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Differential derivation of momentum and energy equations in electroelasticity
acta mechanica solida sinica 30 (2017) 21–26
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journal homepage: www.elsevier.com/locate/CAMSS
Differential derivation of momentum and energy equations in electroelasticity✩ Jiashi Yang a,b,∗ a Piezoelectric
Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China b Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper presents a derivation of the equations of linear momentum, angular momentum,
Received 1 April 2016
and energy of an electroelastic body using a composite particle consisting of two differential
Revised 7 May 2016
elements based on Tiersten’s two-continuum model. The differential derivation shows the
Accepted 17 May 2016
physics involved in a way different from the integral approach in the literature. Like the
Available online 9 February 2017
integral approach, it also produces the expressions of the electric body force, couple, and power which are fundamental to the development of the nonlinear macroscopic theory of
Keywords: Nonlinear
an electroelastic body. © 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.
Electroelasticity Piezoelectricity momentum energy
1.
Introduction
A dielectric material has bound charges. The displacement of these charges is responsible for polarization. Conceivably, these charges experience forces in an electric field. The electric field also does work during the displacement of these charges. When an elastically deformable and electrically polarizable dielectric is subjected to mechanical loads and electric fields, to the lowest order of approximation, a differential element of the material polarizes into a dipole as a consequence of the displacement of the charges. As a simple charge assembly, such a dipole experiences a force and a couple in an electric field. In addition, when the material deforms and polarizes, the electric field does work to the dipole.
Fundamental to the development of the nonlinear theory of electroelasticity is the derivation of the expressions of the electric body force, couple, and power for a polarized body in an electric field. Since the force exerted by an electric field on a charge is directly given by the Maxwellian electric field, to determine the effects of an electric field on a polarized body, a more fundamental model for polarization in terms of charges is needed. This can be achieved by treating the body as charged and interacting particle assemblies, calculating and averaging the fields, force, couple, and power associated with the particle assemblies [1,2]. Tiersten [3] introduced a macroscopic physical model of two mechanically and electrically interacting and interpenetrating continua to describe an electrically polarizable elastic body. One continuum is a lattice continuum which car-
✩
This work was supported by the Y. K. Pao Visiting Professorship at Ningbo University, and the K. C. Wong Magana Fund through Ningbo University. ∗ Correspondence at: Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA. E-mail address: [email protected] http://dx.doi.org/10.1016/j.camss.2016.05.001 0894-9166/© 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.
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acta mechanica solida sinica 30 (2017) 21–26
ries positive charges. The other is an electronic charge continuum which is negatively charged. Electric polarization is modeled by the relative displacement of the electronic continuum with respect to the lattice continuum. The electric interaction between the two continua is described a local electric field. By systematic applications of the basic laws of physics, i.e., the balance of linear and angular momenta as well as energy, to each continuum separately and then combining the resulting equations, Tiersten obtained the expressions for the electric body force, couple and power [3] and constructed the theoretical framework for a nonlinear electroelastic body [3]. The two-continuum model in [3] can be varied or extended to describe more complicated behaviors of nonlinear elasticelectromagnetic interactions in matter [4–6]. Tiersten’s derivation in [3] was an integral approach in which the basic laws of physics were applied to finite bodies and then converted to differential equations. The integral approach is mathematically elegant. Alternatively, as to be shown in the present paper, the same physical laws can be applied to infinitesimal differential elements of the bodies, which offers a differential way for deriving the energy and momentum equations for an electroelastic body. The differential approach shows the physics involved in a different way from the integral approach, and provides different physical insights. The expressions of the electric body force, couple, and power automatically results from the differential derivation.
tinuum is massless and has a negative charge density 0 μe (X). Initially the body is assumed to be electrically neutral with [3] l 0 μ (X)
+ 0 μe (X) = 0.
At time t, the lattice continuum occupies a spatial region v. The current position of the material point of the lattice continuum associated with X is given by y = y(X, t),
yk = yk (X, t).
Two-continuum model
This section is a brief summary of the two-continuum model in [3]. The two-point Cartesian tensor notation, the summation convention for repeated tensor indices, and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index are used. Consider a deformable and polarizable body consisting of a lattice continuum and an electronic charge continuum (see Fig. 1). In the reference state at time t0 , the two continua coincide with each other and occupy a spatial region V. The position of a typical material point of the lattice continuum is denoted by X or XK . The lattice continuum has a mass density ρ 0 (X) and a positive charge density 0 μl (X). The electronic con-
(2)
In this state the mass density of the lattice continuum is ρ and its charge density is μl . The charge density of the electronic continuum is μe . The electronic continuum is permitted to displace with respect to the lattice continuum by an infinitesimal displacement field η(y, t) which accounts for the polarization. It is assumed that η(y, t) preserves the volume of the electronic continuum, i.e. [3], ηk,k (y ) =
∂ ηk (y ) = 0. ∂ yk
(3)
An η(y, t) satisfying (3) is sufficient for describing polarization and (3) is needed to obtain the proper electric charge equation [6]. As a consequence of (1) and (3), at time t we have the following charge neutrality condition [3] μl (y ) + μe (y + η) = 0.
2.
(1)
(4)
With η(y, t), the dipole density per unit volume or the polarization is defined by [3] P = μl (y )(−η) = μe (y + η)η.
(5)
The conservation of mass is given by [7] dρ(y ) + ρ(y )vk,k = 0, dt
(6)
where v = y˙ is the velocity field of the lattice continuum. d/dt or a superimposed dot represents the material time derivative. The two continua are electrically charged. We also have the following equations from conservation of charge for the lattice and electronic continua, respectively, dμl (y ) + μl (y )vk,k = 0, dt
(7)
dμe (y + η) + μe (y + η)(vk + ηk ),k = 0. dt
(8)
Since ηk, k = 0, from (6) and (8) we obtain the following relationship [3] which will be useful later: μ˙ e (y + η) ρ(y ˙ ) = . μe (y + η) ρ(y )
(9)
η(y, t) is assumed to be infinitesimal. For later use, we write the Maxwellian electric field E at y + η through Tayler’s expansion as [3] Fig. 1. – Motion and polarization of an electroelastic body.
