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Constitutive relations for piezoelectric materials in terms of invariants
Meehamcs Research Commumcations, Vol 28, No. 2, pp 179-186, 2001 Copynght© 2001 Elsevier Sctence Ltd Pnnte,d m the USA All nghts reserved 0093-6413D l/S-see front matter
Pergamon
Plh S0093-6413(01)00160-4
C o n s t i t u t i v e Relations for P i e z o e l e c t r i c Materials in Terms of Invariants Pablo Padilla IIMAS-FENOMEC, Universidad Nacional Aut6noma de M@xico Circuito Escolar, Cd. Universitaria M@xico D.F., M@xico Boris E. Pobedria Composites Department, Moscow State University M. V. Lomonosov Moscow, Russia Reinaldo Rodriguez-Ramos Departamento de Ecuaciones Diferenciales y MecAnica, Universidad de la Habana San LAzaro y L, CP. 10400 Habana, Cuba and Instituto de Ingenier~a Universidad Nacional Aut6noma de M@xico Apdo. Postal 70-472 Delegaci6n Coyoac£n, 04510 M@xico D. F. M~xico (Received 2 October 2000. accepted for print 3 January 2001)
1
Introduction
Recently, in industrial applications, some composites have been introduced for which the piezomechanic effect is much more noticeable than for the natural piezoceramic materials, as documented in [1]. We refer the reader to [2], [3], and references therein for more details on linear piezoelectricity or to [4], [5] and [6] for other aspects of nonlinear piezoelectric materials. A recent application of the nonlinear behavior of these materials can be found in [7]. The constitutive relations are one of the main aspects of continuum mechanics. Many attempts have been made in order to write the constitutive relations for anisotropic materials [8]. In this sense, representations of the constitutives equations in terms of invariants have been studied by [9,10,11,12]. In this work, we provide a new approach to the study of constitutive relations of nonlinear piezoelectric materials. We formulate 179
180
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS
them for the transverse isotropic case using an appropriate set of invariants with respect to the group of symmetries of the problem. We can also consider a purely elastic nonlinear material and present an application to plastic-piezoelectricity. The advantage of this formulation lies on the fact that the role of the nonlinearity can be better understood. For instance, by allowing specific types of dependence of the constitutive relations on the invariants, we can deal with the linear, quasilinear and fully nonlinear cases. To the best of our knowledge, there has been no analogous formulation. Another subject in which this treatment of the constitutive relations might be useful is the study of nonlinear composites (see for instance [13] and [14] for the elastic case or [15] for piezoelectric polycrystals and [16] for nonlinear piezoelectric materials). This work is organized as follows. In section 2 we present the mathematical formulation of the invariant relations. We apply this formulation to the transverse isotropic piezoelectric case in terms of the invariants of the involved tensors. Section 3 contains two examples, one for a concrete nonlinearity in piezoelectricity and the other one for a plastic-piezoelectric material.
2
Formulation
in terms
of Invariants
The transverse isotropic medium is the simplest example of a medium where the electric and mechanical effects are coupled. Let us analize this case. The tensorial basis (of the group T, rotation with respect to a fixed axis by an angle a) is the set of second rank tensor "y and the vector ~" [17,18],
~ = 3',:~'z ® ~,
~=c,E,, %:=~,:-Gc:.
(1)
In this case, each vector can be represented as a sum of two mutually orthogonal vectors, which are invariant with respect to the group T. For instance, = Eo~'+ E*
(E, = Eoc, + E~),
(2)
and any vector E has two independent invariants Eo and E*:
E0 =
(3)
Any symmetric tensor of second rank, for instance the stress tensor, tr, can be represented as a sum of four (pairwise) orthogonal tensors, invariant with respect to the group T: 1 . a,3 = -~a %j + aocscj + P,3 + 2Q,3, (4) where
1 .
P,, = a,, - ck(a,~c~ + aj~c,) -- ~ a %3 + aoC, C,,
(5)
CONSTITUTIVE RELATIONS FOR PIEZOELECTRIC MATERIALS
l 81
1 Q . = ~c~(o, kc, + o , ~ ) - ~0~e,,
(6)
P-
(7)
-o.%j,
Q--_
In that case, each tensor tr has five independent invariants (Pobedria (1986)), that is, (7) and (8)
R =- e,#el,,.,QaQ3~P~,~.
A symmetric tensor of second rank has two independent components, for instance, (9)
~k, = ~1%, + ~2CkC4.
