- Email: [email protected]
Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution
Computers and Structures 84 (2006) 1519–1526 www.elsevier.com/locate/compstruc
Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution M. Shakeri *, M.R. Eslami, A. Daneshmehr Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Avenue 15875-4413, Tehran 14838-17173, Iran Received 16 June 2005; accepted 14 January 2006
Abstract Elasticity solution is presented for infinitely long, simply-supported, orthotropic, piezoelectric shell panel under dynamic pressure excitation. The direct and inverse piezoelectric effects are considered. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Numerical examples are presented for [0/90/P] lamination, where P indicates the piezoelectric layer. Finally the results are compared with the published results. 2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Piezoelectric; Panel; Composite; Orthotropic; Shell; Three-dimensional; Sensor
1. Introduction As a result of two characteristics, i.e. the direct piezoelectric effect and the inverse piezoelectric effect, piezoelectric materials are widely used in hydro-and electroacoustics, communications and measurement technology. In recent years the novel usages of them as distributed sensors and actuators in active structures control as noise attenuation, shape control and vibration suppression have attracted serious attention. On the other hand, composite laminates are well known for their high stiffness and strength. Therefore, the integration of piezoelectric materials and structural composites has become the subject of focus in the area of smart materials and structures and numerous papers on this subject have been published. The theories of plates and shells coupled with piezoelectricity theory were applied to piezoelectric sensor and actuator design [1]. System equations for piezoelectric shell vibrations were derived, using Hamilton’s principle and linear piezoelectricity [2,3]. The piezoelastic solutions to long *
Corresponding author. Tel.: +98 21 3790661; fax: +98 21 6419736. E-mail address: [email protected] (M. Shakeri).
cylindrical panel and shell structures were also presented [4–6]. The solution for a homogeneous elastic (or piezoelectric) cylindrical shell was obtained in terms of the six-dimensional (or eight-dimensional) Pseudo-Stroh formalism [7], solution for the smart multilayered cylindrical shell was derived based on the transfer matrix method [7,8]. Recently, various two-dimensional solutions of laminated cylindrical shells with piezoelectric layer have been proposed [9,10]. Modelization and numerical approximation of piezoelectric thin shells has been also obtained [11]. Two finite element models are described to analyze shell structures with axisymmetric and conical panel geometries [12] and a new piezoelectric mixed variational theorem for smart multilayered composites is proposed [13]. Elastic solution of orthotropic thick laminated cylindrical panels subjected to dynamic loading was obtained by the authors [14]. Finally, static elasticity solution for infinite thick laminated shell panel with piezoelectric actuator layer was presented by the authors [15] and dynamic analysis of orthotropic laminated cylindrical panels is published by authors [16]. In this paper, the elastic solution of cross-ply laminated panel with piezoelectric layer is presented. The panel is
0045-7949/$ - see front matter 2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.01.039
1520
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
subjected to dynamic pressure excitation and three dimensional elasticity solution is used. The shell panel is simply supported at circumference edges and infinite at axial direction. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Finally three layered panels with piezoelectric sensor or actuator are solved and the results are compared with latest published results. 2. Basic equations and boundary conditions The equations of motion in the absence of body force for plane–strain (r h) problem are orr rr rh 1 osrh o2 ur þ þ ¼q 2 r oh or r ot 2 osrh 1 orh srh o uh þ þ2 ¼q 2 r oh or r ot
ð1Þ
The charge equation of electrostatics equilibrium in cylindrical coordinates is [4] oDr Dr oDh þ þ ¼0 ð2Þ or r roh The strain–displacement relations and the electric-potential (w) relations of the piezoelectric medium are written as [14] ður þ uh;h Þ 1 ; crh ¼ ður;h uh þ ruh;r Þ ð3Þ r r w;h ð4Þ Er ¼ w;r Eh ¼ r The linear constitutive equations of a piezoelectric medium are given by er ¼ ur;r ;
eh ¼
r ¼ Ce eT E
D ¼ ee þ gE
ð5Þ
where the superscript T denotes the transpose of a matrix. The three-dimensional components of stress r, strain e, electric field E and electric displacement D are given in cylindrical coordinate system (r, h, z), by r ¼ ½ rr
rh
E ¼ ½ Er
rz
eh
Ez ; ez
chz
D ¼ ½ Dr Dh Dz T ; 2 C 11 C 12 C 13 6 6 C 12 C 22 C 23 6 6 6 C 13 C 23 C 33 6 C¼6 6 0 0 0 6 6 6 0 0 0 4 0
srh T ;
srz
T
Eh
e ¼ ½ er
shz
0
0
T
crh ;
crz
ð6Þ 0
0
0
0
0
0
0
0
0
C 44
0
0
0
C 55
0
0
0
C 66
3 7 7 7 7 7 7 7; 7 7 7 7 5
2
e33 6 e¼4 0
e31 0
e32 0
0 0
0 0
0
0
0
0
e24
3 0 7 e15 5; 0
2
g1 6 g¼40
0 g2
0
0
