On the Coupling Effects between Elastic and Electromagnetic Fields from the Perspective of Conservation of Energy

In a natural system, coupling effects among different physical fields substantially reflect the conversion of energy from one form to another. According to the law of conservation of energy (LCE), the loss of energy in one field must equal to the gain of energy in another field. In this paper, this LCE is applied to analyze the reversible processes coupled between elastic and electromagnetic fields. Here, it is called the energy formulation. For simple physical processes such as mechanical movement, diffusion and electrodynamic process, it is shown their governing or constitutive equations all satisfy the LCE. Then, analysis is extended to coupling effects. First, it is found for the linear direct and converse piezoelectric and piezomagnetic effects, their constitutive equations guarantee energy is conserved during the conversion of energies. Second, analyses found for the generalized Villari effects, the electromagnetic energy can be treated as an extra term in the generalized elastic energy. Third, for electrostriction and magnetostriction. It is argued both effects are induced by the Maxwell stress. Their energy is purely electromagnetic, thus both have no converse effects. During these processes, energy can be converted in three ways, i.e., via nonpotential forces, cross dependence of energy terms and directly via the interaction of ions and electrons. In the end, general coupling processes which involve elastic, electromagnetic fields and diffusion are also analyzed. The energy formulation, when combined with the phase-field variational approach, has the potential of being developed into a general approach to analyze coupling effects between reversible and irreversible processes. The advantage of the energy formulation is that it facilitates the discussion of the conversion of energies and provides more physical insights into their mechanisms.


Introduction
The study of coupling effects between electromagnetic and elastic fields dates back to more than a century. Since then, numerous work has been accomplished in these fields. Fundamentals and reviews of the past work can be found in literatures such as these [1,2,3,4] . Here, only a brief and incomprehensive introduction is provided. The first set of such coupling effects are piezoelectric and piezomagnetic effects. Piezoelectricity was reported by the Curie brothers in literatures at 1881 [5] . The direct (converse) piezoelectric effect refers to the phenomenon that in certain substances externally applied stresses (electric fields) result in a change of polarization (arising of strains). It was latter found these effects only exist in materials with microstructures having no inversion of symmetry. Piezoelectric effects were initially discovered in crystals such as tourmaline and quartz, and latter in ceramics like lead zirconate titanate (LZT). Nowadays, they are also found to exist in polymers and even biological materials such as tendons and bones. Due to persistent efforts in searching for better materials and in understanding the effects both microscopically and macroscopically, it eventually led a wide application of these effects in industrial manufacturing, medical instruments and telecommunication devices. Piezomagnetism is the magnetomechanical analogue of piezoelectricity, i.e., externally applied stresses (magnetic fields) results in a change of magnetization (arising of strains). The microstructure of piezomagnetic materials must also have no inversion of symmetry. In 1960, this effect was experimentally observed in antiferromagnetic fluorides of cobalt and manganese [6] . The second set of these coupling effects are electrostriction and magnetostriction. Both of them refer to dimensional change of materials under the influence of externally applied electric or magnetic field. Magnetostriction was reported by Joule in iron in 1842 [7] . There are two specific effects of magnetostriction called the Matteucci effect [8] and the Wiedemann effect [9] . The former refers to the arising of a helical anisotropy of the susceptibility in magnetic materials subjected to a torque; while the latter the arising of a torque within the bulk of materials while subjecting to a helical magnetic field. In addition, there is another effect called the Villari effect, or the inverse magnetostrictive effect. It refers to the phenomenon that an externally applied stress induces a change in the magnetic susceptibility of a magnetic material [10] . The most commonly used electrostrictive materials are lead magnesium niobate (LMN) and its derivatives.
While Tefelon-D and Metglas 2605SC are the most commonly used magnetostrictive materials. The former are widely used in sonars and actuators while the latter in both sonars and sensors. Moreover, it is worth noting that these two sets of coupling effects exhibits very different behaviors. For the first (second) set, strains induced by externally applied fields are linear (quadratic) and thus they are (not) dependent on the field direction.
As shown above, the coupling of the electromagnetic and elastic fields leads to a variety of effects.
