- Email: [email protected]
Piezoelectricity with polarization gradient: homogenization
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 29 (2002) 53–59 www.elsevier.com/locate/mechrescom
Piezoelectricity with polarization gradient: homogenization J.J. Telega *, S. Bytner Institute of Fundamental Technological Research, Polish Academy of Sciences, Swießtokrzyska 21, 00-049 Warsaw, Poland
Abstract The aim of this paper is to perform homogenization of the equation of linear piezoelectricity with the polarization gradient. We assume that the material coefficients are microperiodic. This assumption can be weakened. One can also consider nonuniform homogenization. Then the homogenized (macroscopic) moduli depend on macroscopic variable. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectricity with the polarization gradient; Homogenization; Macroscopic moduli
1. Introduction Performing experimental tests Mead (1962) showed that piezoelectric effects can also appear in centrosymmetric crystals. This and other phenomena cannot be described by the classical Voigt model of piezoelectricity. Mindlin (1968) introduced the gradient of polarization to the expression for the enthalpy in order to take into account the piezoelectric properties of centrosymmetric crystals. Homogenization of a microheterogeneous medium of Mindlin type has been performed in Bytner et al. (2001) for a centrosymmetric material, by the method of two-scale asymptotic expansions, cf. (Bensoussan et al., 1978; Sanchez-Palencia, 1980). The aim of the present contribution is to study the general case of linear piezoelectricity with the polarization gradient. We assume that the material coefficients are microperiodic. To derive the macroscopic (or homogenized) moduli, the mathematically rigorous method of C-convergence has been used, cf. (Attouch, 1984; Dal Maso, 1993; Lewi nski and Telega, 2000). The study of homogenization of classical equations of piezoelectricity started with the paper by the first author (Telega, 1991), where the same method was applied. Recently, Nelli Silva et al. (1999a,b) developed numerical procedures and provided many interesting examples of applications of homogenization to the design of piezoelectric transducers. The approach used in Telega (1991) was extended to nonlinear piezoelectric composites without the polarization gradient in the paper by Telega et al. (1998). Only small deformations were considered. However, such nonlinear theory admits stronger electrical fields. For contributions by other authors to the homogenization the reader is referred to the references cited in Nelli Silva et al. (1999a,b) and Telega et al. (1998). *
Corresponding author. E-mail addresses: [email protected] (J.J. Telega), [email protected] (S. Bytner).
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 2 8 - 8
54
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
The sequence of functionals fJe ge>0 , given by the formula (2), involves the electric field not only in the . The method of C-convergence considered bounded piezoelectric solid X, but also the electric field in R3 n X permits to introduce the so-called perturbation functional U, which does not influence the homogenization procedure and macroscopic moduli. This functional, given by Eq. (7), includes terms with the electrical fields, ‘‘inconvenient’’ from the viewpoint of the method of two-scale asymptotic expansions.
2. Basic relations After Nowacki (1983), the enthalpy is given by 1 H ¼ U L ðe; P; rPÞ 0 u;i u;i þ u;i Pi ; 2
ð1Þ
where e ¼ ðeij Þ is the strain tensor and P ¼ ðPi Þ denotes the polarization vector, moreover rP ¼ ðPi;j Þ. The indices take values 1; 2; 3. We consider a piezoelectric solid with a microstructure, which is characterized by a small parameter e > 0. For a fixed e > 0 the functional of the total potential energy is expressed by Z Z 1 x 1 ; eðuÞ; P; rP 0 ru ru þ ru P dx Je ðu; P; uÞ ¼ UL 0 ru ru dx Lðu; P; uÞ; e 2 2 R3 nX X ð2Þ 3