On the other hand, the symmetric tensors of third rank have three independent components: akv = a1%3Ck + a2(7,k% + %kq) + aaC~C~Ck
(10)
and the tensors of fourth rank with the following symmetry properties, J, jkl = da,kl = J,~k = Jk*~a have five independent components, for instance,
J, jkl
= Al%37kl +
A2(7,jCkCl + 7klCtC3) + A3CtCjCkCl+ A4(TtkTjl + %l%k)
+As(%kcact + 7.CkCa + %kC~C, + %tckc,). Substituting representations (2), (4),
(9)-(11)
(11)
in the electroelastic potential energy
1
X (o', E) - -~(Jz#ta,aak, + 2dk,aEkO',a + t~k,E, Ek).
(12)
we obtain the expresion for the quadratic potential X°(~,/~) for the transverse isotropic medium:
1r
~',.2
X°(tr,/~) = ~[al~
+ a2E~ + (A1 + A4)a .2 + 2A2a*ao + A3ao2 + 2A4P 2 + 4AsQ2
+2dla*Eo + 4d2(E:Q,3c 3 + E ; Q . c , ) + 2aaE0a0],
(13)
where the corresponding invariants and the material constants are given by formulae (3), (7), (8). The quadratic functional (13) can be generalized to the case of any non linear dependence of the invariants:
182
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS X(o', YE) = X ( E * , Eo, a*, ao, P, Q, R, S),
(14)
S = E*Q,3c 3.
(15)
where
3
Constitutive
Relations
Let us analyze another form to give representation of the constitutive relations for electroelastic materials. In what follows we consider two specific examples. The first one deals with a transverse isotropic nonlinear piezoelectric material and the second one with a plastic-piezoelectric material.
3.1
The
nonlinear
piezoelectric
case
If from the number of invariants in (14) we exclude the invariant R given in (8), then the tensorial potential relation will be quasilinear (16)
X ( t r , E) = X ( E * , Eo, a*, ao, P, Q, S).
Similarly to (4)-(7) we can write the strain tensor in the following form: 10, %~ + ~oC~C3+ Pz3 + 2q,3, ~ = -~
(17)
1 , P,3 = % - ck(e,kcj + ejkc,) -- 50 %3 + eoC,C3,
(18)
1 q,j = -~Ck(e,kc3 + e~kC,) -- eoC,C3,
(19)
0*-----ezjT,3, ~o-- e,3c, c3,
p-- ~ ,
q-- ~ .
(20)
The electric displacement vector can be writen in an analogous way to (2), in the form: /) = D06+/9"
(21)
(D, = D0c~ + D:),
and
D0--6.g=
,cz,
Hence, the non linear constitutive relations
6" =
D:.
(22)
CONSTITUTIVE RELATIONS FOR PIEZOELECTRIC MATERIALS
_
e,:
OX(o',ff~) D k - - o x (,,, ~) 0(70 ' OEk
183
(23)
can be writen for the transverse isotropic medium in the form (17), (21). These relations can be substituted, on the one hand, into the relations, between two deviators of the strain tensor p, q and the vector/9", and on the other hand, into the relations for the deviators of the stress tensor P, Q and the vector E*: P,J
P ~Po,
=
q
1
(24)
.
(25)
q,~ = 2 - ~ Q u + -~T(E, cj + E~c~),
D*
(26)
D : = E----TE* + TQ,3%.
The invariants appearing in the expressions (24)-(26) can be calculated according to the formulae: p =
ox(~, g) OP
2ox(~, ~) '
q =
OQ
'
(27) e* ax(,~,~) ,
Oa *
60
ax(~,,g) --
,
Oao
ax(~,g) D* ax(~,~) T -
OS
and Do-
'
=
OE*
ax(¢, g) 0Eo
Therefore, all seven invariants defined by (27), are scalar functions of the seven invariants which are in the right hand side of formula (16).
3.2
T h e p l a s t i c - p i e z o e l e c t r i c case
Let us consider a simple situation. In that case, we can write the constitutive relations where the physical non linearity is related to the mechanical properties only [13] and the linear invariants of strain and stress tensors are linearly coupled [18]. Then, instead of relations (24)-(26), we have: P*J
=
P-P, p .3,
(2s)
184
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS
q,~ = --Q,~ + Q,
(S:c~ + S;c,)
(29)
and
D~ = alE~ + 2d2Qocj,
(30)
and instead of the scalar relation we will have 0* = (A1 + A4)a* + A2a0,
(31)
e0 : A2a* + Aaao + daEo, Do = 2dla* + daffo + ~¢2Eo, p = 2A4[1 + ~'/(P)]P
(32)
q = 2A511 + ¢(Q)]Q.
