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A variational approach to electroelastic analysis of piezoelectric ceramics with surface electrodes
Pergamon
Mechanics Research Communications, Vol. 27, No. 4, pp. 445450, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/(10/S---seefront matter
PII: S0093-6413(00)00116-6
A VARIATIONAL APPROACH TO ELECTROELASTIC ANALYSIS OF PIEZOELECTRIC CERAMICS WITH SURFACE ELECTRODES Ji-Huan He Shanghai University, Shanghai Institute o f Applied Mathematics and Mechanics, Shanghai 200072, China
(Received3 February 1999; acceptedfor print 31 May 2000)
Introduction
Piezoelectric ceramics have been widely used m engineering. A comprehensive list of works in this area may be found in[l,2] and the references cited therein. In order to apply the finite element methods to the num~cal simulation of the piezoelectricity, it is necessary to establish a generalized variational model for the present problem. Due to the coupled constitutive equations of piezoelectricity, it is very difficult to search for a functional whose stationary conditions satisfy all its field equations and boundary conditions by the traditional ways such as the well-known Lagrangc multiplier method, the difficulty, however, can be smoothly overcome by the semi-inverse method[3-11] proposed by the present author. As a f ~ t step, in this paper, we consider the static behavior of the elastic and electric variables in the vicinity of a surface electrode attached to a piezoelectric ceramic.
445
446
J.-H. HE
Statement of The Problem and fundamental Equations We consider a two dimensional piezoelectric medium described by Cartesian coordinates Ix[ < oo and z>0 and assume plane strain perpendicular to the y-axas. The z-axis is assumed to coincide with the six fold axis of symmetry in the class of a 6 mm crystal class, or with the poling axis in the case of poled piezoelectric ceramics. Consider a piezoelectric body with a perfectly conducting electrode of length 2a attached to a portion of its surface as shown in Fig. 1.
1) Equilibrium equations O-xx,x + Crxz~ = 0
(la)
Crzz,z + o-~, z = 0
(lb)
_.
in which o'xx, Crzx = Crxz, crz~ are the stresses, and the comma denotes partial differentiation with respect to the coordinates.
-(t
q-a
Fig. 1 A piezoelectric ceramic with a surface electrode
2) Quasi-staticMaxwell's equations D~, x + D~a = 0
(2)
where D x and Dy are the electric displacements. The electric field components can be written down in the form of an electric potential ~ ( z , x) :
Ex = - ~ , x
(3a)
VARIATIONAL APPROACH TO PIEZOELECTRIC CERAMICS
Ez = -~,z
447
(3b)
3) Strata-displacement relations
e~= = u~,~
(4a)
~z= = Uz,z
(4b)
1
6,= : ~(u~., + uz.,)
(4c)
where u x and u z are the elastic displacements,. 6xx , 6zx = 6xz , ~zz are the strains. 4) Constitutive equations
For linear elasticity, we have ~Yxx = allSx.x +a138zz - e 3 1 E z
(5a)
or= = a446 x= - e l s E x
(5b)
o'=z = a136xx + a338z= - e33E z D x = e156xz + c l l E x
(6a)
D z = e31Sxx + e336zz + c33E z in which. The a 11, a33, a44 ,a13 are the elastic constants
(zero) electric field, e31,e33 , el5 are the
(5e)
(6b) measured at constant
piezoelectric constants measured at
constant (zero) strain, c I 1, c33 are the dielectric permittivity. 5) Boundary cona~tions
We asstune that the mixed boundary conditions are given by O'zz (x,0) = 0
(7a)
,r=(x,O) = o
(to)
~(,~,o) = Vo (o _<}~I< ~ )
(7o)
Dz(x,O ) = O ( a _< {x] < oc)
(7d)
where V0 is a constant voltage.
