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Large deformation finite element analyses of composite structures integrated with piezoelectric sensors and actuators
Finite Elements in Analysis and Design 35 (2000) 1}15
Large deformation "nite element analyses of composite structures integrated with piezoelectric sensors and actuators Sung Yi*, Shih Fu Ling, Ming Ying School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore
Abstract Nonlinear dynamic responses of structures integrated with piezoelectric materials are studied. Finite element formulations are derived on the basis of the updated Lagrangian formulation and the principle of virtual work. Twenty-node solid elements including electrical degrees of freedom are developed to analyze structures with piezoelectric sensors and actuators and multipoint constraints for electrical degrees of freedom are utilized to simulate electrodes. A limited number of numerical examples are presented in order to verify the accuracy and convergence of the present formulation and to demonstrate its usefulness for and applicability to solutions of large deformation dynamic responses of piezoelectric structures. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear FEM; Smart structures
1. Introduction In recent decades, special attention has been paid to integrating piezoelectric materials which can be used for both sensors and actuators into base structures for structural control purpose. As a result of the increasing applications of piezoelectric materials, the accurate prediction of the performance of smart structures with sensing, actuating, and control functions becomes increasingly important. Beam, plate and shell structures with bonded or embedded piezoelectric actuators and sensors have been studied. Crawley and de Luis [1] studied vibration control of cantilever elastic and composite beams with surface-bonded and embedded piezoelectric materials analytically and
* Corresponding author. Tel.: #65-790-6239; fax: #65-791-1859. E-mail address: [email protected] (S. Yi) 0168-874X/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 4 5 - 1
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S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
experimentally. By considering both axial and transverse shear forces, Im and Atluri [2] studied the e!ects of a piezoelectric actuator mounted on a "nitely deformed beam based upon a static analysis. Lee [3] developed a theory for laminated piezoelectric plates based on the classical plate theory. Batra and Liang [4] also analyzed the steady-state vibration of a simply supported rectangular linear elastic laminated plate with embedded PZT layers. For thin laminated shells with piezoelectric layers, motion equations were derived by Tzou and Gadre [5] based upon Love's "rst-approximation shell theory and Hamilton's principle. Since analytical approaches are restricted to relatively simple geometries and boundary conditions, they cannot deal with complex piezoelectric structures. Finite element methods become very important and have been widely employed to model piezoelectric structures. Lammering [6] derived a "nite element formulation based on the shell theory for composite shell structures integrated with sensors and actuators. A shell element covered by the piezoelectric layers at the upper and lower surface symmetrically was developed. Hwang and Park [7] employed the classical laminate theory with the induced actuation and variational principles to formulate equations of motion. The plate element without the electro-mechanical coupling was developed. The total charge on the sensor layer was calculated from the direct piezoelectric equation. In order to model structures with complex geometry, isoparametric hexahedron and tetrahedral solid elements were developed for piezoceramic structures [8,9]. Ha et al. [10] also presented a "nite element formulation with eight-node threedimensional composite brick elements. For thin-walled piezoelectric structures, Tzou and Tseng [11] developed a solid "nite element for thin piezoelectric structures based on an eight-node solid element by introducing the incompatible models into the linear interpolation function. Yi et al. [12] developed a "nite element formulation using a solid element with Allman's rotation in order to analyze composite structures with smart constrained layers. Their study was focused on the use of viscoelastic damping and piezoelectric materials for the active and passive vibration and noise control and the "nite element formulation is developed on the basis of visco-piezoelectricity. Large deformation analyses for beam, plate and shell structures with piezoelectric materials have been reported in the literature. Kulkarni and Hanagud [13] derived a variational formulation for a combined isotropic, linearly elastic body and a piezoelectric body undergoing small or large deformations. A two-dimensional "nite element formulation based on the variational principle was applied to solve static and dynamic problems involving temporally and spatially varying electric "elds. Pai et al. [14] presented a nonlinear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large-rotation and small-strain vibrations. Preiswerk and Venkatesh [15] studied the vibration control of #exible links provided with nearly collocated piezoceramic sensors and actuators undergoing geometrically nonlinear de#ections. Euler}Bernoulli-theory-based beam "nite elements with p-convergence capabilities had been developed using the total Lagrangian formulation in order to represent large rotations of links. Icardi and Sciuva [16] employed the von Karman strain}displacement relations to investigate the stress "eld of multilayered intelligent anisotropic plates with a surfacebonded piezoelectric actuator layer. However, only limited work has been reported in the literature on the large deformation "nite element formulation of smart structures using solid elements. In the present study, dynamic responses of structures integrated with piezoelectric sensors and actuators under large geometrical deformation are considered. Finite element formulations are derived on the basis of the updated Lagrangian formulation and the principle of virtual work. In order to model complex piezoelectric structures, 20-node solid elements including electrical degrees
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
3
of freedom are developed and multipoint constraints for electrical degrees of freedom are employed to simulate electrodes.
2. Analysis 2.1. Mechanical and electrical strain}displacement relationships The strain vector e in material Cartesian coordinates is de"ned in terms of the in"nitesimal and large displacement components as e"[e e e c c c ]T 11 22 33 12 23 13 u 1 [(u )2#(v )2#(w )2] ,1 2 ,1 ,1 ,1 v 1 [(u )2#(v )2#(w )2] ,2 2 ,2 ,2 ,2 w 1 [(u )2#(v )2#(w )2] ,3 ,3 ,3 " # 2 ,3 , u #v u u #v v #w w ,2 ,1 ,1 ,2 ,1 ,2 ,1 ,2 v #w u u #v v #w w ,3 ,2 ,2 ,3 ,2 ,3 ,2 ,3 u #w u u #v v #w w ,3 ,1 ,1 ,3 ,1 ,3 ,1 ,3 where u, v, w are the displacements and the comma denotes the partial di!erentiation. Similarly, the electric "eld vector E is related to the electric potential < by
G HG
H
GH GH
E < 1 ,1 E" E "! < . 2 ,2 E < 3 ,3
(1)
(2)
2.2. Piezoelectric constitutive equations The constitutive relationship for linear piezoelectric materials can be described as r( "CK e(
(3)
with
C
CK "
C
e
D
eT !Z
r( "[r D]T
,
"[p p p q q q D D D ]T 11 22 33 12 23 13 1 2 3 e( "[e!E]T
(4)
(5)
"[e e e c c c !E !E !E ]T, (6) 11 22 33 12 23 13 1 2 3 where the superscript ¹ denotes the matrix transpose, e is the vector of mechanical strains, r is the vector of stresses, D is the vector of electric displacements, and E is the vector of electric "elds.