cf. (Lewi nski and Telega, 2000; Telega, 1991; Telega et al., 1998). Here, by X R we denote a sufficiently regular bounded domain occupied by the piezoelectric solid in the initial configuration, / is the electric field and u ¼ ðui Þ stands for the displacement vector. The functional of external loading L is assumed in the following form: Z Z Z b u dx þ t u ds þ E0 P dx; ð3Þ Lðu; P; uÞ ¼ X
X
S1
where oX ¼ S ¼ S0 [ S1 . The quantities b; t and E0 are prescribed. Obviously, a more general form of the functional L can also be considered. From the condition of vanishing of the first variation of the functional Je , i.e. from dJe ¼ 0, we obtain the equations of the piezoelectricity with the gradient of polarization, cf. (Nowacki, 1983): div r þ b ¼ 0
in X;
L
div E þ E ru þ E0 ¼ 0
ð4Þ
in X;
0 Du þ div P ¼ 0 in X; and Du ¼ 0 in R3 n X
ð5Þ
with the natural boundary conditions rn ¼ t on S1 ;
En ¼ 0
on S;
ð0 sDut þ PÞn ¼ 0
on S:
ð6Þ
As usual, sf t denotes the jump of a function f. After Nowacki (1983), we have introduced the following notation: rij ¼
oU L ; oeij
Eij ¼
oU L ; oPj;i
EiL ¼
oU L : oPi
We are now interested in ‘‘smearing out’’ the hetrogeneities; that means we pass with the parameter e to zero (homogenization). To this end we use the theory of C-convergence; more precisely we use a variant of
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
55
Theorem 1.3.28 formulated in Lewi nski and Telega (2000). An important role is also played by the following lemma, cf. (Dal Maso, 1993; Lewi nski and Telega, 2000). be a sequence of CðsÞ-convergent functionals and let G ¼ CðsÞ lime!0 Ge . If Lemma. Let Ge : ðX ; sÞ ! R U : X ! R is a functional, continuous in the topology s, a so-called perturbation functional, then CðsÞ limðGe þ UÞ ¼ CðsÞ lim Ge þ U ¼ G þ U: e!0
e!0
The last lemma will be used to show that in our case it suffices to find the C-limit of the sequence of functionals Z h i x ; eðuÞ; P; rP dx: Ge ðu; PÞ ¼ UL e X In this paper we shall only consider the case where the energy U L ðxe; ; ; Þ is a quadratic function of its arguments. In the linear case the natural space for displacement fields and polarization fields is the space H 1 ðXÞ3 ¼ ½H 1 ðXÞ3 . The potential u is defined on the whole space R3 ; we assume that u 2 H 1 ðR3 Þ. We define the following (weak) topology: 3 3 s :¼ w–H 1 ðXÞ w–H 1 ðXÞ w–H 1 ðR3 Þ and the perturbation functional Z Z 1 0 ru ru dx þ ru P dx Lðu; P; uÞ: Uðu; P; uÞ ¼ 2 R3 X
ð7Þ
We observe that the functional Z ru P dx X
is continuous in the topology s. The proof of this fact is simple. Let fun gn2N , un * u weakly in H 1 ðR3 Þ when n ! 1, and Pn * P weakly in H 1 ðXÞ
3
when n ! 1:
On account of the Rellich–Kondrashov embedding theorem (Bensoussan et al., 1978; Sanchez-Palencia, 1980; Attouch, 1984; Dal Maso, 1993; Lewi nski and Telega, 2000), Pn ! P in L2 ðXÞ3 strongly if n ! 1. Consequently, we have Z Z run Pn dx ! ru P dx: X
X
Consider now the functional Z 1 0 ru ru dx; 0 > 0: 2 R3 The bilinear form Z aðu; wÞ 0 ru rw dx 6 ckukH 1 ðR3 Þ kwkH 1 ðR3 Þ ; 3
R
where c > 0 denotes a constant, is continuous.
/; w 2 H 1 ðR3 Þ;
56
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
The functional Z 1 U1 ðuÞ ¼ 0 ru ru dx 2 R3 is convex and finite-valued over the space H 1 ðR3 Þ, hence it is continuous. We conclude that functional U given by the formula (7) is continuous in the topology s, provided, that the functional L is also continuous in this topology. In the case where L is given by the formula (3), it is 3 3 3 sufficient to take b 2 L2 ðXÞ , t 2 L2 ðS1 Þ , E0 2 L2 ðXÞ . Thus we get Jh ðu; P; uÞ ¼ CðsÞ lim Je ðu; P; uÞ ¼ CðsÞ lim Ge ðu; PÞ þ Uðu; P; uÞ: e!0
e!0
Now we have to find the CðsÞ-limit of the sequence fGe ge>0 . In the case of quadratic piezoelectric energy function U L ðxe ; e; P; rPÞ we write (Nowacki, 1983): x 1 1 1 e e ; e; P; rP ¼ be0ij Pj;i þ aeij Pi Pj þ beijkl Pj;i Pl;k þ ceijkl eij ekl þ dijkl UeL Pj;i ekl þ fijk Pi ejk þ jeijk Pi Pk;j ; e 2 2 2
ð8Þ
