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Natural frequency analysis of a sandwich panel with soft core based on a refined shear deformation model
Composite Structures 72 (2006) 364–374 www.elsevier.com/locate/compstruct
Natural frequency analysis of a sandwich panel with soft core based on a refined shear deformation model Qunli Liu a, Yi Zhao
b,*
a
b
Department of Mechanical Engineering, University of Nevada, Las Vegas, NV 89154, USA Department of Aerospace Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA Available online 14 March 2005
Abstract The natural frequency of a thick rectangular sandwich panel composed of orthotropic facesheets and a soft core was studied based on a refined shear deformation model. The shear deformation of the sandwich panel was described by a polynomial function. The effect of transverse shear modulus of the facesheets and core on flexural vibration of the panel was investigated. Comparison was made among classical thin plate theory, linear shear (low order) deformation theory and the refined shear (high order) deformation model. Results from finite element analysis were also provided to verify the theoretical predictions. It was shown that the refined shear deformation model provided a better prediction on the natural frequency of vibration of a sandwich panel than thin plate model or low order deformation model. 2005 Elsevier Ltd. All rights reserved. Keywords: Natural frequency; Sandwich panel; Soft core; Refined shear deformation theory
1. Introduction Vibration of sandwich structures is of particular interest to aerospace industry due to its applications in aircraft. It has also been a subject of intensive research during the past decades. Maheri and Adams [1] presented some results from experimental investigation of the dynamic shear property of both Nomex and aluminum honeycomb cores and the damping of composite honeycomb sandwich beams in steady-state flexural vibration. The beams were constructed from either CFRP or GRP facesheets with Nomex or aluminum honeycomb cores. Chen and Sheu [2] studied the dynamic characteristics of a sandwich beam with flexible core based on a model in which two thin beams were
*
Corresponding author. Tel.: +1 386 226 6746; fax: +1 386 226 6747. E-mail address: [email protected] (Y. Zhao).
0263-8223/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.01.006
separated by springs and dashpots uniformly distributed along the beam length. The general damped Timoshenko beam was employed to include the rotary inertia of mass, shear deformation and damping components. Sakiyama et al. [3] proposed an analytical method for free vibration of a three-layer continuous sandwich beam with elastic or viscoelastic core under various support conditions. They derived the characteristic equations for the natural frequency and the associated mode by introducing a discrete Green function. Marur and Kant [4] used high order refined theory to study the transient dynamics of a laminated beam. Most recently, Nilsson [5] applied HamiltonÕs principle to derive a sixth order differential governing equation for threelayered beam with lightweight honeycomb core under various boundary conditions. Transverse shear deformation of a panel subjected to bending has been a topic for many years. Mindlin [6] proposed a model to study transverse shear of an isotropic thick plate, which has since been recognized as
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
Mindlin theory. Dawe and Roufaeil [7] used Rayleigh– Ritz method to study the vibration of a thick plate. Reddy [8] applied a high order theory to model the shear deformation. Unlike the Mindlin plate theory that requires a shear correction factor, the ReddyÕs model does not require this factor, but the governing equations are more complicated. Kant et al. [9,10] studied large amplitude free vibration and transient response of a sandwich plate using Green–Lagrangian strain–displacement relation and FEA. Lee and Fan [11] developed a p-version FEA code to compute the vibration modes of rectangular sandwich panels. A linear shear deformation assumption was adopted in their work. Bardell et al. [12] used a p–h version code to study the vibration of a cantilever T-platform sandwich panel. Lu [13] applied a high order shear deformation theory to nonlinear vibration of coplanar thick composite laminated panels, which were either simply supported or fixed at the edges. Wang [14] compared classical thin plate theory and the Mindlin (low order shear) thick plate theory, and applied low order shear theory for flexural vibration of rectangular sandwich panel composed of isotropic facesheets and core. Recently, Kant and Swaminathan [15] compared the free vibration results on plates from a few high order models including two models by Kant and one by Reddy. For dynamic response of sandwich plate, Pai and Palazotto [16] formulated a finite scheme in which the multi-layered facesheets were modeled as sublaminates and the stress in thickness direction was taken into account. Meunier and Shenoi [17] employed ReddyÕs high order shear deformation theory for dynamic analysis of composite sandwich plate with damping modeled for the viscoelastic PVC foam. Modal loss factor was predicted from the analytical model and experimental results from dynamic mechanical analysis were presented. They concluded that the viscoelasticty of the constitutive materials and their temperature and frequency dependency played significant role in the dynamic properties of sandwich plate. Kakhecha et al. [18] formulated a finite element scheme for vibration and damping analysis of composite plates on the basis of a high order theory similar to that by Kant. A comprehensive survey on modeling of sandwich structures was conducted by Noor et al. [19], which summarized various models, methods, advances and problems. Reviews on analytical models could also be found from the literatures by Allen [20], Mead [21], Pandya and Kant [22], Noor and Burton [23], Ha [24] and Burton and Noor [25]. Although extensive research has been conducted on the topic of vibration of sandwich plates/panels, few results were reported on analytical prediction of the dynamic properties in terms of facesheet and core material as well as structural parameters. Liu and Zhao [26] presented an analytical solution to the vibration of a sandwich panel with isotropic facesheets and orthotro-
365
pic core based on KirchhoffÕs linear shear deformation theory. The rotational effect was taken into consideration and effects of core anisotropy, core density, and the facesheet thickness on natural frequencies were also discussed. A parametric study was also conducted by Liu and Zhao [27] to evaluate the natural frequency of a simply-supported rectangular sandwich panel with orthotropic facesheets and core. Despite of certain improvements in these two studies, the role of soft core was overestimated with KirchhoffÕs thin plate theory or low order shear deformation theory. In this paper, a new model based on refined shear deformation is proposed to better estimate vibration behavior of sandwich panels with soft core.
