A Thermopiezoelectric Mixed Variational Theorem for smart multilayered composites

Computers and Structures 83 (2005) 1266–1276 www.elsevier.com/locate/compstruc

A Thermopiezoelectric Mixed Variational Theorem for smart multilayered composites Ayech Benjeddou *, Orlando Andrianarison Institut Supe´rieur de Me´canique de Paris, LISMMA-Structures, 3 rue Fernand Hainault, 93407 Saint Ouen Cedex, France Accepted 27 August 2004 Available online 2 March 2005

Abstract This work extends for thermopiezoelectric media the Piezoelectric Mixed Variational Theorem (PMVT), proposed recently by the authors, by adding the transverse thermal field-temperature increment relation as a constraint via a Lagrange multiplier. The latter is shown to be the transverse (normal) heat flux, which continuity can now be fulfilled in a natural way as is the case for the transverse stresses and transverse electric field in the PMVT. Hence the resulting Thermopiezoelectric Mixed Variational Theorem (TMVT) will be well suited for implementing new accurate and fully coupled analytical or closed-form and numerical, such as finite element, solutions for thermopiezoelectric multilayered smart composites. Mixed thermopiezoelectric constitutive equations to be used in conjunction with the new TMVT are developed and guidelines for their numerical implementation are also given. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Thermopiezoelectric; Mixed variational theorem; Multilayered composites; Smart structures; Variational formulations

1. Introduction Development of thermopiezoelectric variational principles, as generalisations of either HamiltonÕs principle or the virtual work principle (VWP), has started in the early 1970s mainly for the free-vibration analysis of piezoelectric crystals [1,2]. In particular, Mindlin [2] has derived a three-field thermopiezoelectric variational principle from the conservation of energy in thermopiezoelectricity. It operates on the mechanical displacements, the electric potential, and the temperature. Do¨kmeci [3,4] was the first to present a unified mixed variational principle that produces the complete set of the 3D fundamental linear * Corresponding author. Tel.: +33 1 4945 2979; fax: +33 1 4945 2929. E-mail address: [email protected] (A. Benjeddou).

equations of linear thermopiezoelectricity. He has then extended recently his works to the non-linear thermopiezoelectricity in relation to high frequency vibrations of ceramic materials subjected to strong electric fields and large deflection [5]. These variational principles have been derived using the virtual work principle. An alternative to the energy and VWP approaches is the inverse method proposed recently by He [6]. The latter has derived, without using Lagrange multipliers, HamiltonÕs type thermopiezoelectric variational principles that are equivalent to the generalised thermoelasticity theory for piezoelectric media of Chandrasekhariah [7] which uses the heat flux as in independent variable. Emergence of the relatively new field of smart structures, based on piezoelectric materials, by the end of the seventies has reactivated the researches on thermopiezoelectric composites. In particular, since the 1990s a lot of

0045-7949/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.08.029

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

literature has been published on thermopiezoelectric smart structures and materials as attested by the latest assessment [8], review [9] and survey [10,11] papers. Careful analysis of the latter show that the use of mixed (hybrid) variational principles for analytical or modelling of piezoelectric-based smart structures has received great interest during the last half decade (see [12,13] for a review of this matter). However, to the authors knowledge, above thermopiezoelectric mixed variational principles have never been used. This may be due to the fact that they require assumptions of full mechanical stress, electric displacement and heat flux components. Assumptions of full stresses can be reduced through the use of ReissnerÕs Mixed Variational Theorem (RMVT) [14,15] which restricts them to some components only; i.e. the transverse ones. This allows to fulfil, in a natural way, the interface continuity conditions in multilayered composite structures. RMVT application to thermoelastic multilayered composites has been assessed recently by Carrera [16,17] who has also exploited the RMVT concept for piezoelectric media [18,19]. Constraint of the electric field-potential relation has been proposed first by the authors [12,13] through their Piezoelectric Mixed Variational Theorem (PMVT) which can be seen as the analogue, for piezoelectric media, of RMVT for elastic media. The present work aims to extend, for thermopiezoelectric media, the authors PMVT [12,13] by adding to it the transverse thermal field-temperature relation as a constraint through a Lagrange multiplier. The latter is shown to be the transverse (normal) heat flux. Hence, following ReissnerÕs idea [14,15], this restricts the assumptions of the full heat flux vector to some components also; i.e. the transverse one. As a consequence, the transverse heat flux continuity condition at the interfaces of a layered thermopiezoelectric composite can now be fulfilled in a natural way as is the case for transverse stresses and transverse electric field in the PMVT. The resulting new Thermopiezoelectric Mixed Variational Theorem (TMVT) will be then well suited for the analytical and numerical analyses of thermopiezoelectric multilayered composites. In the following, after recalling the fundamental 3D linear thermopiezoelectric equations, the TMVT is demonstrated; then, the mixed thermopiezoelectric constitutive equations to be used in conjunction with it are derived; finally, guidelines for the numerical implementation of the new TMVT and associated new mixed constitutive equations, are given.