3 0 7 05 g3 ð7Þ
where Cij, eij and gij denote, respectively, the components of the matrices of elastic, piezoelectric and dielectric constants of the piezoelectric materials poled in radial direction. After substitution Eqs. (3)–(6) into Eqs. (1) and (2), the governing equations of equilibrium in terms of displacements for each layer of cylindrical panel are obtained. o2 c11 o c22 c66 o2 2 2 2 ur c11 2 þ or r or r r oh c12 þ c66 o2 c22 þ c66 o þ uh oh r oroh r2 o2 e33 e31 o e15 o2 o2 ur þ 2 2 w¼q 2 þ e33 2 þ or r oh or r ot 2 c66 þ c12 o c22 þ c66 o þ ur oh r oroh r2 o2 c66 o c66 c22 o2 2 2 2 uh þ c66 2 þ or r or r r oh 2 e15 þ e31 o e15 o o2 uh þ 2 þ ð8aÞ w¼q 2 r oroh r oh ot o2 e31 þ e33 o e15 o2 e31 þ e15 o2 e15 o þ 2 2 ur þ e33 2 þ 2 uh or r oh or r r oroh r oh o2 g o g o2 g33 2 þ 33 þ 11 w¼0 ð8bÞ or r or r2 oh2 This equation is derived for a nine termed stiffness matrices material that could be similarly derived for a thirteen termed stiffness matrices material. The simply supported boundary conditions are (The span of panel is hm) ur ¼ rh ¼ szh ¼ w ¼ 0
at h ¼ 0; hm
ð9Þ
The interface conditions (equilibrium and compatibility) that must be met at interfaces of all adjacent layers are expressed as [13] rkr ¼ rkþ1 r
skrh ¼ skþ1 rh
ukr
ukh
¼
ukþ1 r
¼
ukþ1 h
Dkr ¼ Dkþ1 r k
w ¼w
kþ1
ð10aÞ ð10bÞ
where k and k + 1 represent typical adjacent layers. The boundary conditions of the inner and outer surfaces are rr ¼ srh ¼ w ¼ 0 at r ¼ Ri
ð11aÞ
rr ¼ P 0 ðh; tÞ; srh ¼ 0 at r ¼ Ro
ð11bÞ
Dr ¼ 0 ðif piezo: is sensorÞ w ¼ V 0 ðif piezo: is actuatorÞ 3. Solution of governing differential equations The solution satisfying the boundary conditions (9) may be assumed as
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
ur ¼
1 X
/i ðr; tÞ sinðbm hÞ
m¼1
w¼
1 X
uh ¼
1 X m¼1
/h ðr; tÞ cosðbm hÞ ð12Þ
wðr; tÞ sinðbm hÞ
m¼1
where
1521
By integrating the other ordinary differential equations, two similar equations are obtained. Changing Ni to Nj and then repeating the above procedure, three other equations are obtained. The result is written in the following finite element equilibrium equation for each non boundary elements: ½Me fX€ ge þ ½Ke fX ge ¼ fF ge
mp bm ¼ hm
ð16Þ
where [M]6·6, [K]6·6 and {F}6·1 are the mass, stiffness and force matrix respectively and
It should be noted that because of the boundary conditions only the mentioned terms of above equations should be considered. Substituting Eq. (12) into the governing equations of equilibrium in terms of displacements, the partial differential equations reduce to the following ordinary differential equations. d2 c11 d c22 c66 2 2 þ 2 bm /r c11 2 þ dr r dr r r c12 þ c66 d c22 þ c66 þ bm þ bm /h dr r r2 d2 e33 e31 d e15 2 o2 / 2 bm w ¼ q 2 r þ e33 2 þ dr dr r r ot c66 þ c12 d c22 þ c66 þ bm bm /r dr r r2 d2 c66 d c66 c22 2 2 þ 2 bm / h þ c66 2 þ dr r dr r r e15 þ e31 d e15 o2 / bm þ 2 bm w ¼ q 2 h þ ð13aÞ dr r r ot d2 e31 þ e33 d e15 2 2 bm /r e33 2 þ dr dr r r e31 þ e15 d e15 bm þ 2 bm / h þ dr r r 2 d g33 d g11 2 2 bm w ¼ 0 g33 2 þ ð13bÞ dr r dr r This system of equations is solved by considering linear shape functions Ni and Nj for /r, /h and w as " # " # wi /si /s ¼ ½ N i N j s ¼ r; h w ¼ ½ Ni Nj wj /sj ð14Þ and then applying the formal finite element method to the first governing equation (o.d.e.) yields 9 8h i 2 > > c11 drd 2 þ c11r drd cr222 þ cr662 b2m /r > > > > Z rj < =
c12 þc66 d c22 þc66 N i dr ¼ 0 þ bm r dr þ bm r2 /h > > h i ri > > 2 > > 2 : þ e d þ e33 e31 d e15 b2 w q o /r ; 33 dr2 dr r r2 m ot2 ð15Þ
T
fX ge ¼
/ri
/hi
wi
/rj
/hj
wj
Deriving equilibrium condition (10) in term of displacement by using Eqs. (3)–(5), the displacement components on the inner boundaries are obtained in term of values at neighboring nodes. Substituting results into (16) the finite element equilibrium equations for two neighboring elements at interior (k)th and (k + 1)th interfaces are obtained as ½Mk fX€ gk þ ½Kk fX gk ¼ f0g; ½Mkþ1 fX€ gkþ1 þ ½Kkþ1 fX gkþ1 ¼ f0g
ð17Þ
Applying the boundary conditions (11) for the first and last nodes in the inner and outer surfaces and using Eq. (16), the finite element equilibrium equations for the first and last elements become ½M1 fX€ g1 þ ½K1 fX g1 ¼ fF g1 ; ½M fX€ g þ ½K fX g ¼ fF g MI
MI
MI
MI
MI
ð18Þ
where M is the number of layers of panel and I is the number of nodes in a layer. Assembling Eqs. (16)–(18), the general finite element equilibrium equation is obtained as ½MfX€ g þ ½KfX g ¼ fF g
ð19Þ
After establishing the finite element dynamic equation and using the Newmark method as follows X ðt þ DtÞ ¼ X ðtÞ þ DtX_ ðtÞ þ Dt2 fð2 bÞX€ ðtÞ þ bX€ ðt þ 1Þg X_ ðt þ DtÞ ¼ X_ ðtÞ þ Dtfð1 cÞX€ ðtÞ þ cX€ ðt þ 1Þg ð20Þ static finite element matrix equations are obtained, and then, by using the Gauss–Sidel elimination method with a suitable time step, Dt, the equilibrium equation is solved. 4. Numerical results and discussion Three-layered cross-ply panel which its sequence lay-up is (0/90/P) composed of graphite-epoxy and piezoelectric layers is considered. The material properties of isotropic transverse piezoelectric lamina are
1522
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
2
11:5 7:43
7:78
0
0
0
3