E j (y + η) ∼ = E j (y ) + ηi E j,i (y ).
(10)
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3. Linear momentum equation and electric body force For the lattice continuum occupying v at time t, consider a differential element centered at y in the form of a rectangular cuboid (see Fig. 2). Its edges dy1 , dy2 , and dy3 are along the coordinate axes. Its volume is dv= dy1 dy2 dy3 . The corresponding differential element of the electronic continuum dv is centered at y + η. It may become a parallelepiped while preserving the same volume of dv = dv. In this paper, the differential elements of the lattice continuum dv and the electronic continuum dv together are viewed as a composite particle. The charges of the differential elements of the lattice and the electronic continua, μl (y)dv and μe (y + η)dv, are under the action of the Maxwellian electric field E. They also interact with each other electrically through the Coulomb force. In [3], the finite lattice and electronic continua were treated separately initially, and were combined later to obtain equations for the combined continua. The electrical interactions between the two continua were called the local electric fields in [3]. Since the electronic continuum is massless, the Maxwellian electric field and the local electric field acting on the electronic continuum balance each other. In other words, the local electric field and the Maxwellian electric field acting on the electronic continuum are equal in magnitude and opposite in direction. One is the negative vector of the other (see Eq. (3.29) of [3]). It was recently shown [8] that if the basic physical laws are directly applied to the combined finite continua in Tiersten’s two-continuum model, then there is no need to discuss the local field. Similarly, in this paper, since we treat the corresponding differential elements of the lattice and the electronic continua together as a composite particle, their electrical interactions are internal forces within the composite particle. These internal interactions are equal in magnitude and opposite in direction, and do not appear in the momentum equations of the composite particle. Therefore, in the following equation of the balance of linear momentum of the composite particle, we only need to consider contributions from the usual mechanical body force per unit mass f which acts on dv, mechanical tractions on the six faces of the rectangular cuboid of the lattice continuum [3] represented by the Cauchy stress tensor τ kl , and the Maxwellian electric fields on dv and dv :
d (v ρdv ) = ρ fk dv dt k +τ1k y + +τ2k y + +τ3k y +
1 dy1 i1 dy2 dy3 − τ1k y − 2 1 dy2 i2 dy3 dy1 − τ2k y − 2 1 dy3 i3 dy1 dy2 − τ3k y − 2
1 d y 1 i 1 dy 2 d y 3 2 1 d y 2 i 2 dy 3 d y 1 2 1 d y 3 i 3 dy 1 d y 2 2
+μl (y )Ek (y )dv + μe (y + η)Ek (y + η)dv.
(11)
The conservation of mass implies that d(ρdv)/dt = 0 [7], hence d dv k (v ρdv ) = ρdv. dt k dt
(12)
For an infinitesimal dy1 , with Taylor’s expansions at y, we have, approximately, 1 1 τ1k y + dy1 i1 dy2 dy3 − τ1k y − dy1 i1 dy2 dy3 2 2 1 1 ∼ = τ1k (y ) + τ1k,1 (y ) dy1 dy2 dy3 − τ1k (y ) − τ1k,1 (y ) dy1 dy2 dy3 2 2 = τ1k,1 (y )dv,
(13)
and, similarly, 1 1 τ2k y + dy2 i2 dy3 dy1 − τ2k y − dy2 i2 dy3 dy1 ∼ = τ2k,2 (y )dv, 2 2 (14) 1 1 τ3k y + dy3 i3 dy1 dy2 − τ3k y − dy3 i3 dy1 dy2 ∼ = τ3k,3 (y )dv. 2 2 (15) For small η, with the use of (10), (4) and (5), those terms in (11) that are related to the Maxwellian electric field can be written as μl (y )Ek (y )dv + μe (y + η)Ek (y + η)dv ∼ = μl (y )Ek (y )dv + μe (y + η)[Ek (y ) + ηi Ek,i (y )]dv = [μl (y ) + μe (y + η)]Ek (y )dv + μe (y + η)ηi Ek,i (y )dv = 0 + Pi Ek,i dv = FkE dv,
(16)
where we have introduced the electric body force vector as [3] FkE = Pi Ek,i ,
F E = P · ∇E.
(17)
We note that FE is nonlinear in the electric and polarization fields and thus disappears in the linear theory of piezoelectricity. The substitution of (12)–(16) back into (11) gives the linear momentum equation in the following form [3]: ρ
dv k = τik,i + ρ fk + FkE . dt
(18)
The above derivation shows the physical interpretation of τ kl clearly. We note that the body force FE can be expressed by the Maxwell stress tensor TE [9] through Fig. 2. – A composite particle consisting of dv and dv .
TiEj,i = FjE .
(19)
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acta mechanica solida sinica 30 (2017) 21–26
In (19), only the divergence of TiEj is defined. Therefore TiEj is not unique. A common expression for
TiEj
satisfying (19) is [9]
1 TiEj = Pi E j + ε0 Ei E j − Ek Ek δi j . 2
(20)
With TiEj , (18) takes the following form: ρ
dv k = (τik + TikE ),i + ρ fk . dt
(21)
While FE describes a real physical effect of the electric field on a charged or polarized material, the introduction of TiEj is not a necessity. However, it is convenient to have TiEj when treating electric jump conditions at an interface [3].