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In the principal material directions, the material sti!ness matrix C, the piezoelectric stress matrix e and the dielectric constant matrix Z for orthotropic piezoelectric materials can be written as
C
e
e e 12 13 14 e e e e" e 21 22 23 24 e e e e 31 32 33 34 Z 0 0 1 0 Z" 0 Z 2 0 0 Z 3 11
e
C
and
C
D
C 11 C 21 C C" 31 0
D
e 15 e 25 e 35
e 16 e , 26 e 36
(7)
(8)
C 12 C 22 C 32 0
C 13 C 23 C 33 0
0
0
0
44 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C 55 0
0
C
C 66
D
,
(9)
where C "C . ij ji The transformed material properties with respect to the global coordinates can be explained as [12]
C
D C
D
T~1 0 T~T 0 m m CK (10) 0 T~1 0 T e e where T and T are the coordinate transformation matrices from material Cartesian coordinates m e to global Cartesian coordinates for strain and electric "eld vectors, respectively. CK @"
2.3. Updated Lagrangian formulation for piezoelectric materials For the updated Lagrangian formulation, all static and kinematic variables are referred to the con"guration at time t. The principle of virtual work for piezoelectric bodies is de"ned by
P
t
v
dt`*t e( T t`*t r( dv" t`*tR, t t
(11)
where t`*t r( "[ t`*t p t`*t p t`*t p t`*t q t`*t q t`*t q t`*t D t`*t D t`*t D ]T t t xx t yy t zz t xy t yz t xz t x t y t z t`*t e( "[ t`*t e t`*t e t`*t e t`*t c t`*t c t`*t c t t xx t yy t zz t xy t yz t xz ! t`*t E t`*t E t`*t E ]T. t x t y t z
(12)
(13)
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
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In the above, d denotes a small, arbitrary virtual variation, tv is the volume of body at con"guration time t, t`*tR is the external virtual work, t`*t e( and t`*t r( are the vectors of generalized strains and t t stresses at con"guration time t#*t referred to time t, respectively. It is assumed that the loading is deformation independent and can be speci"ed prior to the incremental analysis. The only body force is the inertial D'Alembert force de"ned by 0ot`*tuK with the mass density 0o at t"0. The external virtual work is
P
P
duT 0ot`*tuK dv# (duT t`*t p#d< t`*t Q) dC, (14) 0 0 0C v where u is the displacement vector, t`*tp is the vector of boundary tractions at con"guration time 0 t#*t referred to initial time, t`*t Q is the surface charge, 0v is the original volume of body and 0 0C is the original area on which boundary tractions and electrical charges are prescribed. The generalized stresses and strains can be decomposed into those at time t and incremental ones. Since the generalized Cauchy stresses tq( " t r( and t e( "0, one has t t t`*t r( " tq( # r( , (15) t t t`*t e( " e( " e( (-)# e( (/), (16) t t t t where e( (-) and e( (/) are the linear and nonlinear incremental strains corresponding to Eq. (1). t t Substituting Eqs. (14)}(16) into (11) yields t`*tRK "!
P
t
v
0
P
d e( T r( dv# t t
t
v
P P
d e( (/)T tq( dv"! t
t
v
P
d e( (-)T tq( dv! t
#
0C
0
v
duT 0ot`*tuK dv
(duT t`*tp#d< t`*t Q) dC. 0 0
(17)
By using the approximations r( " CK @ e( (-), t t t d e( "d e( (-). t t Eq. (17) can be linearized as
P
t
v
P
d e( (-)T CK @ e( (-) dv# t t t
t
(18) (19)
v
P
d e( (/)T tq( dv"! t
t
P
d e( (-)T tq( dv! t
0
P
duT 0ot`*tuK dv#
0C
(duT t`*tp 0
v v #d< t`*t Q) dC, (20) 0 where CK @ is the tangent material property matrix at time t in the global coordinate system. t 2.4. Finite element formulations The displacement "eld at time t#*t can be written in the incremental form as t`*tu" tu#u, where tu and u are the displacement "eld at time t and its increment, respectively.
(21)
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The incremental displacement "elds within an element can be explained in terms of the nodal displacements as u"N q(e) 20
(22)
u"[u v w]T,
(23)
with
q(e)"[u v w u v w 2 u v w ]T, (24) 1 1 1 2 2 2 20 20 20 where N is the matrix of interpolation functions for a 20-node solid element and q(e) is the vector 20 of time-dependent nodal displacements. Similarly, the electric potential < within an element can be interpolated using the node electric potential V (e) as <"N V (e), V where
(25)
V (e)"[< < 2 < ]T 1 2 20 N "[N N 2 N ]. V 1 2 20 Then the incremental displacement and electric "elds within an element can be written as
(26)
u( "NK q( (e),
(27)
(28)
where (29)
u( "[u <]T,
C
NK "
N 20 0
0
D
, N V q( (e)"[q(e) V (e)]T.