where be0ij ðxÞ ¼ b0ij e ðxÞ dijkl
x ; ex
¼ dijkl
e
aeij ðxÞ ¼ aij e ðxÞ fijk
;
x
¼ fijk
e x e
x ; ceijkl ðxÞ ¼ cijkl ; e ex jeijk ðxÞ ¼ jijk ; x 2 X: e
beijkl ðxÞ ¼ bijkl
;
;
x
ð9Þ
We observe that for centrosymmetric materials fijk ¼ jijk ¼ 0. Then, however, a linear electromechanical coupling is still present via the coefficients dijkl . The material coefficients b0ij ðyÞ, aij ðyÞ, bijkl ðyÞ, cijkl ðyÞ, dijkl ðyÞ, fijk ðyÞ, jijk ðyÞ, y 2 Y , are Y-periodic. As usual in the case of periodic homogenization, Y denotes the basic cell (Bensoussan et al., 1978; Sanchez-Palencia, 1980; Attouch, 1984; Dal Maso, 1993; Lewi nski and Telega, 2000; Telega, 1991; Nelli Silva et al., 1999a,b). We assume that b0ij 2 L1 ðY Þ; . . . ; jijk 2 L1 ðY Þ. Thus, layered materials are not precluded. It is reasonable to assume that 9c1 > 0 U L ðy; e; P; QÞ P c1 ðjej2 þ jPj2 þ jQj2 ; y 2 Y for every e 2 E3s , P 2 R3 , Q 2 E3 . Here, E3 denotes the set of matrices 3 3, and E3s is the set of symmetric matrices 3 3. By applying Theorem 1.3.28 from Lewi nski and Telega (2000) we have Z Gh ðu; PÞ ¼ CðsÞ lim Ge ðu; PÞ ¼ UhL ðeðuðxÞÞ; PðxÞ; rPðxÞÞ dx; ð10Þ e!0
X
where UhL ðE; P; QÞ
¼ min
1 jY j
Z
L
y
U y; e ðvðyÞÞ þ ; P; ry qðyÞ þ Q dy j v 2 Y
where 2 E3s , P 2 R3 , Q 2 E3 , and n o 3 3 3 1 1 ðY Þ ¼ ½Hper ðY Þ ¼ w 2 H 1 ðY Þ j w is Y -periodic; hwi ¼ 0 : Hper
3 1 Hper ðY Þ ;
q2
3 1 Hper ðY Þ
;
ð11Þ
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
57
By h i we denote the mean value over the basic cell Y. The minimization problem appearing in (11) has a unique solution, which we denote by ðv; qÞ. In this way we may write " ! ! ! Z 1 o q 1 1 o q o q j j l b0ij ðyÞ UhL ð; P; QÞ ¼ þ Qji þ aij ðyÞPi Pj þ bijkl ðyÞ þ Qji þ Qlk jY j Y 2 2 oyi oyi oyk ! 1 oqj y y þ cijkl ðyÞðeij ðvÞ þ ij Þðekl ðvÞ þ kl Þ þ dijkl ðyÞ þ Qji ðeyij ðvÞ þ ij Þ 2 oyi # oqk y þ fijk ðyÞPi ðejk ðv þ ij Þ þ jijk ðyÞPi þ Qkj dy: ð12Þ oyj h h ; fijk and jhijk . On acLet us pass to the determination of the homogenized coefficients bh0ij ; bhijkl ; chijkl ; dijkl count of the linearity of the problem considered we conclude that
v ¼ vðmnÞ mn þ DðmnÞ Qmn þ nðmÞ Pm ;
ð13Þ
q ¼ WðmnÞ mn þ qðmnÞ Qmn þ jðmÞ Pm ; ðmnÞ
where the functions vi
ðmnÞ
; Di
ðmnÞ
; Wmn i ; qi
ðmÞ
; nðmnÞ , and ji i
are Y-periodic, i; m; n ¼ 1; 2; 3: We have
ðmnÞ ðmÞ o qj oWðmnÞ oqj ojj j ¼ mn þ Qnm þ Pm ; oyi oyi oyi oyi eyij ðvÞ ¼ eyij ðvðmnÞ Þmn þ eyij ðDðmnÞ ÞQnm þ eyij ðnðmÞ ÞPm :
ð14Þ
Taking into account (14) in (11), after lengthy calculations we arrive at the homogenized (effective) coefficients: * !+ ðijÞ L oU oq l h ¼ b0kl ðyÞ b0h þ dlj dki ; ð15Þ ij ¼ oQ oy ji ¼0;P¼0;Q¼0
ahij ¼
o2 UhL oPi oPj
k
* + + ðiÞ ðjÞ ojl ojðjÞ ojl y ðiÞ y ðiÞ y ðjÞ n ¼ haij i þ bklmn e ðn Þ þ hcklmn ekl ðn Þemn ðn Þi þ dklmn oyk oym oyk mn * + ðiÞ D E ojðjÞ ojl y ðjÞ ojðiÞ y y ðjÞ ðiÞ m m þ dklmn e ðn Þ þ hfilm elm ðn Þi þ fjlm elm ðn Þ þ jilm þ jjlm ; oyk mn oyl oyl *
o2 UhL oQlk oQji * !+ ðijÞ oqqðklÞ oqn ¼ bmnpq þ dni dmj þ dql dpk þ hcmnpq eymn ðDðijÞ Þeypq ðDðklÞ Þi oym oyp ðijÞ ðklÞ oqn oqn þ dmnpq þ dnj dmi eypq ðDðklÞ Þ þ dmnpq þ dnl dmk eypq ðDðijÞ Þ ; oym oym
ð16Þ
bhijkl ¼
ð17Þ
58
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
chijkl
o2 UhL ¼ ¼ okl oij
*
oWðklÞ oWðijÞ q n bmnpq oym oyp *
þ
h dijkl
h fijk
o2 UhL ¼ ¼ okl oQji
þ hcmnpq ½eymn ðvðijÞ Þ þ dim djn ½eypq ðvðklÞ Þ þ dkp dlq i
oWðijÞ q þ dkm dln oyp
*
+ þ
dmnpq ½eymn ðvðijÞ Þ
oWðklÞ q þ dim djn oyp
! + oqðklÞ oWðijÞ q n bmnpq þ dlq dkp þ hcmnpq ½eymn ðvðklÞ Þ þ dkp dlq eypq ðDðijÞ Þi oyp oym + * ðijÞ oqn oWðklÞ ðijÞ y ðklÞ y n e ðD Þ ; þ dmnpq þ dim djn ½epq ðv Þ þ dkp dlq þ dmnpq oym oym pq
+ ;
ð18Þ
*
*
o2 UhL ¼ ¼ ojk oPi
dmnpq ½eymn ðvðklÞ Þ
+
ðjkÞ ojðiÞ oWq bmnpq oym oyp *
ð19Þ
+ þ hcmnpq ½eymn ðvðjkÞ Þ þ djm dkn eypq ðnðiÞ Þi
ojðiÞ oWðjkÞ n ey ðnðiÞ Þ þ þ djp dks n þ dmnpq oym oym pq * + oWnðjkÞ y ðjkÞ þ hfimn ½emn ðv Þ þ djm dkn i þ jimn ; oym
+
dmnpq ½eypq ðvðjkÞ
jhijk
o2 UhL ¼ ¼ oQkj oPi
*
ð20Þ
