series of \( \boldsymbol{\varepsilon} \) and \( \boldsymbol{ E} \):
where \( \boldsymbol{ C},\boldsymbol{\epsilon}, \boldsymbol{ e},\boldsymbol{ l} \) are the elastic coefficient, permittivity, piezoelectric coefficient, and the electrostrictive coefficient, respectively; \( \boldsymbol{\alpha} \) is a new asymmetric or symmetric electrostrictive coefficient in order to make \( \boldsymbol{ l} \) the same symmetries as that in \( \boldsymbol{ C} \) (Kuang 2007, 2008a). For convenience, the term \( {\alpha_{nm }}{E_m}{E_n}{\varepsilon_{kl }}{\delta_{kl }} \) is also added.
Because \( {g} \) is a state function, constitutive equations can be derived as
In general case in Eqs. (2.2) and (2.3), \( {\varepsilon_{ij }}={u_{i,j }},{\varepsilon_{ij }}\ne {\varepsilon_{ji }} \), and there are nine components for \( {\varepsilon_{ij }} \) and \( {\sigma_{ij }} \). For most practical cases, the body couple is neglected; in this case, \( {\sigma_{ij }} \) and \( {\varepsilon_{ij }}={{{\left( {{u_{i,j }}+{u_{j,i }}} \right)}} \left/ {2} \right.} \) are symmetric and each of them only has six components. Let \( {{\boldsymbol{\sigma}}^{\mathrm{ s}}} \) and \( {{\boldsymbol{\sigma}}^{\mathrm{ a}}} \) be the symmetric and asymmetric parts of \( \boldsymbol{\sigma} \), respectively, we have
where \( \boldsymbol{ D}={\epsilon_0}\boldsymbol{ E}+\boldsymbol{ P} \) was used. In Eq. (2.4), terms containing \( \boldsymbol{ l}:\boldsymbol{\varepsilon}, \boldsymbol{\alpha} :\boldsymbol{\varepsilon}, \boldsymbol{ e}:\boldsymbol{\varepsilon} \) had been neglected. In the usual electromagnetic theory, the electromagnetic body couple is \( \boldsymbol{ P}\times \boldsymbol{ E}=\boldsymbol{ D}\times \boldsymbol{ E} \). In general, \( \boldsymbol{\alpha} \) should be determined by experiments. In this book \( \boldsymbol{\alpha} =-2\boldsymbol{\epsilon} \) is assumed, so Eqs. (2.3) and (2.4) are reduced to
Equation (2.5) shows that the electric body couple is balanced by the moment produced by the asymmetric stresses (Eringen and Maugin 1989). If the electromagnetic body couple is neglected, all the stresses are symmetric. Using Eq. (2.3), Eq. (2.2) is reduced to
where \( {{g}^{\mathrm{ e}}} \) is the part related to the electric field in \( {g} \) or the energy from the total energy minus the pure deformation energy. The value of the term \( {{\boldsymbol{\varDelta}}^{g}}:\boldsymbol{\varepsilon} \) is much less than other terms, so it can be neglected.
In the electroelastic analysis, the dielectric medium, its environment, and their common boundary \( {a^{{\operatorname{int}}}} \) consociate a system and should be considered together, because the electric field exists in every material except the ideal conductor. In this book, 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” (Fig. 2.1). In the environment, Eqs. (2.1), (2.2), (2.3), (2.4), (2.5), and (2.6) are all held.
Dielectric and its environment
2.1.2 Electric Gibbs Free Energy Variational Principle and Governing Equations
Under the assumption that \( \mathbf{u},\varphi, {{\mathbf{u}}^{\mathrm{ env}}},{\varphi^{\mathrm{ env}}} \) satisfy their boundary conditions on their own boundaries \( {a_u},{a_{\varphi }},a_u^{\mathrm{ env}},a_{\varphi}^{\mathrm{ env}} \) and the continuity conditions on the interface \( {a^{{\operatorname{int}}}} \). Given the displacement and electric potential virtual increments, the PVP in terms of the electric Gibbs free energy (which is identical to the electric enthalpy in isothermal case) is (Kuang 2007, 2008a, b, 2011a, c)
where \( \boldsymbol{ f},{{\boldsymbol{ T}}^{*}},{\sigma^{*}} \) are given body force per volume, traction per area, and surface charge density and \( {{\boldsymbol{ f}}^{\mathrm{ env}}},{{\boldsymbol{ T}}^{*\mathrm{ env}}},{\sigma^{*\mathrm{ env}}}, \) and \( {{\boldsymbol{ T}}^{{*\operatorname{int}}}},{\sigma^{{*\operatorname{int}}}} \) are also given values in the environment and on the interface, respectively. \( \boldsymbol{ n}=-{{\boldsymbol{ n}}^{\mathrm{ env}}} \) is the outward normal of the interface. It is noted that in Eq. (2.7) the work done by the electric field has the form \( q\delta\varphi=(\rho_edV)\delta\varphi\) with \( q=\rho_edV=const\)., etc. For small deformation, \( \delta \int\nolimits_V {{g}\mathrm{ d}V} =\int\nolimits_V {\delta {g}\mathrm{ d}V} \) can be used due to small variation of the volume.
The virtual variation of the potential \( \varphi \) is divided into local variation \( {\delta_{\varphi }}\varphi \) and migratory variation \( {\delta_u}\varphi \), and the similar divisions for \(\boldsymbol{ E} \), so we have
Equation (2.8) shows that \( {{{\partial \left( {\delta \varphi } \right)}} \left/ {{\partial {x_j}}} \right.}\ne \delta \left( {{{{\partial \varphi }} \left/ {{\partial {x_j}}} \right.}} \right) \) when \( {\delta_u}\varphi \ne 0 \), and it is discussed also in Eq. (2.130) in Sect. 2.9.1. Using the relation,
where \( a={a_{\sigma }}+{a_u}+{a^{{\operatorname{int}}}}={a_D}+{a_{\varphi }}+{a^{{\operatorname{int}}}} \), \( {\sigma_{ji }}\delta {\varepsilon_{ij }}={\sigma_{ji }}\delta {u_{i,j }} \) for asymmetric \( {\sigma_{ji }} \). It is noted that \( \delta \varphi =0,\;{\delta_{\varphi }}\varphi \ne 0,\;{\delta_u}\varphi \ne 0 \) on \( {a_{\varphi }} \).
Substitution of Eq. (2.9) into \( \delta {\varPi_1} \) in Eq. (2.7) yields
Adding a term \( \int\nolimits_a {{D_i}{n_i}\left( {{E_p}\delta {u_p}+{\delta_u}\varphi } \right)\mathrm{ d}a} =0 \) to Eq. (2.10), we get
In Eq. (2.11), we have
where \( \sigma_{ik}^{\mathrm{ M}} \) is the Maxwell stress, \( \boldsymbol{ S} \) is the pseudo total stress (Jiang and Kuang 2003, 2004) and \( {{\epsilon_{km}}E_{m}=D_{k}} \) has been used. \( \boldsymbol{ S} \) is a symmetric tensor, but \( \boldsymbol{\sigma} \) and \( {{\boldsymbol{\sigma}}^{\mathrm{ M}}} \) are not. Though the adding term \( \int\nolimits_a {{D_k}{n_k}{E_p}\delta {u_p}\mathrm{ d}a}+ \int\nolimits_a {{D_k}{n_k}{\delta_u}\varphi\mathrm{ d}a} \) is zero, the first will be combined with terms of virtual displacements and the second will be combined with terms containing the local variation \( {\delta_{\varphi }}\varphi \).
Equation (2.10) can also be written as
Due to the arbitrariness of \( \delta \varphi \), it is obtained:
Substitution of Eq. (2.13b) into Eq. (2.13a) yields
In Eq. (2.13c), the following relation was used:
Due to the arbitrariness of \( \delta \boldsymbol{ u} \) and \( \delta \varphi \) from Eq. (2.11) or (2.13), it is obtained:
The momentum equation in Eq. (2.14a) in terms of generalized displacements is
where the terms containing \( \boldsymbol{\varepsilon} \) in \( \varphi \) are neglected. Similarly for the environment, we have
Using \( {n_i}=-n_i^{\mathrm{ env}},{u_i}=u_i^{\mathrm{ env}},\varphi ={\varphi^{\mathrm{ env}}} \) and \( \delta {\varPi_1}+\delta {\varPi_2}=\delta {W^{*\mathrm{ int}}} \), we get
The above variational principle requests prior that the generalized displacements satisfy their own boundary conditions and the continuity conditions on the interface, so the following equations should also be added to governing equations:
Equations (2.14), (2.15), (2.16), and (2.17) are the governing equations for the electroelastic analysis.