2. Analytical model A linear elastic stress–strain relation is used to describe the elastic behavior of the orthotropic facesheets and core. A plane stress assumption is made as in other literatures [9–11,26]. Additional assumptions include: • The velocity along thickness of the panel is uniform, and can be represented by that of the mid-plane. • The facesheets and the core are perfectly bonded. • The normal stress in thickness direction can be neglected. With the plane stress assumption, the stress–strain relationship for the orthotropic facesheets can be expressed in terms of engineering constants as 32 3 3 2 Ef 1 2 t12 Ef 2 0 0 0 ex rfx 1t12 t21 1t12 t21 76 7 6r 7 6 t E E 21 f 1 f2 ey 7 6 fy 7 6 0 0 0 7 76 7 6 1t12 t21 1t12 t21 7 6 76 7 6 sfxy 7 ¼ 6 6 0 0 Gfxy 0 0 7 7 6 6 6 cxy 7: 7 6 7 6 7 6 6 7 4 sfxz 5 4 0 0 0 Gfxz 0 54 cxz 5 cyz sfyz 0 0 0 0 Gfyz ð1Þ Since the soft core is also assumed to be orthotropic, its stress–strain relationship has the same form as that for the facesheets with subscript f replaced by c, and m by l, respectively. The structural parameters of the sandwich panels include the size of the panel a · b, the facesheet thickness d, and the thickness of the core 2h. The x-direction is set along a side of the panel, as indicated in Fig. 1. Denoting the transverse displacement of the midplane as w = w(x, y, t), rotations with respect to x-axis and y-axis at the mid-plane as / = /(x, y, t) and u = u(x, y, t), respectively. In the refined shear deformation model, the nonlinear shear deformations of both sandwich core and facesheets are taken into consideration. It is assumed that the shear strains change continuously
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be selected as kn factor. When N = 1 and k1 = 1, the proposed model reduces to ReddyÕs model [8]. When N > 1, the shear function g(z) consists of more than one term with different k factors. The coefficients an become the ‘‘weight factors.’’ Fig. 2(a) and (b) show the behavior of the shear function and its derivative with respect to the panel thickness z, when h + d = 5.5 mm). To demonstrate the derivation process, N = 1 (and therefore a1 = 1) is used for simplicity. The strains at any point in the facesheets or core are Fig. 1. Structural parameters for a sandwich panel.
ex ¼ in the thickness direction, and at the top and bottom surfaces, z = ±(h + d), the shear strains czx = czy = 0. With this assumption, the following displacement functions are proposed: ow ; ox ow ; vðx; y; z; tÞ ¼ ½z gðzÞuðx; y; tÞ gðzÞ oy uðx; y; z; tÞ ¼ ½z gðzÞ/ðx; y; tÞ gðzÞ
ð2Þ
where g(z) is a function describing the nonlinear shear deformation of the sandwich panel gðzÞ ¼
N X n¼1
an
z2kn þ 1 2k ð2k n þ 1Þðh þ dÞ n
ou o/ o2 w ¼ ðz gk Þ gk 2 ; ox ox ox
ov ou o2 w ¼ ðz gk Þ gk 2 ; oy oy oy ou ov o/ ou o2 w þ ¼ ðz gk Þ þ ; cxy ¼ 2gk oy ox oy ox ox oy ow ou og ow cxz ¼ þ ¼ 1 k ; /þ ox oz ox oz ow ov ogk ow þ ¼ 1 cyz ¼ : /þ oy oz oy oz ey ¼
ð4Þ
2.1. Stain energy of the panel ð3Þ
and n = 1, 2, 3, . . . , are weight factors that satisfy a1 + a2 + a3 + = 1. The factor ‘‘kn’’ in the empirical shear function g(z) is selected in such a way that g(z) remains an odd function, which satisfies the following conditions: ex ðh þ dÞ ¼ ex ððh þ dÞÞ; ey ðh þ dÞ ¼ ey ððh þ dÞÞ: The factor kn is not necessary to be an integer—a fraction number such as 1/2, 3/2, 7/4, 5/2, etc., could also
The strain energy of the panel is Z Z Z 1 T U¼ ½r ½edV 2
ð5Þ
which is composed of six components, namely strain energy associated with tension/compression in x and y directions, and with shear strain in the xy plane for the facesheets and core, respectively. The strain energy induced by the tension/compression in the top and bottom facesheets in x direction can be expressed as
Fig. 2. (a) Behavior of shear function with respect to panel thickness. (b) Behavior of the derivative of shear function.