(30) (31)
The linear incremental generalized strains within an element can be represented in terms of the incremental nodal displacements and electric "elds as
C
t B(-) e( (-)" t BK (-)q( (e)" t t t 0
0
D
t B(-) t V
q( (e)
(32)
with
C
N t 1,x 0
0 t B(-)" t N t 1,y 0 N t 1,z
0
0
2
N t 1,y 0
0
2
N t 20,x 0
N t 1,z 0
2
0
2
N t 1,y N t 1,x
2
N t 20,y 0
2
N
N t 1,x N t 1,z 0
t 20,z
0
0
N t 20,y 0
0
N t 20,x N t 20,z 0
D
N t 20,z , 0 N t 20,y N t 20,x
(33)
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
C
N t 1,x t B(-)" N t V t 1,y N t 1,z
N t 2,x N t 2,y N t 2,z
D
N t 20,x , N t 20,y N t 20,z
2 2 2
7
(34)
where N "LN /Ltx, N "LN /Lty and N "LN /Ltz. t i,x i t i,y i t i,z i Consequently, the nonlinear incremental generalized strains become
C
D
1 Ht G 0 q( (e) e( (/)" t BK (/)q( (e)" 2 t t t t 0 0
(35)
where
C D h t x 0
t
H"
0
h t y 0
0
0
h t y 0
0
h t z , 0
h t x h t z 0
(36)
h t y h h t z t x h "[Lu/Ltx Lv/Ltx Lw/Ltx], t x h "[Lu/Lty Lv/Lty Lw/Lty], t y h "[Lu/Ltz Lv/Ltz Lw/Ltz] t z and
C
N t 1,x 0 0
N t 1,y t G" 0 t 0 N t 1,z 0
(37) (38) (39)
0
0
2
N t 1,x 0
0
2
N t 20,x 0
N t 1,x 0
2
0
2
0
2
N t 20,y 0
N t 1,y 0
2
0
2
0
2
N t 20,z 0
0 N t 1,y 0 0 N t 1,z 0
0
N t 20,x 0
0
0 N t 20,y 0 0 N t 20,z 0
N t 20,x 0 0 N t 20,y 0 0
D
.
(40)
N t 1,z 2 t 20,z Note that the incremental displacement gradients are referred to the geometry at con"guration time t. Substitution of Eqs. (32) and (35) into the principle of virtual work leads to the "nite element equilibrium equations for each element 0
N
0
0
m( (e) t`*tq(K (e)#( t kK (-)(e)# t kK (/)(e))q( (e)" t`*tr( (e)! t`*tfK (e), t t
(41)
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where
P C D P P C D P G H P
m( (e)" (e)
0oNTN 0 0
0
t kK (-) " t
v(e) t
v(e)
dv,
0
(42)
t BK (-)T CK @ t BK (-) dv, t t t
(43)
t GT tK t G 0 t t dv, 0 0 t v(e) t`*t p 0 dC, t`*tr( (e)" NK T t`*t Q 0C (e) 0
t kK (/)(e)" t
t`*tfK (e)"
t
(44) (45)
t BK (-)T tq( dv, t
(46)
v(e) and with the 3]3 identity matrix I
C
D
tq I tq I tq I 11 12 13 (47) tK" tq I tq I tq I . 21 22 23 tq I tq I tq I 31 32 33 Then assembling the element matrices into global matrix provides the following system equations: MK t`*tU$K #(t KK (-)# t KK (/))UK " t`*tRK ! t`*tFK . (48) t t For the practical application, the surface of piezoelectric layer is covered by eletrode and the nodal electrical potential is coupled with each other over the surface. To simulate this problem, multipoint constraints are utilized. 3. Results and discussion In order to demonstrate the usefulness for and applicability to the solutions of large deformation dynamic responses of beam, plate and shell structures with piezoelectric patches are studied. PZT materials are used for the piezoelectric patches and their electro-mechanical properties are listed in Table 1. The polarized direction of PZT materials is along the normal direction of the structure surface. An electrode covers the top and bottom surfaces of piezoelectric patches. Table 1 Material properties of PZT E "E (N/m2) 11 22
E (N/m2) 33
l "l "l 12 23 31
o (kg/m3)
6.7]1010
5.0]1010
0.33
7800
e (C/m2) 31
e (C/m2) 33
e (C/m2) 15
Z (F/m) 1
Z (F/m) 3
!11.47
19.35
13.72
2.12]10~8
2.03]10~8
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
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Fig. 1. Central de#ections of the shell.
Fig. 2. A cantilever beam with a piezoelectric patch.
An example is solved to verify the present nonlinear formulation. A circular cylindrical panel is considered. The panel is clamped along all four edges and is subjected to uniform radial inward pressure. Its dimensions are given in Ref. [17]. The material parameters are E"3.10275]109 N/m2 and k"0.3. A mesh of 10]10 20-node elements is used to solve the problem. Due to the symmetry of the geometry and deformation, only one quarter of the panel is
10
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
Fig. 3. Tip dynamic de#ections of the cantilever beam.
Fig. 4. Output voltage of the piezoelectric patch on the cantilever beam.
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
11
Fig. 5. A quarter square plate with a central piezoelectric patch.
Fig. 6. Central dynamic de#ections of the clamped square plate.
analyzed. As shown in Fig. 1, the results calculated by the present method agree very well with those obtained by Chao and Reddy [17]. To illustrate the utility of the present formulation for nonlinear dynamic analysis of smart structures, beams and plates with piezoelectric sensors are considered. Firstly, the de#ections of a beam and the voltage outputs from a piezoelectric sensor are evaluated. Both linear and
12
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
Fig. 7. Output voltage of the piezoelectric patch on the clamped square plate.
Fig. 8. A quarter cylindrical shell with a central piezoelectric patch.