(33)
and The quantity q. which appears in (29) is determinded by
(34) The functions f~(P), ~(Q) are called plasticity functzons. For weak materials from (32) and (33), we have dfl 0 _< 9 _< ~ + P ~ - f < co (35) and
de
0 < ¢ < ¢ + Q-~ < ~.
(36)
When in (32) the involved processes are passive (unloading) we must have 12(P) = 0, • (Q) = 0, and the expressions (28),(29) are replaced by !
p,~ = p . + 2~,(P,~ - P:~)
(37)
qsj = q,j' + 2As(Q,~ - Q:j) + 4d2(E:c~ - E;c,),
(38)
and
where the prime here indicates quantities at the beginning of the unloading process.
4
Conclusions
In the present paper we studied the constitutive relations for piezoelectric materials. These relations were written for the transverse isotropic case using a suitable basis, which is invariant with respect to the symmetry group of the problem.
CONSTITUTIVE RELATIONS FOR PIEZOELECTRIC MATERIALS
185
We have obtained a new formulation for the constitutive relations of nonlinear transverse isotropic piezoelectric materials in terms of the invariants associated to the problem. We have also presented two specific examples.
Acknowledgements This work was sponsored by CoNaCyT Project Numbers G25427-E, 32237-E and 27520-A, DGAPA-UNAM IN114999 and Project High Education Ministry of Cuba No. 03.50, 1999. Thanks are due to Ms. Ana P6rez Arteaga for computational support.
References [1] Gururaja, T.R., Safari, A., Newnham, R.E., and Cross, L.E., "Piezoelectric ceramicpolymer composites for transducer applications, Electronic Ceramics, L. M. Levison," Ed., New York: Marcel Dekker, (1987) 92-128. [2] Maugin, G.A.,"Continuum Mechanics of Electromagnetic Solids," North- Holland, Amsterdam, (1988). [3] Ikeda, T.,"Fundamentals of Piezoelectricity," Oxford University Press, Oxford, (1990). [4] Tiersten, H.F.,"On the Nonlinear Equations of Thermoelectroelasticity," International Journal of Engineering Science, Vol.9, (1971) 587-604. [5] Nelson, D.F, "Theory of Nonlinear Electroacoustics of Dielectric, Piezoelectric and Pyroelectric Crystals," Journal of Acoustic Society of America, Vol. 63, (1978) 17281748. [6] Maugin, G.A., "Nonlinear Electromechanical Effects and Applications," Series in Theoretical and Applied Mechanics, Volume 1, Ed. R. K. T. Hsieh. World Scientific, (1985). [7] Wang, Q.M., Zhang Q., Xu, B., Liu, R. and Cross E., "Nonlinear Piezoelectric Behavior of Ceramic Bending Mode Actuators under Strong Electric Fields," Journal of Applied Physics, Vol.86, (1999) 3352-3360. [8] Kiral, E. and Smith, G.F., "On the constitutive relations for anisotropic materialsTriclinic, monoclinic, rhombic, tetragonal, and hexagonal crystal systems," International Journal of Engineering Science, Vo1.12, (1974) 471-490. [9] Wang, C.C., "A new representation theorem for isotropic functions, Parts I and II," Archive of Rational Mechanics and Analysis, Vol.36, (1970) 166-223. [10] Smith, G.F., "On isotropic functions of symmetric tensors, skew symmetric tensors and vectors," International Journal of Engineering Science, Vo1.19, (1971) 899-916. [11] Spencer, A.J.M., "Theory of invariants, in Continuum Physics," Voh 1, ed. A.C.