2.1.3 A Note of the Maxwell Stress
In the books of Stratton (1941) and Landau and Lifshitz (1959), the formula of the stress in an isotropic electrostrictive material was
where \( {{{\partial {{g}_0}}} \left/ {{\partial {\varepsilon_{ik }}}} \right.} \) is the stress in the media without the electromagnetic field. This formula is just the pseudo total stress \( \boldsymbol{ S} \) in Eq. (2.12) for the electrostrictive material. For the Maxwell stress and its related problems in literatures, different author had different understanding as shown in Sect. 1.2.7. McMeeking et al. (2005, 2007) considered that in the electroelastic theory 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. In their method authors did not distinguish the local and migratory variations.
From Eq. (2.12), it is known that the Maxwell stress is related to the square of \( \boldsymbol{ E} \), i.e., \( \left| \boldsymbol{\sigma} \right|\propto {{\left| \boldsymbol{ E} \right|}^{\;2 }} \), but the stress introduced by the piezoelectric effect is related to \( \boldsymbol{ E} \). So for the piezoelectric material when the electric field is not too large and the piezoelectric coefficient is not too small, the Maxwell stress can be neglected. But the isotropic electrostrictive materials do not have the piezoelectric effect, so in this and similar cases, the Maxwell stress should be considered.
Because the strain is accompanied by the change of the distance between the electric particles, the attraction between electric charges or the Maxwell stress and the stress introduced by strains in the material is produced simultaneously. Though they are produced together, their difference is obvious and important. The strength problem in engineering is determined by the Cauchy stress, which is connected with the constitutive equation. However, the Maxwell stress is an external effective electromagnetic force applied to the body and can be obtained by using the migratory variation of \( \varphi \) in the PVP or in the usual energy principle.
Using \( \boldsymbol{ D}={D_{\mathrm{ n}}}\boldsymbol{ n}+{D_{\mathrm{ t}}}\boldsymbol{ t} \) and similar expressions for \( \boldsymbol{ E} \) and the continuity conditions in Eqs. (2.16) and (2.17) for the \( \boldsymbol{ D},\boldsymbol{ E} \) on the interface, the mechanical continuous condition on the interface can also be rewritten as
where \( \boldsymbol{ n} \) is the unit normal, subscripts n and t mean the normal and tangential direction respectively; and there is no sum on n and t. Equation (2.18) shows that in the small strain case, the boundary traction produced by the Maxwell stress is along the normal direction. The Maxwell stress can be naturally obtained by the migratory variation of \( \varphi \) in PVP.
2.2 Alternative Forms of the Physical Variational Principles
2.2.1 First Alternative Form of the PVP
From Eqs. (2.14), (2.15), (2.16), and (2.17), it is found that if we use \( \boldsymbol{ S} \) instead of \( \boldsymbol{\sigma} \) and \( {{\boldsymbol{ S}}^{\mathrm{ env}}} \) instead of \( {{\boldsymbol{\sigma}}^{\mathrm{ env}}} \) in the governing equations, then the form of governing equations of the physical nonlinear dielectrics is just the same as that in the physical linear electric problem. Therefore, a simpler principle can be obtained: the first alternative form of the variational principle is
In Eq. (2.19), the variations of \( \delta u \) and \( \delta \varphi \) are all local variations or completely independent, i.e., the migratory variations \( {\delta_u}\varphi \) produced by \( \delta \mathbf{u} \) are not needed. \( \delta W,\delta {W^{\mathrm{ env}}}, \) and \( \delta {W^{\mathrm{ int}}} \) are still expressed by Eq. (2.7). An analogous theory was also discussed by Bustamante et al. (2008).
2.2.2 Second Alternative Form of the Physical Variational Principle
Introduce the electric body force \( f_k^{\mathrm{ e}},f_k^{{\mathrm{ e}\;\mathrm{ env}}} \) and traction \( T_k^{\mathrm{ e}},T_k^{{\mathrm{ e}\;\mathrm{ env}}} \) in media as
The second alternative form of the variational principle is
In Eq. (2.21), the variations of \( \delta u \) and \( \delta \varphi \) are also completely independent, i.e., it is also not needed to consider the migratory variation \( {\delta_u}\varphi \). Equation (2.21) is the original form of the PVP Eq. (1.78). The governing equations from Eq. (2.21) are
In many literatures (Pao 1978; Maugin 1988; Moon 1984), the governing equations were expressed in the form of Eq. (2.22) and the electromagnetic force was derived from other methods different with the variational method. In different literatures, \( {{\boldsymbol{ f}}^{\mathrm{ e}}} \) and \( {{\boldsymbol{ T}}^{\mathrm{ e}}} \) may be different.
2.2.3 The Medium Fully Surrounded by the Air
An important engineering problem is that the medium with symmetric material coefficients is fully surrounded by the air. In air, the mechanical stresses and mechanical energy can be neglected and only the electric field and electric energy should be considered. The physical variational formula (2.7) in this case becomes
where the body force is neglected and
2.2.4 Isotropic Materials
For isotropic materials, the constitutive equations are (Kuang 2012)
In isotropic materials, variables \( \boldsymbol{ S} \), \( \boldsymbol{\sigma} \), and \( {{\boldsymbol{\sigma}}^{\mathrm{ M}}} \) are all symmetric. So Eqs. (2.2) and (2.3) are reduced to
If we let \( {l_1}-\epsilon ={a_2},{l_2}-\epsilon =\left( {{1 \left/ {2} \right.}} \right){a_1} \), then Eq. (2.27a) is reduced to
The first formula in Eq. (2.27b) is just the usual form of the constitutive equation, where \( {a_1} \) and \( {a_2} \) are known as electrostrictive coefficients. From Eqs. (2.14), (2.15), (2.16), and (2.17), it is known that solving \( \boldsymbol{ S} \) is easier than that for \( \boldsymbol{\sigma} \), so in experiments, the measured variables usually are \( \left( {\boldsymbol{ S},\boldsymbol{\varepsilon}, \boldsymbol{ E}} \right) \). If the constitutive equation (2.27a) is used, the measured material coefficients are \( {l_1} \) and \( {l_2}-2\epsilon. \) If the constitutive equation (2.27b) is used, the measured material coefficients are \( 2\epsilon -{a_1} \) and \( {a_2}+\epsilon \). Therefore, in experiments, the entire system including the dielectric medium, its environment, and their common boundary should be considered together, and appropriate governing equations should be selected.
2.2.5 The Static Electric Force Acting on the Medium by the Electric Field
Comparing Eqs. (2.7) and (2.21), it is found that the difference between them is that in Eq. (2.7), the local variation and the migratory variation are used simultaneously, however in Eq. (2.21), only the local variation is used, but the electric force introduced by electric field is introduced:
In Eqs. (2.7) and (2.10), the part related to the migratory variations of potential is
Using Eqs. (2.13b) and (2.16) and adding terms \( \int\nolimits_a {{D_i}{n_i}\left( {{E_p}\delta {u_p}+{\delta_u}\varphi } \right)\mathrm{ d}a} =0 \) and \( \int\nolimits_{{{a^{\mathrm{ env}}}}} {D_i^{\mathrm{ env}}n_i^{\mathrm{ env}}\left( {E_p^{\mathrm{ env}}\delta u_p^{\mathrm{ env}}+{\delta_u}{\varphi^{\mathrm{ env}}}} \right)\mathrm{ d}a} =0 \) to Eq. (2.29a), then Eq. (2.29a) can be reduced to
Comparing Eqs. (2.20), (2.28), and (2.29), it is found that the static electric force acting on the medium can be obtained from the general energy migratory variational principle (Kuang 2012):
If the environment is neglected, it is obtained:
2.2.6 Hamilton Principle
In order to use the PVP for moving electroelastic materials, the D’Alembert principle should be used to make the moving media in a state of relative rest. Using D’Alembert principle, the Hamilton principle can easily be obtained from the PVP. Let \( \delta {u_{k0 }} \) and \( \delta {u_{{k\mathrm{ f}}}} \) be displacements at the initial and final times, respectively, in time interval \( [{t_0},{t_{\mathrm{ f}}}] \) and assume \( \delta {u_{k0 }}=\delta {u_{{k\mathrm{ f}}}}=0 \), using
where \( K=\rho {{\dot{u}}_k}{{\dot{u}}_k}/2 \) and \( {K^{\mathrm{ env}}}={\rho^{\mathrm{ env}}}\dot{u}_k^{\mathrm{ env}}\dot{u}_k^{\mathrm{ env}}/2 \) are the kinetic energies in the material and its environment, respectively. Substituting Eq. (2.31) into (2.7) and integrating it from \( {t_0} \) to \( {t_{\mathrm{ f}}} \) then we get the Hamilton principle:
However the energy conservation law is:
Equation (2.33) is held for any time interval. It is noted that the energy principle is held in a real process, but the PVP gives a true process for all virtual possible process satisfying the natural constrained conditions, and it is equivalent to the momentum equation. It is also noted that the Hamilton principle is held in four-dimensional space \( (\boldsymbol{ x},t) \) and the time boundary conditions should be added. But the PVP is held in three-dimensional (3D) space \( (\boldsymbol{ x}) \) and does not consider the time variation.