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
13
Fig. 9. Central dynamic de#ections of the clamped cylindrical shell.
nonlinear analyses are considered. A cantilever beam with a piezoelectric patch is shown in Fig. 2. A total of six elements with 80 nodes are employed for the analysis. Young's modulus and Poisson's ratio of the beam are taken as E"1.97]1011 N/m2 and l"0.33. The mass density is 7900 kg/m3. The constant load P"10 N is applied at the tip of the beam. Fig. 3 shows the dynamic responses of the beam calculated by the present method. Corresponding voltage outputs from the piezoelectric patch are also illustrated in Fig. 4. As shown in Fig. 3, the magnitude of the de#ections and the vibration frequency are signi"cantly a!ected by large deformation. The magnitude of the tip de#ection of the beam calculated by the large deformation analysis is about 71.1% smaller than one by the linear analysis. Fig. 4 shows that the large deformation analysis reduces the magnitude of the output voltage by 27.3%. The results show that the output voltages are not linearly dependent on the of tip de#ections. Secondly, a square plate with a central piezoelectric patch is considered. All four edges of the plate are clamped. Fig. 5 shows a quarter of the plate. A total of 26 elements with 240 nodes are used for this study. Young's modulus and Poisson's ratio of the plate are taken as E"1.97]1011 N/m2 and l"0.33, respectively. The mass density is 7900 kg/m3. The uniform pressure P"2]104 Pa is applied to the surface. Fig. 6 shows the central dynamic de#ections calculated by in"nitesimal and large deformation analyses. Corresponding voltage outputs of the piezoelectric patch are also illustrated in Fig. 7. The results are similar to those for the cantilever beam. Considered next is a composite cylindrical shell with central piezoelectric patch. The dimensions of the cylindrical shell are shown in Fig. 8. All four edges of the cylindrical shell are clamped. A total of 44 elements are employed for a "nite element model. The material properties of the composite shell are taken as E "E "9.653]1010 N/m2, 11 22 E "12.4]1010 N/m2, l "l "l "0.34, and G "G "G "6.205]109 N/m2. The 33 12 23 13 12 23 13
14
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
Fig. 10. Output voltage of the piezoelectric patch on the clamped cylindrical shell.
mass density is 1520 kg/m3. The time step size is *t"1]10~4 s. The principle direction 3 is along the straight edge of the cylindrical shell and the direction 2 is normal to the shell surface. The radial uniform pressure P"6]104 Pa is applied to the surface of the cylindrical shell. Figs. 9 and 10 show the central dynamic de#ections and corresponding output voltages of the piezoelectric patch.
4. Conclusions Based on the updated Lagrangian formulation, the geometrically nonlinear "nite element formulations have been developed for the analyses of structures integrated with piezoelectric materials. Twenty-node solid elements including electrical degrees of freedom have been developed and multipoint constraint equations for electrical degrees of freedom are adopted to simulate electrodes for piezoelectric materials. Numerical results show that de#ection amplitude, vibration frequency and output voltage are signi"cantly in#uenced by the large deformation of structures.
References [1] E.F. Crawley, J. de Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA J. 25 (10) (1987) 1373}1385. [2] S. Im, S.N. Atluri, E!ects of a piezo-actuator on a "nitely deformed beam subjected to general loading, AIAA J. 27 (12) (1989) 1801}1807.
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[3] C.K. Lee, Theory of laminated piezoelectric plates for the design of distributed sensors/actuators, Part I: governing equations and reciprocal relationships, J. Acoust. Soc. Amer. 87 (3) (1990) 1144}1158. [4] R.C. Batra, X.Q. Liang, Vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators, Comput. Struct. 63 (2) (1997) 203}216. [5] H.S. Tzou, M. Gadre, Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls, J. Sound Vibr. 132 (3) (1989) 433}450. [6] R. Lammering, The application of a "nite shell element for composites containing piezo-electric polymers in vibration control, Comput. Struct. 41 (5) (1991) 1101}1109. [7] W.-S Hwang, H.C. Park, Finite element modeling of piezoelectric sensors and actuators, AIAA J. 31 (5) (1993) 930}937. [8] H. Allik, T.J. Hughes, Finite element method for piezoelectric vibration, Int. J. Numer. Methods Eng. 2 (1979) 151}168. [9] M. Nailon, R.H. Coursant, F. Besnier, Analysis of piezoelectric structures by a "nite element method, ACTA Electron. 25 (4) (1983) 341}362. [10] S.K. Ha, C. Keilers, F.-K. Chang, Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators, AIAA J. 30 (3) (1992) 772}780. [11] H.S. Tzou, C.I. Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric "nite element approach, J. Sound Vibr. 138 (1) (1990) 17}34. [12] S. Yi, S.F. Ling, M. Ying, Finite element analysis of composite structures with smart constrained layer damping, Adv. Eng. Software 29 (3.6) (1998) 265}271. [13] G. Kulkarni, S.V. Hanagud, Modeling issues in the vibration control with piezoceramic actuators, Proc. Smart Struct. Mat. ASME, USA 24 (1991) 7}17. [14] P.F. Pai, A.H. Nayfeh, K. Oh, D.T. Mook, Re"ned nonlinear model of composite plates with integrated piezoelectric actuators and sensors, Int. J. Solids Struct. 30 (12) (1993) 1603}1630. [15] M. Preiswerk, A. Venkatesh, Analysis of vibration control using piezoceramics in planar #exible-linkage mechanisms, Smart Mater. Struct. 3 (2) (1994) 190}200. [16] U. Icardi, M.D. Sciuva, Large-de#ection and stress analysis of multilayered plates with induced-strain actuators, Smart Mater. Struct. 5 (2) (1996) 140}164. [17] W.C. Chao, J.N. Reddy, Analysis of laminated composites shells using a degenerated 3-D element, Int. J. Numer. Methods Eng. 20 (1984) 1991}2007.