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
U fx
1 ¼ 2
Z
a 0
1 þ 2 ¼
Z
b
0
Z
a 0
Z
hþd
h
Z
b 0
Z
rfx efx dz dy dx
t12 Dfy2
Z
a
Z
rfx efx dz dy dx
þ t12 Dfy3
ðhþdÞ
ð6aÞ
Ef 1 Ef 1 K f 1 ; Dfx2 ¼ K f 2; 1 t12 t21 1 t12 t21 Ef 1 ¼ K f 3; 1 t12 t21 Ef 2 Ef 2 K f 1 ; Dfy2 ¼ K f 2; 1 t12 t21 1 t12 t21 Ef 2 ¼ K f 3; 1 t12 t21
Dfy1 ¼
Kf 1 ¼ 2
Z
hþd 2
ðz gk Þ dz; K f 2 ¼ 2
h
Kf 3 ¼ 2
Z
a
o/ o2 w ou o2 w þ dy dx ox oy 2 oy ox2 0 ! Z b 2 o w o2 w dy dx ox2 oy 2 0 Z
b
ð6bÞ
b
Dfx1 ¼
Dfy3
Z
0
where Df s are the stiffness of the facesheets with respect to the mid-plane of the panel
Dfx3
a
0
h
2 o/ dy dx ox 0 0 Z a Z b o/ o2 w 2Dfx2 dy dx ox ox2 0 0 Z a Z b 2 2 ow þ 2Dfx3 dy dx ox2 0 0 Z a Z b o/ ou þ t12 Dfy1 dy dx ox oy 0 0 Z a Z b o/ o2 w ou o2 w þ dy dx t12 Dfy2 ox oy 2 oy ox2 0 0 ! Z a Z b 2 o w o2 w þ t12 Dfy3 dy dx ; ox2 oy 2 0 0 1 Dfx1 2
Z
367
Z
and the three components of strain energy associated with shear deformation are Z hþd Z a Z b o/ ou þ U fxy ¼ Gfxy ðz gk Þ oy ox h 0 0 2 2 ow 2gk dxdy dz oxoy 2 Z a Z b 1 o/ ou Dfxy1 þ ¼ dxdy 2 oy ox 0 0 Z a Z b o/ ou o2 w þ dxdy 4Dfxy2 oy ox oxoy 0 0 ! Z a Z b 2 2 ow þ4Dfxy3 dxdy ; oxoy 0 0 2 Z hþd Z a Z b og ow U fxz ¼ Gfxz 1 k dxdy dz /þ ox oz h 0 0 2 Z a Z b 1 ow ¼ S fxz /þ dxdy; 2 ox 0 0 2 Z hþd Z a Z b og ow U fyz ¼ Gfyz 1 k dxdy dz uþ oy oz h 0 0 2 Z a Z b 1 ow ¼ S fyz uþ dxdy; 2 oy 0 0 ð6cÞ where
hþd
gk ðz gk Þdz;
h
Dfxy1 ¼ Gfxy K f 1 ; Dfxy2 ¼ Gfxy K f 2 ; Dfxy3 ¼ Gfxy K f 3 ;
hþd
g2k dz:
h
Similarly, the strain energy of the facesheets associated with stress component in y direction is Z Z Z 1 a b hþd U fy ¼ rfy efy dz dy dx 2 0 0 h Z Z Z 1 a b h þ rfy efy dz dy dx 2 0 0 ðhþdÞ Z a Z b 2 ou ¼ Dfy1 dy dx oy 0 0 Z a Z b ou o2 w 2Dfy2 dy dx oy oy 2 0 0 Z a Z b 2 2 ow dy dx þ 2Dfy3 oy 2 0 0 Z a Z b o/ ou þ t12 Dfy1 dy dx ox oy 0 0
ogk ¼ 2Gfxz 1 dz; S fyz oz h Z hþd og ¼ 2Gfyz 1 k dz: oz h Z
S fxz
hþd
Following the same procedure, the strain energy components of the core are: Z Z Z 1 a b h U cx ¼ ðrcx ecx Þdz dy dx 2 0 0 h Z a Z b 2 1 o/ Dcx1 dy dx ¼ 2 ox 0 0 Z a Z b o/ o2 w 2Dcx2 dy dx ox ox2 0 0 Z a Z b 2 2 ow dy dx þ 2Dcx3 ox2 0 0 Z a Z b o/ ou þ t12 Dcy1 dy dx ox oy 0 0
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Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
t12 Dcy2 þ t12 Dcy3
U cy
1 ¼ 2
Z
a
Z
0
0
o/ o2 w ou o2 w þ dy dx ox oy 2 oy ox2 0 ! Z b 2 o w o2 w dy dx ; ox2 oy 2 0
Z
h
Z Z
b
a 0 a
Z
b
2 ou dy dx oy 0 0 Z a Z b ou o2 w 2Dcy2 dy dx oy oy 2 0 0 Z a Z b 2 2 ow dy dx þ 2Dcy3 oy 2 0 0 Z a Z b o/ ou þ t12 Dcy1 dy dx ox oy 0 0 Z a Z b o/ o2 w ou o2 w t12 Dcy2 þ dydx ox oy 2 oy ox2 0 0 ! Z a Z b 2 o w o2 w dy dx ; þ t12 Dcy3 ox2 oy 2 0 0
1 Dcy1 ¼ 2
U cxy ¼ Gcxy
Z
Z
h
a
Z
h
0 2
Z
a
ð7aÞ
b
S cxz ¼ Gcxz
Z 0
b
S cyz ¼ Gcyz
K c1 ¼ K c3 ¼
U cyz
ð7bÞ
2
ð7cÞ Ec1 Ec1 K c1 ; Dcx2 ¼ K c2 ; ð1 l12 l21 Þ ð1 l12 l21 Þ Ec1 K c3 ; ¼ ð1 l12 l21 Þ
Dcx3
h
og 1 k oz
2 dz;
2 og 1 k dz; oz h h
h Z h
2
ðz gk Þ dz; K c2 ¼
Z
h
gk ðz gk Þdz;
h
g2k dz:
þ U cxy þ U cxz þ U cyz :
ow dx dy dz ox oy 2 Z a Z b o/ ou þ dx dy ¼ Dcxy1 oy ox 0 0 Z a Z b o/ ou o2 w þ dx dy 4Dcxy2 oy ox ox oy 0 0 ! Z a Z b 2 2 ow þ 4Dcxy3 dx dy ; ox oy 0 0 2 Z h Z a Z b og ow ¼ Gcxz 1 k dx dy dz /þ ox oz h 0 0 2 Z a Z b 1 ow ¼ S cxz /þ dx dy; 2 ox 0 0 2 Z h Z a Z b og ow ¼ Gcyz 1 k dx dy dz uþ oy oz h 0 0 2 Z a Z b 1 ow ¼ S cyz uþ dx dy; 2 oy 0 0
Dcx1 ¼
Z
h
h
h
o/ ou þ ð z gk Þ oy ox
where
Z
Z
The total strain energy can thus be expressed as U ¼ U fx þ U fy þ U fxy þ U fxz þ U fyz þ U cx þ U cy
2gk
U cxz
Dcy3
Dcxy1 ¼ Gcxy K c1 ; Dcxy2 ¼ Gcxy K c2 ; Dcxy3 ¼ Gcxy K c3 ;
rcy ecy dz dy dx
h
0
Ec2 Ec2 K c1 ; Dcy2 ¼ K c2 ; ð1 l12 l21 Þ ð1 l12 l21 Þ Ec2 K c3 ; ¼ ð1 l12 l21 Þ
Dcy1 ¼
ð8Þ