2. Fundamental thermopiezoelectric linear equations On account of the classical electromagnetic field and thermoelastic theories, the three-dimensional (3D) linear

1267

fundamental equations for a thermopiezoelectric body of volume X can be summarised as follows [2,4]. 2.1. Divergence equations These equations relate to the linear stress equations of motion, the linear charge equation of electrostatics and the thermal energy balance equation rij;j þ fi ¼ q€ ui

in X

ð1aÞ

Di;i  q ¼ 0 in X

ð1bÞ

hi;i  s ¼ H0 g_ in X

ð1cÞ

where rij stand for the symmetric Cauchy stress tensor components, Di for the electric displacement vector components, hi for the components of heat flux vector. g, H0 are, respectively, the entropy density and the constant positive reference temperature. fi, q and s are body mechanical forces, electric charge and heat source in X, respectively. Notice that standard tensors notation is used with Latin indices running from 1 to 3 and Greek ones from 1 to 2 only. They obey to EinsteinÕs summation convention when repeated. Superposed dots and commas stand for time differentiation and partial one with respect to Cartesian co-ordinates x1, x2, x3 of the 3D Euclidean space. 2.2. Gradient equations The gradient relations correspond to the linear strain-mechanical displacement, linear electric field-electric potential and linear thermal field-temperature change relations. They state as ekl ¼ 12ðuk;l þ ul;k Þ in X

ð2aÞ

Ek ¼ u;k

ð2bÞ

ek ¼ h;k

in X in X

ð2cÞ

where ekl, Ek, ek, u, h are, respectively, the components of the symmetric Lagrange strain tensor, quasi-static electric field vector, thermal field vector, electric potential and temperature change from the reference one H0. 2.3. Constitutive equations In thermodynamics, a fundamental hypothesis is that the state of a material is entirely determined by the values of a certain set of independent variables: the kinematic variables (strains eij, . . .) and the temperature. In addition, the constitutive behaviour is completely defined by the specification of two state functions: a potential energy function of the aforementioned primitive state variables and a dissipation function [4,5].

1268

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

The quadruplet (eij, Ei, ei, h) is chosen here as primal state variables so that the associated conjugate state variables are (rij, Di, hi, g). As a consequence, the constitutive equations are rij ¼

oP oeij

ð3aÞ

in X

i on Su so support imposed mechanical displacements u that Su [ SF = S and, Su \ SF = ;, an electric potential  on Su so that Su [ SQ = S and Su \ SQ = ; and a temu perature  h on Sh so that Sh [ SH = S and Sh \ SH = ;. The mechanical, electric and thermal boundary conditions can then be written as rij nj ¼ F i

oP Di ¼  oEi

in X

ð3bÞ

oP oei

in X

ð3cÞ

oP g¼ oh

in X

hi ¼ 

ð3dÞ

Di ni ¼ Q hi ni ¼ H

ui ¼  ui

on S F ;

 u¼u

on S Q ; on S H ;

Pðeij ; Ei ; ei ; hÞ ¼ Gðeij ; Ei ; hÞ  F ðei Þ

ui ðt0 Þ ¼  u0i ;

G ¼ 12cijkl eij ekl  122ij Ei Ej  eijk eij Ek  12ah2  pi hEi  kij eij h ð5Þ where cijkl, eijk, 2ij are the elastic, piezoelectric-stress and dielectric material constants whereas a, pi, kij are thermal expansion, pyroelectric and thermal stress-temperature material constants. The dissipation function, in Eq. (4), has the following quadratic form F ¼ 12jij ei ej

ð6Þ

with jij denoting the coefficients of heat conduction. Consequent upon Eqs. (4)–(6), the constitutive equations (3) write rij ¼ cijkl ekl  ekij Ek  kij h in X

ð7aÞ

Di ¼ eikl ekl þ 2ik Ek þ pi h

ð7bÞ

in X

g ¼ kkl ekl þ pk Ek þ ah in X

ð7cÞ

hi ¼ jij ej

ð7dÞ

in X

Thanks to the introduction of the temperature gradient, Eq. (2c), the heat conduction relation, Eq. (7d), looks uncoupled from Eqs. (7a)–(7c). 2.4. Boundary conditions The regular boundary oX = S can be loaded with mechanical surface forces Fi on SF, electric surface charge Q on SQ, and a heat flux H on SH. It can also

ð8aÞ ð8bÞ ð8cÞ

where, ni denotes the components of the outward unit normal vector and a bar is used for prescribed quantities. 2.5. Initial conditions

By expanding G with respect to the small quantities (eij,Ei,h), a quadratic form of the electric Gibbs function is obtained

on S u

h¼ h on S h

where use is made here of the so-called thermopiezoelectric potential P which can be expressed in term of the electric Gibbs function G and the dissipation function F by ð4Þ

on S u

The initial conditions to be used for unsteady analysis can be written as 0

_ u_ i ðt0 Þ ¼ u i

in Xðt0 Þ

ð9aÞ

0 uðt0 Þ ¼ u

in Xðt0 Þ

ð9bÞ

hðt0 Þ ¼ H0

in Xðt0 Þ

ð9cÞ

where the superscript 0 states for initial values, at time t0, of the mechanical displacements and velocities, electric potential and temperature. 2.6. Continuity conditions Since a laminated thermopiezoelectric medium is of interest here, perfect bonding at interfaces between laminae is supposed so that the following equations that account for continuity conditions must be verified ½rij nj ¼ 0;

½ui  ¼ 0 on S k;kþ1

ð10aÞ

½Di ni ¼ 0;

½u ¼ 0 on S k;kþ1

ð10bÞ

½hi ni ¼ 0;

½h ¼ 0 on S k;kþ1

ð10cÞ

where the brackets denote the jump, at the interface surface Sk,k+1 between layers k and k + 1, of the enclosed quantities.