Large deformation "nite element analyses of composite structures integrated with piezoelectric sensors and actuators Sung Yi*, Shih Fu Ling, Ming Ying School of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798, Singapore
Abstract Nonlinear dynamic responses of structures integrated with piezoelectric materials are studied. Finite element formulations are derived on the basis of the updated Lagrangian formulation and the principle of virtual work. Twenty-node solid elements including electrical degrees of freedom are developed to analyze structures with piezoelectric sensors and actuators and multipoint constraints for electrical degrees of freedom are utilized to simulate electrodes. A limited number of numerical examples are presented in order to verify the accuracy and convergence of the present formulation and to demonstrate its usefulness for and applicability to solutions of large deformation dynamic responses of piezoelectric structures. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear FEM; Smart structures
1. Introduction In recent decades, special attention has been paid to integrating piezoelectric materials which can be used for both sensors and actuators into base structures for structural control purpose. As a result of the increasing applications of piezoelectric materials, the accurate prediction of the performance of smart structures with sensing, actuating, and control functions becomes increasingly important. Beam, plate and shell structures with bonded or embedded piezoelectric actuators and sensors have been studied. Crawley and de Luis [1] studied vibration control of cantilever elastic and composite beams with surface-bonded and embedded piezoelectric materials analytically and
* Corresponding author. Tel.: #65-790-6239; fax: #65-791-1859. E-mail address: [email protected] (S. Yi) 0168-874X/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 4 5 - 1
2
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
experimentally. By considering both axial and transverse shear forces, Im and Atluri [2] studied the e!ects of a piezoelectric actuator mounted on a "nitely deformed beam based upon a static analysis. Lee [3] developed a theory for laminated piezoelectric plates based on the classical plate theory. Batra and Liang [4] also analyzed the steady-state vibration of a simply supported rectangular linear elastic laminated plate with embedded PZT layers. For thin laminated shells with piezoelectric layers, motion equations were derived by Tzou and Gadre [5] based upon Love's "rst-approximation shell theory and Hamilton's principle. Since analytical approaches are restricted to relatively simple geometries and boundary conditions, they cannot deal with complex piezoelectric structures. Finite element methods become very important and have been widely employed to model piezoelectric structures. Lammering [6] derived a "nite element formulation based on the shell theory for composite shell structures integrated with sensors and actuators. A shell element covered by the piezoelectric layers at the upper and lower surface symmetrically was developed. Hwang and Park [7] employed the classical laminate theory with the induced actuation and variational principles to formulate equations of motion. The plate element without the electro-mechanical coupling was developed. The total charge on the sensor layer was calculated from the direct piezoelectric equation. In order to model structures with complex geometry, isoparametric hexahedron and tetrahedral solid elements were developed for piezoceramic structures [8,9]. Ha et al. [10] also presented a "nite element formulation with eight-node threedimensional composite brick elements. For thin-walled piezoelectric structures, Tzou and Tseng [11] developed a solid "nite element for thin piezoelectric structures based on an eight-node solid element by introducing the incompatible models into the linear interpolation function. Yi et al. [12] developed a "nite element formulation using a solid element with Allman's rotation in order to analyze composite structures with smart constrained layers. Their study was focused on the use of viscoelastic damping and piezoelectric materials for the active and passive vibration and noise control and the "nite element formulation is developed on the basis of visco-piezoelectricity. Large deformation analyses for beam, plate and shell structures with piezoelectric materials have been reported in the literature. Kulkarni and Hanagud [13] derived a variational formulation for a combined isotropic, linearly elastic body and a piezoelectric body undergoing small or large deformations. A two-dimensional "nite element formulation based on the variational principle was applied to solve static and dynamic problems involving temporally and spatially varying electric "elds. Pai et al. [14] presented a nonlinear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large-rotation and small-strain vibrations. Preiswerk and Venkatesh [15] studied the vibration control of #exible links provided with nearly collocated piezoceramic sensors and actuators undergoing geometrically nonlinear de#ections. Euler}Bernoulli-theory-based beam "nite elements with p-convergence capabilities had been developed using the total Lagrangian formulation in order to represent large rotations of links. Icardi and Sciuva [16] employed the von Karman strain}displacement relations to investigate the stress "eld of multilayered intelligent anisotropic plates with a surfacebonded piezoelectric actuator layer. However, only limited work has been reported in the literature on the large deformation "nite element formulation of smart structures using solid elements. In the present study, dynamic responses of structures integrated with piezoelectric sensors and actuators under large geometrical deformation are considered. Finite element formulations are derived on the basis of the updated Lagrangian formulation and the principle of virtual work. In order to model complex piezoelectric structures, 20-node solid elements including electrical degrees
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
3
of freedom are developed and multipoint constraints for electrical degrees of freedom are employed to simulate electrodes.
2. Analysis 2.1. Mechanical and electrical strain}displacement relationships The strain vector e in material Cartesian coordinates is de"ned in terms of the in"nitesimal and large displacement components as e"[e e e c c c ]T 11 22 33 12 23 13 u 1 [(u )2#(v )2#(w )2] ,1 2 ,1 ,1 ,1 v 1 [(u )2#(v )2#(w )2] ,2 2 ,2 ,2 ,2 w 1 [(u )2#(v )2#(w )2] ,3 ,3 ,3 " # 2 ,3 , u #v u u #v v #w w ,2 ,1 ,1 ,2 ,1 ,2 ,1 ,2 v #w u u #v v #w w ,3 ,2 ,2 ,3 ,2 ,3 ,2 ,3 u #w u u #v v #w w ,3 ,1 ,1 ,3 ,1 ,3 ,1 ,3 where u, v, w are the displacements and the comma denotes the partial di!erentiation. Similarly, the electric "eld vector E is related to the electric potential < by
G HG
H
GH GH
E < 1 ,1 E" E "! < . 2 ,2 E < 3 ,3
(1)
(2)
2.2. Piezoelectric constitutive equations The constitutive relationship for linear piezoelectric materials can be described as r( "CK e(
(3)
with
C
CK "
C
e
D
eT !Z
r( "[r D]T
,
"[p p p q q q D D D ]T 11 22 33 12 23 13 1 2 3 e( "[e!E]T
(4)
(5)
"[e e e c c c !E !E !E ]T, (6) 11 22 33 12 23 13 1 2 3 where the superscript ¹ denotes the matrix transpose, e is the vector of mechanical strains, r is the vector of stresses, D is the vector of electric displacements, and E is the vector of electric "elds.
