Conservation laws in linear piezoelectricity

Engineering Fracture Mechanics Vol. 51, No. 6. pp. 1041 1047, 1995

Pergamon

0013-7944(94)00271-1

Copyright '~i 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0013-7944/95 $9.50+ 0.00

C O N S E R V A T I O N LAWS IN L I N E A R PIEZOELECTRICITY J. S. Y A N G ? and R. C. BATRA:~ ?Department of Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, N Y 12180, U.S.A. ++Department of Engineering Science & Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, U.S.A. Abstract--We use Noether's theorem on variational principles invariant under a group of infinitesimal transformations to obtain a class of conservation laws for linear piezoelectric materials and linear elastic dielectrics.

1. I N T R O D U C T I O N CONSERVATION laws and related path-independent integrals are important in studying forces acting on a defect [1]. An important tool in deriving the conservation laws is Noether's theorem [2] on variational principles invariant under a group of infinitesimal transformations. Knowles and Sternberg [3] and Fletcher [4] have used this theorem to derive conservation laws in elastostatics and linear elastodynamics respectively. Li [5] used Noether's theorem in conjunction with the variational principle of complementary energy to obtain another set of conservation laws for linear elastostatics which he called the dual conservation laws. The previously derived conservation laws for piezoelectricity and the more general theory of dielectrics[6-9] are for static deformations of the body. Here we use Noether's theorem in conjunction with four variational formulations of quasistatic piezoelectricity [10] to obtain a class of conservation laws. The quasistatic piezoelectricity theory is defined as one in which inertia forces associated with mechanical deformation are considered but the time-derivative of the magnetic induction in Faraday's law of induction is neglected. These conservation laws represent extensions of Fletcher's and Li's results to quasistatic piezoelectricity. We note that Knowles and Sternberg [3] and Fletcher [4] prove the completeness of conservation laws, but we do not. 2. NOETHER'S THEOREM We state a version of Noether's theorem appropriate for our work. Consider the following functional or action integral

~(y)- fv~:(x,y,~) dV

(1)

where y~, ~ = l, 2 . . . . . N, are functions of xi, i = 1, 2 . . . . . M, and the integration is over the M-dimensional domain V. The Euler-Lagrange equations associated with the functional 7t are

~y~

Oxi\Oy~.~/= 0; y~,~=- ~xi"

(2)

Here and below a repeated index implies summation over the range of the index. Also consider an infinitesimal parametric transformation X: = xi + fiK(X, y ) E K + O(/7),

y~ =y~ + g,x(x, y)/7~+ 0(/7),

(3)

where/7,~, K = 1, 2 . . . . . P are the infinitesimal parameters of the transformation, and

E = (/TK,/Tx)I/2, lira 0(/7__)= 0. (~0 I041

/7

(4)

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J . S . YANG and R. C. BATRA

If under the transformation (3) the functional (I) is invariant, or more specifically,

;v

(5)

Y,-~-~x,)dV'-fvE(X,Y,~x)dV=o(~),

and the Euler-Lagrange eq. (2) is satisfied, then Noether's theorem [2] states that

OF,

~v. g,x] = O.

(6)

Equation (6) is in a zero divergence form and is called a conservation law in physical problems. These can be transformed into path-independent integrals by using the divergence theorem.

3. G O V E R N I N G

EQUATIONS

AND ENERGY DENSITY FUNCTIONS QUASISTATIC PIEZOELECTRICITY

FOR

Throughout this paper we use rectangular Cartesian coordinates. For a piezoelectric material, the internal energy density U is a function of the infinitesimal strain tensor E~j and the electric displacement D~. The stress tensor a~j appropriate for small deformations and the electric field Ei are related to U by

3U

ao = ~ ,

c~U Ee = OD---~"

(7)

Whereas in linear elasticity we have two energy density functions, viz, the strain energy density and the complementary energy density, in linear piezoelectricity, the following three additional energy density functions can be introduced through Legendre transforms [10]. H(~, E) = U - EiDi,

(8)

M(a, D) = U - aoEq,

(9)

G(a, E)

= U - EiDi-

a o E o.

(10)

The corresponding constitutive relations are given by

c~H ao=-, 0%

OH Di= - - - , OE~

E~j-

c~M c3M E, = - c3a~j' OD~ '

('ij--

O(~ij '

dG

(11) (12)

8G

Di:--O--~i i •

(13)

Equations governing the infinitesimal deformations of a piezoelectric body are ffyi,j = P i i i , 1 Eii=5(ui.j+ uzi),

Did = 0, Ei= - O , ,

(14) (15)

where u is the mechanical displacement and ~b the electric potential. Equation (14)j expresses the balance of linear momentum and eq. (14)2 is the Gauss equation. A theory in which one considers the inertia forces associated with mechanical deformations but neglects time-derivatives of the magnetic induction in Faraday's law of induction is usually referred to as quasistatic piezoelectricity.