Trial-Functional and Generalized Variational Princinle In this section we will search for a generalized variational principle with 13
448
J.-H. HE
independent variations(O-x.r, Crxz , Crzz , gxx, ~'xz, £zz, Ux, Uz, D x , D z , E x , E y and @), its stationary conditions with respect to above variables should satisfy 13 field equations and boundary conditions listed above. In general , it is very difficult to establish a generalized variational principle for the above coupled system. Here we will apply the semi-inverse method[3-11] to obtain the required variational principle. The basic idea of which is to establish an energy trial-functional with certain unknowns F and G J = [I L d x d z + IB A where L is a trial-Lagrangian, which can be expressed in the following form L = cr~ex~ + cr~czx + o'zz6zz + F
(8a)
(8b)
and IB is the boundary integral which can be written in the following form IB = f a g d x
(8c)
m which F and G are unknowns. Many other trial-functionals can be established, details can be found in the Refs.[3-11]. We search for such F so that its Euler equations satisfy all its field equations. In view of the stationary conditions with respective to the stresses(Crxx, Crzx, fizz)
anddisplacements(Ux, Uz),weset F = -ux,xcr~, - ½ ( U x , z +Uz,x)Cr = -Uz~Crzz + F 1
(9)
the trial-Lagrangian (8b), therefore, can be rewritten down as follows L = cr~exx + Crzxezx + Crzzezz - Ux.xCr~ - ~(ux, 1 z + Uz.x )Crzx
Uz,z°-zz + Fl
(lO) where F 1 is a newly introduced unknown function, which should be free from the variables of stresses and displacements. It is obvious that the stationary conditions with respective to stresses and displacements satisfy the field equations(4a---4c) and(l a)~(l b) respectively. In view of the field equations(5a)~(5c), we set El = - )-a 1 I l Oexx 2 - al3OezzoOxx - e 3 iEzoexx 1
2
1
2
-~-a44oezx - elsEx6zx - ~-a336"zz - e33Ezezz. + F 2
(11)
so that the stationary conditions with respective to the strams(gxx, gzx, 6zz ) satisfy the field equations(5a)---(5c) . The lxial-Lagrangian(10) can be further renewed as follows
V A R I A T I O N A L A P P R O A C H TO P I E Z O E L E C T R I C C E R A M I C S
449
L = ox.x~xx + crr.xg~ + crzzg,, - U x , x ° x x - l ( U x x + Uz,x)°'zx - Uz,zO'zz
- ~ 1a l l g x x2 - al3gzzgxx _ e31Ezgxx t 2 1 2 - ~a446zx - e l 5 E x t z x - [ a 3 3 # z z - e 3 3 E z s ~ + F 2
where
(12)
F 2 is an unknown function free from the stratus.
The stationary conditions with respect to E x and E z read +~
= 0
(13)
-e31cxr -e336zz + °T2 = 0
(14)
c~Ez
The above equations with an unknown are called as trial-Euler equations , which should satisfy two of the field equations. We set °772 - D~ - c l l E x
(15)
a3F,x
(16)
°T2 = Dz _ e33Ez
gEy
so that the triai-Euler equations satisfy the field equations (6a) and (6b) respectively. From the equations (15) and (16), the unknown can be m ~ i l y identified as follows ! I 1Ex2 - ~1c 3 3 E z2 + F3 F 2 = D x E x + D z E z - ~-c
(17)
in which F 3 is an unknown function free from E x and E z . The substitution of the equation (17) into the trial-Lagrangian (12) results m a new one, now in view of the left field equations(2), (3a) and (3b), we can identify the unknown F 3 as follows (18)
F2 = Dxq~,x + D z @ z
It is obvious that the stationary conditions with respect to ~ , D x , D z are (2), (3a) and (3b) respectively. We , therefore, obtain the following generalized variational principles (19a)
J = ~Ld'~+IB
where L = a=e=
+a=e=
+ a,,6=
- u,,,a=
- ½(u,~ + Uz,x)a=
- uz~=
450
J.-H. HE 1
2
- ~-a 1lO°xx - al3~'zzoOxx - e31Ez~xx 1
2
1
2
- ~-a44~.zx - e l 5 E x ~ z x - ~-a336zz - e33Ezezz + D x E x + D z E z - ~1C l l E x2 - ~1c 3 3 E z2 + D x ~ , x + D z ~ ~
(19b)
the boundary integral can be identified by the same way (19c) Conclusion In the paper, we obtain a generalized variational principle, from which the wellknown Hellinger-Reissner principle and Hu-Washizu principle of elasticity can be obtained if the electrical effect is not taken into consideration. The numerical simulation( f'mite element methods) based on these variational principles will be reported soon.