4
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In the principal material directions, the material sti!ness matrix C, the piezoelectric stress matrix e and the dielectric constant matrix Z for orthotropic piezoelectric materials can be written as
C
e
e e 12 13 14 e e e e" e 21 22 23 24 e e e e 31 32 33 34 Z 0 0 1 0 Z" 0 Z 2 0 0 Z 3 11
e
C
and
C
D
C 11 C 21 C C" 31 0
D
e 15 e 25 e 35
e 16 e , 26 e 36
(7)
(8)
C 12 C 22 C 32 0
C 13 C 23 C 33 0
0
0
0
44 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
C 55 0
0
C
C 66
D
,
(9)
where C "C . ij ji The transformed material properties with respect to the global coordinates can be explained as [12]
C
D C
D
T~1 0 T~T 0 m m CK (10) 0 T~1 0 T e e where T and T are the coordinate transformation matrices from material Cartesian coordinates m e to global Cartesian coordinates for strain and electric "eld vectors, respectively. CK @"
2.3. Updated Lagrangian formulation for piezoelectric materials For the updated Lagrangian formulation, all static and kinematic variables are referred to the con"guration at time t. The principle of virtual work for piezoelectric bodies is de"ned by
P
t
v
dt`*t e( T t`*t r( dv" t`*tR, t t
(11)
where t`*t r( "[ t`*t p t`*t p t`*t p t`*t q t`*t q t`*t q t`*t D t`*t D t`*t D ]T t t xx t yy t zz t xy t yz t xz t x t y t z t`*t e( "[ t`*t e t`*t e t`*t e t`*t c t`*t c t`*t c t t xx t yy t zz t xy t yz t xz ! t`*t E t`*t E t`*t E ]T. t x t y t z
(12)
(13)
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5
In the above, d denotes a small, arbitrary virtual variation, tv is the volume of body at con"guration time t, t`*tR is the external virtual work, t`*t e( and t`*t r( are the vectors of generalized strains and t t stresses at con"guration time t#*t referred to time t, respectively. It is assumed that the loading is deformation independent and can be speci"ed prior to the incremental analysis. The only body force is the inertial D'Alembert force de"ned by 0ot`*tuK with the mass density 0o at t"0. The external virtual work is
P
P
duT 0ot`*tuK dv# (duT t`*t p#d< t`*t Q) dC, (14) 0 0 0C v where u is the displacement vector, t`*tp is the vector of boundary tractions at con"guration time 0 t#*t referred to initial time, t`*t Q is the surface charge, 0v is the original volume of body and 0 0C is the original area on which boundary tractions and electrical charges are prescribed. The generalized stresses and strains can be decomposed into those at time t and incremental ones. Since the generalized Cauchy stresses tq( " t r( and t e( "0, one has t t t`*t r( " tq( # r( , (15) t t t`*t e( " e( " e( (-)# e( (/), (16) t t t t where e( (-) and e( (/) are the linear and nonlinear incremental strains corresponding to Eq. (1). t t Substituting Eqs. (14)}(16) into (11) yields t`*tRK "!
P
t
v
0
P
d e( T r( dv# t t
t
v
P P
d e( (/)T tq( dv"! t
t
v
P
d e( (-)T tq( dv! t
#
0C
0
v
duT 0ot`*tuK dv
(duT t`*tp#d< t`*t Q) dC. 0 0
(17)
By using the approximations r( " CK @ e( (-), t t t d e( "d e( (-). t t Eq. (17) can be linearized as
P
t
v
P
d e( (-)T CK @ e( (-) dv# t t t
t
(18) (19)
v
P
d e( (/)T tq( dv"! t
t
P
d e( (-)T tq( dv! t
0
P
duT 0ot`*tuK dv#
0C
(duT t`*tp 0
v v #d< t`*t Q) dC, (20) 0 where CK @ is the tangent material property matrix at time t in the global coordinate system. t 2.4. Finite element formulations The displacement "eld at time t#*t can be written in the incremental form as t`*tu" tu#u, where tu and u are the displacement "eld at time t and its increment, respectively.
(21)
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The incremental displacement "elds within an element can be explained in terms of the nodal displacements as u"N q(e) 20
(22)
u"[u v w]T,
(23)
with
q(e)"[u v w u v w 2 u v w ]T, (24) 1 1 1 2 2 2 20 20 20 where N is the matrix of interpolation functions for a 20-node solid element and q(e) is the vector 20 of time-dependent nodal displacements. Similarly, the electric potential < within an element can be interpolated using the node electric potential V (e) as <"N V (e), V where
(25)
V (e)"[< < 2 < ]T 1 2 20 N "[N N 2 N ]. V 1 2 20 Then the incremental displacement and electric "elds within an element can be written as
(26)
u( "NK q( (e),
(27)
(28)
where (29)
u( "[u <]T,
C
NK "
N 20 0
0
D
, N V q( (e)"[q(e) V (e)]T.