2.2. Kinetic energy of the panel The kinetic energy consists of five components, corresponding to the transverse motion of the panel and motion in x and y directions of the facesheets and core, respectively. Z Z 2 q a b ow Tz ¼ dy dx; 2 0 0 ot Z a Z b Z hþd 2 2qf ou T fx ¼ dz dy dx ot 2 0 0 h 2 2 Z Z 1 a b o/ o/ ow If1 ¼ 2I f 2 2 0 0 ot ot ox ot 2 2 ! ow þ If3 dy dx; ox ot Z a Z b Z hþd 2 2qf ov T fy ¼ dz dy dx ot 2 0 0 h 2 2 Z Z 1 a b ou ou ow If1 2I f 2 ¼ 2 0 0 ot ot oyot 2 2 ! ow þ If3 dy dx; oyot Z Z Z 2 qc a b h ou T cx ¼ dz dy dx 2 0 0 h ot 2 2 Z Z 1 a b o/ o/ ow I c1 ¼ 2I c2 2 0 0 ot ot ox ot 2 2 ! ow þ I c3 dy dx; ox ot
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
Z Z Z 2 qc a b h ov ¼ dz dy dx 2 0 0 h ot 2 2 Z Z 1 a b ou ou ow I c1 ¼ 2I c2 2 0 0 ot ot oyot 2 2 ! ow þ I c3 dy dx; oyot
T cy
where Dx1 ¼ Dfx1 þ Dcx1 ; Dx2 ¼ Dfx2 þ Dcx2 ; Dx3 ¼ Dfx3 þ Dcx3 ; Dy1 ¼ Dfy1 þ Dcy1 ; Dy2 ¼ Dfy2 þ Dcy2 ; Dy3 ¼ Dfy3 þ Dcy3 ; ð9Þ
where
Eqs. (12a)–(12c) are the governing equations of vibration for a rectangular sandwich panel. For a simply supported panel, these equations can be solved by assuming the following functions:
I f 3 ¼ qf K f 3 ; I c1 ¼ qc K c1 ; I c2 ¼ qc K c2 ; I c3 ¼ qc K c3 ; so the total kinetic energy can be written as T ¼ T z þ T fx þ T fy þ T cx þ T cy :
ð10Þ
Using HamiltonÕs variational principle, Z t2 d ðU T Þdt ¼ 0;
ð11Þ
t1
o3 w o3 w þ t12 Dfy2 þ l12 Dcy2 þ 2 Dfxy2 þ Dcxy2 3 ox ox oy 2
o2 u o2 / Dx1 2 t12 Dfy1 þ l12 Dcy1 þ Dfxy1 þ Dcxy1 ox ox oy 2 o / ow Dfxy1 þ Dcxy1 þ Gxz / þ oy 2 ox o2 / o3 w þ I ; 2 ot2 ox ot2
ðx; yÞ sinðxtÞ; w¼w yÞ sinðxtÞ; / ¼ /ðx;
ð13Þ
ðx; yÞ sinðxtÞ; u¼u where ðx; yÞ ¼ A sinðmpx=aÞ sinðnpy=bÞ; w
a system of differential equations can be obtained
¼ I 1
I 1 ¼ I f 1 þ I c1 ; I 2 ¼ I f 2 þ I c2 ; I 3 ¼ I f 3 þ I c3 ; Gxz ¼ S fxz þ S cxz ; Gyz ¼ S fyz þ S cyz :
q ¼ 2dqf þ 2hqc ; I f 1 ¼ qf K f 1 ; I f 2 ¼ qf K f 2 ;
Dx2
369
ð12aÞ
yÞ ¼ B cosðmpx=aÞ sinðnpy=bÞ; /ðx; ðx; yÞ ¼ C sinðmpx=aÞ cosðnpy=bÞ: u A system of homogeneous equations of constants A, B, and C can be obtained 2 32 3 A A11 þ A12 x2 A21 þ A22 x2 A00 6 7 6 7 6 B11 þ B12 x2 6 7 B00 B21 þ B22 x2 7 4 54 B 5 ¼ 0: C 11 þ C 12 x2
C 21 þ C 22 x2
C 31 þ C 32 x2
ð14Þ
o4 w o4 w Dx3 4 þ Dy3 4 þ 2t12 Dfy3 þ 2l12 Dcy3 ox oy o4 w o3 / o3 u D D þ4 Dfxy3 þ Dcxy3 x2 y2 ox2 oy 2 ox3 oy 3 3 o / o3 u t12 Dfy2 þ l12 Dcy2 þ 2 Dfxy2 þ Dcxy2 þ oxoy 2 ox2 oy o/ o2 w ou o2 w þ 2 Gyz þ 2 Gxz ox ox oy oy 3 4 2 3 ow o/ ou ow o4 w þ þ ¼ q 2 I 2 þ I3 ; ot oxot2 oy ot2 ox2 ot2 oy 2 ot2
For a non-trivial solution, the frequencies of vibration can be determined from the characteristic equation 2 3 A11 þ A12 x2 A21 þ A22 x2 A00 6 7 2 det 6 B00 B21 þ B22 x2 7 4 B11 þ B12 x 5 ¼ 0; C 11 þ C 12 x2 C 21 þ C 22 x2 C 31 þ C 32 x2 ð15Þ where A11 ¼ a2 b2 mpGxz b2 m3 p3 Dx2 a2 mn2 p3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ;
ð12bÞ A12 ¼ a2 b2 mpI 2 ; 3
C
3
o w ow þ t12 Dfy2 þ l12 Dcy2 þ 2 Dfxy2 þ Dcxy2 3 oy ox2 oy 2 o2 / ou D21 2 t12 Dfy1 þ l12 Dcy1 þ Dfxy1 þ Dcxy1 oy oxoy o2 u ow o2 u o3 w Dfxy1 þ Dcxy1 þ G u þ þ I ; ¼ I yz 1 2 ox2 oy ot2 oy ot2
Dy2
ð12cÞ
A21 ¼ a3 b2 Gxz þ ab2 m2 p2 Dx1 þ a3 n2 p2 Dxy1 ; A22 ¼ a3 b2 I 1 ; A00 ¼ a2 bmnp2 ðDxy1 þ t12 Dfy1 þ l12 Dcy1 Þ; B11 ¼ a2 b2 npGyz a2 n3 p3 Dy2 b2 m2 np3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ;
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Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
B12 ¼ pa2 b2 nI 2 ; B00 ¼ ab2 mnp2 ðDxy1 þ t12 Dfy1 þ l12 Dcy1 Þ; B21 ¼ a2 b3 Gyz þ a2 bn2 p2 Dy1 þ b3 m2 p2 Dxy1 ; B22 ¼ a2 b3 I 1 ; C 11 ¼ a2 b4 m2 p2 Gxz þ a4 b2 n2 p2 Gyz þ b4 m4 p4 Dx3 þ a4 n4 p4 Dy3 þ a2 b2 m2 n2 p4 ð4Dxy3 þ 2t12 Dfy3 þ 2l12 Dcy3 Þ; C 12 ¼ a2 b2 ðI 3 p2 ða2 n2 þ b2 m2 Þ þ a2 b2 qÞ; C 21 ¼ a3 b4 mpGxz ab4 m3 p3 Dx2 a3 b2 mn2 p3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ; C 22 ¼ a3 b4 mpI 2 ; C 31 ¼ npa4 b3 Gyz a4 bn3 p3 Dy2 a2 b3 m2 np3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ; C 32 ¼ a4 b3 npI 2 :