186
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS
Eringen, Academic Press, New York, (1971). [12] Eringen, A.C., "Tensor analysis, in Continuum Physics," Vol. I, ed. A.C. Eringen, Academic Press, New York, (1971). "Continuum Physics," Vol. 1, Academic Press, New York, (1971). [13] Ponte-Castafieda, P., "The Effective Mechanical Properties of Nonlinear Isotropic Composites," Journal of the Mechanics and Physics of Solids, Vol.39, No.l, (1991) 45-71. [14] Ponte-Castafieda, P. and Willis, J.R., (1997) "Variational Second-order Estimates for Nonlinear Composites," Proceedings of the Royal Society of London, A Vol.455, (1999) 1799-1811. [15] Olson, T. and AveUaneda, M., "Effective dielectric and elastic constants of piezoelectric polycrystals," Journal of Applied Physics, Vol.71, (1992) 4455-4464. [16] Bisegna, P. and Luciano, R. (1996) "Variational bounds for the overall properties of piezoelectric composites," Journal of the Mechanics and Physics of Solids, Vol.44, (1996) 583-602. [17] Pobedria, B.E., "Lectures on Tensor Analysis," Moscow State University Press, Moscow (in Russian), (1986). [18] Eringen, A.C. and Maugin, G.A., "Electrodynamics of Continua I, Foundations and Solid Media," Springer-Verlag, New York, (1990).
Pergamon
Plh S0093-6413(01)00160-4
C o n s t i t u t i v e Relations for P i e z o e l e c t r i c Materials in Terms of Invariants Pablo Padilla IIMAS-FENOMEC, Universidad Nacional Aut6noma de M@xico Circuito Escolar, Cd. Universitaria M@xico D.F., M@xico Boris E. Pobedria Composites Department, Moscow State University M. V. Lomonosov Moscow, Russia Reinaldo Rodriguez-Ramos Departamento de Ecuaciones Diferenciales y MecAnica, Universidad de la Habana San LAzaro y L, CP. 10400 Habana, Cuba and Instituto de Ingenier~a Universidad Nacional Aut6noma de M@xico Apdo. Postal 70-472 Delegaci6n Coyoac£n, 04510 M@xico D. F. M~xico (Received 2 October 2000. accepted for print 3 January 2001)
1
Introduction
Recently, in industrial applications, some composites have been introduced for which the piezomechanic effect is much more noticeable than for the natural piezoceramic materials, as documented in [1]. We refer the reader to [2], [3], and references therein for more details on linear piezoelectricity or to [4], [5] and [6] for other aspects of nonlinear piezoelectric materials. A recent application of the nonlinear behavior of these materials can be found in [7]. The constitutive relations are one of the main aspects of continuum mechanics. Many attempts have been made in order to write the constitutive relations for anisotropic materials [8]. In this sense, representations of the constitutives equations in terms of invariants have been studied by [9,10,11,12]. In this work, we provide a new approach to the study of constitutive relations of nonlinear piezoelectric materials. We formulate 179
180
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS
them for the transverse isotropic case using an appropriate set of invariants with respect to the group of symmetries of the problem. We can also consider a purely elastic nonlinear material and present an application to plastic-piezoelectricity. The advantage of this formulation lies on the fact that the role of the nonlinearity can be better understood. For instance, by allowing specific types of dependence of the constitutive relations on the invariants, we can deal with the linear, quasilinear and fully nonlinear cases. To the best of our knowledge, there has been no analogous formulation. Another subject in which this treatment of the constitutive relations might be useful is the study of nonlinear composites (see for instance [13] and [14] for the elastic case or [15] for piezoelectric polycrystals and [16] for nonlinear piezoelectric materials). This work is organized as follows. In section 2 we present the mathematical formulation of the invariant relations. We apply this formulation to the transverse isotropic piezoelectric case in terms of the invariants of the involved tensors. Section 3 contains two examples, one for a concrete nonlinearity in piezoelectricity and the other one for a plastic-piezoelectric material.
2
Formulation
in terms
of Invariants
The transverse isotropic medium is the simplest example of a medium where the electric and mechanical effects are coupled. Let us analize this case. The tensorial basis (of the group T, rotation with respect to a fixed axis by an angle a) is the set of second rank tensor "y and the vector ~" [17,18],
~ = 3',:~'z ® ~,
~=c,E,, %:=~,:-Gc:.
(1)
In this case, each vector can be represented as a sum of two mutually orthogonal vectors, which are invariant with respect to the group T. For instance, = Eo~'+ E*
(E, = Eoc, + E~),
(2)
and any vector E has two independent invariants Eo and E*:
E0 =
(3)
Any symmetric tensor of second rank, for instance the stress tensor, tr, can be represented as a sum of four (pairwise) orthogonal tensors, invariant with respect to the group T: 1 . a,3 = -~a %j + aocscj + P,3 + 2Q,3, (4) where
1 .