(30) (31)
The linear incremental generalized strains within an element can be represented in terms of the incremental nodal displacements and electric "elds as
C
t B(-) e( (-)" t BK (-)q( (e)" t t t 0
0
D
t B(-) t V
q( (e)
(32)
with
C
N t 1,x 0
0 t B(-)" t N t 1,y 0 N t 1,z
0
0
2
N t 1,y 0
0
2
N t 20,x 0
N t 1,z 0
2
0
2
N t 1,y N t 1,x
2
N t 20,y 0
2
N
N t 1,x N t 1,z 0
t 20,z
0
0
N t 20,y 0
0
N t 20,x N t 20,z 0
D
N t 20,z , 0 N t 20,y N t 20,x
(33)
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C
N t 1,x t B(-)" N t V t 1,y N t 1,z
N t 2,x N t 2,y N t 2,z
D
N t 20,x , N t 20,y N t 20,z
2 2 2
7
(34)
where N "LN /Ltx, N "LN /Lty and N "LN /Ltz. t i,x i t i,y i t i,z i Consequently, the nonlinear incremental generalized strains become
C
D
1 Ht G 0 q( (e) e( (/)" t BK (/)q( (e)" 2 t t t t 0 0
(35)
where
C D h t x 0
t
H"
0
h t y 0
0
0
h t y 0
0
h t z , 0
h t x h t z 0
(36)
h t y h h t z t x h "[Lu/Ltx Lv/Ltx Lw/Ltx], t x h "[Lu/Lty Lv/Lty Lw/Lty], t y h "[Lu/Ltz Lv/Ltz Lw/Ltz] t z and
C
N t 1,x 0 0
N t 1,y t G" 0 t 0 N t 1,z 0
(37) (38) (39)
0
0
2
N t 1,x 0
0
2
N t 20,x 0
N t 1,x 0
2
0
2
0
2
N t 20,y 0
N t 1,y 0
2
0
2
0
2
N t 20,z 0
0 N t 1,y 0 0 N t 1,z 0
0
N t 20,x 0
0
0 N t 20,y 0 0 N t 20,z 0
N t 20,x 0 0 N t 20,y 0 0
D
.
(40)
N t 1,z 2 t 20,z Note that the incremental displacement gradients are referred to the geometry at con"guration time t. Substitution of Eqs. (32) and (35) into the principle of virtual work leads to the "nite element equilibrium equations for each element 0
N
0
0
m( (e) t`*tq(K (e)#( t kK (-)(e)# t kK (/)(e))q( (e)" t`*tr( (e)! t`*tfK (e), t t
(41)
8
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where
P C D P P C D P G H P
m( (e)" (e)
0oNTN 0 0
0
t kK (-) " t
v(e) t
v(e)
dv,
0
(42)
t BK (-)T CK @ t BK (-) dv, t t t
(43)
t GT tK t G 0 t t dv, 0 0 t v(e) t`*t p 0 dC, t`*tr( (e)" NK T t`*t Q 0C (e) 0
t kK (/)(e)" t
t`*tfK (e)"
t
(44) (45)
t BK (-)T tq( dv, t
(46)
v(e) and with the 3]3 identity matrix I
C
D
tq I tq I tq I 11 12 13 (47) tK" tq I tq I tq I . 21 22 23 tq I tq I tq I 31 32 33 Then assembling the element matrices into global matrix provides the following system equations: MK t`*tU$K #(t KK (-)# t KK (/))UK " t`*tRK ! t`*tFK . (48) t t For the practical application, the surface of piezoelectric layer is covered by eletrode and the nodal electrical potential is coupled with each other over the surface. To simulate this problem, multipoint constraints are utilized. 3. Results and discussion In order to demonstrate the usefulness for and applicability to the solutions of large deformation dynamic responses of beam, plate and shell structures with piezoelectric patches are studied. PZT materials are used for the piezoelectric patches and their electro-mechanical properties are listed in Table 1. The polarized direction of PZT materials is along the normal direction of the structure surface. An electrode covers the top and bottom surfaces of piezoelectric patches. Table 1 Material properties of PZT E "E (N/m2) 11 22
E (N/m2) 33
l "l "l 12 23 31
o (kg/m3)
6.7]1010
5.0]1010
0.33
7800
e (C/m2) 31
e (C/m2) 33
e (C/m2) 15
Z (F/m) 1
Z (F/m) 3
!11.47
19.35
13.72
2.12]10~8
2.03]10~8
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
9
Fig. 1. Central de#ections of the shell.
Fig. 2. A cantilever beam with a piezoelectric patch.
An example is solved to verify the present nonlinear formulation. A circular cylindrical panel is considered. The panel is clamped along all four edges and is subjected to uniform radial inward pressure. Its dimensions are given in Ref. [17]. The material parameters are E"3.10275]109 N/m2 and k"0.3. A mesh of 10]10 20-node elements is used to solve the problem. Due to the symmetry of the geometry and deformation, only one quarter of the panel is
10
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
Fig. 3. Tip dynamic de#ections of the cantilever beam.
Fig. 4. Output voltage of the piezoelectric patch on the cantilever beam.
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
11
Fig. 5. A quarter square plate with a central piezoelectric patch.
Fig. 6. Central dynamic de#ections of the clamped square plate.
analyzed. As shown in Fig. 1, the results calculated by the present method agree very well with those obtained by Chao and Reddy [17]. To illustrate the utility of the present formulation for nonlinear dynamic analysis of smart structures, beams and plates with piezoelectric sensors are considered. Firstly, the de#ections of a beam and the voltage outputs from a piezoelectric sensor are evaluated. Both linear and
12
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
Fig. 7. Output voltage of the piezoelectric patch on the clamped square plate.
Fig. 8. A quarter cylindrical shell with a central piezoelectric patch.