It is obvious that Hamilton principle is also a fundamental principle in the physics and continuum mechanics. Using the local and migratory variation theory, the Maxwell stress can also be obtained automatically.
2.2.7 Physical Variational Principle in Electromagnetic Materials
In this section, the PVP is extended to electromagnetic materials under static electromagnetic field, without the current and the body electromagnetic couple (Kuang 2011a, b, c).
Let constitutive equations be
where \( e_{jkl}^{\mathrm{ e}} \) and \( e_{jkl}^{\mathrm{ m}} \) are piezoelectric and piezomagnetic coefficients, respectively, \( l_{ijkl}^{\mathrm{ e}} \) and \( l_{ijkl}^{\mathrm{ m}} \) are electrostrictive and magnetostrictive coefficients, respectively, and \( {\beta_{ij }}={\beta_{ji }} \) is the magnetoelectric coupling coefficient. The electromagnetic body couple is still balanced by the asymmetric stress. In this case, the electromagnetic Gibbs free energy \( {g} \) is
For the small deformation \( \varDelta:\boldsymbol{\varepsilon} \) can still be neglected. The PVP is
where \( E_i=-{\varphi_i},H_{i}=-\psi_i \). Finishing the variational calculation finally yields
2.3 General Variational Principle
2.3.1 General Variational Principle Not Satisfying Boundary Conditions
This principle does not ask \( \boldsymbol{ u},\varphi \) and \( {{\boldsymbol{ u}}^{\mathrm{ env}}},{\varphi^{\mathrm{ env}}} \) to satisfy boundary conditions on their own boundaries \( {a_u},{a_{\varphi }} \) and \( a_u^{\mathrm{ env}},a_{\varphi}^{\mathrm{ env}} \), respectively, and continuity conditions on the interface prior. For small deformation, this principle is
In Eq. (2.38), some additional virtual work done on the boundary and interface is added, because the displacements and potentials do not satisfy the boundary conditions and the continuity conditions on the interface. As an example, \( \delta \int\nolimits_{{{a^{{\operatorname{int}}}}}} {T_k^{{\mathrm{ S}\;\mathrm{ int}}}({u_k}-u_k^{\mathrm{ env}})\mathrm{ d}a} \) is the virtual work introduced by the difference of the virtual displacement \( ({u_k}-u_k^{\mathrm{ env}}) \) on two sides and the unknown pseudo total traction \( {{\boldsymbol{ T}}^{\mathrm{ S}}} \) on the interface. In Eq. (2.38), \( {{\boldsymbol{ T}}^{\mathrm{ S}}} \) and \( \sigma \) may also be considered as Lagrange multipliers in the mathematical sense (Kuang 1964, 2002).
Equation (2.38) can be proved as follows. Analogous to the derivation of Eq. (2.11), it is obtained:
Due to the arbitrariness of \( \delta \boldsymbol{ u} \) and \( \delta \varphi \) from Eq. (2.39), we get
Similarly for the environment we get
For \( \delta {{{W}}^{\mathrm{ int}}} \) we have
Due to \( \boldsymbol{ n}=-{{\boldsymbol{ n}}^{\mathrm{ env}}} \) and arbitrariness of \( \delta \boldsymbol{ u},\delta \varphi \), \( \delta {{\boldsymbol{ T}}^{\mathrm{ S}}},\delta \sigma \) from \( \delta \varPi =\delta {\varPi_1}+\delta {\varPi_2}=\delta {{{W}}^{{\operatorname{int}}}} \) we get
2.3.2 General Variational Principle in Linear Piezoelectric Materials
Kuang (1964, 2002) proposed a Lagrange multiplier method to derive general variational principle (Hu 1981) from the potential energy principle in linear elasticity. This method is easy extended to the linear electroelastic theory where the Maxwell stress is not considered. So the migratory variation of the electric potential is not needed. Using the Lagrange multiplier method, the general variational principle with independent variables \( \boldsymbol{ u},\varphi, \boldsymbol{\sigma}, \boldsymbol{\varepsilon}, \boldsymbol{ D},\boldsymbol{ E} \) in the small deformation case is easy obtained. The boundary conditions and continuity conditions on the interface do not satisfied prior. The electric Gibbs free energy for the linear piezoelectric material under the small deformation is
Omitting the derivation process, the PVP is
It is easy to show that \( \delta {\varPi_1} \) can be reduced to
Completing the variational calculation and considering the arbitrariness of \( \delta \boldsymbol{ u},\delta \varphi, \delta {{\boldsymbol{ u}}^{\mathrm{ env}}},\delta {\varphi^{\mathrm{ env}}} \) and \( \boldsymbol{ T},\sigma \) finally we get
Equation (2.47) is the complete governing equation.
2.4 Variational Principle in Piezoelectric Materials Under Finite Deformation
2.4.1 The Electric Gibbs Free Energy in Initial Configuration
Some fundamental formulas and notations for finite deformation shown in Sect. 1.3.4 will be used in this chapter. It is emphasized that the same coordinate system is used in the current and initial configurations. Since the isothermal electric Gibbs free energy \( \bar{{g}} \) in the finite deformation state must be invariant in a rigid body rotation, so the \( \bar{{g}} \) for materials without the electric couple problem should be taken in the following form:
where \( {{\bar{C}}_{IJKL }},{{\bar{\epsilon}}_{KL }},{{\bar{e}}_{KIJ }},{{\bar{l}}_{IJKL }} \) are the material coefficients in the initial configuration. It is noted that coefficients in the initial and current configurations are different. From the thermodynamic theory, the constitutive equations are
Using Eq. (2.49), Eq. (2.48) can be reduced to
In \( \bar{{g}} \), the term \( \left( {{1 \left/ {2} \right.}} \right){{\bar{C}}_{IJKL }}{{\bar{\varepsilon}}_{IJ }}{{\bar{\varepsilon}}_{KL }} \) is the mechanical deformation energy, \( \left( {{1 \left/ {2} \right.}} \right){{\bar{D}}_K}{{\bar{E}}_K} \) is the electromagnetic energy, \( \left( {{1 \left/ {2} \right.}} \right){{\bar{e}}_{NKL }}{E_N}{{\bar{\varepsilon}}_{KL }} \) is the mechanical and electromagnetic coupling energy, and \( {{\bar{{g}}}^{\mathrm{ e}}} \) is the sum of the electromagnetic energy and coupling energy. For the small deformation case, \( \left( {{1 \left/ {2} \right.}} \right){{\bar{e}}_{NKL }}{E_N}{{\bar{\varepsilon}}_{KL }} \) can be neglected, so the coupling energy can also be neglected.
2.4.2 Variational Principle with the Electric Gibbs Free Energy Under Finite Deformation
As in Eq. (2–8), variations of \( \varphi, \bar{\boldsymbol{ E}} \) are divided into local variation \( {\delta_{\varphi }}\varphi, {\delta_{\varphi }}\bar{\boldsymbol{ E}} \) and migratory variation \( {\delta_u}\varphi,\;{\delta_u}\bar{\boldsymbol{ E}} \), i.e.,
Let the displacement \( \mathbf{u} \) and the potential \( \varphi \) satisfy their boundary conditions on their own boundaries \( {{\bar{a}}_u},{{\bar{a}}_{\varphi }},{{\bar{a}}_{\psi }} \) and the continuity conditions on their interface \( {{\bar{a}}^{{\operatorname{int}}}} \) (Fig. 2.1). The variational principle with the electric Gibbs free energy under finite deformation for electroelastic media can be expressed in the following form (Kuang 2008b, 2011a):
where \( \bar{T}_k^{*},{{\bar{\sigma}}^{*}};\bar{T}_k^{*\mathrm{ env}},{{\bar{\sigma}}^{*\mathrm{ env}}};\bar{T}_k^{{*\operatorname{int}}},{{\bar{\sigma}}^{*\mathrm{ int}}} \) are the given values on the corresponding surfaces.
Using relations \( {{g}^{\mathrm{ e}}}\mathrm{ d}V={{\bar{{g}}}^{\mathrm{ e}}}\mathrm{ d}\bar{V},\int\nolimits_V {{{g}^{\mathrm{ e}}}\delta {u_{i,i }}\mathrm{ d}V=} \int\nolimits_{\bar{V}} {{{\bar{{g}}}^{\mathrm{ e}}}\delta {u_{i,i }}\mathrm{ d}\bar{V}} \) and \( \int\nolimits_V {\delta {g}\mathrm{ d}V} =\int\nolimits_{\bar{V}} {\delta \bar{{g}}\mathrm{ d}\bar{V}} \) (Kuang 2008b, 2009a) yields
where \( \delta {u_{k,k }}=\delta {u_{k,J }}{X_{J,k }} \) was used. Substitution of Eq. (2.53) into Eq. (2.52) yields
The last three terms in (2.54a) can be reduced to
where \( {X_{L,p }}\delta {u_p}{{\bar{E}}_L}={E_p}\delta {u_p} \) was used. So Eq. (2.54a) can be reduced to
where
\( {{\bar{S}}_{IJ }} \) is called the pseudo total stress in the initial configuration, \( \bar{\sigma}_{IJ}^{\mathrm{ M}} \) may be called the second kind of the Maxwell stress defined in initial configurations, and \( {X_{L,k }}\bar{\sigma}_{JL}^{\mathrm{ M}} \) may be called the first kind of the Maxwell stress defined in current and initial configurations. From Eq. (2.55), it is known that when the initial configuration is used as the reference configuration, the Maxwell stress is related to strain. But for isotropic materials, the Maxwell stress is still not related to strain due to \( {{\bar{e}}_{NML }}=0 \).