Acknowled~ment.~ The work is supported by Shanghai Education Foundation for Young Scientists ( 9 8 Q N 4 7 ) and National Key Basic Research Special Fund of China (No. G1998020318). References [1] Shindo,Y., Narita,F. & Sosa, lnt. J. Engineering Science, 36(9), 1998, 1001-1009 [2] Zhou, S.A., Hsien, R.K.T. & Maugin, G.A. Int. J. Solids & Structures, 22(12), 1986,14111422 [3] He,J.H. ,International Journal of Nonlinear Sciences & Numerical Simulation, 1(2),2000, 133138 [4] He,J.H., Communications in Nonlinear Science & Numerical Simulation, 3(3), 1998, 179-183 [5] He, J.H, J. Shanghai University (English Edition), 1(2), 1997, 117-122. [6] He, J.H., Int. J. Turbo &Jet-Engines, 14(1),1997, 23-28 [7] He, J.H., Int. J. Turbo &Jet-Engines, 14(1),1997, 17-22 [8] He,J.H., Applied Math. Modelling, 22(1998), 395-403 [9] He, J.H., lnt. d. Turbo & Jet-Engines, 15(2),1998, 95-100 [10] He, J.H., Int. d. Turbo &Jet-Engines, 15(2),1998, 101-107 [1 I I He,LH. Generallized Hellinger-Reissnerprinciple, ASME J. Appl. Mech. (accepted)
Mechanics Research Communications, Vol. 27, No. 4, pp. 445450, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/(10/S---seefront matter
PII: S0093-6413(00)00116-6
A VARIATIONAL APPROACH TO ELECTROELASTIC ANALYSIS OF PIEZOELECTRIC CERAMICS WITH SURFACE ELECTRODES Ji-Huan He Shanghai University, Shanghai Institute o f Applied Mathematics and Mechanics, Shanghai 200072, China
(Received3 February 1999; acceptedfor print 31 May 2000)
Introduction
Piezoelectric ceramics have been widely used m engineering. A comprehensive list of works in this area may be found in[l,2] and the references cited therein. In order to apply the finite element methods to the num~cal simulation of the piezoelectricity, it is necessary to establish a generalized variational model for the present problem. Due to the coupled constitutive equations of piezoelectricity, it is very difficult to search for a functional whose stationary conditions satisfy all its field equations and boundary conditions by the traditional ways such as the well-known Lagrangc multiplier method, the difficulty, however, can be smoothly overcome by the semi-inverse method[3-11] proposed by the present author. As a f ~ t step, in this paper, we consider the static behavior of the elastic and electric variables in the vicinity of a surface electrode attached to a piezoelectric ceramic.
445
446
J.-H. HE
Statement of The Problem and fundamental Equations We consider a two dimensional piezoelectric medium described by Cartesian coordinates Ix[ < oo and z>0 and assume plane strain perpendicular to the y-axas. The z-axis is assumed to coincide with the six fold axis of symmetry in the class of a 6 mm crystal class, or with the poling axis in the case of poled piezoelectric ceramics. Consider a piezoelectric body with a perfectly conducting electrode of length 2a attached to a portion of its surface as shown in Fig. 1.
1) Equilibrium equations O-xx,x + Crxz~ = 0
(la)
Crzz,z + o-~, z = 0
(lb)
_.
in which o'xx, Crzx = Crxz, crz~ are the stresses, and the comma denotes partial differentiation with respect to the coordinates.