The underlying mechanisms of these effects are both important and interesting. Many models and theoretical approaches were developed over the past century to study them both microscopically and macroscopically. To be very brief here also, the microscopic understanding mainly focuses on the relations between the crystal structures and properties of the materials, while the macroscopic understanding mainly uses phenomenological models to obtain the constitutive equations of these coupling effects. One major theoretical approach at the macroscopic level is the Lagrangian formulation. The Lagrangian formulation was initially developed in analytical mechanics and latter extended to continuum mechanics to derive the equations of motion. In electrodynamics, Lagrangian formulation can also be used to derive governing equations for the scalar and vector potentials for the electric and magnetic fields. These equations are in fact equivalent to the set of Maxwell equations. Using the Lagrangian formulation, governing equations are derived to ensure that the first variation of the integral of Lagrangian, which is defined to be the difference between the kinetic and potential energies of the system, over a time interval is zero. A comprehensive treatment of the electro-mechanical effects using the Lagrangian formulation with nonlinear finite deformation can be found in literatures such as here [3] . A detailed theory of piezoelectricity and its application in piezoelectric devices can be found in literatures such as here [1,2] . However, given the fact that the coupling effects between electromagnetic and elastic fields essentially arises from the energy conversion between the electromagnetic fields and the elastic field, it appears that one more important issue is still left unaddressed. That is, how is energy converted between these two fields and how can this process be formulated?
In this paper, the fundamental law of conservation of energy is applied to analyze these coupling effects. For brevity, this approach is hereafter called the energy formulation. It is shown this energy formulation enables the governing equations of these coupling effects to strictly satisfy the law of conservation of energy. The purpose of this paper is two-fold: first, using the law of conservation of energy to analyze these coupling effects and thus, to contribute to a straightforward understanding of the conversion of energies during these effects and to provide more physical insights into their mechanisms; second, using these examples to show that the energy formulation can be developed into a general approach to analyze the coupling effects among reversible and irreversible processes.
Organization of this paper is as follows. In the second section, a general discussion on the relations between the governing equations and the law of conservation of energy for simple processes is provided. In the third and fourth sections, the piezoelectric and piezomagnetic effects, as well as the Villari effect and its analogue in the electric field are analyzed, respectively. In the fifth section, electrostriction and magnetostriction are studied. In the sixth section, general coupling processes involving elastic, electromagnetic fields and diffusion are analyzed. In the seventh section, some comments on the energy conversion is provided and a summary is given in the end.

Governing Equations and the Law of Conservation of Energy
In this section, the connection between the governing equations and the law of conservation of energy for uncoupled and simple processes is discussed. It is shown that not only in classic mechanics but also in thermodynamics and electrodynamics, the governing or constitutive equations must satisfy the law of conservation of energy.
For discussions in simple processes, to begin with, the equation of motion in classic mechanics is discussed. In a natural system with the kinetic energy T (q) being a homogenous quadratic function of the generalized velocityq, the Lagrangian is defined to be is the potential energy and the generalized mass m ij is a symmetric tensor. According to analytical mechanics, in this system, Lagrange's equations or the equations of motion, d dt ( ∂L ∂q k − ∂L ∂q k ) = 0, lead to the law of conservation of energy directly. A brief outline of the proof [11] is presented below.
As a result, the equations of motion satisfy the law of conservation of energy in a natural system. This in fact provide an alternative approach to determine the equations of motion for a mechanical system. That is, when the total energy E of a mechanical system is determined, the time derivative of E gives the equations of motion directly. In the above derivation, following the steps backwards, the time derivative of the total energy can be written as a linear combination of the generalized velocities as shown in Eqn (1). Then the coefficients associated with eachq k must be zero, which give the equations of motion, because the seriesq k are linearly independent.