Due to the arbitrariness of \( \delta {{\bar{u}}_i},\;\delta \bar{\varphi} \), from Eq. (2.54b) we get
and
Similarly for the environment, we have
Due to the arbitrariness of \( \delta \bar{u}_i^{\mathrm{ env}},\delta {{\bar{\varphi}}^{\mathrm{ env}}} \), from Eq. (2.58), we get
and
Noting \( {{\bar{n}}_I}=-\bar{n}_I^{\mathrm{ env}},{{\bar{u}}_I}=\bar{u}_I^{\mathrm{ env}},\varphi ={\varphi^{\mathrm{ env}}} \) on the interface, we get
So on the interface, it is obtained:
The above variational principle requests prior that the displacements and the potential satisfy their own boundary conditions and the continuity conditions on the interface, so the following equations should also be added to governing equations:
Equations (2.55), (2.56), (2.59), (2.57b), (2.60), and (2.61) are the governing equations under the finite deformation. It is noted that for the elastic material, these formulas are reduced to the usual elastic governing equations for elasticity. If in Eq. (2.52) we use \( \delta \int\nolimits_{\bar{V}} {\bar{{g}}\mathrm{ d}\bar{V}} \) instead of \( \int\nolimits_{\bar{V}} {\delta \bar{{g}}\mathrm{ d}\bar{V}} +\int\nolimits_{\bar{V}} {{{\bar{{g}}}^{\mathrm{ e}}}\delta {u_{i,i }}\mathrm{ d}\bar{V}} \), Eq. (2.52) cannot be reduced to the usual elastic variational formula.
2.5 Internal Energy Variational Principle in Piezoelectric Materials
2.5.1 Internal Energy
It is noted that the constitutive equations of the general electroelastic materials are linear in the elastic part, but are nonlinear in the electric part for small deformation. The internal energy \( \mathfrak{A} \) for materials without electric couple is assumed in the following form under small deformation:
where \( {h_{kij }},{\beta_{kl }},{k_{ijkl }}, \) and \( {C_{ijkl }} \) are material constants. The constitutive equations are
Equation (2.62a) can be rewritten as
2.5.2 Internal Energy Variational Principle under Small Deformation
Let \( \mathbf{u},\boldsymbol{ D},{{\mathbf{u}}^{\mathrm{ env}}},{{\boldsymbol{ D}}^{\mathrm{ env}}} \) satisfy their boundary conditions on their own boundaries \( {a_u},{a_D},a_u^{\mathrm{ env}},a_D^{\mathrm{ env}} \) and the continuity conditions on the interface \( {a^{{\operatorname{int}}}} \), i.e.,
where \( \boldsymbol{ n} \) is the outward normal of the body. Inside the body and environment, it is assumed that
Under the above conditions, given the displacement and electric charge virtual increments, the PVP in term of the internal energy is (Kuang 2009a)
where \( a={a_{\sigma }}+{a_u}+{a^{{\operatorname{int}}}}={a_D}+{a_{\varphi }}+{a^{{\operatorname{int}}}} \), \( {f_k}{}, T_k^{*}{}, \varphi \), and \( {\varphi^{*}} \) are given body force,traction, potential, and surface potential, respectively. It is noted that the work done by the electric field in Eq. (2.66) has the form \( \varphi \delta q\), i.e. the potential is kept constant, but the electric charge \( {\rho_{\mathrm{ e}}}\mathrm{ d}V\), \( {\sigma da} \) etc. have virtual increment. \( {\sigma^{*}} \) and \( {\sigma^{*\mathrm{ env}}} \) are given constants and do not change when virtual displacements happen, so terms \( {\sigma^{*}}\mathrm{ d}a \) and \( {\sigma^{*\mathrm{ env}}}\mathrm{ d}{a^{\mathrm{ env}}} \) will not be constants. Thus, terms \( \int\nolimits_{{{a_D}}} {\varphi \delta ({\sigma^{*}}\mathrm{ d}a)} \) and \( \int\nolimits_{{a_D^{env }}} {{\varphi^{\mathrm{ env}}}\delta ({\sigma^{*\mathrm{ env}}}\mathrm{ d}a} ) \) etc. should be added to the variational formula. \( T_k^{{*\operatorname{int}}} \) and \( {\varphi^{{*\operatorname{int}}}} \) are given surface force and the jump of electric potential on the interface, respectively. Similar to Eq. (2.8),
The variation of the differential volume and area, etc. are
Neglecting terms containing \( ({\sigma_{kl }}{\varepsilon_{kl }}+{k_{ijkl }}{D_i}{D_j}{\varepsilon_{kl }})/2 \), it is obtained:
and
In Eq. (2.70b), \( {D_i}{n_i}=-\sigma^* \) is a given value on \( {a_D} \), so its variation vanishes on \( {a_D} \).
Substituting above equations into \( \delta {\varPi_1} \) in Eq. (2.66), it is obtained:
From Eqs. (2.66)and (2.71) and the arbitrariness of \( \delta {u_i},\;\delta {D_i} \) we get
where \( {{\boldsymbol{\sigma}}^{\mathrm{ M}}} \) is the Maxwell stress. Using Eq. (2.72), \( \delta {\varPi_1} \) is reduced to
Similarly for the environment, we have
\( \delta {W^{{\operatorname{int}}}} \) can be reduced to
Substituting Eqs. (2.73), (2.74) and (2.75) into Eq. (2.66) and noting \( {{\boldsymbol{ n}}^{\mathrm{ env}}}=-\boldsymbol{ n} \), \( \boldsymbol{ u}={{\boldsymbol{ u}}^{\mathrm{ env}}} \) we have
Due to the arbitrariness of \( \delta {u_i},\;\delta {D_i} \), we get
Equations (2.72), (2.74), (2.77), (2.64), and (2.65) form the complete governing equations.
2.5.3 The Force Acting on the Dielectric in a Plate Capacitor
As an application of the PVP, we discuss the force acting on the dielectric in a plate capacitor filled the dielectric with permittivity \( \epsilon \) as shown in Fig. 2.2. Assume both the length and width of the plates are infinitely long, the distance \( h \) between two electrodes is small. There is no external mechanical force. The electric field inside the dielectric of the capacitor is homogeneous and \( \boldsymbol{ E}={E_2}\boldsymbol{ n} \), where \( \boldsymbol{ n} \) is along the positive direction of the axis \( {x_2} \). The electric field inside the electrode is zero. In this simple case, the static electric force can be directly derived from the Maxwell stress and the general equation of the PVP.
A plate capacitor
1. The Maxwell Stress Method
Using the Maxwell stress in Eqs. (2.12) and (2.18), the force acting on the dielectric is
2. The Internal Energy Variational Principle
Let the upper plate electrode possesses negative charge and the lower electrode possesses positive charge. Because on the electrobe the electric charge is given in the internal energy variational principle, the boundary conditions on the whole boundary of the dielectric are known. In Eq. (2.66) we only need to discuss \(\delta\Pi_1 \). Given the upper electrode a virtual displacement \(\delta u_2=\delta h \), the virtual strain of the dielectric is \( {\varepsilon_{22 }}={u_{2,2 }}={{{\delta h}} \left/ {h} \right.} \). Because only \( \delta u_2 \) is considered, the surface integrals in \( \delta \Pi_1 \) can be neglected due to that the surface area keeps constant in the virtual displacement process. Therefore the variational principle for the volume between unit surfaces of electrodes is
Because electric charge \( q \) on the electrode is constant, so \( {\delta_D}{D_2}=0 \). In volume \( {D_2}=\mathrm{ const}. \) due to \( {D_{2,p }}=0 \), so \( {\delta_u}{D_2}=0 \). Therefore, it is obtained:
The result is identical with that in Eq. (2.78).
3. The Electric Gibbs Free Energy Variational Principle
Let the lower electrode possesses positive potential and the upper plate electrode grounded. Analogous to the above problem, but on the electrode the electric potential is given in the electric Gibbs free energy variational principle now. In Eq. (2.7) we only need to discuss \( \delta \Pi_1 \).
Give a virtual displacement under the constant electric potential on the electrode plate. It is noted that though \( \varphi \) is constant on the plate, but after virtual displacement, \( \varphi \) is changed for the point inside the dielectric. For a fixed \( \boldsymbol{ x} \), the change of the electric field due to changed \( \varphi \) is
The potential \( \varphi \) on the electrode is constant, \( {E_{2,p }}=0 \), so \( \delta {E_2}={\delta_{\varphi }}{E_2} \). Therefore, we have
The result is identical with that in Eq. (2.78).