3. Thermopiezoelectric mixed variational theorem The set of equations (1)–(7) completely describe the linear motions of a thermopiezoelectric body submitted to the mixed boundary conditions Eq. (8) and initial conditions Eq. (9). The purpose of this section is to express these fundamental equations in a variational form in order to obtain approximate solutions. The VWP is taken as a basis for deriving the weak formulation

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

associated to the local equations (1)–(9). The Thermopiezoelectric Virtual Work Principle (TVWP), which extends the classical Virtual Displacement Principle (VDP) to thermopiezoelectric media, can be stated for Eqs. (1)–(9) as Z Z Z deij rij dX þ dui ðq€ui  fi ÞdX  dui F i dS X

X

Z



dEi Di dX þ

Z

X

Z



duq dX þ

X

dei hi dX 

Z

X

SF

Z

Z

deij rij dX þ X

þ

dhðH0 g_  sÞdX 

þ

Z

X

dhH dS ¼ 0 SH



Z dui F i dS SF

 ½ðdua;3 þ du3;a Þ  dca3 la3 þ ðdu3;3  de33 Þl33 dX

X

 dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðdu3;3  de33 Þ dX X Z Z Z duQdS  dEi Di dX þ duqdX þ þ

duQ dS SQ

dui ðq€ ui  fi ÞdX 

X

Z Z

Z

1269



Z Z



X

X

X

SQ

Z

ðdu;3 þ dE3 ÞkdX þ dkðu;3 þ E3 ÞdX X Z Z 0 dei hi dX  dhðH g_  sÞdX  dhH dS ¼ 0

X

X

SH

ð11Þ

ð13Þ

where the gradient and constitutive equations (2) and (7), and the essential boundary conditions (8)2 are constraints to be added to (11). One simple way to deal with variational constraints consists of using the well known Lagrange multipliers method. Hence, constraining the mechanical stress–strain gradient relation (2a), but relative to the transverse strains only, via Lagrange multipliers li3, transforms Eq. (11) to Z Z Z deij rij dX þ dui ðq€ ui  fi ÞdX  dui F i dS

It can be shown that the Lagrange multipliers k and li3 correspond, respectively, to the transverse electric displacement and transverse stresses (see the demonstration below). With this in mind, a new five-field partial mixed variational equation results. It can be seen as an application of the PMVT [12,13] to Eq. (11) and can be applied for refining layered thermopiezoelectric models in order to fulfil the transverse stresses and electric displacement continuity through the interfaces. The second new idea proposed here consists of constraining mechanical, electric and thermal gradient relations (2a)–(2c), but for the transverse strain, electric and thermal fields components only, via the Lagrange multipliers li3, k and v. This transforms Eq. (11) to Z Z Z deij rij dX þ dui ðq€ ui  fi ÞdX  dui F i dS X X SF Z   ½ðdua;3 þ du3;a Þ  dca3 la3 þ ðdu3;3  de33 Þl33 þ ZX   þ dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðu3;3  e33 Þ dX Z Z ZX duQ dS  dEi Di dX þ duq dX þ

X

þ

Z

X

SF



 ½ðdua;3 þ du3;a Þ  dca3 la3 þ ðdu3;3  de33 Þl33 dX



 dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðu3;3  e33 Þ dX

X

þ

Z

X



Z

dEi Di dX þ

Z

X



Z X

duqdX þ X

dei hi dX 

Z X

Z duQdS SQ

dhðH0 g_  sÞdX 

Z

dhH dS ¼ 0

SH

ð12Þ

X

where ca3 = 2ea3 and the in-plane strain (2a), electric and thermal fields (2b), (2c) gradient relations, constitutive equation (7) and essential boundary conditions (8)2 are still to be added to Eq. (12). Usual procedures of variational calculus allow us to identify the Lagrange multipliers li3 as the transverse stresses (see the demonstration below). With this in mind, a new four-fields partial mixed variational equation is constructed. It can be seen as an application of the RMVT to Eq. (11) and can be applied for refining layered thermopiezoelectric models in order to fulfil the transverse stresses continuity at the interfaces. Another idea proposed here is to apply ReissnerÕs idea also to the electric part of the Eq. (12). This can be achieved by constraining the gradient relations (2b), but for the transverse electric field only, via a Lagrange multipliers k, so that Eq. (12) becomes

Z

X

Z

SQ

ðdu;3 þ dE3 Þk dX þ dkðu;3 þ E3 ÞdX X Z Z 0 dhH dS  dei hi dX  dhðH g_  sÞdX  X X SH Z Z ð14Þ þ ðdh;3 þ de3 Þv dX þ dvðh;3 þ e3 ÞdX ¼ 0

þ

ZX

X

X

The Lagrange multipliers li3, k and v being identified as the transverse stresses, electric displacement and heat flux, this equation will result in a new six-field variational one, called the Thermopiezoelectric Mixed Variational Theorem (TMVT). It can be applied for refining layered thermopiezoelectric models in order to fulfil the transverse stresses, electric displacement and heat flux continuity conditions at the interfaces. To identify the aforementioned Lagrange multipliers and to show that above local thermopiezoelectric

1270

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

equations are fulfilled, in-plane and transverse summations in Eq. (14) are separated so that it now reads Z Z deab rab dX þ dua ðq€ua  fa ÞdX X Z XZ Z dua F a dS þ dca3 ra3 dX þ de33 r33 dX  X X ZS F Z þ du3 ðq€u3  f3 ÞdX  du3 F 3 dS SF ZX   ½ðdua;3 þ du3;a Þ  dca3 la3 þ ðdu3;3  de33 Þl33 dX þ ZX   dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðu3;3  e33 Þ dX þ Z Z ZX  dEa Da dX  dE3 D3 dX þ duqdX X Z X Z ZX duQdS þ ðdu;3 þ dE3 ÞkdX þ dkðu;3 þ E3 ÞdX þ X ZX Z ZS Q 0  dea ha dX  de3 h3 dX  dhðH g_  sÞdX X ZX ZX dhH dS þ ðdh;3 þ de3 ÞvdX  X ZS H þ dvðh;3 þ e3 ÞdX ¼ 0 ð15Þ X