1 1 εi jk y j + dy2 δ2 j τ2k y + dy2 i2 dy3 dy1 2 2 1 1 −εi jk y j − dy2 δ2 j τ2k y − dy2 i2 dy3 dy1 2 2 ∼ = εi jk y j τ2k,2 (y )dv + εi jk δ2 j τ2k (y )dv,
(25)
1 1 εi jk y j + dy3 δ3 j τ3k y + dy3 i3 dy1 dy2 2 2 1 1 −εi jk y j − dy3 δ3 j τ3k y − dy3 i3 dy1 dy2 2 2 ∼ εi jk y j τ3k,3 (y )dv + εi jk δ3 j τ3k (y )dv, =
(26)
εi jk y j μl (y )Ek (y )dv + εi jk (y j + η j )μe (y + η)Ek (y + η)dv ∼ = εi jk y j μl (y )Ek (y )dv + εi jk (y j + η j )μe (y + η)[Ek (y ) + Ek,m (y )ηm ]dv ∼ = εi jk y j μl (y )Ek (y )dv + εi jk μe (y + η)[y j Ek (y ) + y j Ek,m (y )ηm +η j Ek (y )]dv
4. Angular momentum equation and electric body couple For the composite particle in Fig. 2 consisting of the lattice continuum dv at y and the electronic continuum of dv at y + η, the angular momentum equation about the origin of the reference frame is d (ε y v ρdv ) = εi jk y j ρ fk dv dt i jk j k 1 +εi jk y j + dy1 δ1 j τ1k y + 2 1 −εi jk y j − dy1 δ1 j τ1k y − 2 1 +εi jk y j + dy2 δ2 j τ2k y + 2 1 −εi jk y j − dy2 δ2 j τ2k y − 2 1 +εi jk y j + dy3 δ3 j τ3k y + 2 1 −εi jk y j − dy3 δ3 j τ3k y − 2
1 d y 1 i 1 dy 2 d y 3 2 1 d y 1 i 1 dy 2 d y 3 2 1 d y 2 i 2 dy 3 d y 1 2 1 d y 2 i 2 dy 3 d y 1 2 1 d y 3 i 3 dy 1 d y 2 2 1 d y 3 i 3 dy 1 d y 2 2
+εi jk y j μl (y )Ek (y )dv + εi jk (y j + η j )μe (y + η)Ek (y + η)dv,
+εi jk μe (y + η)y j Ek,m (y )ηm dv + εi jk μe (y + η)η j Ek (y )dv = 0 + εi jk y j Ek,m (y )Pm dv + εi jk Pj Ek (y )dv = εi jk y j FkE dv + CiE dv,
(27)
where (4), (5) and (17) have been used, and we have introduced the electric body couple CE by [3] CiE = εi jk Pj Ek (y ),
CE = P × E(y ).
(28)
Obviously, CE is nonlinear and is irrelevant to the linear theory of piezoelectricity. The substitution of (23)-(27) into (22) yields εi jk y j
dv k ρ = εi jk y j ρ fk +εi jk y j τmk,m (y )+εi jk δm j τmk (y )+εi jk y j FkE +CiE , dt (29)
or (22)
where the moment of inertia of ρdv about its own mass center has been neglected as a higher-order infinitesimal. For the terms in (22), from the left-hand side to the right-hand side, we have d d (ε y v ρdv ) = εi jk (y j vk )ρdv dt i jk j k dt dv dv = εi jk v j vk + y j k ρdv = εi jk y j k ρdv, (23) dt dt 1 1 εi jk y j + dy1 δ1 j τ1k y + dy1 i1 dy2 dy3 2 2 1 1 −εi jk y j − dy1 δ1 j τ1k y − dy1 i1 dy2 dy3 2 2 1 1 ∼ = εi jk y j + dy1 δ1 j τ1k (y ) + τ1k,1 (y ) dy1 dy2 dy3 2 2 1 1 −εi jk y j − dy1 δ1 j τ1k (y ) − τ1k,1 (y ) dy1 dy2 dy3 2 2 ∼ ε y τ (y ) d v + ε δ τ (y ) d v, = i jk j 1k,1 i jk 1 j 1k
= εi jk y j [μl (y ) + μe (y + η)]Ek (y )dv
(24)
dv εi jk y j ρ k − τmk,m − ρ fk − FkE = εi jk τ jk (y ) + CiE . dt
(30)
The left-hand side of (30) vanishes because of the linear momentum equation in (18). Then (30) reduces to [3] εi jk τ jk (y ) + CiE = 0,
(31)
or εi jk (τ jk + Pj Ek ) = 0.
(32)
An important implication of (31) is that the Cauchy stress tensor τ kl is asymmetric. Also note that in the above derivation the electric body couple CE results from the derivation automatically. This is fundamentally different from [3] in which CE and its expression were artificially inserted. Alternatively, the angular momentum equation can be written with respect to the center of mass of the composite particle which is the same as the center of mass of the differential element of the lattice continuum ρdv because the electronic continuum is massless. In this case the moment of
acta mechanica solida sinica 30 (2017) 21–26
inertia of ρdv about its own mass center is a higher-order infinitesimal and can be neglected. In addition, for a differential element the body force and the electric force on the lattice continuum essentially go through the mass center of ρdv and do not contribute to the moment about its mass center. Hence the angular momentum equation is simply 1 1 εi jk dy1 δ1 j τ1k y + dy1 i1 dy2 dy3 2 2 1 1 −εi jk − dy1 δ1 j τ1k y − dy1 i1 dy2 dy3 2 2 1 1 +εi jk dy2 δ2 j τ2k y + dy2 i2 dy3 dy1 2 2 1 1 −εi jk − dy2 δ2 j τ2k y − dy2 i2 dy3 dy1 2 2 1 1 +εi jk dy3 δ3 j τ3k y + dy3 i3 dy1 dy2 2 2 1 1 −εi jk − dy3 δ3 j τ3k y − dy3 i3 dy1 dy2 2 2 +εi jk η j μe (y + η)Ek (y + η)dv = 0.
(33)
With the use of (5), (10) and the Taylor’s expansions of the stress components, we can write (33) as 1 1 εi jk dy1 δ1 j τ1k (y )dy2 dy3 − εi jk − dy1 δ1 j τ1k (y )dy2 dy3 2 2 1 1 +εi jk dy2 δ2 j τ2k (y )dy3 dy1 − εi jk − dy2 δ2 j τ2k (y )dy3 dy1 2 2 1 1 +εi jk dy3 δ3 j τ3k (y )dy1 dy2 − εi jk − dy3 δ3 j τ3k (y )dy1 dy2 2 2 +εi jk η j μe (y + η)Ek (y )dv = εi jk δ1 j τ1k dv + εi jk δ2 j τ2k dv + εi jk δ3 j τ3k dv + εi jk Pj Ek dv = εi jk δm j τmk dv + εi jk Pj Ek dv = εi jk (τ jk + Pj Ek )dv = 0,
(34)
which is exactly (32). It is worth mentioning that in the derivation of the angular momentum equation about the center of mass of ρdv, the linear momentum equation in (18) was not directly used. This is different from the derivation of (31).