Equation (2.78) shows that the force acting on the dielectric is compressive. It is just the attractive force between two electrodes. This result is identical with that in usual textbooks.
2.6 Constitutive Equations in Electroelasticity
2.6.1 Constitutive Equations
In this section, we only discuss the case with symmetric stresses. When the thermal effect is omitted in Eq. (1.59), there are only four thermodynamic character functions: the internal energy \( \mathfrak{A}(\boldsymbol{\varepsilon}, \boldsymbol{ D}) \) is equivalent to the free energy \( f \), the electric Gibbs function \( {g}(\boldsymbol{\varepsilon}, \boldsymbol{ E}) \) is equivalent to the electric enthalpy \( {h^{\mathrm{ e}}} \), the enthalpy \( h(\boldsymbol{\sigma}, \boldsymbol{ E}) \) is equivalent to the Gibbs function \( {{g}^{g}} \), and the elastic Gibbs function \( {{g}^{\mathrm{ el}}}(\boldsymbol{\sigma}, \boldsymbol{ D}) \) is equivalent to the elastic enthalpy \( {h^{\mathrm{ el}}} \). In general case, there are two groups with four variables: \( (\boldsymbol{\sigma}, \boldsymbol{\varepsilon} ),(\boldsymbol{ E},\boldsymbol{ D}) \) in electroelasticity. Because each variable in two groups can be used as the independent variable, there are four group constitutive equations corresponding to four thermodynamic character functions \( \mathfrak{A} \) (\( f \)), \( {g} \) (\( {h^{\mathrm{ e}}} \)), \( h \) (\( {{g}^{g}} \)), and \( {h^{\mathrm{ el}}} \) (\( {{g}^{\mathrm{ el}}} \)). Constitutive equation (2.3) is derived from \( {g} \); Eq. (2.63) is derived from \( \mathfrak{A} \). The enthalpy \( h \) and the elastic Gibbs function \( {{g}^{\mathrm{ el}}} \) can, respectively, be assumed in the following forms:
where \( \boldsymbol{ s} \) is the flexibleness coefficient tensor. From Eqs. (2.79) and (2.80), the following constitutive equations are obtained, respectively:
Equations (2.3), (2.63), (2.81), and (2.82) are four kinds of constitutive equations for general ferroelectric materials. In these equations, it has been assumed that \( \boldsymbol{\sigma} =\boldsymbol{\varepsilon} =\boldsymbol{ E}=\boldsymbol{ D}=\mathbf{0} \) at the natural state. Constitutive equations of some simpler materials are as follows.
Linear piezoelectric materials
The constitutive equations of the first, second, third, and fourth types of linear piezoelectric materials are
where the superscript letter “\( \varsigma \)” of a material constant means that the constant is measured at \( \varsigma =\mathrm{ const}. \) As an example, \( \boldsymbol{ C}_{ijkl}^E \) means that the constant \( \boldsymbol{ C}_{ijkl}^E \) is measured at \( {E_i}=\mathrm{ const}. \) Usually the coefficient measured at constant \( \boldsymbol{ E} \) is called the closed circuit coefficient, and the coefficient measured at constant \( \boldsymbol D \) is called the open circuit coefficient. Usually \( e\cdot E=e^{\varepsilon}_{ijk} E_k \) is more convenient than \( E\cdot e=e^{\varepsilon}_{kij} E_k \) in use. If the Voigt notation (see Eq. (1.37) is used, Eq. (2.83) can be rewritten as
It is noted that though some coefficients have the same notation in different kind of constitutive equations, they should be measured in different conditions.
Electrostrictive materials with symmetric center
For all electrostrictive materials with symmetric center, the material coefficients with odd number subscript are all zero, so the piezoelectric effect disappeared. In Eq. (2.3), if terms containing \( \boldsymbol{\alpha} \) are omitted, the constitutive equations have following forms:
Under the high electric field, usually the electric hysteretic loop of the electrostrictive material, like \( \mathrm{ PMN} \), is smaller than that of the piezoelectric material, like \( \mathrm{ PZT} \).
2.6.2 Relations Between Material Constants of the Linear Piezoelectric Materials
Equation (2.83) is the four kinds of constitutive equations for the linear piezoelectric materials. Substitution of \( \boldsymbol{ D} \) in the second equation into the fourth equation in Eq. (2.83) yields
and in the similar discussion, we finally get
For the nonlinear ferroelectric materials, relations between material coefficients of different constitutive equations are difficult expressed in simple unique forms.
2.7 Variational Principle in Pyroelectric Materials and Its Governing Equations
2.7.1 Internal Energy and Electric Gibbs Function
According to the continuum thermodynamics in Sect. 1.5, the electric Gibbs function \( {g} \), the electric complementary dissipative energy rate \( {{\dot{h}}_{g}} \), the internal energy \( \mathfrak{A} \), and the dissipative energy rate \( {{\dot{h}}_{\mathfrak{A}}} \) can be assumed as
Where \( \vartheta =T-{T_0} \), \( {T_0} \) is the temperature of the environment. It is noted that in Eq. (2.87), \( s=0 \) when \( T={T_0} \), if \( \varepsilon_{ij} =E_i=0 \), but in Eq. (2.88), \( s=0 \) when \( T=0 \) and \( s={s_0} \) when \( T={T_0} \); if \( \varepsilon_{ij }= D_i=0 \); \( {{\tilde{\alpha}}_{ij }},{{\tilde{\tau}}_i},\tilde{C},{{\tilde{\lambda}}_{ij }} \) are all material constants. In the later sections, this rule will be adopted. Constitutive and evolution equations corresponding to Eq. (2.87) are
where the evolution equation of temperature has been shown in Eq. (1.71). Corresponding to Eq. (2.88) the constitutive and evolution equations are
It is obvious that there is \( {T_{,j }}={\vartheta_{,j }},\dot{T}=\dot{\vartheta} \). Using Eqs. (2.89) and (2.90), Eqs. (2.87) and (2.88) can be rewritten, respectively, as
In Eq. (2.91), \( {\varDelta_{kl }}{\varepsilon_{kl }} \) and \( {{\varDelta^{\prime}}_{kl }}{\varepsilon_{kl }} \) can be neglected for the case of small strain.
Using the inertial entropy theory given in Sect. 1.7.2, from Eqs. (2.89) and (1.74), the thermal conductive or energy equation can be obtained:
If \( \vartheta \) is much less than \( {T_0} \), \( \vartheta \ll {T_0} \), then the above equation is reduced to
Equations (2.92) and (2.93) are temperature wave equations with finite phase velocity.
2.7.2 Electric Gibbs Function Variational Principle
In this section, we only discuss the PVP of the pyroelectric material with linear elasticity under small deformation. For simplicity, it is assumed that the environment is air. It is also assumed that in the air, the temperature is constant or \( {\vartheta^{\mathrm{ env}}}=0 \) and at infinity, \( {{\boldsymbol{ E}}^{\infty }}=\mathbf{0},\ {\sigma^{{*\infty }}}=0 \). The interface is heat insulated. The heat input and heat output by heat conduction may be occured at some internal boundaries. Analogous to Eq. (2.8), the variation of the temperature \( \vartheta \) can also be divided into \( {\delta_{\vartheta }}\vartheta \) and \( {\delta_u}\vartheta \), but it is not needed because the final result shows that terms containing \( {\delta_u}\vartheta \) are countervailed each other. So the body and air only have electric connection. However, the contribution of the heat due to the variation of the volume seems to be considered.