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
13
Fig. 9. Central dynamic de#ections of the clamped cylindrical shell.
nonlinear analyses are considered. A cantilever beam with a piezoelectric patch is shown in Fig. 2. A total of six elements with 80 nodes are employed for the analysis. Young's modulus and Poisson's ratio of the beam are taken as E"1.97]1011 N/m2 and l"0.33. The mass density is 7900 kg/m3. The constant load P"10 N is applied at the tip of the beam. Fig. 3 shows the dynamic responses of the beam calculated by the present method. Corresponding voltage outputs from the piezoelectric patch are also illustrated in Fig. 4. As shown in Fig. 3, the magnitude of the de#ections and the vibration frequency are signi"cantly a!ected by large deformation. The magnitude of the tip de#ection of the beam calculated by the large deformation analysis is about 71.1% smaller than one by the linear analysis. Fig. 4 shows that the large deformation analysis reduces the magnitude of the output voltage by 27.3%. The results show that the output voltages are not linearly dependent on the of tip de#ections. Secondly, a square plate with a central piezoelectric patch is considered. All four edges of the plate are clamped. Fig. 5 shows a quarter of the plate. A total of 26 elements with 240 nodes are used for this study. Young's modulus and Poisson's ratio of the plate are taken as E"1.97]1011 N/m2 and l"0.33, respectively. The mass density is 7900 kg/m3. The uniform pressure P"2]104 Pa is applied to the surface. Fig. 6 shows the central dynamic de#ections calculated by in"nitesimal and large deformation analyses. Corresponding voltage outputs of the piezoelectric patch are also illustrated in Fig. 7. The results are similar to those for the cantilever beam. Considered next is a composite cylindrical shell with central piezoelectric patch. The dimensions of the cylindrical shell are shown in Fig. 8. All four edges of the cylindrical shell are clamped. A total of 44 elements are employed for a "nite element model. The material properties of the composite shell are taken as E "E "9.653]1010 N/m2, 11 22 E "12.4]1010 N/m2, l "l "l "0.34, and G "G "G "6.205]109 N/m2. The 33 12 23 13 12 23 13
14
S. Yi et al. / Finite Elements in Analysis and Design 35 (2000) 1}15
Fig. 10. Output voltage of the piezoelectric patch on the clamped cylindrical shell.
mass density is 1520 kg/m3. The time step size is *t"1]10~4 s. The principle direction 3 is along the straight edge of the cylindrical shell and the direction 2 is normal to the shell surface. The radial uniform pressure P"6]104 Pa is applied to the surface of the cylindrical shell. Figs. 9 and 10 show the central dynamic de#ections and corresponding output voltages of the piezoelectric patch.
4. Conclusions Based on the updated Lagrangian formulation, the geometrically nonlinear "nite element formulations have been developed for the analyses of structures integrated with piezoelectric materials. Twenty-node solid elements including electrical degrees of freedom have been developed and multipoint constraint equations for electrical degrees of freedom are adopted to simulate electrodes for piezoelectric materials. Numerical results show that de#ection amplitude, vibration frequency and output voltage are signi"cantly in#uenced by the large deformation of structures.
References [1] E.F. Crawley, J. de Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA J. 25 (10) (1987) 1373}1385. [2] S. Im, S.N. Atluri, E!ects of a piezo-actuator on a "nitely deformed beam subjected to general loading, AIAA J. 27 (12) (1989) 1801}1807.
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[3] C.K. Lee, Theory of laminated piezoelectric plates for the design of distributed sensors/actuators, Part I: governing equations and reciprocal relationships, J. Acoust. Soc. Amer. 87 (3) (1990) 1144}1158. [4] R.C. Batra, X.Q. Liang, Vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators, Comput. Struct. 63 (2) (1997) 203}216. [5] H.S. Tzou, M. Gadre, Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls, J. Sound Vibr. 132 (3) (1989) 433}450. [6] R. Lammering, The application of a "nite shell element for composites containing piezo-electric polymers in vibration control, Comput. Struct. 41 (5) (1991) 1101}1109. [7] W.-S Hwang, H.C. Park, Finite element modeling of piezoelectric sensors and actuators, AIAA J. 31 (5) (1993) 930}937. [8] H. Allik, T.J. Hughes, Finite element method for piezoelectric vibration, Int. J. Numer. Methods Eng. 2 (1979) 151}168. [9] M. Nailon, R.H. Coursant, F. Besnier, Analysis of piezoelectric structures by a "nite element method, ACTA Electron. 25 (4) (1983) 341}362. [10] S.K. Ha, C. Keilers, F.-K. Chang, Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators, AIAA J. 30 (3) (1992) 772}780. [11] H.S. Tzou, C.I. Tseng, Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric "nite element approach, J. Sound Vibr. 138 (1) (1990) 17}34. [12] S. Yi, S.F. Ling, M. Ying, Finite element analysis of composite structures with smart constrained layer damping, Adv. Eng. Software 29 (3.6) (1998) 265}271. [13] G. Kulkarni, S.V. Hanagud, Modeling issues in the vibration control with piezoceramic actuators, Proc. Smart Struct. Mat. ASME, USA 24 (1991) 7}17. [14] P.F. Pai, A.H. Nayfeh, K. Oh, D.T. Mook, Re"ned nonlinear model of composite plates with integrated piezoelectric actuators and sensors, Int. J. Solids Struct. 30 (12) (1993) 1603}1630. [15] M. Preiswerk, A. Venkatesh, Analysis of vibration control using piezoceramics in planar #exible-linkage mechanisms, Smart Mater. Struct. 3 (2) (1994) 190}200. [16] U. Icardi, M.D. Sciuva, Large-de#ection and stress analysis of multilayered plates with induced-strain actuators, Smart Mater. Struct. 5 (2) (1996) 140}164. [17] W.C. Chao, J.N. Reddy, Analysis of laminated composites shells using a degenerated 3-D element, Int. J. Numer. Methods Eng. 20 (1984) 1991}2007.