In analytical mechanics, the Lagrangian formulation is a standard approach to determine the equations of motion in a mechanical system. But there is one major disadvantage with this formulation. The equations of motion are derived to minimize a scalar integral of the Lagrangian over a time interval. Thus, they do not offer any physical insights into the issue of conversion of energies. In the following discussions, it is straightforward to show that the energy formulation can conveniently achieve this purpose. In Eqn (1), the time derivative of E gives where in the last step the equations of motion are found by setting the coefficients in the square bracket to be zero. Note that, in the first term of the second step, m ijqi is the inertia force and a positive product of this inertia force with the generalized velocityq j (which indicates an increase of the kinetic energy T ) gives the rate of conversion of the potential energy V into T ; in the second term, − ∂V ∂q j is the potential force and a positive product of this potential force with the generalized velocityq j (which indicates a positive work and a decrease in the potential energy V ) also gives the rate of conversion of the potential energy V into T . These two terms must cancel each other, which leads to the law of conservation of energy. It is evident that, with the energy formulation, the conversion of energies can be discussed more straightforwardly.
The energy formulation can also be extended to continuum mechanics to give the Cauchy's equation of motion. In a continuum media, the total energy can be defined as V [ 1 2 ρv i v i + u(e ij )]dv where ρ is the mass density, v i is the velocity and u(e ij ) is the elastic energy associated with the infinitesimal elastic strain e ij . Then, where the elastic stress σ ij = ∂u ∂e ij . Thus, to guarantee that the energy is conserved in a continuum media with a stress-free boundary, the terms in the parenthesis of the second volume integral at the last step must be zero, which leads to Cauchy's equation of motion directly. Here, the advantage of the energy formulation is also evident when it comes to the discussion of conversion of energies using the terms at the last step. The surface integral indicates the work done on the continuum media via surface traction t i = σ ij n j and the surface density of work being t i v i . Within the volume integral, a positive product of the inertia force ρ ∂v i ∂t and the potential force σ ij,j with the velocity v i both indicate the rate of conversion of the elastic potential energy u el into the kinetic energy and thus, they are canceled.
In the above discussions, it is shown that in both classic and continuum mechanics, the law of conservation of energy can be used to determine the equation of motion. Compared with the Lagrangian formulation, the energy formulation directly facilitate the discussion of conversion of energies. In the following discussions, the energy formulation is extended to study the governing equations in both thermodynamics and electrodynamics.
The governing equations in thermodynamics are diffusion-type equations. Recently, the phasefield variational approach (PFVA) has become a popular tool to construct governing equations which abide by the corollary of the second law of thermodynamics, i.e., the total free energy is non-increasing during the evolution of a system [12,13] . A brief outline of this approach is presented here. Suppose in a thermodynamic system, the chemical free energy density is f (c) and the chemical potential is µ = ∂f ∂c . Then, where the law of conservation of mass is applied in the second step. Then to guarantee in an insulated system the total free energy is non-increasing as time evolves, i.e., the satisfaction of the corollary of the second law of thermodynamics, the compositional flux J c i needs be defined as J c i = −m∂ i µ, where m is the mobility of diffusion. Considering that during the thermodynamic evolution, the chemical free energy is usually dissipated as heat, then a slight modification in the above derivation with the introduction of a dissipation function can guarantee the first law of thermodynamics is also satisfied. Let the rate of change of the free energy dissipation function be 1 m J c i J c i , which is in analogue with that considered in classic mechanics [11] , then the variation of the total energy is Thus in an insulated system, to guarantee the satisfaction of the law of conservation of energy, the constitutive flux equation is found to be J c i = −m∂ i µ. As a result, the energy formulation combined with PFVA can be used to determine the equation of flux, which satisfy the law of conservation of energy, for a simple and uncoupled thermodynamic process.
In electrodynamics, the governing equations are the set of Maxwell equations: where field; ξ ijk is the permutation symbol; J e i is the electric current density. Among them, Eqns (6) and (8) are time-independent, i.e., static equations; while Eqns (7) and (9) are time-dependent, i.e., dynamic equations. Here, The law of conservation of energy in electrodynamics is called the Poynting's theorem. A brief outline [14] is presented below. Beginning with the work done by the electric field J e i E i , At the first step, both the dynamics equations are substituted. Here, on the right-hand side, the first term indicates the work done by the electric field; the second term, analogously, can be considered as the work done by the magnetic field. Since in the expression of the Lorentz force, the magnetic force in fact does no work, then this term is zero. Note, here the magnetic force is the potential force resulting from the vector potential of the magnetic field. The nonpotential damping force during the magnetization process is not considered here. Note that at the third step and hereafter, the permittivity ǫ ij and permeability µ ij are assumed to be diagonal tensors, so that U em can be explicitly defined. At the last step, the physical content of Poynting's theorem states the energy transferred into the boundary of a continuum media by the Poynting vector (usually from the electric power source) equals the sum of the increase in the electromagnetic energy and the work done by the electric field (which is usually dissipated into heat or converted into other energies via electronic devices) within the bulk of the material. Shown by the above derivation, it is evident that the two dynamic equations of Maxwell equations must satisfy the law of conservation of energy.