where H is the thickness of panel, and
6 7:43 13:9 7:43 0 0 0 7 7 6 7 6 6 7:78 7:43 13:9 0 0 0 7 7 1010 Pa 6 C¼6 0 0 3:06 0 0 7 7 6 0 7 6 4 0 0 0 0 2:56 0 5 0 0 0 0 0 2:56 2 3 15:1 5:20 5:20 0 0 0 6 7 e¼4 0 0 0 0 0 12:7 5 C m2 0 0 0 0 12:7 0 2 3 5:62 0 0 6 7 g¼4 0 6:46 0 5 109 F m1 0 0 6:46
Rm ¼ 1;
hm ¼ p=3 rad:
P 0 ¼ 1 Pa;
S ¼ 8; 20; 52:
The piezoelectric layer in this research is considered as a sensor (direct effect), and as an actuator (inverse effect). As this research is mainly based on direct effect of piezoelectric layer, this effect is discussed in detail, and finally few examples of inverse effect is added. The variation of external load versus time is presented in Fig. 1. It is shown that after T = 0.001 s the curve flattens and the load reaches to its peak value, P0 = 1 Pa. In this respect all of the results are presented for T = 0.0075 s, as an example. To verify the validity of code and formulation, the piezoelectric layer is dropped and the static state results are compared with those of Chen et al. [4], and Ren [17],
The material properties of the graphite-epoxy composite are EL ¼ 25ET ; GTT ¼ 0:2ET ;
ET ¼ 6:85 GPa;
GLT ¼ 0:5ET ;
mLT ¼ mTT ¼ 0:25
where subscripts ‘‘L’’ and ‘‘T’’ denote the fiber and transverse directions, respectively. The forcing function is chosen as 1 X P 0 ðh; tÞ ¼ P 0 ð1 eat Þ sinðbm hÞ where a ¼ 13; 100 m¼1
ð21Þ The numerical results are described in the form of maximum non dimensional displacements and stresses as follow 100ET r Rm ; ður ; uh Þ; r ¼ 4 H HS ðrr ; rh ; srh Þ ¼ ðrr ; rh ; srh Þ=P 0 ð ur ; uh Þ ¼
S¼
Rm ; H
ð22Þ
Fig. 1. Variation in external pressure with time P0(t) = P0 (1 eat) where a = 13,100.
Table 1 Maximum displacement and stress in static state S
[4] ur ð0; hm =2Þ
Case 1 2 0.997 10
0.115
100
0.0755
Case 2 2 1.436 10
0.144
100
0.0786
rh 12 ; hm =2 2.455 1.907 0.890 0.807 0.758 0.751
3.647 2.463 0.995 0.897 0.787 0.779
[17] srh ð0; 0Þ
ur ð0; hm =2Þ
0.555
0.999
0.579
0.115
0.565
0.0755
0.394
1.436
0.525
0.144
0.523
0.0787
rh 12 ; hm =2 2.455 1.907 0.890 0.807 0.758 0.751
3.647 2.463 0.995 0.897 0.786 0.781
Present srh ð0; 0Þ
ur ð0; hm =2Þ
0.555
1.011
0.579
0.115
0.565
0.0755
0.394
1.443
0.525
1.144
0.523
0.0788
rh 12 ; hm =2 2.279 2.029 0.873 0.814 0.755 0.753
3.653 2.571 0.997 0.899 0.792 0.783
srh ð0; 0Þ 5.56 0.572 0.561
0.394 0.525 0.524
Case 1: Single-layered substrate with fiber parallel to the h direction. Case 2: Three-layered substrate with fiber of the center layer and of top and bottom layers oriented in the z direction and the h direction, respectively, each layer being of equal thickness.
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
1523
in Table 1. It is seen that a good agreement stands between the results and the deviations are negligible. Additionally all the presented results satisfy the boundary conditions. 4.1. Direct effect Figs. 2–4 illustrate the circumferential, in plane shear and radial stresses across the thickness with S, respectively. The boundary and inter laminar conditions are satisfied in them. The stresses depend strongly on the radial coordinate. Fig. 5 illustrates the radial stress across the thickness for static loading. The difference of results in Figs. 4 and 5 is because of the inertia effect in dynamic analysis. Fig. 3 shows the through-thickness distribution of transverse shear stress (srh), which in each layer is very close to a parabola form. In Figs. 6 and 7, the distributions of the mechanical displacement in the radial direction for different S are presented. The boundary and inter laminar continuity
Fig. 4. Distribution of rr across the r (0/90/P, h = hm/2).
Fig. 2. Distribution of rh across the r (0/90/P, h = hm/2).
Fig. 5. Distribution of rr across the r in static load P0 = 1 Pa (0/90/P, h = hm/2).
Fig. 3. Distribution of srh across the r (0/90/P, h = 0).
Fig. 6. Distribution of uh across the r (0/90/P, h = 0).
1524
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
Fig. 7. Distribution of ur across the r (0/90/P, h = hm/2).
Fig. 9. Distribution of w across the r (0/90/P, h = hm/2).
Fig. 10. Variation in ur with time (0/90/P, S = 20, h = hm/2). Fig. 8. Distribution of ur across the r in static load P0 = 1 Pa (0/90/P, h = hm/2).
conditions are satisfied. In these figures slopes imply inplane strains which are discontinuous in the interfaces. Fig. 8 shows the distribution of the mechanical displacement for static loading P0 = 1 Pa, which can be used to find the inertia effect in dynamic analysis, when compared with Fig. 7. Fig. 9 shows the electric potential distribution corresponding to different S. It is shown that the distribution of the electric potential depends on S. When S is less than 4, the electric potential in the sensor will change linearly. This is different from the results of [4]. The time histories of the radial and circumferential displacements, ur , uh in the middle layer of panel, are shown in Figs. 10 and 11 respectively. The variations in amplitude and period of motion in radial direction are similar to those for the circumferential direction. The time histories of the electric potential, w, for the middle layer of sensor, is also shown in Fig. 12.