It is worth noting that constitutive equations and essential boundary conditions are still to be added to Eq. (15); dca3, de33, dla3, dl33, dE3, dk, dv are arbitrary whereas dua, du and dh obey the in-plane gradient relations (2a)–(2c), so that deab ¼ 12ðdua;b þ dub;a Þ in X

ð16aÞ

dEa ¼ du;a

ð16bÞ

dea ¼ dh;a

in X

ð16cÞ

in X

Using these relations for the virtual in-plane strain, electric field and heat flux components, Eq. (15) writes Z Z Z dua;b rab dX þ dua ðq€ua  fa ÞdX  dua F a dS X Z X SF Z þ dca3 ðra3  la3 ÞdX þ de33 ðr33  l33 ÞdX Z X ZX du3 F 3 dS þ du3 ðq€u3  f3 ÞdX  SF ZX Z Z þ dua;3 la3 dX þ du3;a la3 dX þ du3;3 l33 dX X X ZX   dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðu3;3  e33 Þ dX þ Z Z ZX þ du;a Da dX  dE3 ðD3  kÞdX þ duqdX X Z X Z ZX duQdS þ du;3 kdX þ dkðu;3 þ E3 ÞdX þ X ZS Q ZX Z þ dh;a ha dX  de3 ðh3  vÞdX  dhðH0 g_  sÞdX X ZX Z X Z  dhH dS þ dh;3 vdX þ dvðh;3 þ e3 ÞdX ¼ 0 SH

X

X

ð17Þ

After integrating by parts the terms containing differentiation of virtual displacements, electric potential and temperature, the following equation is obtained Z Z  dua rab;b dX þ dua rab nb dS X S Z Z þ dua ðq€ ua  fa ÞdX  dua F a dS SF Z ZX þ dca3 ðra3  la3 ÞdX þ de33 ðr33  l33 ÞdX Z X Z ZX u3  f3 ÞdS  du3 F 3 dS  dua la3;3 dX þ du3 ðq€ SF Z Z X ZX  du3 la3;a dX  du3 l33;3 dX þ dua la3 n3 dS S ZX ZX þ du3 la3 na dS þ du3 l33 n3 dS S ZS   þ dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðu3;3  e33 Þ dX Z Z ZX  duDa;a dX þ duDa na dS  dE3 ðD3  kÞdX X ZX Z S Z þ duqdX þ duQdS  duk;3 dX SQ X Z Z ZX þ dukn3 dS þ dkðu;3 þ E3 ÞdX  dhha;a dX ZX ZX ZS þ dhha na dS  de3 ðh3  vÞdX  dhðH0 g_  sÞdX X ZS Z X Z  dhH dS  dhv;3 dX þ dhvn3 dS X S ZS H þ dvðh;3 þ e3 ÞdX ¼ 0 ð18Þ X

Grouping together all the terms relative to the same virtual variables, the previous equation is turned into Z  dua ½rab;b þ la3;3  ðq€ ua  fa ÞdX X Z Z dua F a dS þ dua ðrab nb þ la3 n3 ÞdS  SF ZS u3  f3 ÞdX  du3 ½la3;a þ l33;3  ðq€ Z ZX du3 F 3 dS þ du3 ðla3 na þ l33 n3 ÞdS  Z SF ZS þ dca3 ðra3  la3 ÞdX þ de33 ðr33  l33 ÞdX X ZX   dla3 ½ðua;3 þ u3;a Þ  ca3  þ dl33 ðu3;3  e33 Þ dX þ Z ZX  duðDa;a þ k;3  qÞdX þ duðDa na þ kn3 ÞdS S Z Z ZX duQdS  dE3 ðD3  kÞdX þ dkðu;3 þ E3 ÞdX þ X Z X ZS Q 0  dhðha;a þ v;3 þ H g_  sÞdX þ dhðha na þ vn3 ÞdS S ZX Z  dhH dS  de3 ðh3  vÞdX X ZS H ð19Þ þ dvðh;3 þ e3 ÞdX ¼ 0 X

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

Since dca3, de33, dla3, dl33, dE3, dk, de3 and dv are arbitrary, their corresponding coefficients vanish, leading to the following relations la3 ¼ ra3

in X

ð20aÞ

l33 ¼ r33

in X

ð20bÞ

ca3 ¼ ua;3 þ u3;a

ð20cÞ

in X



1271

Z

dua ðrab;b þ ra3;3 þ fa  q€ ua ÞdX Z dua ðrab nb þ ra3 n3  F a ÞdS þ X

SF



Z

du3 ðra3;a þ r33;3 þ f3  q€ u3 ÞdX

X

þ

Z

du3 ðra3 na þ r33 n3  F 3 ÞdS

SF

e33 ¼ u3;3 k ¼ D3

ð20dÞ

in X

v ¼ h3

ð20fÞ

in X

in X

e3 ¼ h;3

in X

Z

duðDa;a þ D3;3  qÞdX

X

ð20eÞ

in X

E3 ¼ u;3

 þ

Z

duðDa na þ D3 n3 þ QÞdS

SQ

ð20gÞ



ð20hÞ

þ

Z

dhðha;a þ h3;3 þ H0 g_  sÞdX

X

Z

dhðha na þ h3 n3  H ÞdS ¼ 0

ð22Þ

SH

It is clear from Eqs. (20a), (20b), (20e) and (20g) that the Lagrange multipliers li3, k and v identify themselves, respectively, to the transverse stresses ri3, electric displacement D3 and heat flux h3 components. Eqs. (20c), (20d), (20f) and (20h) complement the in-plane gradient relations (16a)–(16c), respectively. Substituting back Eqs. (20a)–(20h) into Eq. (19) transforms it to Z  dua ½rab;b þ ra3;3 þ fa  q€ua dX X