P,, = a,, - ck(a,~c~ + aj~c,) -- ~ a %3 + aoC, C,,
(5)
CONSTITUTIVE RELATIONS FOR PIEZOELECTRIC MATERIALS
l 81
1 Q . = ~c~(o, kc, + o , ~ ) - ~0~e,,
(6)
P-
(7)
-o.%j,
Q--_
In that case, each tensor tr has five independent invariants (Pobedria (1986)), that is, (7) and (8)
R =- e,#el,,.,QaQ3~P~,~.
A symmetric tensor of second rank has two independent components, for instance, (9)
~k, = ~1%, + ~2CkC4.
On the other hand, the symmetric tensors of third rank have three independent components: akv = a1%3Ck + a2(7,k% + %kq) + aaC~C~Ck
(10)
and the tensors of fourth rank with the following symmetry properties, J, jkl = da,kl = J,~k = Jk*~a have five independent components, for instance,
J, jkl
= Al%37kl +
A2(7,jCkCl + 7klCtC3) + A3CtCjCkCl+ A4(TtkTjl + %l%k)
+As(%kcact + 7.CkCa + %kC~C, + %tckc,). Substituting representations (2), (4),
(9)-(11)
(11)
in the electroelastic potential energy
1
X (o', E) - -~(Jz#ta,aak, + 2dk,aEkO',a + t~k,E, Ek).
(12)
we obtain the expresion for the quadratic potential X°(~,/~) for the transverse isotropic medium:
1r
~',.2
X°(tr,/~) = ~[al~
+ a2E~ + (A1 + A4)a .2 + 2A2a*ao + A3ao2 + 2A4P 2 + 4AsQ2
+2dla*Eo + 4d2(E:Q,3c 3 + E ; Q . c , ) + 2aaE0a0],
(13)
where the corresponding invariants and the material constants are given by formulae (3), (7), (8). The quadratic functional (13) can be generalized to the case of any non linear dependence of the invariants:
182
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS X(o', YE) = X ( E * , Eo, a*, ao, P, Q, R, S),
(14)
S = E*Q,3c 3.
(15)
where
3
Constitutive
Relations
Let us analyze another form to give representation of the constitutive relations for electroelastic materials. In what follows we consider two specific examples. The first one deals with a transverse isotropic nonlinear piezoelectric material and the second one with a plastic-piezoelectric material.
3.1
The
nonlinear
piezoelectric
case
If from the number of invariants in (14) we exclude the invariant R given in (8), then the tensorial potential relation will be quasilinear (16)
X ( t r , E) = X ( E * , Eo, a*, ao, P, Q, S).
Similarly to (4)-(7) we can write the strain tensor in the following form: 10, %~ + ~oC~C3+ Pz3 + 2q,3, ~ = -~
(17)
1 , P,3 = % - ck(e,kcj + ejkc,) -- 50 %3 + eoC,C3,
(18)
1 q,j = -~Ck(e,kc3 + e~kC,) -- eoC,C3,
(19)
0*-----ezjT,3, ~o-- e,3c, c3,
p-- ~ ,
q-- ~ .
(20)
The electric displacement vector can be writen in an analogous way to (2), in the form: /) = D06+/9"
(21)
(D, = D0c~ + D:),
and
D0--6.g=
,cz,
Hence, the non linear constitutive relations
6" =
D:.
(22)
CONSTITUTIVE RELATIONS FOR PIEZOELECTRIC MATERIALS
_
e,:
OX(o',ff~) D k - - o x (,,, ~) 0(70 ' OEk
183
(23)
can be writen for the transverse isotropic medium in the form (17), (21). These relations can be substituted, on the one hand, into the relations, between two deviators of the strain tensor p, q and the vector/9", and on the other hand, into the relations for the deviators of the stress tensor P, Q and the vector E*: P,J
P ~Po,
=
q
1
(24)
.
(25)
q,~ = 2 - ~ Q u + -~T(E, cj + E~c~),
D*
(26)
D : = E----TE* + TQ,3%.
The invariants appearing in the expressions (24)-(26) can be calculated according to the formulae: p =
ox(~, g) OP
2ox(~, ~) '
q =
OQ
'
(27) e* ax(,~,~) ,
Oa *
60
ax(~,,g) --
,
Oao
ax(~,g) D* ax(~,~) T -
OS
and Do-
'
=
OE*
ax(¢, g) 0Eo
Therefore, all seven invariants defined by (27), are scalar functions of the seven invariants which are in the right hand side of formula (16).