ðiÞ + ojq oqðkjÞ n bmnpq þ djn dkm þ hcmnpq eymn ðnðiÞ Þeypq ðDðjkÞ Þi oym oyp ðjkÞ oqn ojðiÞ þ dmnpq þ dkn djm eypq ðnðiÞ Þ þ dmnpq n eypq ðDðjkÞ Þ oym oym ðjkÞ oqn þ hfimn eymn ðDðjkÞ Þi þ jimn þ dkn djm : oym
ð21Þ
Now we have to determine the Y -periodic functions WðijÞ , qðijÞ , jðiÞ , vðijÞ , DðijÞ , and nðiÞ . The minimization problem appearing in the r.h.s. of (11) is a convex problem, which is equivalent to ! ! Z Z Z Z o qj o qj oql y y 0 oqj bij dy þ bijkl dy þ cijkl ½eij ðvÞ þ ij ekl ðvÞ dy þ dijkl þ Qji þ Qji eykl ðvÞ dy oyi oyi oyk oyi Y Y Y Y Z Z Z Z oqj oqk y y y dy þ fijk Pi ejk ðvÞ dy þ jijk Pi ejk ðvÞ dy þ jijk Pi dy ¼ 0 ð22Þ þ dijkl ½ekl ðvÞ þ kl oy oyj i Y Y Y Y 1 for every q; v 2 Hper ðY Þ3 and for every 2 E3s ; Q 2 E3 ; P 2 R3 . The periodic functions WðmnÞ ; vðmnÞ ; . . . ; nðmÞ are solutions to the following three variational problems. 3
3
1 1 Problem P1 : Find functions WðmnÞ 2 Hper ðY Þ ; vðmnÞ 2 Hper ðY Þ ðm; n ¼ 1; 2; 3Þ such that ) Z ( ðmnÞ oWj oql 3 y ðmnÞ 1 bijkl dy ¼ 0 8q 2 Hper ðY Þ ; þ dklij ½eij ðv Þ þ dim djn oy oyk i Y
Z ( Y
)
ðmnÞ
oWj dijkl oyi
þ
cijkl ½eyij ðvðmnÞ Þ
þ dim djn eykl ðvÞ dy ¼ 0
3
1 8v 2 Hper ðY Þ :
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59 3
59
3
1 1 Problem P2 : Find functions qðmnÞ 2 Hper ðY Þ ; DðmnÞ 2 Hper ðY Þ such that ( ! ) Z ðmnÞ oqj oql 3 1 þ dim djn þ dklij eyij ðDðmnÞ Þ bijkl dy ¼ 0 8q 2 Hper ðY Þ ; oy oy i k Y
Z ( dijkl Y
)
!
ðmnÞ
oqj yi
þ dim djn
þ
cijkl eyij ðDðmnÞ Þ
eYkl ðvÞ dy ¼ 0
3
1 8v 2 Hper ðY Þ :
1 1 Problem P3 : Find functions jðmÞ 2 Hper ðY Þ3 ; nðmÞ 2 Hper ðY Þ3 such that ) Z ( ðmÞ ojj oql y ðmÞ 1 þ dklij eij ðn Þ þ jmkl bijkl dy ¼ 0 8q 2 Hper ðY Þ3 ; oy oy i k Y
Z ( Y
) ðmÞ ojj dijkl þ cijkl eyij ðnðmÞ Þ þ fmkl ekkl ðvÞ dy ¼ 0 oyi
3
1 8v 2 Hper ðY Þ :
3. Final remarks The assumption about periodicity of moduli can be weakened. One can consider the so-called nonuniform homogenization, cf. (Lewi nski and Telega, 2000). In such a more general case the homogenized h h h moduli b0h ij ; aij ; bijkl ; . . . ; jijk depend additionally on macroscopic variable x 2 X. More detailed presentation and examples will be given in a separate paper. Acknowledgements The first author was supported by the State Committee for Scientific Research (KBN, Poland) through the grant no. 7 T 07A 043 18. References Attouch, H., 1984. Variational Convergence for Functions and Operators. Pitman, Boston. Bensoussan, A., Lions, J.L., Papanicolau, G., 1978. Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam. Bytner, S., Gałka, A., Gambin, B., 2001. Effective properties of layered piezoelectric material with gradient of polarization. J. Tech. Phys. 42, 437–447. Dal Maso, G., 1993. An Introduction to C-convergence. Birkh€auser, Boston. Lewi nski, T., Telega, J.J., 2000. Plates, Laminates and Shells Asympotic Analysis and Homogenization. World Scientific, Singapore. Mead, C.A., 1962. Electron transport mechanism in thin insulating films. Phys. Rev. 128, 2088–2095. Mindlin, R.D., 1968. Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–652. Nelli Silva, E.C., Nishiwaki, S., Kikuchi, N., 1999a. Design of piezoelectric materials and piezoelectric transducers using topology optimization. Part II. Arch. Comp. Meth. Eng. 6, 191–222. Nelli Silva, E.C., Ono Fonseca, J.S., Montero de Espinosa, F., Crumm, A.T., Brady, G.A., Halloran, J.W., Kikuchi, N., 1999b. Design of piezocomposite materials and piezoelectric transducers using topology optimization. Part I. Arch. Comp. Meth. Eng. 6, 117–182. Nowacki, W., 1983. Electromagnetic Effects in Deformable Solids. PWN, Warsaw (in Polish). Sanchez-Palencia, E., 1980. In: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin. Telega, J.J., 1991. Piezoelectricity and homogenization. Application to biomechanics. In: Maugin, G.A. (Ed.), Continuum Models and Discrete Systems, vol. 2. Longman, Harlow, Essex, pp. 220–229. Telega, J.J., Gałka, A., Gambin, B., 1998. Effective properties of physically nonlinear piezoelectric composites. Arch. Mech. 50, 321– 340.