25
For the terms in (35), from the left-hand side to the righthand side, we have d 1 d 1 vk vk ρdv + ερdv = ρdv vk vk + ε dt 2 dt 2 = vk
dv k dε ρdv + ρdv, dt dt
(36)
1 1 τ1k y + dy1 i1 vk y + dy1 i1 dy2 dy3 2 2 1 1 −τ1k y − dy1 i1 vk y − dy1 i1 dy2 dy3 2 2 d y dy 1 1 ∼ vk (y ) + vk,1 (y ) d y 2 dy 3 = τ1k (y ) + τ1k,1 (y ) 2 2 dy 1 dy 1 vk (y ) − vk,1 (y ) dy2 dy3 − τ1k (y ) − τ1k,1 (y ) 2 2 ∼ = τ1k (y )vk,1 (y )dv + τ1k,1 (y )vk (y )dv,
(37)
1 1 τ2k y + dy2 i2 vk y + dy2 i2 dy3 dy1 2 2 1 1 −τ2k y − dy2 i2 vk y − dy2 i2 dy3 dy1 2 2 ∼ τ2k (y )vk,2 (y )dv + τ2k,2 (y )vk (y )dv, =
(38)
1 1 τ3k y + dy3 i3 vk y + dy3 i3 dy1 dy2 2 2 1 1 −τ3k y − dy3 i3 vk y − dy3 i3 dy1 dy2 2 2 ∼ = τ3k (y )vk,3 (y )dv + τ3k,3 (y )vk (y )dv
(39)
μl (y )Ek (y )vk (y )dv + μe (y + η)Ek (y + η)[vk (y ) + η˙ k ]dv ∼ = μl (y )Ek (y )vk (y )dv + μe (y + η) Ek (y ) + Ek,i (y )ηi [vk (y ) + η˙ k ]dv ∼ = μl (y )Ek (y )vk (y )dv + μe (y + η)[Ek (y )vk (y ) + Ek,i (y )ηi vk (y ) +Ek (y )η˙ k ]dv = μl (y ) + μe (y + η) Ek (y )vk (y )dv +μe (y + η)Ek,i (y )ηi vk (y )dv + μe (y + η)Ek (y )η˙ k dv = 0 + Pi Ek,i (y )vk (y )dv + μe (y + η)Ek (y )η˙ k dv
5.
Energy equation and electric body power
Let the internal energy density per unit mass of the lattice continuum be ε. For the composite particle in Fig. 2, the energy equation can be written as d 1 vk vk ρdv + ερdv = ρ fk vk dv dt 2 1 1 +τ1k y + dy1 i1 vk y + dy1 i1 dy2 dy3 2 2 1 1 −τ1k y − dy1 i1 vk y − dy1 i1 dy2 dy3 2 2 1 1 +τ2k y + dy2 i2 vk y + dy2 i2 dy3 dy1 2 2 1 1 −τ2k y − dy2 i2 vk y − dy2 i2 dy3 dy1 2 2 1 1 +τ3k y + dy3 i3 vk y + dy3 i3 dy1 dy2 2 2 1 1 −τ3k y − dy3 i3 vk y − dy3 i3 dy1 dy2 2 2 +μl (y )Ek (y )vk (y )dv + μe (y + η)Ek (y + η)[vk (y ) + η˙ k ]dv.
(35)
= FkE vk (y )dv + wE dv,
(40)
where (4), (5), (10) and (17) have been used. In (40), we have introduced the electric body power wE by [3] wE = μe (y + η)Ek (y )η˙ k
d e dμe (y + η) = Ek (y ) ηk [μ (y + η)ηk ] − dt dt μ˙ e (y + η) e = Ek (y ) P˙k − μ˙ (y + η)ηk = Ek P˙k − μe (y + η) e η E (y ) μ (y + η) k k = Ek P˙k −
μ˙ e (y + η) ρ˙ P E = Ek P˙k − Pk Ek , μe (y + η) k k ρ
(41)
where (5) and (9) have been used. With the introduction of the polarization per unit mass π i by [3] πi =
Pi , ρ
(42)
(41) becomes [3] ρ˙ PE ρ j j = (ρπ ˙ j + ρ π˙ j )E j − ρπ ˙ j E j = ρ π˙ j E j .
wE = P˙ j E j −
(43)
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The substitution of (36)–(40) into (35) gives dv dε ρ k − τmk,m − ρ fk − FkE vk dv + ρ dv = τmk vk,m dv + wE dv dt dt (44)
The left-hand side of (44) vanishes because of the linear momentum equation in (18). Then the energy equation in (44) becomes [3] ρ
dε = τmk vk,m + wE . dt
(45)
The constitutive relations of electroelastic materials follow (45) in the usual manner [3].
6.
Conclusion
The equations of liner momentum, angular momentum, and energy of an electroelastic body can be systematically obtained by applying the basic laws of physics to a composite particle based on Tiersten’s two-continuum model. The expressions of electric body force, couple and power automatically result from the derivation. The differential derivation exhibits the physics involved in a different way from the integral derivation. It shows the effect of the Cauchy stress tensor directly. The electric body force and couple are nonlinear and disappear in the linear theory of piezoelectricity. Because of the electric body couple, the Cauchy stress tensor is asym-
metric. The local electric field represents the electric interaction between the differential element of the lattice continuum and the corresponding differential element of the electronic continuum.
references
[1] S.R. De Groot, L.G. Suttorp, Foundations of Electrodynamics, Noord-Hollandsche U.M., Amsterdam, 1972. [2] A.C. Eringen, G.A. Maugin, Electrodynamics of Continua, Springer-Verlag, New York, 1990. [3] H.F. Tiersten, On the nonlinear equations of thermoelectroelasticity, Int. J. Eng. Sci. 9 (1971) 587–604. [4] H.F. Tiersten, Coupled magnetomechanical equation for magnetically saturated insulators, J. Math. Phys. 5 (1964) 1298–1318. [5] H.F. Tiersten, C.F. Tsai, On the interaction of the electromagnetic field with heat conducting deformable insulators, J. Math. Phys. 13 (1972) 361–378. [6] H.G. De Lorenzi, H.F. Tiersten, On the interaction of the electromagnetic field with heat conducting deformable semiconductors, J. Math. Phys. 16 (1975) 938–957. [7] A.C. Eringen, Mechanics of Continua, Robert E. Krieger Publishing Company, Huntington, New York, 1980. [8] J.S. Yang, On the derivation of electric body force, couple and power in an electroelastic body, Acta Mech. Solida Sin. 28 (2015) 613–617. [9] J.C. Baumhauer, H.F. Tiersten, Nonlinear electroelastic equations for small fields superposed on a bias, J. Acoust. Soc. Am. 54 (1973) 1017–1034.