The Lagrangian formulation has already been extended to study electrodynamics and the Maxwell equations can be derived from it. However, the Lagrangian formulation also does not provide any physical insight into the conversion of energies here. Furthermore, the form of the Lagrangian is also a bit puzzling since the electrical and magnetic potentials are associated with different signs.
Compared with the law of conservation of energy in the above two cases, Poynting's theorem is of a very different form. However, if starting with the time derivative of u em and the divergence of the Poynting vector as shown in the third step of Eqn (10), then the two dynamic equations can still be found by identifying the electric and magnetic work as shown in the first step of Eqn (10).
That is, in electrodynamics, the energy formulation can also be used to determine the two dynamic equations of the Maxwell equations via Poynting's theorem.
In the above discussions, it is argued that the governing equations of uncoupled and simple processes in mechanics, thermodynamics and electrodynamics can be determined using the law of conservation of energy, i.e., the energy formulations. However, most natural processes are coupled and complex processes, which usually involves two or even more physical fields. During these couple processes, variation of one field usually leads to variation of another field simultaneously and thus, results in conversion of energies from one field to another. In our previous work, it was shown that for coupled irreversible thermodynamic processes, the energy is in fact conserved during the process of conversion of energies using the flux equations found via PFVA. In the following sections, using concrete examples such as piezoelectric and piezomagnetic effects, further discussions are provided for coupled reversible processes between elastic and electromagnetic fields. It is shown that governing equations found with the energy formulation strictly satisfy the law of conservation of energy. Thus, discussion of conversion of energies during these couple processes becomes straightforward.

Linear Piezoelectric and Piezomagnetic Effects
For the direct piezoelectric and piezomagnetic effects which behave linearly, their constitutive equations are where β ijk (γ ijk ) is the piezoelectric (piezomagnetic) coefficient and e jk is the infinitesimal strain tensor. For the converse effect, the constitutive equation is where C jklm is the stiffness tensor.
In the following analyzes, it is shown that, beginning with the constitutive equations for the direct effects, the law of conservation of energy leads to the constitutive equations for the converse effects directly and furthermore, discussion on the conversion of energies becomes straightforward.
Note that, again, ǫ ij and µ ij are assumed to be diagonal tensors. Beginning with the work done by the electric field, where at the second step, the constitutive equations of the direct piezoelectric and peizomagnetic effects are substituted. At the last step, it is straightforward to see that the last two terms representing the conversion of energies due to the coupling effects. Considering the converse effects during which the energy, transferred by the Poynting's vector from the power source, is converted into the elastic energy u el . Then the rate of the increase in u el is E i β ijk ∂ t e jk + H i γ ijk ∂ t e jk according to Eqn (16). Thus, in this case, the conservation of the total mechanic energy is where the total kinetic energy K = V 1 2 ρv j v j dv and the total elastic energy U el = V u el dv. Then, Thus, the equation of motion, taking into account the coupling effects, is and the generalized stress is found to be which is exactly the constitutive equation (14) for the converse effects. That is, the constitutive equations for the direct and converse effects guarantees the satisfaction of the law of conservation of energy.
The terms indicating the conversion of energies between the elastic and electromagnetic fields which consists of a bulk contribution and a surface contribution. Assuming a homogeneous distribution of the electromagnetic fields within the bulk of the devices, the surface contribution becomes dominant. Then, the above equation can be used to obtain the rate of conversion of energies for a certain device with a specific shape.