Mechanics Research Communications 29 (2002) 53–59 www.elsevier.com/locate/mechrescom
Piezoelectricity with polarization gradient: homogenization J.J. Telega *, S. Bytner Institute of Fundamental Technological Research, Polish Academy of Sciences, Swießtokrzyska 21, 00-049 Warsaw, Poland
Abstract The aim of this paper is to perform homogenization of the equation of linear piezoelectricity with the polarization gradient. We assume that the material coefficients are microperiodic. This assumption can be weakened. One can also consider nonuniform homogenization. Then the homogenized (macroscopic) moduli depend on macroscopic variable. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Piezoelectricity with the polarization gradient; Homogenization; Macroscopic moduli
1. Introduction Performing experimental tests Mead (1962) showed that piezoelectric effects can also appear in centrosymmetric crystals. This and other phenomena cannot be described by the classical Voigt model of piezoelectricity. Mindlin (1968) introduced the gradient of polarization to the expression for the enthalpy in order to take into account the piezoelectric properties of centrosymmetric crystals. Homogenization of a microheterogeneous medium of Mindlin type has been performed in Bytner et al. (2001) for a centrosymmetric material, by the method of two-scale asymptotic expansions, cf. (Bensoussan et al., 1978; Sanchez-Palencia, 1980). The aim of the present contribution is to study the general case of linear piezoelectricity with the polarization gradient. We assume that the material coefficients are microperiodic. To derive the macroscopic (or homogenized) moduli, the mathematically rigorous method of C-convergence has been used, cf. (Attouch, 1984; Dal Maso, 1993; Lewi nski and Telega, 2000). The study of homogenization of classical equations of piezoelectricity started with the paper by the first author (Telega, 1991), where the same method was applied. Recently, Nelli Silva et al. (1999a,b) developed numerical procedures and provided many interesting examples of applications of homogenization to the design of piezoelectric transducers. The approach used in Telega (1991) was extended to nonlinear piezoelectric composites without the polarization gradient in the paper by Telega et al. (1998). Only small deformations were considered. However, such nonlinear theory admits stronger electrical fields. For contributions by other authors to the homogenization the reader is referred to the references cited in Nelli Silva et al. (1999a,b) and Telega et al. (1998). *
Corresponding author. E-mail addresses: [email protected] (J.J. Telega), [email protected] (S. Bytner).
0093-6413/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 3 - 6 4 1 3 ( 0 2 ) 0 0 2 2 8 - 8
54
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
The sequence of functionals fJe ge>0 , given by the formula (2), involves the electric field not only in the . The method of C-convergence considered bounded piezoelectric solid X, but also the electric field in R3 n X permits to introduce the so-called perturbation functional U, which does not influence the homogenization procedure and macroscopic moduli. This functional, given by Eq. (7), includes terms with the electrical fields, ‘‘inconvenient’’ from the viewpoint of the method of two-scale asymptotic expansions.
2. Basic relations After Nowacki (1983), the enthalpy is given by 1 H ¼ U L ðe; P; rPÞ 0 u;i u;i þ u;i Pi ; 2
ð1Þ
where e ¼ ðeij Þ is the strain tensor and P ¼ ðPi Þ denotes the polarization vector, moreover rP ¼ ðPi;j Þ. The indices take values 1; 2; 3. We consider a piezoelectric solid with a microstructure, which is characterized by a small parameter e > 0. For a fixed e > 0 the functional of the total potential energy is expressed by Z Z 1 x 1 ; eðuÞ; P; rP 0 ru ru þ ru P dx Je ðu; P; uÞ ¼ UL 0 ru ru dx Lðu; P; uÞ; e 2 2 R3 nX X ð2Þ 3