Fig. 11. Variation in uh with time (0/90/P, S = 20, h = 0).
4.2. Inverse effect Fig. 13 shows the through-thickness distribution of transverse shear stress, (srh). The boundary and inter
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
1525
Fig. 12. Variation in w with time (0/90/P, S = 20, h = hm/2). Fig. 14. Distribution of ur across the r in static load P0 = 1 Pa (0/90/P, h = hm/2).
Fig. 13. Distribution of srh across the r (0/90/P, h = 0). Fig. 15. Distribution of w across the r (0/90/P, h = hm/2).
laminar continuity conditions are satisfied. This figure has a near-parabola form for each layer. According to this figure the electrical effect becomes much stronger than the mechanical loading with increasing the applied voltage. In Fig. 14, the distribution of the mechanical displacement in the radial direction due to the applied voltage is presented. The boundary and inter laminar continuity conditions are satisfied. Fig. 15 shows the electric potential distribution corresponding to different applied voltages. The distribution of the electric potential depends on the applied voltage. Only if the applied voltage is much stronger than the induced electric potential by strain, the electric potential in the actuator will change linearly. From the above studies, we can conclude that the linear variation assumption of the electric potential in the piezoelectric layer must be treated with caution and the threedimensional methods for the piezoelastic response analysis are shown to be the preferred ones.
5. Conclusion A study on the three-dimensional elasticity solution of shell panel piezoelectric sensor or actuator is presented. In this paper, the structure is infinitely long, simply-supported, orthotropic and under pressure excitation. The present study has shown that the Fourier series expansion method is suitable for the mechanical displacement and electric potential analysis in panel-type actuators. The direct piezoelectric effect of the piezoelectric layer subjected to outer pressure is investigated in detail. It is always assumed in the analytical analysis of the piezoelectric structures that the electric potential in the piezoelectric layers varies linearly and the displacements change in the form of prescribed functions across its thickness [10]. However, it has been shown that the distributions of the mechanical displacements and electric potential of piezoelectric response are very complicated and cannot be
1526
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
treated as pure elastic structures or piezoelectric structures. Therefore, three-dimensional analysis of piezoelastic behavior of structures is recommended even for thin laminated structures. Since a comprehensive and exact study of active piezoelectric structures is still unavailable, the present work provides an enhanced insight to the mechanical and electric behaviors of this type of smart structure. Results presented in this paper are also useful for assessing approximate analysis. References [1] Lee PCY, Yu JD. Governing equations of piezoelectric plates with graded properties across the thickness. Proc Annu IEEE Int Freq Control Symp 1996:623–31. [2] Tzou HS, Zhong JP. Electromechanics and vibrations of piezoelectric shell distributed systems. J Dyn Syst Meas Control 1993;115(3): 506–17. [3] Tzou HS, Zhong JP. Linear theory of piezoelectric shell vibrations. J Sound Vib 1994;175(1):77–88. [4] Chen C-Q, Shen Y-P, Wang X-M. Exact solution for orthotropic cylindrical shell with piezoelectric layers under cylindrical bending. Int J Solids Struct 1996;33(30):4481–94. [5] Dumir PC, Dube GP, Kapuria S. Exact piezoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending. Int J Solids Struct 1997;34(6):685–702. [6] Kapuria S, Sengupta S, Dumir PC. Three-dimensional solution for simply-supported piezoelectric cylindrical shell for axisymmetric load. Comput Methods Appl Mech Eng 1997;140(1–2):139–55.
[7] Pan E. Exact solution for simply supported and multilayered magneto-electro-elastic plates. ASME J Appl Mech 2001;68: 169–87. [8] Yue ZQ, Yin JH. Backward transfer-matrix method for elastic analysis of layered solids with imperfect bonding. J Elasticity 1998;50:109–28. [9] Wang X, Zhong Z. Three-dimensional solution of smart laminated anisotropic circular cylindrical shell with imperfect bonding. Int J Solids Struct 2003;40:5901–21. [10] Chen CQ, Shen YP. Piezothermoelasticity analysis for a cylindrical shell under the state of axisymmetric deformation. Int J Eng Sci 2003;34(14). [11] Bernadou M, Haenel C. Modelization and numerical approximation of piezoelectric thin shells. Part 1: The continuous problems. Comput Methods Appl Mech Eng 2003;192:4003–43. [12] Mota Soares CA, Mota Soares CM, Pinto Correia IF. Modeling of laminated shells with integrated sensors and actuators. In: Topping BHV, Mota Soares CA, editors. Progress in computational structures technology. Saxe-Coburg Publications; 2004. p. 281–309. [13] Benjeddou A, Andrianarison O. A piezoelectric mixed variational theorem for smart multilayered composites. Mech Adv Mater Struct 2005;12:1–11. [14] Shakeri M, Alibiglu A, Eslami MR. Elasticity solution for thick laminated anisotropic cylindrical panels under dynamic load. J Mech Eng Sci 2002;216(Part C). I Mech E. [15] Shakeri M, Daneshmehr A, Alibiglu A. Elasticity solution for thick laminated shell panel with piezoelectric layer. EASEC-9 Conf., Bali, Indonesia; 2003. [16] Shakeri M, Alibeiglooo A. Dynamic analysis of orthotropic laminated cylindrical panel. Mech Adv Mater Struct 2005;12:67–75. [17] Ren JG. Exact solution for laminated cylindrical shell in cylindrical bending. Comput Sci Technol 1987;29:169–87.