The generalized Villari Effects
In this section, the Villari effect and its analogue in electric field are analyzed. It is reasonable to assume that there is an analogue in the electric field, i.e., the permittivity tensors also vary under the influence of an externally applied stresses. These effects are hereafter called the generalized Villari effects. To facilitate the discussion, here the permittivity and permeability tensors (note: already assumed diagonal) are treated as functions of strains, i.e., ǫ ij (e kl ) and µ ij (e kl ). Then according to Poynting's Theorem, Evidently, the last term indicates the energy conversion between the elastic and electromagnetic fields.
Analogous to the analyzes in the above section, to guarantee the conservation of the total mechanic energy, then Thus, the generalized stress is found to be As a result, the constitutive equation (25) guarantees the satisfaction of the law of conservation of energy. Furthermore, since externally applied stresses lead to variation of the electromagnetic energy, thus the generalized Villari effects can be applied in sensors. In addition, from the above analysis, the electromagnetic energy can be treated as an extra contribution in the generalized elastic energy. In previous work [15] , there were similar approaches of using the elastic energy as an extra contribution to the generalized chemical energy while studying diffusion under the influence of elastic fields. The generalized Villari effects in fact exemplify the conversion of energies owing to a cross-dependence in the energy terms. However, in previous section, the conversion of energies for piezoelectric and piezomagnetic effects is different. In Eqn (16), due to the lack of terms, Moreover, these terms introduce a dependence of energy terms on the directions of physical fields, which is also not very reasonable. Thus, it is argued here, for piezoelectric and piezomagnetic effects, there are no specific energies associated with these two processes. Note that, in classic and fluid mechanics, not all processes are associated with specific energies, e.g., the processes due to friction and viscosity which dissipate the kinetic energy of mechanic movements into heat. These forces, usually called nonpotential forces (i.e., they can not be derived from potential functions), are associated with work which convert energies from one form into another. As a result, it is argued here, the variation of energies owing to the piezoelectric and piezomagnetic effects are in fact work terms, (β ijk E i + γ ijk H i ) ,k v j , as shown in Eqn (22), with their nonpotential body forces being (β ijk E i ) ,k and (γ ijk H i ) ,k . Note that, these forces are not proportional to velocities as in the case of the viscous force, because they are responsible for conversion of energies rather than dissipation of energies into heat. In brief, the piezoelectric and piezomagnetic effects in fact exemplify the conversion of energies owing to the work done by nonpotential forces. Moreover, the generalized Villari effects in fact arise from the dependence of the permittivity and permeability on the elastic stresses or strains. Since the Mauttecci effect refers to the arising of a helical anisotropy of the susceptibility of materials under the influence of a torque, then it should be classified as a specific example of the generalized Villari effects.

Electrostriction and Magnetostriction
It is known electrostriction and magnetostriction are induced by polarization and magnetization of the materials. Both of them are quadratic effects with constitutive equations being where Q e ijkl and Q m ijkl are electrostriction and magnetostriction coefficients, P k and M k are polarization and magnetization vectors.
Among the mechanisms of polarization in dielectric materials, it is believed the ionic and electronic polarization contribute mostly to electrostriction [4] . That is, under the influence of an externally applied electric field, electrostriction mainly arises from the displacements of ions away from their equilibrium positions at the lattice and the distortion of the electronic distribution around ions. While for magnetostriction, boundary movements of the magnetic domains are believed to be the major cause [16] . Under the influence of an externally applied magnetic field H i , domains with magnetic moments aligned with H i will grow. That is, magnetic moments around the domain boundary will rotate to the direction of H i and thus help the domain boundaries move forward.
Such a process also simultaneously result in a dimensional change for the materials. During these processes, there are increases in energy and the energy increased in fact comes from the electric power source. According to the conservation of energy in electrodynamics as shown by Eqn (11), the energy is in fact transferred into the bulk of the materials via the Poynting vector. Note that, the transfer of energy is always companied by the transfer of momentum simultaneously. Momentum is also needed during the processes of polarization and magnetization, e.g., when an ion is pushed out of its equilibrium position during the ionic polarization and when the magnetic moments start to rotate to help the domain grow. Furthermore, at the macroscopic level, materials under the influence of alternative fields usually vibrates, which indicates a gain of mechanic momentum. Then, where does these momentum (which is substantially the cause of electrostriction and magnetostriction) gained at both the microscopic and macroscopic levels come from? Evidently, it would be very beneficial if the law of conservation of momentum in electrodynamics is examined.