cf. (Lewi nski and Telega, 2000; Telega, 1991; Telega et al., 1998). Here, by X R we denote a sufficiently regular bounded domain occupied by the piezoelectric solid in the initial configuration, / is the electric field and u ¼ ðui Þ stands for the displacement vector. The functional of external loading L is assumed in the following form: Z Z Z b u dx þ t u ds þ E0 P dx; ð3Þ Lðu; P; uÞ ¼ X
X
S1
where oX ¼ S ¼ S0 [ S1 . The quantities b; t and E0 are prescribed. Obviously, a more general form of the functional L can also be considered. From the condition of vanishing of the first variation of the functional Je , i.e. from dJe ¼ 0, we obtain the equations of the piezoelectricity with the gradient of polarization, cf. (Nowacki, 1983): div r þ b ¼ 0
in X;
L
div E þ E ru þ E0 ¼ 0
ð4Þ
in X;
0 Du þ div P ¼ 0 in X; and Du ¼ 0 in R3 n X
ð5Þ
with the natural boundary conditions rn ¼ t on S1 ;
En ¼ 0
on S;
ð0 sDut þ PÞn ¼ 0
on S:
ð6Þ
As usual, sf t denotes the jump of a function f. After Nowacki (1983), we have introduced the following notation: rij ¼
oU L ; oeij
Eij ¼
oU L ; oPj;i
EiL ¼
oU L : oPi
We are now interested in ‘‘smearing out’’ the hetrogeneities; that means we pass with the parameter e to zero (homogenization). To this end we use the theory of C-convergence; more precisely we use a variant of
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
55
Theorem 1.3.28 formulated in Lewi nski and Telega (2000). An important role is also played by the following lemma, cf. (Dal Maso, 1993; Lewi nski and Telega, 2000). be a sequence of CðsÞ-convergent functionals and let G ¼ CðsÞ lime!0 Ge . If Lemma. Let Ge : ðX ; sÞ ! R U : X ! R is a functional, continuous in the topology s, a so-called perturbation functional, then CðsÞ limðGe þ UÞ ¼ CðsÞ lim Ge þ U ¼ G þ U: e!0
e!0
The last lemma will be used to show that in our case it suffices to find the C-limit of the sequence of functionals Z h i x ; eðuÞ; P; rP dx: Ge ðu; PÞ ¼ UL e X In this paper we shall only consider the case where the energy U L ðxe; ; ; Þ is a quadratic function of its arguments. In the linear case the natural space for displacement fields and polarization fields is the space H 1 ðXÞ3 ¼ ½H 1 ðXÞ3 . The potential u is defined on the whole space R3 ; we assume that u 2 H 1 ðR3 Þ. We define the following (weak) topology: 3 3 s :¼ w–H 1 ðXÞ w–H 1 ðXÞ w–H 1 ðR3 Þ and the perturbation functional Z Z 1 0 ru ru dx þ ru P dx Lðu; P; uÞ: Uðu; P; uÞ ¼ 2 R3 X
ð7Þ
We observe that the functional Z ru P dx X
is continuous in the topology s. The proof of this fact is simple. Let fun gn2N , un * u weakly in H 1 ðR3 Þ when n ! 1, and Pn * P weakly in H 1 ðXÞ
3
when n ! 1:
On account of the Rellich–Kondrashov embedding theorem (Bensoussan et al., 1978; Sanchez-Palencia, 1980; Attouch, 1984; Dal Maso, 1993; Lewi nski and Telega, 2000), Pn ! P in L2 ðXÞ3 strongly if n ! 1. Consequently, we have Z Z run Pn dx ! ru P dx: X
X
Consider now the functional Z 1 0 ru ru dx; 0 > 0: 2 R3 The bilinear form Z aðu; wÞ 0 ru rw dx 6 ckukH 1 ðR3 Þ kwkH 1 ðR3 Þ ; 3
R
where c > 0 denotes a constant, is continuous.
/; w 2 H 1 ðR3 Þ;
56
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
The functional Z 1 U1 ðuÞ ¼ 0 ru ru dx 2 R3 is convex and finite-valued over the space H 1 ðR3 Þ, hence it is continuous. We conclude that functional U given by the formula (7) is continuous in the topology s, provided, that the functional L is also continuous in this topology. In the case where L is given by the formula (3), it is 3 3 3 sufficient to take b 2 L2 ðXÞ , t 2 L2 ðS1 Þ , E0 2 L2 ðXÞ . Thus we get Jh ðu; P; uÞ ¼ CðsÞ lim Je ðu; P; uÞ ¼ CðsÞ lim Ge ðu; PÞ þ Uðu; P; uÞ: e!0
e!0
Now we have to find the CðsÞ-limit of the sequence fGe ge>0 . In the case of quadratic piezoelectric energy function U L ðxe ; e; P; rPÞ we write (Nowacki, 1983): x 1 1 1 e e ; e; P; rP ¼ be0ij Pj;i þ aeij Pi Pj þ beijkl Pj;i Pl;k þ ceijkl eij ekl þ dijkl UeL Pj;i ekl þ fijk Pi ejk þ jeijk Pi Pk;j ; e 2 2 2
ð8Þ
where be0ij ðxÞ ¼ b0ij e ðxÞ dijkl
x ; ex
¼ dijkl
e
aeij ðxÞ ¼ aij e ðxÞ fijk