Dynamic analysis of thick laminated shell panel with piezoelectric layer based on three dimensional elasticity solution M. Shakeri *, M.R. Eslami, A. Daneshmehr Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Avenue 15875-4413, Tehran 14838-17173, Iran Received 16 June 2005; accepted 14 January 2006
Abstract Elasticity solution is presented for infinitely long, simply-supported, orthotropic, piezoelectric shell panel under dynamic pressure excitation. The direct and inverse piezoelectric effects are considered. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Numerical examples are presented for [0/90/P] lamination, where P indicates the piezoelectric layer. Finally the results are compared with the published results. 2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Piezoelectric; Panel; Composite; Orthotropic; Shell; Three-dimensional; Sensor
1. Introduction As a result of two characteristics, i.e. the direct piezoelectric effect and the inverse piezoelectric effect, piezoelectric materials are widely used in hydro-and electroacoustics, communications and measurement technology. In recent years the novel usages of them as distributed sensors and actuators in active structures control as noise attenuation, shape control and vibration suppression have attracted serious attention. On the other hand, composite laminates are well known for their high stiffness and strength. Therefore, the integration of piezoelectric materials and structural composites has become the subject of focus in the area of smart materials and structures and numerous papers on this subject have been published. The theories of plates and shells coupled with piezoelectricity theory were applied to piezoelectric sensor and actuator design [1]. System equations for piezoelectric shell vibrations were derived, using Hamilton’s principle and linear piezoelectricity [2,3]. The piezoelastic solutions to long *
Corresponding author. Tel.: +98 21 3790661; fax: +98 21 6419736. E-mail address: [email protected] (M. Shakeri).
cylindrical panel and shell structures were also presented [4–6]. The solution for a homogeneous elastic (or piezoelectric) cylindrical shell was obtained in terms of the six-dimensional (or eight-dimensional) Pseudo-Stroh formalism [7], solution for the smart multilayered cylindrical shell was derived based on the transfer matrix method [7,8]. Recently, various two-dimensional solutions of laminated cylindrical shells with piezoelectric layer have been proposed [9,10]. Modelization and numerical approximation of piezoelectric thin shells has been also obtained [11]. Two finite element models are described to analyze shell structures with axisymmetric and conical panel geometries [12] and a new piezoelectric mixed variational theorem for smart multilayered composites is proposed [13]. Elastic solution of orthotropic thick laminated cylindrical panels subjected to dynamic loading was obtained by the authors [14]. Finally, static elasticity solution for infinite thick laminated shell panel with piezoelectric actuator layer was presented by the authors [15] and dynamic analysis of orthotropic laminated cylindrical panels is published by authors [16]. In this paper, the elastic solution of cross-ply laminated panel with piezoelectric layer is presented. The panel is
0045-7949/$ - see front matter 2006 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2006.01.039
1520
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
subjected to dynamic pressure excitation and three dimensional elasticity solution is used. The shell panel is simply supported at circumference edges and infinite at axial direction. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients by means of trigonometric function expansion in circumferential direction. The resulting ordinary differential equations are solved by the finite element method. Finally three layered panels with piezoelectric sensor or actuator are solved and the results are compared with latest published results. 2. Basic equations and boundary conditions The equations of motion in the absence of body force for plane–strain (r h) problem are orr rr rh 1 osrh o2 ur þ þ ¼q 2 r oh or r ot 2 osrh 1 orh srh o uh þ þ2 ¼q 2 r oh or r ot
ð1Þ
The charge equation of electrostatics equilibrium in cylindrical coordinates is [4] oDr Dr oDh þ þ ¼0 ð2Þ or r roh The strain–displacement relations and the electric-potential (w) relations of the piezoelectric medium are written as [14] ður þ uh;h Þ 1 ; crh ¼ ður;h uh þ ruh;r Þ ð3Þ r r w;h ð4Þ Er ¼ w;r Eh ¼ r The linear constitutive equations of a piezoelectric medium are given by er ¼ ur;r ;
eh ¼
r ¼ Ce eT E
D ¼ ee þ gE
ð5Þ
where the superscript T denotes the transpose of a matrix. The three-dimensional components of stress r, strain e, electric field E and electric displacement D are given in cylindrical coordinate system (r, h, z), by r ¼ ½ rr
rh
E ¼ ½ Er
rz
eh
Ez ; ez
chz
D ¼ ½ Dr Dh Dz T ; 2 C 11 C 12 C 13 6 6 C 12 C 22 C 23 6 6 6 C 13 C 23 C 33 6 C¼6 6 0 0 0 6 6 6 0 0 0 4 0
srh T ;
srz
T
Eh
e ¼ ½ er
shz
0
0
T
crh ;
crz
ð6Þ 0
0
0
0
0
0
0
0
0
C 44
0
0
0
C 55
0
0
0
C 66
3 7 7 7 7 7 7 7; 7 7 7 7 5
2
e33 6 e¼4 0
e31 0
e32 0
0 0
0 0
0