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journal homepage: www.elsevier.com/locate/CAMSS
Differential derivation of momentum and energy equations in electroelasticity✩ Jiashi Yang a,b,∗ a Piezoelectric
Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China b Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
This paper presents a derivation of the equations of linear momentum, angular momentum,
Received 1 April 2016
and energy of an electroelastic body using a composite particle consisting of two differential
Revised 7 May 2016
elements based on Tiersten’s two-continuum model. The differential derivation shows the
Accepted 17 May 2016
physics involved in a way different from the integral approach in the literature. Like the
Available online 9 February 2017
integral approach, it also produces the expressions of the electric body force, couple, and power which are fundamental to the development of the nonlinear macroscopic theory of
Keywords: Nonlinear
an electroelastic body. © 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.
Electroelasticity Piezoelectricity momentum energy
1.
Introduction
A dielectric material has bound charges. The displacement of these charges is responsible for polarization. Conceivably, these charges experience forces in an electric field. The electric field also does work during the displacement of these charges. When an elastically deformable and electrically polarizable dielectric is subjected to mechanical loads and electric fields, to the lowest order of approximation, a differential element of the material polarizes into a dipole as a consequence of the displacement of the charges. As a simple charge assembly, such a dipole experiences a force and a couple in an electric field. In addition, when the material deforms and polarizes, the electric field does work to the dipole.
Fundamental to the development of the nonlinear theory of electroelasticity is the derivation of the expressions of the electric body force, couple, and power for a polarized body in an electric field. Since the force exerted by an electric field on a charge is directly given by the Maxwellian electric field, to determine the effects of an electric field on a polarized body, a more fundamental model for polarization in terms of charges is needed. This can be achieved by treating the body as charged and interacting particle assemblies, calculating and averaging the fields, force, couple, and power associated with the particle assemblies [1,2]. Tiersten [3] introduced a macroscopic physical model of two mechanically and electrically interacting and interpenetrating continua to describe an electrically polarizable elastic body. One continuum is a lattice continuum which car-
✩
This work was supported by the Y. K. Pao Visiting Professorship at Ningbo University, and the K. C. Wong Magana Fund through Ningbo University. ∗ Correspondence at: Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA. E-mail address: [email protected] http://dx.doi.org/10.1016/j.camss.2016.05.001 0894-9166/© 2017 Published by Elsevier Ltd on behalf of Chinese Society of Theoretical and Applied Mechanics.
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ries positive charges. The other is an electronic charge continuum which is negatively charged. Electric polarization is modeled by the relative displacement of the electronic continuum with respect to the lattice continuum. The electric interaction between the two continua is described a local electric field. By systematic applications of the basic laws of physics, i.e., the balance of linear and angular momenta as well as energy, to each continuum separately and then combining the resulting equations, Tiersten obtained the expressions for the electric body force, couple and power [3] and constructed the theoretical framework for a nonlinear electroelastic body [3]. The two-continuum model in [3] can be varied or extended to describe more complicated behaviors of nonlinear elasticelectromagnetic interactions in matter [4–6]. Tiersten’s derivation in [3] was an integral approach in which the basic laws of physics were applied to finite bodies and then converted to differential equations. The integral approach is mathematically elegant. Alternatively, as to be shown in the present paper, the same physical laws can be applied to infinitesimal differential elements of the bodies, which offers a differential way for deriving the energy and momentum equations for an electroelastic body. The differential approach shows the physics involved in a different way from the integral approach, and provides different physical insights. The expressions of the electric body force, couple, and power automatically results from the differential derivation.
tinuum is massless and has a negative charge density 0 μe (X). Initially the body is assumed to be electrically neutral with [3] l 0 μ (X)
+ 0 μe (X) = 0.
At time t, the lattice continuum occupies a spatial region v. The current position of the material point of the lattice continuum associated with X is given by y = y(X, t),
yk = yk (X, t).
Two-continuum model
This section is a brief summary of the two-continuum model in [3]. The two-point Cartesian tensor notation, the summation convention for repeated tensor indices, and the convention that a comma followed by an index denotes partial differentiation with respect to the coordinate associated with the index are used. Consider a deformable and polarizable body consisting of a lattice continuum and an electronic charge continuum (see Fig. 1). In the reference state at time t0 , the two continua coincide with each other and occupy a spatial region V. The position of a typical material point of the lattice continuum is denoted by X or XK . The lattice continuum has a mass density ρ 0 (X) and a positive charge density 0 μl (X). The electronic con-
(2)
In this state the mass density of the lattice continuum is ρ and its charge density is μl . The charge density of the electronic continuum is μe . The electronic continuum is permitted to displace with respect to the lattice continuum by an infinitesimal displacement field η(y, t) which accounts for the polarization. It is assumed that η(y, t) preserves the volume of the electronic continuum, i.e. [3], ηk,k (y ) =
∂ ηk (y ) = 0. ∂ yk
(3)
An η(y, t) satisfying (3) is sufficient for describing polarization and (3) is needed to obtain the proper electric charge equation [6]. As a consequence of (1) and (3), at time t we have the following charge neutrality condition [3] μl (y ) + μe (y + η) = 0.
2.
(1)
(4)
With η(y, t), the dipole density per unit volume or the polarization is defined by [3] P = μl (y )(−η) = μe (y + η)η.