þ  þ  þ  þ

Z

Z Z Z Z Z Z

du3 ½ra3;a þ r33;3 þ f3  q€u3 dX X

du3 ðra3 na þ r33 n3 ÞdS S

duðDa;a þ D3;3  qÞdX X

in X

ð23aÞ

ra3;a þ r33;3 þ f3 ¼ q€ u3

in X

ð23bÞ

Da;a þ D3;3  q ¼ 0 in X

ð23cÞ

ha;a þ h3;3  s ¼ H0 g_ in X

ð23dÞ

duðDa na þ D3 n3 ÞdS S

þ dhðha;a þ h3;3 þ H g_  sÞdX

X

dhðha na þ h3 n3 ÞdS 

Z

du3 F 3 dS þ

Z

Z

Z

SQ

dhðha na þ h3 n3  H ÞdS ¼ 0

ð24Þ

SH

0

SF



rab;b þ ra3;3 þ fa ¼ q€ ua

These equations are identical to the divergence ones ((1a–c)) where, as can be seen, in-plane and transverse contributions are separated. Substituting Eqs. (23a)– (23d) back into Eq. (22) gives Z dua ðrab nb þ ra3 n3  F a ÞdS SF Z du3 ðra3 na þ r33 n3  F 3 ÞdS þ S ZF duðDa na þ D3 n3 þ QÞdS þ

dua ðrab nb þ ra3 n3 ÞdS S

S



With dua = du3 = 0 on SF, du = 0 on SQ and dh = 0 on SH, their corresponding coefficients in Eq. (22) vanish, leading to

Due to the arbitrariness of dua, du3, du and dh, the following relations are obtained from Eq. (24)

Z dua F a dS SF

duQ dS

rab nb þ ra3 n3 ¼ F a

on S F

ð25aÞ

ra3 na þ r33 n3 ¼ F 3

on S F

ð25bÞ

Da na þ D3 n3 ¼ Q

on S Q

ð25cÞ

SQ

dhH dS ¼ 0

ð21Þ

SH

Supposing dui = 0 on Su, du = 0 on Su and dh = 0 on Sh, the previous equation reduces to

ha na þ h3 n3 ¼ H

on S H

ð25dÞ

Eqs. (25ab), (25c) and (25d) correspond to the boundary conditions (8a)1, (8b)1 and (8c)1 respectively.

1272

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

The new Thermopiezoelectric Mixed Variational Theorem (TMVT) can now be formulated, from Eqs. (17), (20a), (20b), (20e), (20g), as Z Z Z dua;b rab dX þ ðdua;3 þ du3;a Þra3 dX þ du3;3 r33 dX X X X Z Z þ dr33 ðu3;3  e33 ÞdX þ dra3 ½ðua;3 þ u3;a Þ  ca3 dX X X Z Z Z dui F i dS þ du;a Da dX þ dui ðq€ui  fi ÞdX  X

X

SF

Z du;3 D3 dX þ dD3 ðu;3 þ E3 ÞdX þ duqdX X X X Z Z Z duQdS þ dh;a ha dX þ dh;3 h3 dX þ þ

Z

Z

X

SQ

þ

Z

dh3 ðh;3 þ e3 ÞdX 

X



X

Z

dhðH0 g_  sÞdX X

Z

dhH dS ¼ 0

ð26Þ

SH

4. Mixed thermopiezoelectric constitutive equations

where in-plane stresses, electric displacement and heat flux, as well as transverse strains, electric and thermal fields are to be computed from a special mixed thermopiezoelectric constitutive equations to be presented in the subsequent section. However, transverse stresses, electric displacement and heat flux, as well as the mechanical displacements, electrical potential and temperature increment are to be assumed in the framework of this TMVT. The above TMVT can also be written in the following matrix form Z Z Z T T C GT M G C deG r dX þ de r dX þ drM p t p t t ðet  et ÞdX X

þ

Z

X

duT ðq€u  fÞdX 

X



Z

M dEG t Dt dX 

X

þ

Z

duQ dS 

SQ



Z

Z

Z

duT F dS 

Z

G C dDM t ðE t  Et ÞdX þ

X T

C deG p hp dX 

T

C dEG p Dp dX

X

SF

G C dhM t ðet  et ÞdX 

Z

X

Z

X

X



potential and increment temperature. Underlined bold quantities indicate vectors whereas non-underlined ones indicate scalars. Notice that, in this approach, the in-plane strain-displacement, electric in-plane field-potential and thermal in-plane field-temperature relations, essential boundary conditions (8)2, and the mixed thermopiezoelectric constitutive equations, to be developed in the subsequent section, are the subsidiary conditions (satisfied exactly); whereas, the remaining governing equations are weakly fulfilled by the TMVT, i.e. divergence equation (1), natural boundary conditions (8)1, and the transverse straindisplacement, electric transverse field-potential and transverse thermal vector-temperature relations. These equations will be satisfied approximately by the adopted numerical method, such as the FE method, implementing the TMVT.