-(t
q-a
Fig. 1 A piezoelectric ceramic with a surface electrode
2) Quasi-staticMaxwell's equations D~, x + D~a = 0
(2)
where D x and Dy are the electric displacements. The electric field components can be written down in the form of an electric potential ~ ( z , x) :
Ex = - ~ , x
(3a)
VARIATIONAL APPROACH TO PIEZOELECTRIC CERAMICS
Ez = -~,z
447
(3b)
3) Strata-displacement relations
e~= = u~,~
(4a)
~z= = Uz,z
(4b)
1
6,= : ~(u~., + uz.,)
(4c)
where u x and u z are the elastic displacements,. 6xx , 6zx = 6xz , ~zz are the strains. 4) Constitutive equations
For linear elasticity, we have ~Yxx = allSx.x +a138zz - e 3 1 E z
(5a)
or= = a446 x= - e l s E x
(5b)
o'=z = a136xx + a338z= - e33E z D x = e156xz + c l l E x
(6a)
D z = e31Sxx + e336zz + c33E z in which. The a 11, a33, a44 ,a13 are the elastic constants
(zero) electric field, e31,e33 , el5 are the
(5e)
(6b) measured at constant
piezoelectric constants measured at
constant (zero) strain, c I 1, c33 are the dielectric permittivity. 5) Boundary cona~tions
We asstune that the mixed boundary conditions are given by O'zz (x,0) = 0
(7a)
,r=(x,O) = o
(to)
~(,~,o) = Vo (o _<}~I< ~ )
(7o)
Dz(x,O ) = O ( a _< {x] < oc)
(7d)
where V0 is a constant voltage.
Trial-Functional and Generalized Variational Princinle In this section we will search for a generalized variational principle with 13
448
J.-H. HE
independent variations(O-x.r, Crxz , Crzz , gxx, ~'xz, £zz, Ux, Uz, D x , D z , E x , E y and @), its stationary conditions with respect to above variables should satisfy 13 field equations and boundary conditions listed above. In general , it is very difficult to establish a generalized variational principle for the above coupled system. Here we will apply the semi-inverse method[3-11] to obtain the required variational principle. The basic idea of which is to establish an energy trial-functional with certain unknowns F and G J = [I L d x d z + IB A where L is a trial-Lagrangian, which can be expressed in the following form L = cr~ex~ + cr~czx + o'zz6zz + F
(8a)
(8b)
and IB is the boundary integral which can be written in the following form IB = f a g d x
(8c)
m which F and G are unknowns. Many other trial-functionals can be established, details can be found in the Refs.[3-11]. We search for such F so that its Euler equations satisfy all its field equations. In view of the stationary conditions with respective to the stresses(Crxx, Crzx, fizz)
anddisplacements(Ux, Uz),weset F = -ux,xcr~, - ½ ( U x , z +Uz,x)Cr = -Uz~Crzz + F 1
(9)
the trial-Lagrangian (8b), therefore, can be rewritten down as follows L = cr~exx + Crzxezx + Crzzezz - Ux.xCr~ - ~(ux, 1 z + Uz.x )Crzx
Uz,z°-zz + Fl
(lO) where F 1 is a newly introduced unknown function, which should be free from the variables of stresses and displacements. It is obvious that the stationary conditions with respective to stresses and displacements satisfy the field equations(4a---4c) and(l a)~(l b) respectively. In view of the field equations(5a)~(5c), we set El = - )-a 1 I l Oexx 2 - al3OezzoOxx - e 3 iEzoexx 1
2
1
2
-~-a44oezx - elsEx6zx - ~-a336"zz - e33Ezezz. + F 2
(11)
so that the stationary conditions with respective to the strams(gxx, gzx, 6zz ) satisfy the field equations(5a)---(5c) . The lxial-Lagrangian(10) can be further renewed as follows
V A R I A T I O N A L A P P R O A C H TO P I E Z O E L E C T R I C C E R A M I C S
449
L = ox.x~xx + crr.xg~ + crzzg,, - U x , x ° x x - l ( U x x + Uz,x)°'zx - Uz,zO'zz
- ~ 1a l l g x x2 - al3gzzgxx _ e31Ezgxx t 2 1 2 - ~a446zx - e l 5 E x t z x - [ a 3 3 # z z - e 3 3 E z s ~ + F 2
where
(12)
F 2 is an unknown function free from the stratus.