Conservation laws in linear piezoelectricity

1043

4. VARIATIONAL PRINCIPLES FOR QUASISTATIC PIEZOELECTRICITY

Corresponding to each energy density function, there is a variational principle for quasistatic piezoelectricity [10]. We list below the functional and the corresponding stationary conditions rq (u, e, a, ~b, E, D) = f ' d z f Z , d to

(16)

dV,

V

Z, = H ( t , E ) - aij(~ij- 51( u , . j + u j , ) ) + D , ( E i + 4 ) , , ) . - ~ pu,"u,," a/i.j = pii~,

(17)

D,, i = O,

1 c,j = 5(u,.j+ Uj.3, E~ = -49,,,

OH f l u - 0%

gH 8Ei

(18)

E2 d V,

(19)

Di

n2 (u, o, (h, D) =

;tf dr

o

V

E 2 = M ( a , D) - aj,.ju,- D,.,c]) - ½ pd, i6, f f ji, j = p i i , ,

½(u,,j + uj,,) =

(20)

D i , i = O,

¢?M aa,j

c3M OD,

(21)

]E]3 d V,

(22)

c~,,

or

;f

/'g3(U, O', (~, E, D) =

dr

o

V

E3 = G(o, E) - aj,.ju, + D,(E, + c~,,) - ½pti, zi,, ffji.j = piii,

OG

½(u,j+ujA-

OG

a,rij' O'=-aE---~;

7~4(U, ~, G, ~ , D) =

;f dr

0

~j~ j = pill,

(25)

E4dV , V

Z~= U(~,D)-a~A~ij-½(u,j+uj3 • , ) -- D J )

eii=½(ui.i+uj.i),

(24)

Ei = --qS, i,

Di. i = O,

(23)

~ "•

-- 5 pUiUi,

(26)

Di. i = 0,

~U aij=-~%,

¢,i-

~U OD,"

(27)

These mixed variational principles are generalizations of the Hellinger-Reissner and the Hu-Washizu principles in linear elasticity. Each principle gives a different but equivalent set of equations as stationary conditions. The details of deriving the stationary conditions have been omitted and can be found in the cited references, e.g. see [3, 4]. Functions E~, Y~2,E3, and E4 depend implicity upon time t through the dependence of their arguments on t.

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J.S. YANG and R. C. BATRA 5. I N V A R I A N C E U N D E R T R A N S L A T I O N S

For a homogeneous piezoelectric body, energy density functions U, H, M, and G do not depend explicitly upon the Cartesian coordinates x, and the functionais n~, n2, n3, and n4 are invariant under the translational transformation t' = t,

x~ = x i + E,

(28)

since u; = . , ,

E;/=

~/,

~;j = o,j,

4'=c~,

E:=E~.

D:=D,,

(29)

under the transformation (28). The application of Noether's theorem (6)-(17), (20), (23) and (26) with x = (xl, x2, x3, t), y = (u, ~, a, q~, E, D) for (17), y = (u, a, ~b, D) for (20), y = (u, a, q~, E, D) for (23) and y = (u, ~, o, ~b, D) for (26) gives the following set of conservation laws Of + 0gl.~I 8t -~Xk= 0 '

C¢=1,2,3,4,

(30)

where

f,

=

pu~uj,,,

g(])ik ~

'~,1(~ik - - l'tj, i(Tjk - - ~ , i a k ,

g(2)ik =

Z 2 O i k q_ 6kj, iU / q._ Dk.i~b '

g(3) ik

E3(~ik .jr_ (Tkj, iUj - - ~),iDk '

g(4) ik

=

E4(~ik - - Uj, i¢~jk "~ Dk , i49,

(31)

and 6~: is the Kronecker delta. These conservation laws can be verified by direct differentiation, and reduce to Fletcher's [4] results when quantities associated with the electric field are neglected, and the Li's [5] results when dynamic terms are also ignored.

6. I N V A R I A N C E U N D E R R O T A T I O N S

For a homogeneous and isotropic material, the energy density functions are isotropic functions. For example, the energy density function H satisfies H ( Q t Q T, QE) = H(t, E)

(32)

for any orthogonal tensor Q(E) satisfying Q(0) = 1. The functions Z~, E2, E3, and Z4 are invariant under the transformations x : = Qgjxj, u: = Qijuj,

t' = t,

e:j = Qi.,Q:.~.,.,

O' = c~,

E; = Q~:E:,

a;j = QimQjn~mn, D ; = QijDj.