3. Results and discussions The nonlinear transverse shear deformation of a sandwich core is described by the proposed displacement function, Eqs. (2) and (3). For demonstration purpose, the results presented thereafter are corresponding to the case in which n = 1, a1 = 1 and therefore, gk ¼
1 ð2k þ 1Þðh þ dÞ2k
z2kþ1 :
ð16Þ
For the cases illustrated in the following, the panel size is a = b = 200 mm. The influence of different values of k is shown in Fig. 3 in which h = 5 mm and d = 0.5 mm. The material properties includes density 2.8 · 103 kg/m3, elastic modulus 70 GPa, PoisonÕs ratio 0.33 for face-
Fig. 3. Effect of index k on frequency prediction.
sheets and density 40 kg/m3, elastic modulus (in applicable directions) 10 MPa, PoisonÕs ratio 0.3 for core if not explicitly stated otherwise. The results from the classical thin plate theory as well as low order (or linear) shear deformation theory are also presented. The result from thin plate theory was used as the reference for comparison. It is seen that both thin plate theory and linear shear deformation theory overestimate the mode frequency of a sandwich panel, particularly when the transverse shear modulus of core (Gxz and Gyz) is low. In comparison, the proposed refined model offers the flexibility of using selected high order terms by adjusting weight distribution and, therefore, has the versatility to handle the constitutive parameters in a relatively wide range. As illustrated in Fig. 2 of the shear function g(z), it becomes clear now that a smaller value for parameter k may be employed for low modulus core and a larger value for high modulus core. Combination of the terms with various k values and appropriate weights may be used for cores with intermediate shear modulus. In practical applications, the terms and the associate parameters may be selected based on preliminary experimental investigation. A comparison between the refined shear deformation model (for k = 1) and the low order (linear) shear deformation theory as well as the classic thin plate theory is provided in Fig. 4(a)–(d). The transverse shear modulus of the core is assumed to be Gcxz = Gcyz with four different values used: Gcxz = Gcyz = 50, 200, 500 and 1000 MPa. Results from finite element analysis are also plotted in Fig. 4(b). The FEA software code used was ABAQUS. The facesheests are represented by one-layer of solid hexahedral brick elements at the top and bottom respectively. The soft core is also modeled as equivalent solid brick elements that have same size as the facesheet elements in panelÕs length and width directions. A few scenarios were established where the core has 2, 5 and 10 layers elements of the same size (small cubic). The results were quite similar and the frequency from smaller element case was slight lower than in the bigger element case. The results presented were from the scenario where the core has 10 layers of elements (1 mm by 1 mm by 1 mm) and the facesheets have one-layer of elements (1 mm by 1 mm by 0.5 mm). It can be seen from the four figures that the differences among the three models increase as the core thickness increases. It can also be found that the transverse shear modulus of the core has noticeable effect. The lower the shear modulus, the greater the difference is. When Gcxz = Gcyz = 1000 MPa, the differences among the three models are insignificant; as the shear modulus of the core increases, the difference becomes negligible. For the facesheet with 0.5 mm thickness (4.5% of total thickness of the panel), the prediction from linear shear deformation theory is very close to that from thin plate theory; as the facesheet thickness increases, the
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
371
Fig. 4. Comparison for effects of core thickness on frequency prediction.