Under the assumption that \( \boldsymbol{ u},\varphi, \vartheta \) satisfy their own boundary conditions \( {u_i}=u_i^{*},\varphi ={\varphi^{*}} \) and \( \vartheta ={\vartheta^{*}} \) on \( {a_u},{a_{\varphi }} \) and \( {a_T} \), respectively. \( \varphi ={\varphi^{\mathrm{ env}}},\vartheta ={\vartheta^{\mathrm{ env}}}=0 \) on the interface except at some heat source and sink places. In the medium \( {\varepsilon_{ij }}=({u_{i,j }}+{u_{j,i }})/2,{E_i}=-{\varphi_{,i }} \), \( T{{\dot{\eta}}_j}=-{\lambda_{ij }}{\vartheta_{,i }} \) and the constitutive equation (2.89) are held. Noting Eq. (1.59), \( {g}=\mathfrak{A}-\boldsymbol{ E}\cdot \boldsymbol{ D}-Ts \), the PVP in terms of the electric Gibbs function for the pyroelectricity can be written as (Kuang 2009b)
where \( {f_k},T_k^{*},{\rho_{\mathrm{ e}}},{\sigma^{*}},\rho_{\mathrm{ e}}^{\mathrm{ e}\mathrm{ nv}} \) and \( \dot{\eta}_i^{*}\ ({{\dot{\eta}}^{*}}=\dot{\eta}_i^{*}{n_i}) \) are the given mechanical body force, traction, body electric charge density, surface electric charge density, body electric charge density in the air, and surface entropy flow, respectively, and \( {a_q} \) is the surface given thermal flow, \( {{g}^{\mathrm{ env}}}={{g}^{{{E}\ T\;\mathrm{ env}}}}=-\left( {{1 \left/ {2} \right.}} \right)D_k^{\mathrm{ env}}E_k^{\mathrm{ env}},D_k^{\mathrm{ env}}={\epsilon_0}E_k^{\mathrm{ env}} \). In Eq. (2.94), the term \( \int\nolimits_0^t {{{\dot{s}}^{{(\mathrm{ i})}}}\delta \vartheta\mathrm{ d}\tau } \) is the electric complement heat rate per unit volume corresponding to the inner electric complement dissipation energy rate \( \delta {h_{g}} \). This is consistent with the laws of the thermodynamics. In order to obtain the heat conduction equation and the boundary condition of the heat flow from the variational principle, the electric complement dissipation energy \( \int\nolimits_V {\delta {h_{g}}\mathrm{ d}V} \) in \( \delta \varPi \) and the inner irreversible electric complement heat \( \int\nolimits_V {\int\nolimits_0^t {{{\dot{s}}^{{(\mathrm{ i})}}}\delta \vartheta\mathrm{ d}\tau}\mathrm{ d}V} \) in \( \delta {Q}^{\prime} \) should be simultaneously included in the variational functional. In Eq. (2.94), the integrands contain the time derivatives of variables and need to integrate with time t, because in the irreversible process the integral is dependent to the integral path. But the time is a parameter and does not join the virtual variation.
It is noted that
Substituting Eq. (2.95) into \( \delta {\varPi_1} \) of Eq. (2.94) and adding a term \( \int\nolimits_a {{D_k}{n_k}\left( {{E_p}\delta {u_p}+{\delta_u}\varphi } \right)\mathrm{ d}a} \) to it, similar to the derivation in Sect. 2.1.2, finally we get
where
where \( {{\boldsymbol{\sigma}}^{{\mathrm{ M}\;T}}} \) is the general Maxwell stress. Whether \( {{\boldsymbol{\sigma}}^{{\mathrm{ M}\;T}}} \) includes the term \( s\vartheta \), it should still be proved by experiments; \( \boldsymbol{ S} \) is the pseudo total stress (Jiang and Kuang 2003, 2004). In pyroelectric materials, when the electric field and temperature are not too large and the piezoelectric coefficient is not too small, the general Maxwell stress can be neglected.
Due to the arbitrariness of \( \delta {u_i},\;\delta \varphi \) and \( \delta \vartheta \), from Eq. (2.96), it is obtained:
Analogously in the air,
Substituting Eqs. (2.98) and (2.99) into Eq. (2.94) and noting \( {{\boldsymbol{ n}}^{\mathrm{ env}}}=-\boldsymbol{ n} \) we get
The governing equations must contain the prior conditions of the variational principle:
If \( \vartheta \ll {T_0} \), the integral can be integrated in Eq. (2.94), and \( \delta {\varPi_1} \) in Eq. (2.94) is reduced to
where \( \eta_0^{*}=({1 \left/ {{{T_0}}} \right.})\int\nolimits_0^t {{q^{*}}\mathrm{ d}t} \).
There are eight thermodynamic character functions in pyroelectric materials, so there are eight fundamental variational principles. However, the electric Gibbs function only contains five independent variables \( \boldsymbol{ u},\varphi, T \) and is convenient in practical application.
2.7.3 An Example for Purely Thermal Conduction
When \( \vartheta \ll {T_0} \) for the purely thermal conduction problem without the internal heat source in an isotropic material, Eq. (2.93) is reduced to
Now discuss a simple problem in which the wave propagates along the \( {x_1} \) direction, i.e.,
where \( x \) is the dimensionless coordinate, \( \tau \) is the dimensionless time, and \( c \) is the phase velocity. Let
where \( H(t) \) is the Heaviside function and \( {\varTheta_0} \) is a constant. The solution of the above problem is
where \( {I_1}\left[ \cdot \right] \) is the modified first kind of the Bessel function. Equation (2.105) shows that \( \vartheta \) is an attenuated advanced wave. At the wave front \( x=\tau \) or \( {x_1}=ct \), \( \vartheta \) is interrupted with value \( {e^{{-{x \left/ {2} \right.}}}}={e^{{-{{{{x_1}}} \left/ {{2c{\rho_{\mathrm{ s}0}}}} \right.}}}} \) which is decreased with time.
For a problem without initial conditions, let
where \( {\varTheta_0} \) is the amplitude of the wave. Substituting the above equation into Eq. (2.103) yields
so
Equation (2.106) shows that the temperature wave is an attenuated dispersive wave. When \( {\rho_{\mathrm{ s}0}}\to 0 \), \( c\to \sqrt{{{{{2\omega \lambda }} \left/ {C} \right.}}} \) which is just the result of the classical heat conduction theory. It shows that when \( {\rho_{\mathrm{ s}0}} \) is small, the heat inertial effect can be neglected for the problem without initial conditions.
2.8 Variational Principle and Governing Equations in Pyroelectric Materials with Diffusion
2.8.1 Internal Energy, Electrochemical Gibbs Function, and Electric Gibbs Function
In the diffusion theory, mechanical and electrical processes are reversible, but thermal and diffuse processes are irreversible. The internal energy and entropy are all state functions. The Gibbs equation and evolution equation are still expressed by Eqs. (1.72) and (1.77), respectively. According to Eqs. (1.72), (1.77) and (1.69) the internal energy can be given by
where \( \mathfrak{A},{{g}_{\mathrm{ c}}}, \) and \( {g} \) are the internal energy, electrochemical Gibbs function, and electric Gibbs function. \( {{\dot{h}}_{\mathfrak{A}}},{{\dot{h}}_{{{g}\mathrm{ c}}}} \) and \( {{\dot{h}}_{g}} \) are the corresponding dissipative or complementary dissipative energy rates. In this section, we only discuss electrochemical Gibbs function and electric Gibbs function variational principles. \( {{g}_{\mathrm{ c}}},{{\dot{h}}_{{{g}\mathrm{ c}}}} \) and \( {g},{{\dot{h}}_{g}} \) can be assumed as
where \( C,{C_{ijkl }},{e_{kij }},{\epsilon_{ij }},{\alpha_{ij }},{\tau_i},b,{b_{ij }},{b_i},a,\hat{b},{{\hat{b}}_{ij }},{{\hat{b}}_i},\hat{a} \) are all material constants.
Constitutive and evolution equations corresponding to \( {{g}_{\mathrm{ c}}} \) and \( {g} \) are, respectively,
where the evolution equations of temperature and concentration have been given in Eq. (1.77).
Using Eqs. (2.110) and (2.111), \( {{g}_{\mathrm{ c}}} \) and \( {g} \) can be rewritten as
2.8.2 The Electrochemical Gibbs Function Variational Principle
In this section and the following Sect. 2.8.3, we only discuss the pyroelectric material with linear elasticity under small deformation and small variation of the temperature; the environment is air. It is assumed that on the interface, there is no diffusion and heat flow, but the electric coupling is allowed, i.e. \( \varphi=\varphi^{\rm env} , q=q^{\rm env}=0 \ {\rm and}\ d=d^{\rm env}=0\). The heat and input and output may be occurred at some internal boundaries. The temperature and concentration problems do not considered in air, but the electric field is discussed and at infinity \( \varphi^{*\rm env}=0\).
Under assumptions that \( \boldsymbol{ u} \), \( \varphi \), \( \vartheta \), and \( \mu \) satisfy their own boundary conditions \( \boldsymbol{ u}={{\boldsymbol{ u}}^{*}} \), \( \varphi ={\varphi^{*}} \), \( \vartheta ={\vartheta^{*}} \), and \( \mu ={\mu^{*}} \) on \( {a_u},{a_{\varphi }} \), \( {a_T} \), and \( {a_{\mu }} \), respectively and on the interface \( \varphi=\varphi^{\rm env}\) are satisfied prior. Analogous to Sect. 2.7, the PVP in terms of the electro-chemical Gibbs function is (Kuang 2010, 2011c)
In Eq. (2.114), \( {f_k}{}, T_k^{*} \), \( T_k^{*},{\sigma^{*}},\rho_{\mathrm{ e}}^{\mathrm{ e}\mathrm{ nv}},{\sigma^{*\mathrm{ env}}},T_k^{{*\operatorname{int}}},{\sigma^{{*\operatorname{int}}}} \), \( {{\dot{\eta}}^{*}}=\dot{\eta}_i^{*}{n_i} \), and \( {{\dot{\xi}}^{*}}=\dot{\xi}_i^{*}{n_i} \) are given values; \( \delta {Q}^{\prime} \) is related to heat (including the irreversible heat produced by the irreversible process in the material and the inertial heat); \( \delta \varPhi \) is related to the diffusion energy. Equation (2.108) shows that there is no term in \( \int\nolimits_V {\delta {h_{\mathrm{ c}}}\mathrm{ d}V} \) corresponding to the term \( -\int\nolimits_a {\int\nolimits_0^t {{T^{-1 }}\mu {{\dot{\xi}}_i}{n_i}\delta \vartheta\mathrm{ d}\tau\mathrm{ d}\mathrm{ a}} } \), so it should not be included in \( \delta {Q}^{\prime} \), as shown in Eq. (2.114). It is also noted that
Finishing the variational calculation yields
where
From the second and third equations of Eq. (2.116) we find \( T\left( {\dot{s}+{\rho_{\mathrm{ s}}}\ddot{\vartheta}} \right)+\mu \left( {\dot{c}+{\rho_{\mu }}\ddot{\mu}} \right) = \dot{r}-{q_{i,i }} \), which is identical with Eq. (1.62).