Z duq dX X

Z

In order to facilitate the derivation of the mixed linear thermopiezoelectric constitutive equations, required by the above TMVT, the following enlarged displacement, strain, stress and load vectors are introduced       u e r ; S¼ U¼ ; T¼ ; u E D ð28Þ     f F g¼ ; G¼ q Q where, the negative sign before the electric field, in the enlarged strain, is introduced in order to get symmetric piezoelectric constitutive equations. With these notations, the TMVT can also be written in matrix form as Z Z Z T T GT M G C C dSG T dX þ dS T dX þ dTM p t p t t ðSt  St ÞdX X



M deG t ht dX



dhðH0 g_ C  sÞdX

dUT g dX 

Z

Z

ð27Þ

þ

Z

T

C deG p hp dX 

dhs dX 

X

SH

where the subscripts p and t stand for in-plane and transverse quantities, respectively; the superscripts G, C, M and T denote Gradient, Constitutive, Mixed and Transpose operation, respectively. The former two ones indicate that the relative quantities fulfil the gradient relations (2a)–(2c) and the constitutive equations respectively. The latter have to be in a mixed form and are detailed in the next section. Mixed (M) quantities are to be assumed as should be done for the displacement, electric

dUT G dS þ

Z SH

Z

X

X

S

X

X

dhH dS ¼ 0

X

X

X

Z

Z

Z

M deG t ht dX  X

dhH dS 

€ dX dUT qU Z

G C dhM t ðet  et ÞdX X

Z

dhH0 g_ C dX ¼ 0 X

ð29Þ With notations (28) and engineering ones (using two indices), the constitutive equations (7a)–(7c) can be written in the system frame reference, as [12,13,20] T ¼ C S  A h

ð30aÞ

g ¼ A T S þ ah

ð30bÞ

h ¼ j e

ð30cÞ

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

1273

where

Notice that matrix C* is symmetric and can describe all three cases of thermopiezoelectric materials poled separately in the 1, 2, 3 or in-plane material directions. It can be applied for unidirectional composites with elastic or piezoelectric fibres embedded in the (1, 2) plane. When decomposed into in-plane and transverse contributions, Eqs. (30a)–(30c) become, respectively,   " C C #    Ap Tp Sp pp pt ¼  h ð31aÞ A t Tt S C T C t pt tt  g¼ 

hp ht

A p A t

T  "

 ¼

Sp St



ð31bÞ

þ ah

j pp

j pt

j T pt

j tt

#

ep et

ð31cÞ

T Tp ¼ h r11 r22 r12 jD1 D2 i; TTt ¼ h r13 r23 r33 jD3 i; hTp ¼ h h1

STt

¼ h c13

e22 c23

Tp ¼ C pp Sp þ C pt St  A p h Tt ¼ C T pt Sp þ Ctt St  At h T g ¼ A T p Sp þ At St þ ah

hp ¼ j pp ep þ j pt et



where

S Tp ¼ h e11

Eqs. (31a)–(31c) can be written in explicit form, respectively, as

c12 j  E1

E2 i;

e33 j  E3 i;

eTp

¼ h e1

h2 i;

ht ¼ h3

ht ¼ j T pt ep þ jtt et

et ¼ e3

ð32bÞ

ð32cÞ

The transverse enlarged strains and temperature gradient are first extracted from the second lines of Eqs. (32a) and (32c), then substituted in their first lines and in Eq. (32b) so that Eqs. (32a)–(32c) become, respectively, Tp ¼ Cpp Sp þ Cpt Tt  Ap h

e2 i;

ð32aÞ

St ¼ CTpt Sp þ Ctt Tt þ At h

ð33aÞ

1274

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

g ¼ ATp Sp þ ATt Tt þ Ah h

ð33bÞ

hp ¼ jpp ep þ jpt ht

ð33cÞ

et ¼ jTpt ep þ jtt ht where T Cpp ¼ C pp  C pt C 1 tt Cpt ;

Ap ¼ A p  C pt C 1 tt At ;

Cpt ¼ C pt C 1 tt ;

Ctt ¼ C 1 tt

At ¼ C 1 tt At ;

1 Ah ¼ a þ A T t Ctt At T jpp ¼ j pp  j pt j 1 tt jpt ;

jpt ¼ j pt j 1 tt ;

jtt ¼ j 1 tt

Notice that Eqs. (33a) and (33c) are anti-symmetric since T T T T Ctp ¼ C 1 and jtp ¼ j 1 tt Cpt ¼ Cpt tt jpt ¼ jpt . Eqs. (33a)–(33c) define the mixed linear thermopiezoelectric constitutive equations required by the TMVT; i.e. ( C) " #( G )   Cpp Cpt Tp Sp Ap ¼ þ h ð34aÞ CTpt Ctt At TM SCt t T M h gC ¼ ATp SG p þ At T t þ A h

(

hCp eCt

)

" ¼

jpp jTpt

jpt jtt

#(

eG p hM t

ð34bÞ ) ð34cÞ

where the in-plane enlarged strains and temperature gradients are defined by Eqs. (16a)–(16c), and the mixed transverse enlarged stresses and heat flux are to be assumed (postulated). Recall that superscript G indicates that the relative quantity is to be computed from the gradient relations; whereas the superscript M denotes a quantity to be assumed. Substitution of Eqs. (34a)– (34c) in the variational equation (29) leads to Z Z Z T T G GT M M dSG C S dX þ dS C T dX þ dSG pp p pt t p p t Tt dX X X X Z Z T T T G M C S dX  dTM þ dTM t t Ctt Tt dX pt p X X Z Z Z T G dUT g dX  dUT G dS þ dTM t St dX  X X SG Z Z Z € dX  dSGT Ap h dX  dTMT At h dX þ dUT qU p t X Z X ZX G M  dhH0 ATp S_ p dX  dhH0 ATt T_ t dX X X Z Z T G  dhH0 Ah h_ dX  deG p jpp ep dX X X Z Z Z T M M T G deG dhM  deG t ht dX  t jpt ep dX p jpt ht dX  ZX ZX ZX M G dhM dhs dX þ dhM t jtt ht dX  t et dX þ X X X Z dhH dS ¼ 0 ð35Þ  SH