difference between the linear shear deformation theory and the thin plate theory increases. Apparently, the classic thin plate theory overestimates the natural frequency. The low order linear shear deformation theory only slightly improves the estimation of natural frequency. The prediction from the refined shear model is considerably different from that from either low order shear theory or thin plate theory, especially when the transverse
shear modulus of core is low. The proposed refined (high order) shear deformation model provides a more agreeable prediction of the vibration frequency of the sandwich panel than the other models considered. The effect of transverse shear modulus of the core on natural frequency is presented in Fig. 5(a) and the effect of transverse shear modulus of facesheets in Fig. 5(b). In both cases the thickness of the core is fixed at h = 5 mm,
Fig. 5. (a) Effect of transverse shear modulus of core (Gcxz = Gcyz). (b) Effect of transverse shear modulus of facesheets (Gfxz = Gfyz).
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and the parameter k is the ratio of facesheet thickness to core height. Clearly, the larger the shear modulus, the higher the vibration frequency will be. This result is quite expected because of the correlation between shear modulus and panel stiffness. This effect is also related to the relative thickness of the facesheets with respect to the core thickness. Fig. 6 shows the effect of core orthotropy for k = 0.1 (thin facesheet) and 0.4 (thick facesheet). With Gcxz specified, the natural frequency increases as Gcyz increases and the relation is nonlinear. Fig. 7 shows the effect of facesheet orthotropy with k = 0.1 and 0.4. With Gfxz specified, the natural frequency increases as Gfyz increases. The relation is almost linear for the case of thin facesheets, but is nonlinear for the thick facesheets. A special case is when the sandwich panel becomes a solid plate, i.e., the thickness of core is set at zero. The results are shown in Fig. 8. When the thickness of the plate is small, all three models provide the identical results (this provides another validation of the proposed
Fig. 8. Comparison of mode frequency from various models for isotropic solid plate.
refined shear deformation model). As the thickness of the plate increases, the predictions from both low order and high order theories deviate gradually from that of
Fig. 6. Effect of orthotropy of transverse shear modulus of core in refined model: (a) k = 0.1 (thin facesheet); (b) k = 0.4 (thick facesheet).
Fig. 7. Effect of orthotropy of transverse shear modulus of facesheet: (a) k = 0.1 (thin facesheet); (b) k = 0.4 (thick facesheet).
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
thin plate theory. The results also show that the difference between the low order and high order theories is negligible when they are used for a solid plate. This suggests that a linear transverse theory provides sufficient accuracy in studying a solid plate with various thicknesses. Although high order shear deformation model partially corrects the overestimation of shear deformation effect so that the natural frequency predicted is lower than those by classical thin plate theory and low order theory, the natural frequency is still higher than those obtained from FEA, in which the modulus of both the facesheets and core in thickness direction are taken into account. Further computation verifies that the higher the modulus in thickness direction, the smaller the difference between analytical results and FEA results. Either low order or high order theories, though correct the effect of shear deformation to certain extent, still overestimate the effect of modulus in thickness direction. One of the assumptions made in this model is that the deformation in thickness direction is small so that strain energy associated with this deformation can be neglected. This implies that the modulus of the panel in the thickness direction is assumed to be infinite. However, a theoretical model with the deformation in thickness direction considered would be extremely difficult and was therefore not attempted in this paper. If the shear modulus of facesheets is much higher than that of the core, and the thickness is much smaller, only the shear modulus of the core may be taken into account—in this case, predictions by both low order and high order models are very close to those from FEA. The two analytical solutions are useful to parametrically study the effect of all the structural and materials parameters in sandwich panel, except the modulus of facesheet and core in thickness direction.
4. Conclusions A refined shear deformation model was proposed to investigate the natural frequency of a thick rectangular sandwich panel with orthotropic facesheets and deformable soft core. The high order model is based on a nonlinear function with multiple and adjustable weights to describe the transverse shear deformation. A noticeable improvement over the linear shear theory can be achieved using this model, especially for the sandwich core with a low transverse shear modulus. The predictions on the modal frequency of a thick sandwich panel from this model are in better agreement with results from finite element analysis. This model provides an analytical approach to quantitatively investigate the effect of the constitutive parameters of a sandwich panel on its modal frequency.