3.2
T h e p l a s t i c - p i e z o e l e c t r i c case
Let us consider a simple situation. In that case, we can write the constitutive relations where the physical non linearity is related to the mechanical properties only [13] and the linear invariants of strain and stress tensors are linearly coupled [18]. Then, instead of relations (24)-(26), we have: P*J
=
P-P, p .3,
(2s)
184
P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS
q,~ = --Q,~ + Q,
(S:c~ + S;c,)
(29)
and
D~ = alE~ + 2d2Qocj,
(30)
and instead of the scalar relation we will have 0* = (A1 + A4)a* + A2a0,
(31)
e0 : A2a* + Aaao + daEo, Do = 2dla* + daffo + ~¢2Eo, p = 2A4[1 + ~'/(P)]P
(32)
q = 2A511 + ¢(Q)]Q.
(33)
and The quantity q. which appears in (29) is determinded by
(34) The functions f~(P), ~(Q) are called plasticity functzons. For weak materials from (32) and (33), we have dfl 0 _< 9 _< ~ + P ~ - f < co (35) and
de
0 < ¢ < ¢ + Q-~ < ~.
(36)
When in (32) the involved processes are passive (unloading) we must have 12(P) = 0, • (Q) = 0, and the expressions (28),(29) are replaced by !
p,~ = p . + 2~,(P,~ - P:~)
(37)
qsj = q,j' + 2As(Q,~ - Q:j) + 4d2(E:c~ - E;c,),
(38)
and
where the prime here indicates quantities at the beginning of the unloading process.
4
Conclusions
In the present paper we studied the constitutive relations for piezoelectric materials. These relations were written for the transverse isotropic case using a suitable basis, which is invariant with respect to the symmetry group of the problem.
CONSTITUTIVE RELATIONS FOR PIEZOELECTRIC MATERIALS
185
We have obtained a new formulation for the constitutive relations of nonlinear transverse isotropic piezoelectric materials in terms of the invariants associated to the problem. We have also presented two specific examples.
Acknowledgements This work was sponsored by CoNaCyT Project Numbers G25427-E, 32237-E and 27520-A, DGAPA-UNAM IN114999 and Project High Education Ministry of Cuba No. 03.50, 1999. Thanks are due to Ms. Ana P6rez Arteaga for computational support.
References [1] Gururaja, T.R., Safari, A., Newnham, R.E., and Cross, L.E., "Piezoelectric ceramicpolymer composites for transducer applications, Electronic Ceramics, L. M. Levison," Ed., New York: Marcel Dekker, (1987) 92-128. [2] Maugin, G.A.,"Continuum Mechanics of Electromagnetic Solids," North- Holland, Amsterdam, (1988). [3] Ikeda, T.,"Fundamentals of Piezoelectricity," Oxford University Press, Oxford, (1990). [4] Tiersten, H.F.,"On the Nonlinear Equations of Thermoelectroelasticity," International Journal of Engineering Science, Vol.9, (1971) 587-604. [5] Nelson, D.F, "Theory of Nonlinear Electroacoustics of Dielectric, Piezoelectric and Pyroelectric Crystals," Journal of Acoustic Society of America, Vol. 63, (1978) 17281748. [6] Maugin, G.A., "Nonlinear Electromechanical Effects and Applications," Series in Theoretical and Applied Mechanics, Volume 1, Ed. R. K. T. Hsieh. World Scientific, (1985). [7] Wang, Q.M., Zhang Q., Xu, B., Liu, R. and Cross E., "Nonlinear Piezoelectric Behavior of Ceramic Bending Mode Actuators under Strong Electric Fields," Journal of Applied Physics, Vol.86, (1999) 3352-3360. [8] Kiral, E. and Smith, G.F., "On the constitutive relations for anisotropic materialsTriclinic, monoclinic, rhombic, tetragonal, and hexagonal crystal systems," International Journal of Engineering Science, Vo1.12, (1974) 471-490. [9] Wang, C.C., "A new representation theorem for isotropic functions, Parts I and II," Archive of Rational Mechanics and Analysis, Vol.36, (1970) 166-223. [10] Smith, G.F., "On isotropic functions of symmetric tensors, skew symmetric tensors and vectors," International Journal of Engineering Science, Vo1.19, (1971) 899-916. [11] Spencer, A.J.M., "Theory of invariants, in Continuum Physics," Voh 1, ed. A.C.
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P. PADILLA, B. E. POBEDRIA and R. RODRIGUEZ-RAMOS
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