Here, electrostriction is mainly analyzed while magnetostriction merely treated as an analogue.
where the integrand at the right-hand side is the Lorentz force. In the following discussions, for simplicity, the permittivity and permeability are assumed constant, i.e. ǫ and µ. Substitution of Eqns (6) and (9) into the above equation and further mathematical manipulations lead to the law of conservation of momentum in electrodynamics [14] , where the field momentum P F i = ǫµξ ijk E j H k and T ij , the Maxwell stress, is Evidently, as shown by Eqn (29), both the mechanical momentum P M i and the field momentum P F i come from the momentum transferred by the Maxwell Stress. Note that P F i is the momentum associated with the electromagnetic fields, while P M i is the momentum obtained by ions and electrons during polarization and magnetization. As argued above, it is this mechanical momentum P M i that causes dimensional changes, i.e., electrostriction and magnetostriction. Since this P M i is transferred by the Maxwell stress T ij , it is reasonable to argue that both electrostriction and magnetostriction are induced by T ij . To be more specific, they are in fact induced by a portion of T ij .
First, assume the space considered is vacuum, then the mechanical momentum is zero since there are no particles; furthermore, the Maxwell stress acting on the surface, T 0 ij (with ǫ and µ in Eqn (30) replaced by ǫ 0 and µ 0 in the vacuum), then is totally responsible for the transfer of the field momentum ǫ 0 µ 0 ξ ijk E j H k . Suppose now the space is occupied by materials. Then the variation in P F i is (ǫµ − ǫ 0 µ 0 )ξ ijk E j H k which can be assumed minor for typical dielectric and paramagnetic substances. However, the mechanical momentum changes from 0 to P M i . Then the difference between T ij and T 0 ij can be argued to be roughly the portion of stress being responsible of the transfer of P M i . Using ǫ = ǫ 0 (1 + χ e ); µ = µ 0 (1 + χ m ) and P i = ǫ 0 χ e E i ; M i = µ 0 χ m H i , then this portion of stress is Since T m ij induces both electrostriction and magnetostriction, then the corresponding strain is found to be Here, strains are assumed to be infinitesimal strains. Compared with Eqns (26) and (27), the electrostriction and magnetostriction coefficients are found to be which qualitatively explains the resemblance of Q e ijkl to S ijkl in a cubic crystal system [4] . The Wiedemann effect found in 1858 is one specific example of magnetostriction. During the experiment, a ferromagnetic rod is placed within a solenoid. Then, a longitudinal magnetic field is induced by electric currents passing through the solenoid, and simultaneously, a circular magnetic field by the electric currents passing through the rod. As a result, a torsion of the rod can be observed in this helical magnetic field. Previous analysis found that the angle of torsion is proportional to the longitudinal magnetic field and the magnitude of the current density passing through the rod, and inversely proportional to the shear modulus G [17] . Assume the rod is elastically isotropic.
Then, using Eqn (33), it is straight forward to show that the angle of torsion, e zφ = In addition, from the analyzed above, the energy associated with electrostriction and magnetostriction can be assumed to be 1 2 T m ij e kl = 1 2 S ijkl T m ij T m kl . Here, the value of T m ij is of the order of magnitude of the electromagnetic energy u em stored within the media. Since the strain resulted from electrostriction and magnetostriction is usually around 0.1%, then the kinetic energy obtained by mechanical vibrations is only roughly one thousands of u em . Furthermore, according to Poynting's theorem, this energy is also transferred into the bulk of materials via the Poynting vector.
Then when electrostriction and magnetostriction are considered, Poynting's theorem becomes For simplicity, Eqn (31) is used for the substitution of T m kl . Mathematical manipulation, straightforward however tedious, found that the second and fourth equations of the Maxwell equations become Permutation of indices 1 → 2 → 3 leads to the rest two sets of equations. Detailed derivations can be found in the Appendix. It is evident these equations are highly nonlinear equations. Also note that, during the derivations above, the strains induced by electrostriction and magnetostriction are assumed to be infinitesimal strains. Thus Eqns (38) and (39) are only suitable for metallic or ceramic strictive materials. Nowadays, it is found that electrostriction and magnetostriction can induce definite and even large strains in polymer strictive materials. Then, the above analyzes must be combined with the theory of large deformations to tackle this problem.