;
x
¼ fijk
e x e
x ; ceijkl ðxÞ ¼ cijkl ; e ex jeijk ðxÞ ¼ jijk ; x 2 X: e
beijkl ðxÞ ¼ bijkl
;
;
x
ð9Þ
We observe that for centrosymmetric materials fijk ¼ jijk ¼ 0. Then, however, a linear electromechanical coupling is still present via the coefficients dijkl . The material coefficients b0ij ðyÞ, aij ðyÞ, bijkl ðyÞ, cijkl ðyÞ, dijkl ðyÞ, fijk ðyÞ, jijk ðyÞ, y 2 Y , are Y-periodic. As usual in the case of periodic homogenization, Y denotes the basic cell (Bensoussan et al., 1978; Sanchez-Palencia, 1980; Attouch, 1984; Dal Maso, 1993; Lewi nski and Telega, 2000; Telega, 1991; Nelli Silva et al., 1999a,b). We assume that b0ij 2 L1 ðY Þ; . . . ; jijk 2 L1 ðY Þ. Thus, layered materials are not precluded. It is reasonable to assume that 9c1 > 0 U L ðy; e; P; QÞ P c1 ðjej2 þ jPj2 þ jQj2 ; y 2 Y for every e 2 E3s , P 2 R3 , Q 2 E3 . Here, E3 denotes the set of matrices 3 3, and E3s is the set of symmetric matrices 3 3. By applying Theorem 1.3.28 from Lewi nski and Telega (2000) we have Z Gh ðu; PÞ ¼ CðsÞ lim Ge ðu; PÞ ¼ UhL ðeðuðxÞÞ; PðxÞ; rPðxÞÞ dx; ð10Þ e!0
X
where UhL ðE; P; QÞ
¼ min
1 jY j
Z
L
y
U y; e ðvðyÞÞ þ ; P; ry qðyÞ þ Q dy j v 2 Y
where 2 E3s , P 2 R3 , Q 2 E3 , and n o 3 3 3 1 1 ðY Þ ¼ ½Hper ðY Þ ¼ w 2 H 1 ðY Þ j w is Y -periodic; hwi ¼ 0 : Hper
3 1 Hper ðY Þ ;
q2
3 1 Hper ðY Þ
;
ð11Þ
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
57
By h i we denote the mean value over the basic cell Y. The minimization problem appearing in (11) has a unique solution, which we denote by ðv; qÞ. In this way we may write " ! ! ! Z 1 o q 1 1 o q o q j j l b0ij ðyÞ UhL ð; P; QÞ ¼ þ Qji þ aij ðyÞPi Pj þ bijkl ðyÞ þ Qji þ Qlk jY j Y 2 2 oyi oyi oyk ! 1 oqj y y þ cijkl ðyÞðeij ðvÞ þ ij Þðekl ðvÞ þ kl Þ þ dijkl ðyÞ þ Qji ðeyij ðvÞ þ ij Þ 2 oyi # oqk y þ fijk ðyÞPi ðejk ðv þ ij Þ þ jijk ðyÞPi þ Qkj dy: ð12Þ oyj h h ; fijk and jhijk . On acLet us pass to the determination of the homogenized coefficients bh0ij ; bhijkl ; chijkl ; dijkl count of the linearity of the problem considered we conclude that
v ¼ vðmnÞ mn þ DðmnÞ Qmn þ nðmÞ Pm ;
ð13Þ
q ¼ WðmnÞ mn þ qðmnÞ Qmn þ jðmÞ Pm ; ðmnÞ
where the functions vi
ðmnÞ
; Di
ðmnÞ
; Wmn i ; qi
ðmÞ
; nðmnÞ , and ji i
are Y-periodic, i; m; n ¼ 1; 2; 3: We have
ðmnÞ ðmÞ o qj oWðmnÞ oqj ojj j ¼ mn þ Qnm þ Pm ; oyi oyi oyi oyi eyij ðvÞ ¼ eyij ðvðmnÞ Þmn þ eyij ðDðmnÞ ÞQnm þ eyij ðnðmÞ ÞPm :
ð14Þ
Taking into account (14) in (11), after lengthy calculations we arrive at the homogenized (effective) coefficients: * !+ ðijÞ L oU oq l h ¼ b0kl ðyÞ b0h þ dlj dki ; ð15Þ ij ¼ oQ oy ji ¼0;P¼0;Q¼0
ahij ¼
o2 UhL oPi oPj
k
* + + ðiÞ ðjÞ ojl ojðjÞ ojl y ðiÞ y ðiÞ y ðjÞ n ¼ haij i þ bklmn e ðn Þ þ hcklmn ekl ðn Þemn ðn Þi þ dklmn oyk oym oyk mn * + ðiÞ D E ojðjÞ ojl y ðjÞ ojðiÞ y y ðjÞ ðiÞ m m þ dklmn e ðn Þ þ hfilm elm ðn Þi þ fjlm elm ðn Þ þ jilm þ jjlm ; oyk mn oyl oyl *
o2 UhL oQlk oQji * !+ ðijÞ oqqðklÞ oqn ¼ bmnpq þ dni dmj þ dql dpk þ hcmnpq eymn ðDðijÞ Þeypq ðDðklÞ Þi oym oyp ðijÞ ðklÞ oqn oqn þ dmnpq þ dnj dmi eypq ðDðklÞ Þ þ dmnpq þ dnl dmk eypq ðDðijÞ Þ ; oym oym
ð16Þ
bhijkl ¼
ð17Þ
58
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59
chijkl
o2 UhL ¼ ¼ okl oij
*
oWðklÞ oWðijÞ q n bmnpq oym oyp *
þ
h dijkl
h fijk
o2 UhL ¼ ¼ okl oQji
þ hcmnpq ½eymn ðvðijÞ Þ þ dim djn ½eypq ðvðklÞ Þ þ dkp dlq i
oWðijÞ q þ dkm dln oyp
*
+ þ
dmnpq ½eymn ðvðijÞ Þ
oWðklÞ q þ dim djn oyp
! + oqðklÞ oWðijÞ q n bmnpq þ dlq dkp þ hcmnpq ½eymn ðvðklÞ Þ þ dkp dlq eypq ðDðijÞ Þi oyp oym + * ðijÞ oqn oWðklÞ ðijÞ y ðklÞ y n e ðD Þ ; þ dmnpq þ dim djn ½epq ðv Þ þ dkp dlq þ dmnpq oym oym pq
+ ;
ð18Þ
*
*
o2 UhL ¼ ¼ ojk oPi
dmnpq ½eymn ðvðklÞ Þ
+
ðjkÞ ojðiÞ oWq bmnpq oym oyp *
ð19Þ
+ þ hcmnpq ½eymn ðvðjkÞ Þ þ djm dkn eypq ðnðiÞ Þi
ojðiÞ oWðjkÞ n ey ðnðiÞ Þ þ þ djp dks n þ dmnpq oym oym pq * + oWnðjkÞ y ðjkÞ þ hfimn ½emn ðv Þ þ djm dkn i þ jimn ; oym
+
dmnpq ½eypq ðvðjkÞ
jhijk
o2 UhL ¼ ¼ oQkj oPi
*
ð20Þ