In the air and on the interface, there are
The above variational principle requests prior that the \( \boldsymbol{ u},\varphi, \vartheta \) and \( \mu \) satisfy their own boundary conditions, so in governing equations, the following equations should also be added:
Equations (2.116), (2.117), (2.118), and (2.119) are the governing equations of the generalized thermodiffusion theory.
If we neglect the term \( \mu \left( {\dot{c}+{{\dot{c}}^{{(\mathrm{ a})}}}} \right) \) in Eq. (1.77), or let \( T\left( {\dot{s}+{{\dot{s}}^{{(\mathrm{ a})}}}} \right)=\dot{r}-{q_{i,i }} \), then we get
If we also assume that \( {T_{,i }} \) and \( {\mu_{,j }} \) are not dependent with each other, for \( \dot{r}=0 \), Eq. (2.120) becomes
The formulas in literatures analogous to Eq. (2.121) can be found, such as in the paper of Sherief et al. (2004), where they used the Maxwell-Cattaneo formula. Genin and Xu (1999) discussed the thermoelastic plastic metals with mass diffusion.
2.8.3 The Electric Gibbs Function Variational Principle
Under assumptions that \( \boldsymbol{ u} \), \( \varphi \), \( \vartheta \), and c satisfy their own boundary conditions \( \boldsymbol{ u}={{\boldsymbol{ u}}^{*}} \), \( \varphi ={\varphi^{*}} \), \( \vartheta ={\vartheta^{*}} \), and \( c={c^{*}} \) on \( {a_u},{a_{\varphi }} \), \( {a_T} \), and \( {a_{\mathrm{ c}}} \), respectively. The PVP in terms of the electric Gibbs function for the thermo-electro-elasto-diffusive problem is (Kuang 2010)
where the symbols are the same as that in Sect. 2.8.2, but the gradient of the inertial chemical potential \( \mu_{,i}^{{\left( \mathrm{ a} \right)}}={\rho_{\mathrm{ c}}}{{\ddot{\xi}}_i} \) is introduced, and \( {\mu^{*}} \) is given value.
Finishing the variational calculation finally yields
If differentiating the equation of the chemical potential with \( \boldsymbol{ x} \) in Eq. (2.123), it is obtained:
If \( {{\hat{D}}_{ij }}=\hat{D}{\delta_{ij }},{\lambda_{ij }}=\lambda {\delta_{ij }},{{\hat{L}}_{ij }}=0 \) from Eq. (2.124), a simpler diffusion equation can be obtained:
Governing equations in the air are the same as that in Eq. (2.118).
2.8.4 Constitutive Equations
In general case, there are three groups with six variables: \( (\boldsymbol{\sigma}, \boldsymbol{\varepsilon} ),(\boldsymbol{ E},\boldsymbol{ D}),(\vartheta, s) \) for pyroelectric materials. Because each variable in three groups can be used as the independent variable, there are eight group constitutive equations which just correspond to eight thermodynamic character functions in Eq. (1.59). Equations (2.89) and (2.90) are the constitutive equations corresponding to electric Gibbs function \( {g}\) and internal energy \( \mathfrak A \). However for pyroelectric materials with diffusion there are four groups with eight variables: \( (\boldsymbol{\sigma}, \boldsymbol{\varepsilon} ),(\boldsymbol{ E},\boldsymbol{ D}),(\vartheta, s), (\mu ,c) \). So there are sixteen group constitutive equations. Equations (2.110) and (2.111) are the constitutive equations corresponding to electrochemical Gibbs function \( {{g}_{\mathrm{ c}}} \) and electric Gibbs function \( {g} \).
2.9 Conservation Integrals in Piezoelectric Materials
2.9.1 Noether Theory
In previous sections of this chapter, it is found that the electroelastic governing equations can be obtained from the extreme value of a variational functional. The governing equation is just the Euler-Lagrange equation of that functional. Based on the theory of Noether’s invariant variational problem (1918), conservation laws (integrals) can be easily obtained (Fletcher 1976; Honein and Herrmann 1997). These conservation integrals are very useful in fracture mechanics due to their path independence property. Here, some conservation laws for inhomogeneous materials (Shi and Kuang 2003) will be obtained by using the Noether’s invariant variational principle.
Let the variational functional in the continuum mechanics be
where \( L({x_i},{\psi_{{\alpha, j}}}) \) is the Lagrange density function and \( \boldsymbol{ x},\boldsymbol{\psi} \) are the independent and dependent variables, respectively. The Euler- Lagrange equation of ℑ is
Give an infinitesimal transform as
Using
from Eq. (2.128) yields
Equation (2.130a) can also be reduced to
Equation (2.130b) is identical with that in Eq. (2.8) in Sect. 2.1.2.
Because \( \mathtt{\delta}{\psi_{\alpha }}({x_i},{\psi_{\beta }}) \) is the function of \( {x_i},{\psi_{\beta }} \), so
where the notation \( {{\bar{\partial}} \left/ {{\partial {x_i}}} \right.} \) is the partial derivative with respect to explicit \( {x_i} \) in \( {\psi_{\alpha }} \). If under the transform, Eq. (2.128), on the accuracy of the first order of \( \delta {x_i},\delta {\psi_{\alpha }},\delta {\psi_{{\alpha, j}}} \), the following equality holds:
the group of transform Eq. (2.128) is called the symmetric group of a system. From Eq. (2.132), some conservation laws can be found.
Appling Eqs. (2.128) and (2.130), the following relation can be obtained:
So that
Substituting the identity
into Eq. (2.133) yields
\( {P_{ij }} \) is called the energy-momentum tensor of matter. Substitution of Eq. (2.134) into Eq. (2.132) yields
Equations (2.134) and (2.135) are the invariant conditions under the infinitesimal transform. The second equation in Eq. (2.135) is a path independence integral. Due to the arbitrariness of the volume from Eq. (2.135), the invariant condition in the differential form is
In the above discussion, it is assumed that there is no body force, body electric charge, etc.
For the electroelastic problem without body couple let \( \boldsymbol{\psi} =\left[ {{u_i},\varphi } \right]^T,L\equiv {g} \), we have
where \( {\varSigma_{{\alpha j}}} \) is the generalized stress and \( {\psi_{{\alpha, j}}} \) is the generalized strain and the Greek indices take 1–4.
2.9.2 Conservation Integral in a Homogeneous Material
For a homogeneous material, \( L \) is independent to \( \boldsymbol{ x} \), so \( L=L({\psi_{{\alpha, j}}}),\ {{\bar{\partial}} \left/ {{\partial {x_i}}} \right.}=0 \).
1. Infinitesimal translation of general displacement. Let
where \( {c_{\alpha }} \) is a constant and \( \varepsilon \) is an infinitesimal parameter. Substitution of Eq. (2.138) into Eq. (2.136) yields the invariant condition:
Equation (2.139) is just the generalized momentum equation.
2. Infinitesimal translation of coordinate. \( \boldsymbol{ x} \) Let
Substitution of Eq. (2.140) into Eq. (2.136) yields
Equation (2.141) is just the energy conservative equation.
3. Infinitesimal translation of coordinate and generalized displacement. Let
where \( {c_i},{\varOmega_{\alpha }} \) are constants and \( \varepsilon \) is an infinitesimal parameter. Substituting Eq. (2.142) into Eq. (2.136) and noting \( {\varSigma_{{\alpha j,j}}}=0 \) we find
Equation (2.143) shows that the integral value is zero along a closed surface for the integrated function \( \left( {{g}{\delta_{ij }}-{\varSigma_{{\alpha j}}}{\psi_{{\alpha, i}}}} \right){n_j} \). For two open surfaces initiated from a same closed curve, Eq. (2.143) shows that the integral values for two open surfaces are the same. In the two-dimensional (2D) problem, it represents the path independence \( J \) integral, \( {J_i} \). Equation (2.143) can also be obtained by taking the divergence of \( {g} \). In fact using \( \nabla {g}=\left( {{{{\partial {g}}} \left/ {{\partial {x_i}}} \right.}} \right){{\boldsymbol{ e}}_i} \) and the equilibrium equation, we have
This method was adopted by many authors (Delph 1982; Pak 1992; Wang and Shen 1996).