The stationary conditions with respect to E x and E z read +~
= 0
(13)
-e31cxr -e336zz + °T2 = 0
(14)
c~Ez
The above equations with an unknown are called as trial-Euler equations , which should satisfy two of the field equations. We set °772 - D~ - c l l E x
(15)
a3F,x
(16)
°T2 = Dz _ e33Ez
gEy
so that the triai-Euler equations satisfy the field equations (6a) and (6b) respectively. From the equations (15) and (16), the unknown can be m ~ i l y identified as follows ! I 1Ex2 - ~1c 3 3 E z2 + F3 F 2 = D x E x + D z E z - ~-c
(17)
in which F 3 is an unknown function free from E x and E z . The substitution of the equation (17) into the trial-Lagrangian (12) results m a new one, now in view of the left field equations(2), (3a) and (3b), we can identify the unknown F 3 as follows (18)
F2 = Dxq~,x + D z @ z
It is obvious that the stationary conditions with respect to ~ , D x , D z are (2), (3a) and (3b) respectively. We , therefore, obtain the following generalized variational principles (19a)
J = ~Ld'~+IB
where L = a=e=
+a=e=
+ a,,6=
- u,,,a=
- ½(u,~ + Uz,x)a=
- uz~=
450
J.-H. HE 1
2
- ~-a 1lO°xx - al3~'zzoOxx - e31Ez~xx 1
2
1
2
- ~-a44~.zx - e l 5 E x ~ z x - ~-a336zz - e33Ezezz + D x E x + D z E z - ~1C l l E x2 - ~1c 3 3 E z2 + D x ~ , x + D z ~ ~
(19b)
the boundary integral can be identified by the same way (19c) Conclusion In the paper, we obtain a generalized variational principle, from which the wellknown Hellinger-Reissner principle and Hu-Washizu principle of elasticity can be obtained if the electrical effect is not taken into consideration. The numerical simulation( f'mite element methods) based on these variational principles will be reported soon.
Acknowled~ment.~ The work is supported by Shanghai Education Foundation for Young Scientists ( 9 8 Q N 4 7 ) and National Key Basic Research Special Fund of China (No. G1998020318). References [1] Shindo,Y., Narita,F. & Sosa, lnt. J. Engineering Science, 36(9), 1998, 1001-1009 [2] Zhou, S.A., Hsien, R.K.T. & Maugin, G.A. Int. J. Solids & Structures, 22(12), 1986,14111422 [3] He,J.H. ,International Journal of Nonlinear Sciences & Numerical Simulation, 1(2),2000, 133138 [4] He,J.H., Communications in Nonlinear Science & Numerical Simulation, 3(3), 1998, 179-183 [5] He, J.H, J. Shanghai University (English Edition), 1(2), 1997, 117-122. [6] He, J.H., Int. J. Turbo &Jet-Engines, 14(1),1997, 23-28 [7] He, J.H., Int. J. Turbo &Jet-Engines, 14(1),1997, 17-22 [8] He,J.H., Applied Math. Modelling, 22(1998), 395-403 [9] He, J.H., lnt. d. Turbo & Jet-Engines, 15(2),1998, 95-100 [10] He, J.H., Int. d. Turbo &Jet-Engines, 15(2),1998, 101-107 [1 I I He,LH. Generallized Hellinger-Reissnerprinciple, ASME J. Appl. Mech. (accepted)