(5)
The conservation of mass is given by [7] dρ(y ) + ρ(y )vk,k = 0, dt
(6)
where v = y˙ is the velocity field of the lattice continuum. d/dt or a superimposed dot represents the material time derivative. The two continua are electrically charged. We also have the following equations from conservation of charge for the lattice and electronic continua, respectively, dμl (y ) + μl (y )vk,k = 0, dt
(7)
dμe (y + η) + μe (y + η)(vk + ηk ),k = 0. dt
(8)
Since ηk, k = 0, from (6) and (8) we obtain the following relationship [3] which will be useful later: μ˙ e (y + η) ρ(y ˙ ) = . μe (y + η) ρ(y )
(9)
η(y, t) is assumed to be infinitesimal. For later use, we write the Maxwellian electric field E at y + η through Tayler’s expansion as [3] Fig. 1. – Motion and polarization of an electroelastic body.
E j (y + η) ∼ = E j (y ) + ηi E j,i (y ).
(10)
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3. Linear momentum equation and electric body force For the lattice continuum occupying v at time t, consider a differential element centered at y in the form of a rectangular cuboid (see Fig. 2). Its edges dy1 , dy2 , and dy3 are along the coordinate axes. Its volume is dv= dy1 dy2 dy3 . The corresponding differential element of the electronic continuum dv is centered at y + η. It may become a parallelepiped while preserving the same volume of dv = dv. In this paper, the differential elements of the lattice continuum dv and the electronic continuum dv together are viewed as a composite particle. The charges of the differential elements of the lattice and the electronic continua, μl (y)dv and μe (y + η)dv, are under the action of the Maxwellian electric field E. They also interact with each other electrically through the Coulomb force. In [3], the finite lattice and electronic continua were treated separately initially, and were combined later to obtain equations for the combined continua. The electrical interactions between the two continua were called the local electric fields in [3]. Since the electronic continuum is massless, the Maxwellian electric field and the local electric field acting on the electronic continuum balance each other. In other words, the local electric field and the Maxwellian electric field acting on the electronic continuum are equal in magnitude and opposite in direction. One is the negative vector of the other (see Eq. (3.29) of [3]). It was recently shown [8] that if the basic physical laws are directly applied to the combined finite continua in Tiersten’s two-continuum model, then there is no need to discuss the local field. Similarly, in this paper, since we treat the corresponding differential elements of the lattice and the electronic continua together as a composite particle, their electrical interactions are internal forces within the composite particle. These internal interactions are equal in magnitude and opposite in direction, and do not appear in the momentum equations of the composite particle. Therefore, in the following equation of the balance of linear momentum of the composite particle, we only need to consider contributions from the usual mechanical body force per unit mass f which acts on dv, mechanical tractions on the six faces of the rectangular cuboid of the lattice continuum [3] represented by the Cauchy stress tensor τ kl , and the Maxwellian electric fields on dv and dv :
d (v ρdv ) = ρ fk dv dt k +τ1k y + +τ2k y + +τ3k y +
1 dy1 i1 dy2 dy3 − τ1k y − 2 1 dy2 i2 dy3 dy1 − τ2k y − 2 1 dy3 i3 dy1 dy2 − τ3k y − 2
1 d y 1 i 1 dy 2 d y 3 2 1 d y 2 i 2 dy 3 d y 1 2 1 d y 3 i 3 dy 1 d y 2 2
+μl (y )Ek (y )dv + μe (y + η)Ek (y + η)dv.
(11)
The conservation of mass implies that d(ρdv)/dt = 0 [7], hence d dv k (v ρdv ) = ρdv. dt k dt
(12)
For an infinitesimal dy1 , with Taylor’s expansions at y, we have, approximately, 1 1 τ1k y + dy1 i1 dy2 dy3 − τ1k y − dy1 i1 dy2 dy3 2 2 1 1 ∼ = τ1k (y ) + τ1k,1 (y ) dy1 dy2 dy3 − τ1k (y ) − τ1k,1 (y ) dy1 dy2 dy3 2 2 = τ1k,1 (y )dv,
(13)
and, similarly, 1 1 τ2k y + dy2 i2 dy3 dy1 − τ2k y − dy2 i2 dy3 dy1 ∼ = τ2k,2 (y )dv, 2 2 (14) 1 1 τ3k y + dy3 i3 dy1 dy2 − τ3k y − dy3 i3 dy1 dy2 ∼ = τ3k,3 (y )dv. 2 2 (15) For small η, with the use of (10), (4) and (5), those terms in (11) that are related to the Maxwellian electric field can be written as μl (y )Ek (y )dv + μe (y + η)Ek (y + η)dv ∼ = μl (y )Ek (y )dv + μe (y + η)[Ek (y ) + ηi Ek,i (y )]dv = [μl (y ) + μe (y + η)]Ek (y )dv + μe (y + η)ηi Ek,i (y )dv = 0 + Pi Ek,i dv = FkE dv,
(16)
where we have introduced the electric body force vector as [3] FkE = Pi Ek,i ,
F E = P · ∇E.
(17)
We note that FE is nonlinear in the electric and polarization fields and thus disappears in the linear theory of piezoelectricity. The substitution of (12)–(16) back into (11) gives the linear momentum equation in the following form [3]: ρ
dv k = τik,i + ρ fk + FkE . dt
(18)
The above derivation shows the physical interpretation of τ kl clearly. We note that the body force FE can be expressed by the Maxwell stress tensor TE [9] through Fig. 2. – A composite particle consisting of dv and dv .
TiEj,i = FjE .
(19)
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In (19), only the divergence of TiEj is defined. Therefore TiEj is not unique. A common expression for
TiEj
satisfying (19) is [9]
1 TiEj = Pi E j + ε0 Ei E j − Ek Ek δi j . 2
(20)
With TiEj , (18) takes the following form: ρ
dv k = (τik + TikE ),i + ρ fk . dt
(21)
While FE describes a real physical effect of the electric field on a charged or polarized material, the introduction of TiEj is not a necessity. However, it is convenient to have TiEj when treating electric jump conditions at an interface [3].