373
References [1] Maheri MR, Adams RD. Steady-state flexural vibration damping of honeycomb sandwich beams. Compos Sci Technol (UK) 1994; 52(3):333–47. [2] Chen YH, Sheu JT. Dynamic characteristics of layered beam with flexible core. J Vib Accoust 1994;116(3):350–6. [3] Sakiyama T, Matsuda H, Morita C. Free vibration analysis of continuous sandwich beams with elastic or viscoelastic cores by applying the discrete Green function. J Sound Vib 1996;198(4): 439–54. [4] Marur SR, Kant T. Transient dynamics of laminated Beams: An evaluation with a high-order refined theory. Compos Struct 1998; 11:1–11. [5] Nilsson E, Nilsson AC. Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores. J Sound Vib 2002;251(3):409–30. [6] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 1951;18:31–8. [7] Dawe DJ, Roufaeil OL. Rayleigh–Rits vibration analysis of Mindlin plates. J Sound Vib 1980;69(3):345–59. [8] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51(4):745–52. [9] Kant T, Arora CP, Varaiya JH. Finite element transient analysis of composite and sandwich plates based on a refined theory and a mode superposition method. Compos Struct 1992;22(2):109–20. [10] Kant T, Kommineni JR. Large amplitude free vibration analysis of cross-Ply composite and sandwich laminates with a refined theory and C0 finite elements. Comput Struct 1994;50(1):123–34. [11] Lee LJ, Fan YJ. Bending and vibration analysis of composite sandwich plates. Comput Struct 1996;60(1):103–12. [12] Bardell NS, Dunsdon JM, Langley RS. Free vibration analysis of coplanar sandwich panels. Compos Struct 1997;38(1–4):463–75. [13] Lu C-H, Libove C. Beamlike harmonic vibration of variablethickness sandwich plates. AIAA J 1991;29(2):299–305. [14] Wang CM. Deducing thick plate solutions from classical thin plate solutions. Struct Eng Mech 2001;11(1):89–104. [15] Kant T, Swaminathan K. Free vibration of isotropic, orthotropic, and multilayer plates based on higher order refined theories. J Sound Vib 2001;241(2):319–27. [16] Pai PF, Palazotto AN. A higher-order sandwich plate theory accounting for 3-D stress. Int J Solids Struct 2001;38(21–32): 5045–62. [17] Meunier M, Shenoi RA. Dynamic analysis of composite sandwich plates with damping modelled using high-order shear deformation theory. Compos Struct 2001;54(2):243–54. [18] Makhecha DP, Ganapathi M, Patel BP. Vibration and damping analysis of laminated/sandwich composite plates using higherorder theory. J Reinf Plast Compos 2002;21(6):559–75. [19] Noor AK, Burton WS, Bert CW. Computational models for sandwich panels and shells. Appl Mech Rev 1996;49(3):155–99. [20] Allen HG. Analysis of sandwich panels: The significance of shear deformation. Composites 1970;1(6):215–20. [21] Mead DJ. A comparison of some equations for the flexural vibration of damped sandwich beams. J Sound Vib 1982;83(3): 363–77. [22] Pandya BN, Kant T. Higher-order shear deformable theories for flexure of sandwich plates—finite element evaluation. Int J Solids Struct 1988;24(12):1267–86. [23] Noor AK, Burton WS. Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 1989;42(1): 1–13. [24] Ha KH. Finite element analysis of sandwich plates: An overview. Comput Struct 1990;37(4):397–403. [25] Burton WS, Noor AK. Assessment of computational models for sandwich panels and shells. Comput Methods Appl Mech Eng 1995;124(1–2):125–51.
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[26] Liu Q, Zhao Y. Prediction of natural frequencies of vibration for a sandwich panel using thick plate theory. J Sandwich Struct Mater 2001;3(4):289–309.
[27] Liu Q, Zhao Y. A parametric study on natural frequency of sandwich panel using refined shear model, ETCE 2002, Houston, TX, 4–5 February 2002, vol. 2. p. 535–40.
Natural frequency analysis of a sandwich panel with soft core based on a refined shear deformation model Qunli Liu a, Yi Zhao
b,*
a
b
Department of Mechanical Engineering, University of Nevada, Las Vegas, NV 89154, USA Department of Aerospace Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, USA Available online 14 March 2005
Abstract The natural frequency of a thick rectangular sandwich panel composed of orthotropic facesheets and a soft core was studied based on a refined shear deformation model. The shear deformation of the sandwich panel was described by a polynomial function. The effect of transverse shear modulus of the facesheets and core on flexural vibration of the panel was investigated. Comparison was made among classical thin plate theory, linear shear (low order) deformation theory and the refined shear (high order) deformation model. Results from finite element analysis were also provided to verify the theoretical predictions. It was shown that the refined shear deformation model provided a better prediction on the natural frequency of vibration of a sandwich panel than thin plate model or low order deformation model. 2005 Elsevier Ltd. All rights reserved. Keywords: Natural frequency; Sandwich panel; Soft core; Refined shear deformation theory
1. Introduction Vibration of sandwich structures is of particular interest to aerospace industry due to its applications in aircraft. It has also been a subject of intensive research during the past decades. Maheri and Adams [1] presented some results from experimental investigation of the dynamic shear property of both Nomex and aluminum honeycomb cores and the damping of composite honeycomb sandwich beams in steady-state flexural vibration. The beams were constructed from either CFRP or GRP facesheets with Nomex or aluminum honeycomb cores. Chen and Sheu [2] studied the dynamic characteristics of a sandwich beam with flexible core based on a model in which two thin beams were
*
Corresponding author. Tel.: +1 386 226 6746; fax: +1 386 226 6747. E-mail address: [email protected] (Y. Zhao).