0
0
0
e24
3 0 7 e15 5; 0
2
g1 6 g¼40
0 g2
0
0
3 0 7 05 g3 ð7Þ
where Cij, eij and gij denote, respectively, the components of the matrices of elastic, piezoelectric and dielectric constants of the piezoelectric materials poled in radial direction. After substitution Eqs. (3)–(6) into Eqs. (1) and (2), the governing equations of equilibrium in terms of displacements for each layer of cylindrical panel are obtained. o2 c11 o c22 c66 o2 2 2 2 ur c11 2 þ or r or r r oh c12 þ c66 o2 c22 þ c66 o þ uh oh r oroh r2 o2 e33 e31 o e15 o2 o2 ur þ 2 2 w¼q 2 þ e33 2 þ or r oh or r ot 2 c66 þ c12 o c22 þ c66 o þ ur oh r oroh r2 o2 c66 o c66 c22 o2 2 2 2 uh þ c66 2 þ or r or r r oh 2 e15 þ e31 o e15 o o2 uh þ 2 þ ð8aÞ w¼q 2 r oroh r oh ot o2 e31 þ e33 o e15 o2 e31 þ e15 o2 e15 o þ 2 2 ur þ e33 2 þ 2 uh or r oh or r r oroh r oh o2 g o g o2 g33 2 þ 33 þ 11 w¼0 ð8bÞ or r or r2 oh2 This equation is derived for a nine termed stiffness matrices material that could be similarly derived for a thirteen termed stiffness matrices material. The simply supported boundary conditions are (The span of panel is hm) ur ¼ rh ¼ szh ¼ w ¼ 0
at h ¼ 0; hm
ð9Þ
The interface conditions (equilibrium and compatibility) that must be met at interfaces of all adjacent layers are expressed as [13] rkr ¼ rkþ1 r
skrh ¼ skþ1 rh
ukr
ukh
¼
ukþ1 r
¼
ukþ1 h
Dkr ¼ Dkþ1 r k
w ¼w
kþ1
ð10aÞ ð10bÞ
where k and k + 1 represent typical adjacent layers. The boundary conditions of the inner and outer surfaces are rr ¼ srh ¼ w ¼ 0 at r ¼ Ri
ð11aÞ
rr ¼ P 0 ðh; tÞ; srh ¼ 0 at r ¼ Ro
ð11bÞ
Dr ¼ 0 ðif piezo: is sensorÞ w ¼ V 0 ðif piezo: is actuatorÞ 3. Solution of governing differential equations The solution satisfying the boundary conditions (9) may be assumed as
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
ur ¼
1 X
/i ðr; tÞ sinðbm hÞ
m¼1
w¼
1 X
uh ¼
1 X m¼1
/h ðr; tÞ cosðbm hÞ ð12Þ
wðr; tÞ sinðbm hÞ
m¼1
where
1521
By integrating the other ordinary differential equations, two similar equations are obtained. Changing Ni to Nj and then repeating the above procedure, three other equations are obtained. The result is written in the following finite element equilibrium equation for each non boundary elements: ½Me fX€ ge þ ½Ke fX ge ¼ fF ge
mp bm ¼ hm
ð16Þ
where [M]6·6, [K]6·6 and {F}6·1 are the mass, stiffness and force matrix respectively and
It should be noted that because of the boundary conditions only the mentioned terms of above equations should be considered. Substituting Eq. (12) into the governing equations of equilibrium in terms of displacements, the partial differential equations reduce to the following ordinary differential equations. d2 c11 d c22 c66 2 2 þ 2 bm /r c11 2 þ dr r dr r r c12 þ c66 d c22 þ c66 þ bm þ bm /h dr r r2 d2 e33 e31 d e15 2 o2 / 2 bm w ¼ q 2 r þ e33 2 þ dr dr r r ot c66 þ c12 d c22 þ c66 þ bm bm /r dr r r2 d2 c66 d c66 c22 2 2 þ 2 bm / h þ c66 2 þ dr r dr r r e15 þ e31 d e15 o2 / bm þ 2 bm w ¼ q 2 h þ ð13aÞ dr r r ot d2 e31 þ e33 d e15 2 2 bm /r e33 2 þ dr dr r r e31 þ e15 d e15 bm þ 2 bm / h þ dr r r 2 d g33 d g11 2 2 bm w ¼ 0 g33 2 þ ð13bÞ dr r dr r This system of equations is solved by considering linear shape functions Ni and Nj for /r, /h and w as " # " # wi /si /s ¼ ½ N i N j s ¼ r; h w ¼ ½ Ni Nj wj /sj ð14Þ and then applying the formal finite element method to the first governing equation (o.d.e.) yields 9 8h i 2 > > c11 drd 2 þ c11r drd cr222 þ cr662 b2m /r > > > > Z rj < =
c12 þc66 d c22 þc66 N i dr ¼ 0 þ bm r dr þ bm r2 /h > > h i ri > > 2 > > 2 : þ e d þ e33 e31 d e15 b2 w q o /r ; 33 dr2 dr r r2 m ot2 ð15Þ
T
fX ge ¼
/ri
/hi
wi
/rj
/hj
wj
Deriving equilibrium condition (10) in term of displacement by using Eqs. (3)–(5), the displacement components on the inner boundaries are obtained in term of values at neighboring nodes. Substituting results into (16) the finite element equilibrium equations for two neighboring elements at interior (k)th and (k + 1)th interfaces are obtained as ½Mk fX€ gk þ ½Kk fX gk ¼ f0g; ½Mkþ1 fX€ gkþ1 þ ½Kkþ1 fX gkþ1 ¼ f0g
ð17Þ
Applying the boundary conditions (11) for the first and last nodes in the inner and outer surfaces and using Eq. (16), the finite element equilibrium equations for the first and last elements become ½M1 fX€ g1 þ ½K1 fX g1 ¼ fF g1 ; ½M fX€ g þ ½K fX g ¼ fF g MI
MI
MI
MI
MI
ð18Þ
where M is the number of layers of panel and I is the number of nodes in a layer. Assembling Eqs. (16)–(18), the general finite element equilibrium equation is obtained as ½MfX€ g þ ½KfX g ¼ fF g
ð19Þ
After establishing the finite element dynamic equation and using the Newmark method as follows X ðt þ DtÞ ¼ X ðtÞ þ DtX_ ðtÞ þ Dt2 fð2 bÞX€ ðtÞ þ bX€ ðt þ 1Þg X_ ðt þ DtÞ ¼ X_ ðtÞ þ Dtfð1 cÞX€ ðtÞ þ cX€ ðt þ 1Þg ð20Þ static finite element matrix equations are obtained, and then, by using the Gauss–Sidel elimination method with a suitable time step, Dt, the equilibrium equation is solved. 4. Numerical results and discussion Three-layered cross-ply panel which its sequence lay-up is (0/90/P) composed of graphite-epoxy and piezoelectric layers is considered. The material properties of isotropic transverse piezoelectric lamina are
1522
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
2
11:5 7:43