ðiÞ + ojq oqðkjÞ n bmnpq þ djn dkm þ hcmnpq eymn ðnðiÞ Þeypq ðDðjkÞ Þi oym oyp ðjkÞ oqn ojðiÞ þ dmnpq þ dkn djm eypq ðnðiÞ Þ þ dmnpq n eypq ðDðjkÞ Þ oym oym ðjkÞ oqn þ hfimn eymn ðDðjkÞ Þi þ jimn þ dkn djm : oym
ð21Þ
Now we have to determine the Y -periodic functions WðijÞ , qðijÞ , jðiÞ , vðijÞ , DðijÞ , and nðiÞ . The minimization problem appearing in the r.h.s. of (11) is a convex problem, which is equivalent to ! ! Z Z Z Z o qj o qj oql y y 0 oqj bij dy þ bijkl dy þ cijkl ½eij ðvÞ þ ij ekl ðvÞ dy þ dijkl þ Qji þ Qji eykl ðvÞ dy oyi oyi oyk oyi Y Y Y Y Z Z Z Z oqj oqk y y y dy þ fijk Pi ejk ðvÞ dy þ jijk Pi ejk ðvÞ dy þ jijk Pi dy ¼ 0 ð22Þ þ dijkl ½ekl ðvÞ þ kl oy oyj i Y Y Y Y 1 for every q; v 2 Hper ðY Þ3 and for every 2 E3s ; Q 2 E3 ; P 2 R3 . The periodic functions WðmnÞ ; vðmnÞ ; . . . ; nðmÞ are solutions to the following three variational problems. 3
3
1 1 Problem P1 : Find functions WðmnÞ 2 Hper ðY Þ ; vðmnÞ 2 Hper ðY Þ ðm; n ¼ 1; 2; 3Þ such that ) Z ( ðmnÞ oWj oql 3 y ðmnÞ 1 bijkl dy ¼ 0 8q 2 Hper ðY Þ ; þ dklij ½eij ðv Þ þ dim djn oy oyk i Y
Z ( Y
)
ðmnÞ
oWj dijkl oyi
þ
cijkl ½eyij ðvðmnÞ Þ
þ dim djn eykl ðvÞ dy ¼ 0
3
1 8v 2 Hper ðY Þ :
J.J. Telega, S. Bytner / Mechanics Research Communications 29 (2002) 53–59 3
59
3
1 1 Problem P2 : Find functions qðmnÞ 2 Hper ðY Þ ; DðmnÞ 2 Hper ðY Þ such that ( ! ) Z ðmnÞ oqj oql 3 1 þ dim djn þ dklij eyij ðDðmnÞ Þ bijkl dy ¼ 0 8q 2 Hper ðY Þ ; oy oy i k Y
Z ( dijkl Y
)
!
ðmnÞ
oqj yi
þ dim djn
þ
cijkl eyij ðDðmnÞ Þ
eYkl ðvÞ dy ¼ 0
3
1 8v 2 Hper ðY Þ :
1 1 Problem P3 : Find functions jðmÞ 2 Hper ðY Þ3 ; nðmÞ 2 Hper ðY Þ3 such that ) Z ( ðmÞ ojj oql y ðmÞ 1 þ dklij eij ðn Þ þ jmkl bijkl dy ¼ 0 8q 2 Hper ðY Þ3 ; oy oy i k Y
Z ( Y
) ðmÞ ojj dijkl þ cijkl eyij ðnðmÞ Þ þ fmkl ekkl ðvÞ dy ¼ 0 oyi
3
1 8v 2 Hper ðY Þ :
3. Final remarks The assumption about periodicity of moduli can be weakened. One can consider the so-called nonuniform homogenization, cf. (Lewi nski and Telega, 2000). In such a more general case the homogenized h h h moduli b0h ij ; aij ; bijkl ; . . . ; jijk depend additionally on macroscopic variable x 2 X. More detailed presentation and examples will be given in a separate paper. Acknowledgements The first author was supported by the State Committee for Scientific Research (KBN, Poland) through the grant no. 7 T 07A 043 18. References Attouch, H., 1984. Variational Convergence for Functions and Operators. Pitman, Boston. Bensoussan, A., Lions, J.L., Papanicolau, G., 1978. Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam. Bytner, S., Gałka, A., Gambin, B., 2001. Effective properties of layered piezoelectric material with gradient of polarization. J. Tech. Phys. 42, 437–447. Dal Maso, G., 1993. An Introduction to C-convergence. Birkh€auser, Boston. Lewi nski, T., Telega, J.J., 2000. Plates, Laminates and Shells Asympotic Analysis and Homogenization. World Scientific, Singapore. Mead, C.A., 1962. Electron transport mechanism in thin insulating films. Phys. Rev. 128, 2088–2095. Mindlin, R.D., 1968. Polarization gradient in elastic dielectrics. Int. J. Solids Struct. 4, 637–652. Nelli Silva, E.C., Nishiwaki, S., Kikuchi, N., 1999a. Design of piezoelectric materials and piezoelectric transducers using topology optimization. Part II. Arch. Comp. Meth. Eng. 6, 191–222. Nelli Silva, E.C., Ono Fonseca, J.S., Montero de Espinosa, F., Crumm, A.T., Brady, G.A., Halloran, J.W., Kikuchi, N., 1999b. Design of piezocomposite materials and piezoelectric transducers using topology optimization. Part I. Arch. Comp. Meth. Eng. 6, 117–182. Nowacki, W., 1983. Electromagnetic Effects in Deformable Solids. PWN, Warsaw (in Polish). Sanchez-Palencia, E., 1980. In: Nonhomogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin. Telega, J.J., 1991. Piezoelectricity and homogenization. Application to biomechanics. In: Maugin, G.A. (Ed.), Continuum Models and Discrete Systems, vol. 2. Longman, Harlow, Essex, pp. 220–229. Telega, J.J., Gałka, A., Gambin, B., 1998. Effective properties of physically nonlinear piezoelectric composites. Arch. Mech. 50, 321– 340.