1 1 εi jk y j + dy2 δ2 j τ2k y + dy2 i2 dy3 dy1 2 2 1 1 −εi jk y j − dy2 δ2 j τ2k y − dy2 i2 dy3 dy1 2 2 ∼ = εi jk y j τ2k,2 (y )dv + εi jk δ2 j τ2k (y )dv,
(25)
1 1 εi jk y j + dy3 δ3 j τ3k y + dy3 i3 dy1 dy2 2 2 1 1 −εi jk y j − dy3 δ3 j τ3k y − dy3 i3 dy1 dy2 2 2 ∼ εi jk y j τ3k,3 (y )dv + εi jk δ3 j τ3k (y )dv, =
(26)
εi jk y j μl (y )Ek (y )dv + εi jk (y j + η j )μe (y + η)Ek (y + η)dv ∼ = εi jk y j μl (y )Ek (y )dv + εi jk (y j + η j )μe (y + η)[Ek (y ) + Ek,m (y )ηm ]dv ∼ = εi jk y j μl (y )Ek (y )dv + εi jk μe (y + η)[y j Ek (y ) + y j Ek,m (y )ηm +η j Ek (y )]dv
4. Angular momentum equation and electric body couple For the composite particle in Fig. 2 consisting of the lattice continuum dv at y and the electronic continuum of dv at y + η, the angular momentum equation about the origin of the reference frame is d (ε y v ρdv ) = εi jk y j ρ fk dv dt i jk j k 1 +εi jk y j + dy1 δ1 j τ1k y + 2 1 −εi jk y j − dy1 δ1 j τ1k y − 2 1 +εi jk y j + dy2 δ2 j τ2k y + 2 1 −εi jk y j − dy2 δ2 j τ2k y − 2 1 +εi jk y j + dy3 δ3 j τ3k y + 2 1 −εi jk y j − dy3 δ3 j τ3k y − 2
1 d y 1 i 1 dy 2 d y 3 2 1 d y 1 i 1 dy 2 d y 3 2 1 d y 2 i 2 dy 3 d y 1 2 1 d y 2 i 2 dy 3 d y 1 2 1 d y 3 i 3 dy 1 d y 2 2 1 d y 3 i 3 dy 1 d y 2 2
+εi jk y j μl (y )Ek (y )dv + εi jk (y j + η j )μe (y + η)Ek (y + η)dv,
+εi jk μe (y + η)y j Ek,m (y )ηm dv + εi jk μe (y + η)η j Ek (y )dv = 0 + εi jk y j Ek,m (y )Pm dv + εi jk Pj Ek (y )dv = εi jk y j FkE dv + CiE dv,
(27)
where (4), (5) and (17) have been used, and we have introduced the electric body couple CE by [3] CiE = εi jk Pj Ek (y ),
CE = P × E(y ).
(28)
Obviously, CE is nonlinear and is irrelevant to the linear theory of piezoelectricity. The substitution of (23)-(27) into (22) yields εi jk y j
dv k ρ = εi jk y j ρ fk +εi jk y j τmk,m (y )+εi jk δm j τmk (y )+εi jk y j FkE +CiE , dt (29)
or (22)
where the moment of inertia of ρdv about its own mass center has been neglected as a higher-order infinitesimal. For the terms in (22), from the left-hand side to the right-hand side, we have d d (ε y v ρdv ) = εi jk (y j vk )ρdv dt i jk j k dt dv dv = εi jk v j vk + y j k ρdv = εi jk y j k ρdv, (23) dt dt 1 1 εi jk y j + dy1 δ1 j τ1k y + dy1 i1 dy2 dy3 2 2 1 1 −εi jk y j − dy1 δ1 j τ1k y − dy1 i1 dy2 dy3 2 2 1 1 ∼ = εi jk y j + dy1 δ1 j τ1k (y ) + τ1k,1 (y ) dy1 dy2 dy3 2 2 1 1 −εi jk y j − dy1 δ1 j τ1k (y ) − τ1k,1 (y ) dy1 dy2 dy3 2 2 ∼ ε y τ (y ) d v + ε δ τ (y ) d v, = i jk j 1k,1 i jk 1 j 1k
= εi jk y j [μl (y ) + μe (y + η)]Ek (y )dv
(24)
dv εi jk y j ρ k − τmk,m − ρ fk − FkE = εi jk τ jk (y ) + CiE . dt
(30)
The left-hand side of (30) vanishes because of the linear momentum equation in (18). Then (30) reduces to [3] εi jk τ jk (y ) + CiE = 0,
(31)
or εi jk (τ jk + Pj Ek ) = 0.
(32)
An important implication of (31) is that the Cauchy stress tensor τ kl is asymmetric. Also note that in the above derivation the electric body couple CE results from the derivation automatically. This is fundamentally different from [3] in which CE and its expression were artificially inserted. Alternatively, the angular momentum equation can be written with respect to the center of mass of the composite particle which is the same as the center of mass of the differential element of the lattice continuum ρdv because the electronic continuum is massless. In this case the moment of
acta mechanica solida sinica 30 (2017) 21–26
inertia of ρdv about its own mass center is a higher-order infinitesimal and can be neglected. In addition, for a differential element the body force and the electric force on the lattice continuum essentially go through the mass center of ρdv and do not contribute to the moment about its mass center. Hence the angular momentum equation is simply 1 1 εi jk dy1 δ1 j τ1k y + dy1 i1 dy2 dy3 2 2 1 1 −εi jk − dy1 δ1 j τ1k y − dy1 i1 dy2 dy3 2 2 1 1 +εi jk dy2 δ2 j τ2k y + dy2 i2 dy3 dy1 2 2 1 1 −εi jk − dy2 δ2 j τ2k y − dy2 i2 dy3 dy1 2 2 1 1 +εi jk dy3 δ3 j τ3k y + dy3 i3 dy1 dy2 2 2 1 1 −εi jk − dy3 δ3 j τ3k y − dy3 i3 dy1 dy2 2 2 +εi jk η j μe (y + η)Ek (y + η)dv = 0.