This variational equation can then be used to develop the corresponding numerical formulation by postulating the distributions of the two-generalised fields (enlarged displacements and transverse stresses) and the temperature and transverse heat flux. To show explicitly the mechanical, thermal and electric contributions and their couplings, as can be seen from Eqs. (31a), the bloc matrices can be decomposed further into mechanical (m), electric (e) and thermal (h) contributions " mm # " mm # C me Cpp C me Cpt pp pt Cpp ¼ ; Cpt ¼ ; T C em C ee C me C ee pt pt pp pp " mm # Ctt C me tt ð36aÞ Ctt ¼ T C me C ee tt tt ( A p

¼

Ap mh

)

( ;

Ap eh

A t

¼

At mh At eh

) ð36bÞ

From the relations following Eq. (33c), it is clear that the in-plane and transverse bloc matrices remain symmetric. Decomposition (36a) can then be applied to the matrices of Eq. (34a) so that they read now " mm # " mm # Cpp Cme Cme Cpt pp pt ; Cpt ¼ ; Cpp ¼ T Cem Cee Cme Cee pt pt pp pp " mm # Ctt Cme tt ð37aÞ Ctt ¼ T ee Cme C tt tt ( Ap ¼

Amh p

)

Aeh p

( ;

At ¼

Amh t Aeh t

) ð37bÞ

Using these matrices and definitions (28), then separating the mechanical–mechanical, mechanical–electric, mechanical–thermal, electric–electric and thermal–thermal contributions, the variational equation (35) can be written, in terms of the internal (int), external (ext), inertia (in) and thermal(h)-induced equivalent damping (d) virtual works, as W int  W ext ¼ W in þ W hd

ð38Þ

with W int ¼ W mm  W me  W em þ W ee  W mh  W eh  W hh ; W ext ¼ W m  W e þ W h ; and W mm ¼

he hh W hd ¼ W hm d Wd þWd

Z

Z T T mm G mm M deG deG p Cpp ep dX þ p Cpt rt dX X X Z Z T T mmT G mm M þ drM drM t Cpt ep dX  t Ctt rt dX X X Z Z T T M G r dX þ drM þ deG t t t et dX X

X

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

Z

Z T T me G emT G deG drM p Cpp Ep dX þ t Cpt Ep dX X X Z Z T me M GT me M  dep Cpt Dt dX þ drM t Ctt Dt dX

W me ¼

X

X

Z

T

Z

T

T

me G em M dEG dEG p Cpp ep dX þ p Cpt rt dX X X Z Z meT G meT M dDM  dDM t Cpt ep dX þ t Ctt rt dX

W em ¼

X

W ee ¼

X

Z

Z

T

X

W mh ¼

Z

X T

mh deG p Ap h dX þ

Z

X

W

eh

T

mh drM t At h dX;

X

Z

¼

T eh dEG p Ap h dX

þ

Z

X

W hh ¼ þ

Z

þ þ

T

G deG p jpp ep dX þ

ZX Z

eh dDM t At h dX

X

T G dhM t jpt ep dX X M deG t ht dX þ

X

Wm ¼

T

ee G ee M dEG dEG p Cpp Ep dX  p Cpt Dt dX X X Z Z eeT G ee M C E dX  dDM  dDM t t Ctt Dt dX p pt X X Z Z M G  dEG D dX  dDM t t t E t dX

Z

 Z

Z

T

M deG p jpt ht dX

ZX

M dhM t jtt ht dX

X G dhM t et dX

Z

duT f dX þ duT F dS; X S F Z Z Z duq dX þ duQ dS; W in ¼  duT q€u dX We ¼ X

Z

dhH dS; W ¼  dhs dX þ X SH Z Z T 0 mhT G _ e r_ M ¼ dhH A dX þ dhH0 Amh W hm p t dX d p t X

W he d ¼

Z

X

T _G dhH0 Aeh p Ep dX 

X

W hh d ¼

Z

W int  W ext ¼ W in

ð39Þ

where the different contributions are those given after Eq. (38). The most studied case in the literature is that when only the effect of a temperature load on the piezocomposite is considered. Here, the mechanical–thermal and electric–thermal contributions to the internal virtual work become additional loads so that they have to be moved to the external virtual work. Also, the thermal– thermal and thermal contributions to the internal and external virtual works, respectively, vanish in this case. The TMVT can then be written as ð40Þ

with W int ¼ W mm  W me  W em þ W ee ; W ext ¼ W m  W e  W mh  W eh

X

SQ

Z

h

temperature fields, when Eq. (38) is used, are to be postulated. An advantage of the variational formulation (35) is that the same distributions, as for the RMVT ones [16–19], can be retained for the two-enlarged fields; whereas, the variational formulation (38) keeps the freedom of postulating different distributions for the electromechanical fields [21–23]. Under isentropic conditions, which is the common framework of most studies on thermopiezocomposites, the entropy density contribution can be eliminated, so that the temperature and heat flux are sufficient to characterise the thermal state of the material. For this, Eq. (7c) is generally dropped and the thermal induced equivalent damping vanishes in Eq. (38); in this case the TMVT reduces to

W int  W ext ¼ W in

X

1275

Z

M

_ dhH0 Aeh t Dt dX;