(33)

Noether's theorem (6) when applied to (17), (20), (23), and (26), with x and y in (6) identified as in the previous section, gives the following set of conservation laws.

ahjk -~c~l Oq ik - -

e~jk~

+~

-

U,

~¢= I, 2, 3, 4,

(34)

Conservation laws in linear piezoelectricity

1045

where

hjk = peuk(XjtmUm.~ + f~jUk), qO) ik

- - eimj(XmEl(~j

k

- - Xm Ul, j f f lk - -

x,.dp,jDk + ajku,.),

(2) q~k - e~.~(x.,X26jk + x . , ~ j u ~ + xmDkj~ + ~ou~6.,~ + ~Um -- 6~ ~D.,), _

q(3) ik = ein,i(x,,,X36j k +

X m ~kl, j Ul

-- x,.dp.~Dk + aout6,.~ + a~ju..).

q~a) ei.,i(x,.X46j~ - XmU~.~a~ + x.,D~,j49 + a~u,. - 6~D,.ck ), ik

(35)

and eu~ is the permutation symbol. These conservation laws also reduce to Fletcher's [4] results for linear elastodynamics and to Li's [5] results for linear elastostatics.

7. I N V A R I A N C E U N D E R C H A N G E S OF SCALE

For homogeneous linear piezoelectric body, the energy density functions are quadratic functions of their arguments. The functions E i, E2, E3, and E 4 a r e invariant under the scale change x~ = e'xi,

u; = e-'ui,

t' = e't,

~b' = e-'q~,

E~j = e-2'Eu,

E; = e 2'Ei,

tr~j = e 2'tru,

D~ = e-2'Di .

(36)

(37)

By using Noether's theorem (6) we obtain

c~--~-+-~x =0, ~=1,2,3,4,

(38)

where ~

= paj(uj + X.,Uj.m + ti,j) + tZ~,

g~') = xkE, -- trjk(Uj + X,,Uj.,, + tfij) -- Dk(c~ + XmCb,m + t~) ~2) = Xk~, 2 + Uj(2ffj k + XmtTjk.,,,+ tdjk) + ~b(2Dk + XmOk. m + t19k), ~ 3 ) = XkE3 + u~(2trjk + x,~ajk.,, + tdrjk) -- Ok(dp

+

Xml~),m-{- tc~),

g~4) = XkZ4 -- ajk(Uj + XmUj.,, + tf~j) + ~b(2Ok + XmOk.,, + tbk).

(39)

These conservation laws can be verified by carrying out the indicated differentiations, and they also reduce to Fletcher's [4] results when quantities associated with the electric field are neglected. However, these cannot be reduced to Li's [5] results for linear elastostatics because the scale change for the static case [9] is different from that given in (32). For static problems the functionals are invariant under the following scale change x; = e-2'xi,

u; = e'ui,

~b' = e'~

(40)

which results in

(.~j = e3'f.ij,

E; = e3'Ei,

tr~j = e3'~rij, D; = e3'Di.

(41)

The application of Noether's theorem (6) with x = ( x ~ , x z , x3), y = ( u , ~ , o , ~ , E , D ) for (17), y = (u, o, ~, D) for (20), y = (u, a, ~b, E, D) for (23) and y = (u, ~, a, ~b, D) for (26) results in the EFM Sl/6---L

1046

J.S. YANG and R. C. BATRA

following conservation laws (3 (3x ( x i z , -

xjD,49.

- ½, jiuj- ½D,49) = O,

(3 (3x---~(x,Z2 + Xjaik.jUk + xjDi, j49 + ~aj, u i + 3 D , 4 9 ) = O,

(3xi

(xiZ3 + X iU,k, jUk - xjDi49,i + ~ ajiui -- ½Dick) = O,

(3

~X--i (Xi'~4 -- XjUk,jO'ki + xjDi..i49 -- ½ajiuj + 3 Did?) = 0

(42)

which reduce to Li's [5] results when quantities associated with the electric field are neglected.