(33)
With the use of (5), (10) and the Taylor’s expansions of the stress components, we can write (33) as 1 1 εi jk dy1 δ1 j τ1k (y )dy2 dy3 − εi jk − dy1 δ1 j τ1k (y )dy2 dy3 2 2 1 1 +εi jk dy2 δ2 j τ2k (y )dy3 dy1 − εi jk − dy2 δ2 j τ2k (y )dy3 dy1 2 2 1 1 +εi jk dy3 δ3 j τ3k (y )dy1 dy2 − εi jk − dy3 δ3 j τ3k (y )dy1 dy2 2 2 +εi jk η j μe (y + η)Ek (y )dv = εi jk δ1 j τ1k dv + εi jk δ2 j τ2k dv + εi jk δ3 j τ3k dv + εi jk Pj Ek dv = εi jk δm j τmk dv + εi jk Pj Ek dv = εi jk (τ jk + Pj Ek )dv = 0,
(34)
which is exactly (32). It is worth mentioning that in the derivation of the angular momentum equation about the center of mass of ρdv, the linear momentum equation in (18) was not directly used. This is different from the derivation of (31).
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For the terms in (35), from the left-hand side to the righthand side, we have d 1 d 1 vk vk ρdv + ερdv = ρdv vk vk + ε dt 2 dt 2 = vk
dv k dε ρdv + ρdv, dt dt
(36)
1 1 τ1k y + dy1 i1 vk y + dy1 i1 dy2 dy3 2 2 1 1 −τ1k y − dy1 i1 vk y − dy1 i1 dy2 dy3 2 2 d y dy 1 1 ∼ vk (y ) + vk,1 (y ) d y 2 dy 3 = τ1k (y ) + τ1k,1 (y ) 2 2 dy 1 dy 1 vk (y ) − vk,1 (y ) dy2 dy3 − τ1k (y ) − τ1k,1 (y ) 2 2 ∼ = τ1k (y )vk,1 (y )dv + τ1k,1 (y )vk (y )dv,
(37)
1 1 τ2k y + dy2 i2 vk y + dy2 i2 dy3 dy1 2 2 1 1 −τ2k y − dy2 i2 vk y − dy2 i2 dy3 dy1 2 2 ∼ τ2k (y )vk,2 (y )dv + τ2k,2 (y )vk (y )dv, =
(38)
1 1 τ3k y + dy3 i3 vk y + dy3 i3 dy1 dy2 2 2 1 1 −τ3k y − dy3 i3 vk y − dy3 i3 dy1 dy2 2 2 ∼ = τ3k (y )vk,3 (y )dv + τ3k,3 (y )vk (y )dv
(39)
μl (y )Ek (y )vk (y )dv + μe (y + η)Ek (y + η)[vk (y ) + η˙ k ]dv ∼ = μl (y )Ek (y )vk (y )dv + μe (y + η) Ek (y ) + Ek,i (y )ηi [vk (y ) + η˙ k ]dv ∼ = μl (y )Ek (y )vk (y )dv + μe (y + η)[Ek (y )vk (y ) + Ek,i (y )ηi vk (y ) +Ek (y )η˙ k ]dv = μl (y ) + μe (y + η) Ek (y )vk (y )dv +μe (y + η)Ek,i (y )ηi vk (y )dv + μe (y + η)Ek (y )η˙ k dv = 0 + Pi Ek,i (y )vk (y )dv + μe (y + η)Ek (y )η˙ k dv
5.
Energy equation and electric body power
Let the internal energy density per unit mass of the lattice continuum be ε. For the composite particle in Fig. 2, the energy equation can be written as d 1 vk vk ρdv + ερdv = ρ fk vk dv dt 2 1 1 +τ1k y + dy1 i1 vk y + dy1 i1 dy2 dy3 2 2 1 1 −τ1k y − dy1 i1 vk y − dy1 i1 dy2 dy3 2 2 1 1 +τ2k y + dy2 i2 vk y + dy2 i2 dy3 dy1 2 2 1 1 −τ2k y − dy2 i2 vk y − dy2 i2 dy3 dy1 2 2 1 1 +τ3k y + dy3 i3 vk y + dy3 i3 dy1 dy2 2 2 1 1 −τ3k y − dy3 i3 vk y − dy3 i3 dy1 dy2 2 2 +μl (y )Ek (y )vk (y )dv + μe (y + η)Ek (y + η)[vk (y ) + η˙ k ]dv.
(35)
= FkE vk (y )dv + wE dv,
(40)
where (4), (5), (10) and (17) have been used. In (40), we have introduced the electric body power wE by [3] wE = μe (y + η)Ek (y )η˙ k
d e dμe (y + η) = Ek (y ) ηk [μ (y + η)ηk ] − dt dt μ˙ e (y + η) e = Ek (y ) P˙k − μ˙ (y + η)ηk = Ek P˙k − μe (y + η) e η E (y ) μ (y + η) k k = Ek P˙k −
μ˙ e (y + η) ρ˙ P E = Ek P˙k − Pk Ek , μe (y + η) k k ρ
(41)
where (5) and (9) have been used. With the introduction of the polarization per unit mass π i by [3] πi =
Pi , ρ
(42)
(41) becomes [3] ρ˙ PE ρ j j = (ρπ ˙ j + ρ π˙ j )E j − ρπ ˙ j E j = ρ π˙ j E j .
wE = P˙ j E j −
(43)
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The substitution of (36)–(40) into (35) gives dv dε ρ k − τmk,m − ρ fk − FkE vk dv + ρ dv = τmk vk,m dv + wE dv dt dt (44)
The left-hand side of (44) vanishes because of the linear momentum equation in (18). Then the energy equation in (44) becomes [3] ρ
dε = τmk vk,m + wE . dt
(45)
The constitutive relations of electroelastic materials follow (45) in the usual manner [3].
6.
Conclusion
The equations of liner momentum, angular momentum, and energy of an electroelastic body can be systematically obtained by applying the basic laws of physics to a composite particle based on Tiersten’s two-continuum model. The expressions of electric body force, couple and power automatically result from the derivation. The differential derivation exhibits the physics involved in a different way from the integral derivation. It shows the effect of the Cauchy stress tensor directly. The electric body force and couple are nonlinear and disappear in the linear theory of piezoelectricity. Because of the electric body couple, the Cauchy stress tensor is asym-
metric. The local electric field represents the electric interaction between the differential element of the lattice continuum and the corresponding differential element of the electronic continuum.
references
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