7:78
0
0
0
3
where H is the thickness of panel, and
6 7:43 13:9 7:43 0 0 0 7 7 6 7 6 6 7:78 7:43 13:9 0 0 0 7 7 1010 Pa 6 C¼6 0 0 3:06 0 0 7 7 6 0 7 6 4 0 0 0 0 2:56 0 5 0 0 0 0 0 2:56 2 3 15:1 5:20 5:20 0 0 0 6 7 e¼4 0 0 0 0 0 12:7 5 C m2 0 0 0 0 12:7 0 2 3 5:62 0 0 6 7 g¼4 0 6:46 0 5 109 F m1 0 0 6:46
Rm ¼ 1;
hm ¼ p=3 rad:
P 0 ¼ 1 Pa;
S ¼ 8; 20; 52:
The piezoelectric layer in this research is considered as a sensor (direct effect), and as an actuator (inverse effect). As this research is mainly based on direct effect of piezoelectric layer, this effect is discussed in detail, and finally few examples of inverse effect is added. The variation of external load versus time is presented in Fig. 1. It is shown that after T = 0.001 s the curve flattens and the load reaches to its peak value, P0 = 1 Pa. In this respect all of the results are presented for T = 0.0075 s, as an example. To verify the validity of code and formulation, the piezoelectric layer is dropped and the static state results are compared with those of Chen et al. [4], and Ren [17],
The material properties of the graphite-epoxy composite are EL ¼ 25ET ; GTT ¼ 0:2ET ;
ET ¼ 6:85 GPa;
GLT ¼ 0:5ET ;
mLT ¼ mTT ¼ 0:25
where subscripts ‘‘L’’ and ‘‘T’’ denote the fiber and transverse directions, respectively. The forcing function is chosen as 1 X P 0 ðh; tÞ ¼ P 0 ð1 eat Þ sinðbm hÞ where a ¼ 13; 100 m¼1
ð21Þ The numerical results are described in the form of maximum non dimensional displacements and stresses as follow 100ET r Rm ; ður ; uh Þ; r ¼ 4 H HS ðrr ; rh ; srh Þ ¼ ðrr ; rh ; srh Þ=P 0 ð ur ; uh Þ ¼
S¼
Rm ; H
ð22Þ
Fig. 1. Variation in external pressure with time P0(t) = P0 (1 eat) where a = 13,100.
Table 1 Maximum displacement and stress in static state S
[4] ur ð0; hm =2Þ
Case 1 2 0.997 10
0.115
100
0.0755
Case 2 2 1.436 10
0.144
100
0.0786
rh 12 ; hm =2 2.455 1.907 0.890 0.807 0.758 0.751
3.647 2.463 0.995 0.897 0.787 0.779
[17] srh ð0; 0Þ
ur ð0; hm =2Þ
0.555
0.999
0.579
0.115
0.565
0.0755
0.394
1.436
0.525
0.144
0.523
0.0787
rh 12 ; hm =2 2.455 1.907 0.890 0.807 0.758 0.751
3.647 2.463 0.995 0.897 0.786 0.781
Present srh ð0; 0Þ
ur ð0; hm =2Þ
0.555
1.011
0.579
0.115
0.565
0.0755
0.394
1.443
0.525
1.144
0.523
0.0788
rh 12 ; hm =2 2.279 2.029 0.873 0.814 0.755 0.753
3.653 2.571 0.997 0.899 0.792 0.783
srh ð0; 0Þ 5.56 0.572 0.561
0.394 0.525 0.524
Case 1: Single-layered substrate with fiber parallel to the h direction. Case 2: Three-layered substrate with fiber of the center layer and of top and bottom layers oriented in the z direction and the h direction, respectively, each layer being of equal thickness.
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
1523
in Table 1. It is seen that a good agreement stands between the results and the deviations are negligible. Additionally all the presented results satisfy the boundary conditions. 4.1. Direct effect Figs. 2–4 illustrate the circumferential, in plane shear and radial stresses across the thickness with S, respectively. The boundary and inter laminar conditions are satisfied in them. The stresses depend strongly on the radial coordinate. Fig. 5 illustrates the radial stress across the thickness for static loading. The difference of results in Figs. 4 and 5 is because of the inertia effect in dynamic analysis. Fig. 3 shows the through-thickness distribution of transverse shear stress (srh), which in each layer is very close to a parabola form. In Figs. 6 and 7, the distributions of the mechanical displacement in the radial direction for different S are presented. The boundary and inter laminar continuity
Fig. 4. Distribution of rr across the r (0/90/P, h = hm/2).
Fig. 2. Distribution of rh across the r (0/90/P, h = hm/2).
Fig. 5. Distribution of rr across the r in static load P0 = 1 Pa (0/90/P, h = hm/2).
Fig. 3. Distribution of srh across the r (0/90/P, h = 0).
Fig. 6. Distribution of uh across the r (0/90/P, h = 0).
1524
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
Fig. 7. Distribution of ur across the r (0/90/P, h = hm/2).
Fig. 9. Distribution of w across the r (0/90/P, h = hm/2).
Fig. 10. Variation in ur with time (0/90/P, S = 20, h = hm/2). Fig. 8. Distribution of ur across the r in static load P0 = 1 Pa (0/90/P, h = hm/2).
conditions are satisfied. In these figures slopes imply inplane strains which are discontinuous in the interfaces. Fig. 8 shows the distribution of the mechanical displacement for static loading P0 = 1 Pa, which can be used to find the inertia effect in dynamic analysis, when compared with Fig. 7. Fig. 9 shows the electric potential distribution corresponding to different S. It is shown that the distribution of the electric potential depends on S. When S is less than 4, the electric potential in the sensor will change linearly. This is different from the results of [4]. The time histories of the radial and circumferential displacements, ur , uh in the middle layer of panel, are shown in Figs. 10 and 11 respectively. The variations in amplitude and period of motion in radial direction are similar to those for the circumferential direction. The time histories of the electric potential, w, for the middle layer of sensor, is also shown in Fig. 12.