Furthermore, note that the energy, 1 2 S ijkl T m ij T m kl , associated with electrostriction and magnetostriction, in fact depends on the electromagnetic fields only. Thus, it is a pure electromagnetic energy. However, it is this pure electromagnetic energy results in strains which are pure mechanical responses. In this case, energy associated with electromagnetic fields is directly converted into elastic energy, which is very different from the previously mentioned two cases (conversion of energies via the nonpotential forces or the cross-dependence of energy terms). When electromagnetic fields are applied, the charged particles within the substance are subject to the Lorentz force immediately.
This leads to a distribution of the Maxwell Stress across the lattice of the substance and induces strains there, i.e., infinitesimal displacements of the charged particles away from their equilibrium positions. As a result, momentum is directly transferred by the Maxwell stress to the substance lattice. Simultaneously, the energy transferred from the power source by the Poynting vector is directly converted into elastic energy stored by the lattice. However, in this case, the energy associated with electrostriction and magnetostriction has no dependence on the elastic fields. Furthermore, since there is no inverse mechanisms, by which pure mechanic movements can result in a distribution of the Maxwell stress and induce the Lorentz force at the absence of electromagnetic fields, then there are no converse effects for electrostriction and magnetostriction discussed above. Note that, here electrostriction and magnetostriction only refer to the effects caused by the Maxwell stress, which do not include the generalized Villari effects.
In our previous work [18] , it is assumed that the Maxwell stresses and strains can be linearly superposed with the elastic stresses and strains. The tensor product between the Maxwell stresses and the elastic strains, which include compositional strains as well, is the energy giving rise to electromigration. Here, it is argued the the tensor product, or a portion of the product, between the Maxwell stress and strain is the energy giving rise to electrostriction and magnetostriction.
In this section, both laws of conservation of momentum and energy are used to analyzes electrostriction and magnetostriction. It is argued that they are both induced by the Maxwell stress.
The two dynamic equations of the Maxwell equations are found using Poynting's theorem with the energy associate with electrostriction and magnetostriction being 1 2 S ijkl T M ij T M kl . In the following section, general coupling processes, which involves not only electromagnetic and elastic fields, but also a compositional field, are analyzed.

Formulation of General Coupling Processes in Electronic Devices
Most physical processes usually involve several physical fields. The evolution of them leads to dissipation and conversion of energies simultaneously. For example, in certain electronic devices which use functionally gradient materials, electromagnetic, elastic and compositional fields evolve simultaneously. Thus, it is beneficial to analyze such general coupling processes here.
In our previous work, the diffusion process under the influence of elastic field is analyzed [19] .
The analysis there is extended to include the electromagnetic energy with permittivity and permeability being both dependent on the compositional field and strains, i.e., u em = 1 2 ǫ ij (c, e lm )E i E j + 1 2 µ ij (c, e lm )H i H j . Consider a binary system with c denotes the composition of one species. Then, in this system, the coupling processes cause the variation of following energies: the kinetic energy k = 1 2 ρv l v l , the elastic energy of u el (c, e lm ), the chemical free energy of diffusion f (c, e lm ) and the electromagnetic energy u em . In addition, a dissipation function 1 m J c i J c i is also needed for the diffusion process. Here, the law of conservation of energy is applied for each process respectively. For the mechanical movements, we have As a result, the equation of motion and the generalized stress tensor are found to be For the diffusion process, we have Thus, the generalized flux equation is found to be Then, substitution of the above equation into the law of conservation of mass leads to the governing equation of the diffusion process. For the electromagnetic fields, we have As a result, the two dynamic equations of the Maxwell equations are found by setting the terms in the square brackets in the above equation equal to zero, and they are where D i = ǫ ij E j + β ilm e lm and B i = µ ij H j + γ ilm e lm , and extra terms are present taking into account the generalized Villari effects and the dependence of both ǫ and µ on the composition c.