4. Infinitesimal expansion of coordinate and generalized displacement. Let
where \( \varepsilon \) is an infinitesimal parameter. Substitution of Eq. (2.145) into (2.136) yields
In the two-dimensional problem Eq. (2.146) represents the path independence \( M \) integral
5. Infinitesimal rotation about the axis \( {x_3} \). Let
Substitution of Eq. (2.148) into Eq. (2.136) yields
6. The conservative integral in pyroelectric material. Wang and Kuang (2001) discussed the conservative integral in pyroelectric material by Noether theory and got
where \( {{\dot{\eta}}_j}={{{{q_j}}} \left/ {{{T_0}}} \right.},\vartheta =T-{T_0} \).
2.9.3 The Force Acting on a Defect in an Inhomogeneous Material
For an inhomogeneous material, \( L \) is dependent to \( \boldsymbol{ x} \), so \( L=L({x_i},{\psi_{{\alpha, j}}}),{{\bar{\partial}} \left/ {{\partial {x_i}}} \right.}\ne 0 \).
1. Infinitesimal translation of \( \boldsymbol{ x} \) and generalized displacement. \( \delta {x_i},\delta {u_{\alpha }} \) are also given in Eq. (2.142). The invariant condition under infinitesimal transformation is still \( \varepsilon {P_{ij,j }}{c_i}=0 \), but
So the integral in Eq. (2.143) in an inhomogeneous material becomes
Though Eq. (2.151) is not a conservative integral, it still has important meaning. Eshelby (1956, 1975) pointed out that \( P_{ij, j}-{{{\bar{\partial}{g}}} \left/ {{\partial {x_i}}} \right.}=0 \), so the negative derivative of the electric Gibbs function with \( \boldsymbol{ x} \), \( -{{{\bar{\partial}{g}}} \left/ {{\partial {x_i}}} \right.} \), is the so-called material inhomogeneity force with the dimension of force.
2. Infinitesimal expansion of coordinate and general displacement. \( \delta {x_i},\delta {u_{\alpha }} \) are also given in Eq. (2.145). The invariant condition under infinitesimal transformation is still \( \varepsilon {{\left[ {-\left( {{1 \left/ {2} \right.}} \right){\varSigma_{{\alpha j}}}{\psi_{\alpha }}+{P_{ij }}{x_i}} \right]}_{,j }}=0 \), but
So the integral in Eq. (2.146) in an inhomogeneous material becomes
where \( -{x_i}{{{\bar{\partial}{g}}} \left/ {{\partial {x_i}}} \right.} \) is the so-called material inhomogeneity moment.
3. Infinitesimal rotation about the axis. \( {x_3} \) \( \delta {x_i},\delta {u_{\alpha }} \) are also given in Eq. (2.148). The invariant condition under the infinitesimal transformation is still expressed by \( {{\left( {{P_{1k }}{x_2}-{P_{2k }}{x_1}+{\sigma_{1k }}{u_2}-{\sigma_{2k }}{u_1}} \right)}_{,k }}=0 \), but
2.9.4 Conservation Integral in an Inhomogeneous Material
1. Infinitesimal translation of coordinate and general displacement is related to an undetermined function. Let
where \( {c_i} \) is a constant, \( {\varOmega_{\alpha }} \) is an undetermined function, and \( \varepsilon \) is an infinitesimal parameter. Substituting Eq. (2.154) into Eq. (2.136) and noting \( {P_{ij,j }}={{({g}{\delta_{ij }}-{\varSigma_{{\alpha j}}}{\psi_{{\alpha, i}}})}_{,j }}=({{{\bar{\partial}{g}}} \left/ {{\partial {x_i}}} \right.}) \), \( {\varSigma_{{\alpha j,j}}}=0 \), we find
2. Infinitesimal translation of coordinate, infinitesimal translation, and expansion of general displacement. Let
where \( {\varOmega_{\alpha }} \) is an undetermined function and \( \varepsilon \) is an infinitesimal parameter. Substituting Eq. (2.156) into (2.136) yields
From Eqs. (2.156) and (2.157), it is known that when material constants obey definite distribution and select appropriate \( {c_i},{\varOmega_{\alpha }} \), the conservative integrals can be obtained; otherwise, the conservative integrals do not exist. In the following sections, some examples will be given to illustrate the above theory.
2.9.5 One-Directional Gradient Material
In engineering, the combined material is often used to improve the material behavior, such as on the substrate covering a surface heat-resisting layer to defense the high temperature environment. In order to reduce the stress on the interface between the substrate and heat-resisting layer, a transient layer constituted of the gradient material is added. One-directional gradient material is often used.
1. Material constants varied as exponential function. Assume material constants varied as \( C_{ijkl}^0{e^{{\lambda {x_1}}}},e_{kij}^0{e^{{\lambda {x_1}}}},\epsilon_{ij}^0{e^{{\lambda {x_1}}}} \), where \( C_{ijkl}^0,e_{kij}^0,\epsilon_{ij}^0,\lambda \) are all constants. Let
Substituting Eq. (2.158) into (2.136) yields
Because \( {\varSigma_{{\alpha j}}}{\psi_{{\alpha, j}}}=2{g} \), \( {{{\bar{\partial}{g}}} \left/ {{\partial {x_1}}} \right.}=\lambda {g} \), from Eq. (2.159), we get \( 2b+\lambda =0 \) or \( b=-{\lambda \left/ {2} \right.} \). Substituting these results into Eq. (2.158) and then into Eq. (2.159) we get
Using the relation
It is easy to get the path independence integral
2. Material constants varied as power function. Assume material constants varied as \( {C_{ijkl }}=C_{ijkl}^0{{(1+p{x_1})}^q} \), \( {e_{kij }}=e_{kij}^0{{(1+p{x_1})}^q} \), and \( {\epsilon_{ij }}=\epsilon_{ij}^0{{(1+p{x_1})}^q} \), where \( C_{ijkl}^0,e_{kij}^0,\epsilon_{ij}^0,p,q \) are constants. Let
Substitution of Eq. (2.162) into Eq. (2.136) yields
The relations between material constants are
Substitution of Eq. (2.164) into Eq. (2.163) and using (2.164), we find
Using the relation \( p{P_{ij }}{{{\partial {x_i}}} \left/ {{\partial {x_j}}} \right.}=-p{g} \) and Eq. (2.161), Eq. (2.165) is reduced to
Finally, the path independence integral is obtained:
2.9.6 Transversely Isotropic Materials
For transversely isotropic materials, the infinitesimal transformation is taken as
where ω i is undetermined function, \( {A_{{\alpha \beta }}} \) is an undetermined constant. Substitution of Eq. (2.168) into (2.136) yields
From Eq. (2.169), we obtain
where
From Eq. (2.170) we have
As an example, we discuss the transversely isotropic piezoelectric ceramic (such as PZT). Assume \( {x_3} \) is the poling direction, so \( {x_1}{x_2} \) is the isotropic plane. Now the plane \( {x_1}{x_3} \) is discussed. Appling Voigt notation, the constitutive equation is
where \( {C_{11 }}={C_{1111 }},{C_{12 }}={C_{1122 }},{C_{13 }}={C_{1133 }},{C_{33 }}={C_{3333 }},{C_{44 }}={C_{1313 }};{e_{15}}={e_{113}},{e_{31 }}={e_{311 }},{e_{33 }}={e_{333 }} \).
Though the number of the undetermined constants in Eq. (2.168) is less than the number of the equation in Eq. (2.170), the undetermined constants can still be determined by special selection of constants. Finally, we get
where \( {C_i} \) is a new arbitrary constant. When coefficients in Eq. (2.168) take values given in Eq. (2.174), we can get a group of linear partial differential equation to determine the unknown coefficients by using the invariant conditions Eq. (2.172). This group linear partial differential equation is
If Eq. (2.175) has solution, the infinitesimal transform given by Eq. (2.168) can be obtained. Substitution of Eq. (2.168) into Eq. (2.136) yields the conservative integral:
For a homogeneous material, Eq. (2.175) is reduced to linear equations and its solutions are
where \( {C_1},{C_2},{C_3},{A_{12 }}, \) and \( {A_{11 }} \) are arbitrary constants. In this case, Eq. (2.176) is reduced to
Equation (2.178) can be divided into five group independent conservative integrals due to the arbitrariness of constants. The independent conservative integrals corresponding to \( {C_1},{A_{11 }},{A_{12 }} \) are identical with Eqs. (2.143), (2.147), and (2.149). There are no new conservative integrals corresponding \( {C_2},{C_3} \) because \( {P_{2j,j }}={P_{3j,j }}=0 \) are the special cases of \( {P_{ij,j }}=0 \).
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Kuang, ZB. (2014). Physical Variational Principle and Governing Equations. In: Theory of Electroelasticity. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36291-0_2
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