Fig. 11. Variation in uh with time (0/90/P, S = 20, h = 0).
4.2. Inverse effect Fig. 13 shows the through-thickness distribution of transverse shear stress, (srh). The boundary and inter
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
1525
Fig. 12. Variation in w with time (0/90/P, S = 20, h = hm/2). Fig. 14. Distribution of ur across the r in static load P0 = 1 Pa (0/90/P, h = hm/2).
Fig. 13. Distribution of srh across the r (0/90/P, h = 0). Fig. 15. Distribution of w across the r (0/90/P, h = hm/2).
laminar continuity conditions are satisfied. This figure has a near-parabola form for each layer. According to this figure the electrical effect becomes much stronger than the mechanical loading with increasing the applied voltage. In Fig. 14, the distribution of the mechanical displacement in the radial direction due to the applied voltage is presented. The boundary and inter laminar continuity conditions are satisfied. Fig. 15 shows the electric potential distribution corresponding to different applied voltages. The distribution of the electric potential depends on the applied voltage. Only if the applied voltage is much stronger than the induced electric potential by strain, the electric potential in the actuator will change linearly. From the above studies, we can conclude that the linear variation assumption of the electric potential in the piezoelectric layer must be treated with caution and the threedimensional methods for the piezoelastic response analysis are shown to be the preferred ones.
5. Conclusion A study on the three-dimensional elasticity solution of shell panel piezoelectric sensor or actuator is presented. In this paper, the structure is infinitely long, simply-supported, orthotropic and under pressure excitation. The present study has shown that the Fourier series expansion method is suitable for the mechanical displacement and electric potential analysis in panel-type actuators. The direct piezoelectric effect of the piezoelectric layer subjected to outer pressure is investigated in detail. It is always assumed in the analytical analysis of the piezoelectric structures that the electric potential in the piezoelectric layers varies linearly and the displacements change in the form of prescribed functions across its thickness [10]. However, it has been shown that the distributions of the mechanical displacements and electric potential of piezoelectric response are very complicated and cannot be
1526
M. Shakeri et al. / Computers and Structures 84 (2006) 1519–1526
treated as pure elastic structures or piezoelectric structures. Therefore, three-dimensional analysis of piezoelastic behavior of structures is recommended even for thin laminated structures. Since a comprehensive and exact study of active piezoelectric structures is still unavailable, the present work provides an enhanced insight to the mechanical and electric behaviors of this type of smart structure. Results presented in this paper are also useful for assessing approximate analysis. References [1] Lee PCY, Yu JD. Governing equations of piezoelectric plates with graded properties across the thickness. Proc Annu IEEE Int Freq Control Symp 1996:623–31. [2] Tzou HS, Zhong JP. Electromechanics and vibrations of piezoelectric shell distributed systems. J Dyn Syst Meas Control 1993;115(3): 506–17. [3] Tzou HS, Zhong JP. Linear theory of piezoelectric shell vibrations. J Sound Vib 1994;175(1):77–88. [4] Chen C-Q, Shen Y-P, Wang X-M. Exact solution for orthotropic cylindrical shell with piezoelectric layers under cylindrical bending. Int J Solids Struct 1996;33(30):4481–94. [5] Dumir PC, Dube GP, Kapuria S. Exact piezoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending. Int J Solids Struct 1997;34(6):685–702. [6] Kapuria S, Sengupta S, Dumir PC. Three-dimensional solution for simply-supported piezoelectric cylindrical shell for axisymmetric load. Comput Methods Appl Mech Eng 1997;140(1–2):139–55.
[7] Pan E. Exact solution for simply supported and multilayered magneto-electro-elastic plates. ASME J Appl Mech 2001;68: 169–87. [8] Yue ZQ, Yin JH. Backward transfer-matrix method for elastic analysis of layered solids with imperfect bonding. J Elasticity 1998;50:109–28. [9] Wang X, Zhong Z. Three-dimensional solution of smart laminated anisotropic circular cylindrical shell with imperfect bonding. Int J Solids Struct 2003;40:5901–21. [10] Chen CQ, Shen YP. Piezothermoelasticity analysis for a cylindrical shell under the state of axisymmetric deformation. Int J Eng Sci 2003;34(14). [11] Bernadou M, Haenel C. Modelization and numerical approximation of piezoelectric thin shells. Part 1: The continuous problems. Comput Methods Appl Mech Eng 2003;192:4003–43. [12] Mota Soares CA, Mota Soares CM, Pinto Correia IF. Modeling of laminated shells with integrated sensors and actuators. In: Topping BHV, Mota Soares CA, editors. Progress in computational structures technology. Saxe-Coburg Publications; 2004. p. 281–309. [13] Benjeddou A, Andrianarison O. A piezoelectric mixed variational theorem for smart multilayered composites. Mech Adv Mater Struct 2005;12:1–11. [14] Shakeri M, Alibiglu A, Eslami MR. Elasticity solution for thick laminated anisotropic cylindrical panels under dynamic load. J Mech Eng Sci 2002;216(Part C). I Mech E. [15] Shakeri M, Daneshmehr A, Alibiglu A. Elasticity solution for thick laminated shell panel with piezoelectric layer. EASEC-9 Conf., Bali, Indonesia; 2003. [16] Shakeri M, Alibeiglooo A. Dynamic analysis of orthotropic laminated cylindrical panel. Mech Adv Mater Struct 2005;12:67–75. [17] Ren JG. Exact solution for laminated cylindrical shell in cylindrical bending. Comput Sci Technol 1987;29:169–87.