8. E L A S T I C D I E L E C T R I C S W I T H P O L A R I Z A T I O N

GRADIENT

We now study deformations of a homogeneous linear elastic dielectric material for which the energy density W of deformation and polarization depends upon the infinitesimal strain tensor Eij, the polarization vector Pi and its gradients P~j. Mindlin [1 1] generalized Toupin's [12] theory of piezoelectricity by incorporating the dependence of the energy density of deformation and polarization upon the gradients Pij of the polarization vector Pi and studied linear elastic dielectrics. Governing equations for such a material in the absence of external body forces are [1 1] a~.i = piij, .Ej + E~.i - 49,j = O, --eo 49, ii + Pi.i = 0,

(43)

where dW a i i - aEii =a/i,

~W (3W Ei = - aP----~' Eij ==-pj.i,

(44)

and e0 is the permittivity of a vacuum. For W, Mindlin [1 1] proposed the following expression, t

1

W = b° Pj,, + ½ai)P, Pj + ib~jktPy.ie,.k + iCuktEi/Ck,+ diiktej.iekt+ fjk~.iiek + gijkPiPk.j.

(45)

We note that eqs (43) are Euler-Lagrange equations for the functional (~, P, 49) --

f'd, fZ, dV,

30

(46)

where Zs(~,P, 49)= W(~,P, 49) - ieo49,i49,i ~ + 4 9 , i P i - ½priorii.

(47)

The invariance under translations (28), rotations (33) and the application of Noether's theorem (6) with x = (xt, x2, x3, t) and y = (u, P, 49) give Ot (pftjuj.,) + eUkp(xjf~''u''k + ftjuk) +

(Z, 6,k - uj.,ajk - 49, i(Pk - eo49,k) -- rj.iEkj) = 0,

(48)

eimj(X,,,Z5 5jk -- X,,,Ut.fltk -- X,,49,j(ek -- eo49,k) -- x,,Pt.jEkl+ ajkU,, + EkjP,,,)=O.

(49)

These conservation laws can be verified by performing the indicated differentiation and they reduce to conservation laws for static problems once the time dependent terms are omitted. For static problems, there is no need to perform the integration with respect to time in eq. (46).

Conservation laws in linear piezoelectricity

1047

9. CONCLUSIONS We have derived a class of conservation laws for linear piezoelectric materials by using Noether's theorem in conjunction with the invariance of various functionals under translations, rotations, and changes of scale. Our results generalize conservation laws for linear elastodynamics derived by Fletcher to linear piezoelectricity and Mindlin's elastic dielectrics. Acknowledgement--This work was supported by the U.S. Army Research Office grant DAAH04-93-G-0214 to the University of Missouri-Rolla, and a matching grant from the Missouri Research and Training Center.

REFERENCES [1] J. R. Rice, A path independent integral and the approximate analysis of strain concentrations by notches and cracks. J. appl. Mech. 35(2), 379-386 (1968). [2] J. D. Logan, Invariant Variational Principles. pp. 70-71. Academic Press, New York (1977). [3] J. K. Knowles and E. Sternberg, On a class of conservation laws in linearized and finite elastostatics. Arch. Rat. Mech. Anal. 44, 187 211 (1972). [4] D. C. Fletcher, Conservation laws in linear elastodynamics. Arch. Rat. Mech. Anal. 60, 329-353 (1975). [5] X. Li, Dual conservation laws in elastostatics. Engng Fracture Mech. 19(2), 233-241 (1988). [6] Y. E. Pak and G. Herrmann, Conservation laws and material momentum tensor for the elastic dielectric. Int. J. Engng Sci. 24(8), 1365 1374 (1986). [7] R. M. McMeeking, A J-integral for the analysis of electrically induced mechanical stress at cracks in elastic dielectrics. Int. J. Engng Sci. 28(7), 605-613 (1990). [8] G. A. Maugin and M. Epstein, The electroelastic energy-momentum tensor. Proc. R. Soc. Lond. A. 433, 299-312 (1991). [9] C.-S. Yeh, Y.-C. Shu and K.-C Wu, On a class of conservation laws for linearized and nonlinear elastic dielectrics. Mechanics of Electromagnetic Materials and Structures, (Edited by J. S. Lee, G. A. Maugin, and Y. Shindo). AMD-Vol. 161, MD-Vol. 42, pp. 35-39. ASME, New York (1993). [10] J. S. Yang, Mixed variational principles for piezoelectric elasticity. Developments in Theoretical and Applied Mechanics, (Edited by B. Antar, R. Engels, A. A. Prinaris and T. H. Moulden). Vol. XVI, II.1, pp. 31-38. The University of Tennessee Space Institute (1992). [11] R. D. Mindlin, Polarization gradient in elastic dielectrics. Int. J. Solids Structures 4, 637-642 (1968). [12] R. A. Toupin, The elastic dielectric. J. Rational Mech. Anal. 5, 849-915 (1956). [13] X. Li, Conservation laws and dual conservation laws for crack problems in homogeneous isotropic conductive solids. Engng Fracture Mech. 42(1), 51-57 (1992). (Received 21 February 1994)