X

The different remaining contributions are those defined after Eq. (38). It is worthy to notice that the resulting variational formulation, Eq. (40), leads to only uncoupled thermopiezoelectric behaviour since there are no stiffness contributions. Besides, no heat flux can be applied and the interface continuity of the transverse heat flux cannot be fulfilled in this case.

dhH0 Ah h_ dX

X

5. Discussion Either of Eqs. (35) or (38) can be used to develop numerical formulations based on the proposed TMVT. For this, the enlarged mixed displacement and stress vectors together with the temperature and transverse heat flux, when Eq. (35) is used, and the mixed transverse stresses, electric displacement and heat flux together with mechanical displacement, electric potential and

6. Conclusions and perspectives A new Thermopiezoelectric Mixed Variational Theorem (TMVT) and its corresponding mixed thermopiezoelectric constitutive equations have been proposed for the variational-based modelling of thermopiezoelectric multilayered composites. It is an extension of the Piezoelectric Mixed Variational Theorem (PMVT) proposed recently [12,13] by the authors and can be seen also as an analogue of the ReissnerÕs Mixed Variational Theorem (RMVT), initially imagined for layered elastic composites. The proposed variational formulations can be

1276

A. Benjeddou, O. Andrianarison / Computers and Structures 83 (2005) 1266–1276

used, for example, for developing new 3D closed-form or 2D numerical solutions for layered smart structures. In particular, they were written in two forms suitable for direct derivation of finite elements. This could be done through the discretization of either four-fields (enlarged mixed displacement and stress vectors, similar to RMVT and PMVT, and the temperature and transverse heat flux) or six-fields (mechanical displacement and transverse stress vectors, electric potential and transverse displacement, temperature and transverse heat flux scalars). In order to judge the quality of the proposed TMVT, its finite element implementation for thermopiezoelectric multilayered composite plates is currently in progress. The corresponding theoretical formulation and numerical results will be published in an upcoming paper.

References [1] Tiersten HF. On the non-linear equations of thermoelectroelasticity. Int J Engng Sci 1971;9:587–604. [2] Mindlin RD. Equations of high-frequency vibrations of thermopiezoelectric crystal plates. Int J Solids Struct 1974;10:625–37. [3] Do¨kmeci MC. Recent advances: vibrations of piezoelectric crystals. Int J Engng Sci 1980;18:431–48. [4] Altay GA, Do¨kmeci MC. Fundamental variational equations of discontinuous thermopiezoelectric fields. Int J Engng Sci 1996;34(7):769–82. [5] Altay GA, Do¨kmeci MC. Electroelastic plate equations for high frequency vibrations of thermo-piezoelectric materials. Thin-Walled Struct 2001;39:95–109. [6] He J-H. Hamilton principle and generalized variational principles of linear thermopiezoelectricity. ASME J Appl Mech 2001;68:666–7. [7] Chandrasekhariah DS. A generalized linear thermoelasticity theory for piezoelectric media. Acta Mech 1988;71: 39–49. [8] Tang YY, Noor AK, Xu K. Assessment of computational models for thermoelectroelastic multilayered plates. Comput Struct 1996;61(5):915–33. [9] Saravanos DA, Heyliger P. Mechanics and computational models for laminated piezoelectric beams, plates and shells. Appl Mech Rev 1999;52(10):305–20.

[10] Tauchert TR, Ashida F, Noda N, Adali S, Verijenko V. Developments in thermopiezoelasticity with relevance to smart composite structures. Compos Struct 2000;48: 31–8. [11] Benjeddou A. Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Comput Struct 2000;76(1–3):347–63. [12] Benjeddou A, Andrianarison O. Extension of ReissnerÕs mixed variational theorem to piezoelectric multilayered composites. In: Proc 6th Hellenic Europ Res Comput Math Appl, Athens, Greece, 25–27 September 2003. [13] Benjeddou A, Andrianarison O. A piezoelectric mixed variational theorem for smart multilayered composites. Mech Adv Mater Struct 2005;12(1):1–11. [14] Reissner E. On a certain mixed variational theorem and a proposed application. Int J Numer Methods Engng 1984;20:1366–8. [15] Reissner E. On a mixed variational theorem and on shear deformable plate theory. Int J Numer Meth Engng 1986;23:193–8. [16] Carrera E. An assessment of mixed and classical theories for the thermal stress analysis of orthotropic multilayered plates. J Thermal Stress 2000;23:797–831. [17] Carrera E. Temperature profile influence on layered plates response considering classical and advanced theories. AIAA J 2002;40(9):1885–96. [18] Carrera E. An improved Reissner-Mindlin-type model for the electro-mechanical analysis of multilayered plates including piezo-layers. J Intell Mater Syst Struct 1997;8(3):232–48. [19] Ballhause D, DÕOttavio M, Carrera E. Partially mixed multilayered theories for piezoelectric plates. In: Proc 6th Hellenic Europ Res Comput Math Appl, Athens, Greece, 25–27 September 2003. [20] Tzou HS, Bao Y. A theory on anisotropic piezothermoelastic shell laminates. J Sound Vib 1995;184(3):453–73. [21] Ye R, Tzou HS. Control of adaptive shells with thermal and mechanical excitations. J Sound Vib 2000;231(5): 1321–38. [22] Zhou X, Chattopadhyay A, Gu H. Dynamic responses of smart composites using a coupled thermo-piezoelectricmechanical model. AIAA J 2000;38(10):1939–48. [23] Gu H, Chattopadhyay A, Zhou X. A higher order temperature theory for coupled thermo-piezoelectricmechanical modeling of smart composites. Int J Solids Struct 2000;37:6479–97.