Furthermore, the following terms in Eqn (44) can be rewritten as Note that at the last step of the above equation, the two terms on the right-hand side are shown in Eqns (40) and (42) respectively. This guarantees that energy is conserved during the conversion of energies between the electromagnetic fields with the elastic and compositional fields, respectively. In previous work [19] , it has already been shown energy is conserved during the conversion of energies between the elastic and compositional fields and thus it is omitted here.
In this section, general coupling processes, which are usually present in electronic devices and involve three physical fields, are discussed. With the energy formulation, governing equations for these fields are found. More importantly, it is shown these equations guarantee that energy is conserved during the process of energy conversions. This is no doubt an evident advantage of the energy formulation.

Some Comments on Energy Conversions
Most natural processes are in fact processes coupled among several physical fields. Thus, energy conversion is very common during these processes. According to the analyses above, energy can be converted via the following three different ways, i.e., work done by the nonpotential forces, cross dependence of energy terms and direct conversion as shown by electrostriction and magnetostriction.
Compared with the first two ways, direct conversion is rare but it in fact reflects the substantial

Summary
Mechanical movements, diffusion and electrodynamic process are all natural processes. For these simple, i.e., uncoupled processes, it is shown their governing or constitutive equations all satisfy the law of conservation of energy. Processes coupled among them substantially reflect the conversion of energy from one form into another. According to the law of conservation of energy, the loss of energy in one field must equal the gain of energy in another. This principle is used to analyze the reversible processes coupled between elastic and electromagnetic fields. It is found for the direct and converse piezoelectric and piezomagnetic effects which behave linearly, their constitutive equations guarantee that energy is conserved during the conversion of energies. Furthermore, it is argued that rather than associate with any specific energy terms, these processes are in fact associated with the work done by nonpotential forces. For the generalized Villari effects, using the law of conservation of energy, it is found while deriving the equation of motion for the elastic fields, the electromagnetic energy can be treated as a extra term in the generalize elastic energy. Thus, in this case, energy is converted via the cross dependence of the energy terms. While for electrostriction and magnetostriction, both the laws of conservation of momentum and energy are used to analyze them. It is found that they are induced by the Maxwell stress and their energy is purely electromagnetic. Since there are no inverse mechanism that pure mechanical movements can give rise to a distribution of the Maxwell stress across the lattice of substance, then it is argued that there are no converse effects for both of them.
General coupling processes involving elastic, electromagnetic fields and diffusion which are usually present in electronic devices are also analyzed. Both the equation of motion and the governing equation for diffusion are determined. Using Poynting's theorem, the two dynamic equations of the Maxwell equations are found with extra terms taking into account of the generalized Villari effects and the compositional dependence of permittivity and permeability. It is also shown these extra terms guarantee that energy is conserved during the energy conversion between the electromagnetic fields with the elastic and compositional fields.
Though the Lagrangian formulation is a traditional approach to determine the constitutive equations and construct governing equations for coupling processes. However, this formulation does not offer any physical insights into the conversion of energies during these processes. In this paper, the law of conservation of energy is used to construct governing equations directly for reversible processes coupled between electromagnetic fields and elastic fields. It begins with the law of conservation of energy in one field and thus, the terms responsible for the conversion of energy can be identified. Then, they are substituted into the the law of conservation of energy in another field to help determine its governing or constitutive equation. It is shown with this energy formulation, not only the conservation of energy is guaranteed for these coupling processes, but also the discussion of conversion of energies becomes straightforward. Furthermore, it also contribute to a better understanding of the underlying physical mechanisms of these processes. In a previous paper, PFVA has been used to study coupled irreversible processes and energy is also found to be conserved during the process of energy conversions [19] . It is argued here this energy formulation when combined with PFVA can be extended to study most natural reversible and irreversible physical processes, which probably includes chemical reactions, phase transitions and so on. For coupled natural processes, if the law of conservation of energy associated with each simple process can be found and furthermore, the energy or work terms responsible for energy conversions can be identified, then governing equations can be constructed to strictly satisfy the law of conservation of energy and the second law of thermodynamics using this energy formulation.
Thus, Then the extra terms in Eqns (38) and (39) can be found and permutation of indices leads to the rest two sets of equations.