0263-8223/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.01.006
separated by springs and dashpots uniformly distributed along the beam length. The general damped Timoshenko beam was employed to include the rotary inertia of mass, shear deformation and damping components. Sakiyama et al. [3] proposed an analytical method for free vibration of a three-layer continuous sandwich beam with elastic or viscoelastic core under various support conditions. They derived the characteristic equations for the natural frequency and the associated mode by introducing a discrete Green function. Marur and Kant [4] used high order refined theory to study the transient dynamics of a laminated beam. Most recently, Nilsson [5] applied HamiltonÕs principle to derive a sixth order differential governing equation for threelayered beam with lightweight honeycomb core under various boundary conditions. Transverse shear deformation of a panel subjected to bending has been a topic for many years. Mindlin [6] proposed a model to study transverse shear of an isotropic thick plate, which has since been recognized as
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
Mindlin theory. Dawe and Roufaeil [7] used Rayleigh– Ritz method to study the vibration of a thick plate. Reddy [8] applied a high order theory to model the shear deformation. Unlike the Mindlin plate theory that requires a shear correction factor, the ReddyÕs model does not require this factor, but the governing equations are more complicated. Kant et al. [9,10] studied large amplitude free vibration and transient response of a sandwich plate using Green–Lagrangian strain–displacement relation and FEA. Lee and Fan [11] developed a p-version FEA code to compute the vibration modes of rectangular sandwich panels. A linear shear deformation assumption was adopted in their work. Bardell et al. [12] used a p–h version code to study the vibration of a cantilever T-platform sandwich panel. Lu [13] applied a high order shear deformation theory to nonlinear vibration of coplanar thick composite laminated panels, which were either simply supported or fixed at the edges. Wang [14] compared classical thin plate theory and the Mindlin (low order shear) thick plate theory, and applied low order shear theory for flexural vibration of rectangular sandwich panel composed of isotropic facesheets and core. Recently, Kant and Swaminathan [15] compared the free vibration results on plates from a few high order models including two models by Kant and one by Reddy. For dynamic response of sandwich plate, Pai and Palazotto [16] formulated a finite scheme in which the multi-layered facesheets were modeled as sublaminates and the stress in thickness direction was taken into account. Meunier and Shenoi [17] employed ReddyÕs high order shear deformation theory for dynamic analysis of composite sandwich plate with damping modeled for the viscoelastic PVC foam. Modal loss factor was predicted from the analytical model and experimental results from dynamic mechanical analysis were presented. They concluded that the viscoelasticty of the constitutive materials and their temperature and frequency dependency played significant role in the dynamic properties of sandwich plate. Kakhecha et al. [18] formulated a finite element scheme for vibration and damping analysis of composite plates on the basis of a high order theory similar to that by Kant. A comprehensive survey on modeling of sandwich structures was conducted by Noor et al. [19], which summarized various models, methods, advances and problems. Reviews on analytical models could also be found from the literatures by Allen [20], Mead [21], Pandya and Kant [22], Noor and Burton [23], Ha [24] and Burton and Noor [25]. Although extensive research has been conducted on the topic of vibration of sandwich plates/panels, few results were reported on analytical prediction of the dynamic properties in terms of facesheet and core material as well as structural parameters. Liu and Zhao [26] presented an analytical solution to the vibration of a sandwich panel with isotropic facesheets and orthotro-
365
pic core based on KirchhoffÕs linear shear deformation theory. The rotational effect was taken into consideration and effects of core anisotropy, core density, and the facesheet thickness on natural frequencies were also discussed. A parametric study was also conducted by Liu and Zhao [27] to evaluate the natural frequency of a simply-supported rectangular sandwich panel with orthotropic facesheets and core. Despite of certain improvements in these two studies, the role of soft core was overestimated with KirchhoffÕs thin plate theory or low order shear deformation theory. In this paper, a new model based on refined shear deformation is proposed to better estimate vibration behavior of sandwich panels with soft core.
2. Analytical model A linear elastic stress–strain relation is used to describe the elastic behavior of the orthotropic facesheets and core. A plane stress assumption is made as in other literatures [9–11,26]. Additional assumptions include: • The velocity along thickness of the panel is uniform, and can be represented by that of the mid-plane. • The facesheets and the core are perfectly bonded. • The normal stress in thickness direction can be neglected. With the plane stress assumption, the stress–strain relationship for the orthotropic facesheets can be expressed in terms of engineering constants as 32 3 3 2 Ef 1 2 t12 Ef 2 0 0 0 ex rfx 1t12 t21 1t12 t21 76 7 6r 7 6 t E E 21 f 1 f2 ey 7 6 fy 7 6 0 0 0 7 76 7 6 1t12 t21 1t12 t21 7 6 76 7 6 sfxy 7 ¼ 6 6 0 0 Gfxy 0 0 7 7 6 6 6 cxy 7: 7 6 7 6 7 6 6 7 4 sfxz 5 4 0 0 0 Gfxz 0 54 cxz 5 cyz sfyz 0 0 0 0 Gfyz ð1Þ Since the soft core is also assumed to be orthotropic, its stress–strain relationship has the same form as that for the facesheets with subscript f replaced by c, and m by l, respectively. The structural parameters of the sandwich panels include the size of the panel a · b, the facesheet thickness d, and the thickness of the core 2h. The x-direction is set along a side of the panel, as indicated in Fig. 1. Denoting the transverse displacement of the midplane as w = w(x, y, t), rotations with respect to x-axis and y-axis at the mid-plane as / = /(x, y, t) and u = u(x, y, t), respectively. In the refined shear deformation model, the nonlinear shear deformations of both sandwich core and facesheets are taken into consideration. It is assumed that the shear strains change continuously
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be selected as kn factor. When N = 1 and k1 = 1, the proposed model reduces to ReddyÕs model [8]. When N > 1, the shear function g(z) consists of more than one term with different k factors. The coefficients an become the ‘‘weight factors.’’ Fig. 2(a) and (b) show the behavior of the shear function and its derivative with respect to the panel thickness z, when h + d = 5.5 mm). To demonstrate the derivation process, N = 1 (and therefore a1 = 1) is used for simplicity. The strains at any point in the facesheets or core are Fig. 1. Structural parameters for a sandwich panel.
ex ¼ in the thickness direction, and at the top and bottom surfaces, z = ±(h + d), the shear strains czx = czy = 0. With this assumption, the following displacement functions are proposed: ow ; ox ow ; vðx; y; z; tÞ ¼ ½z gðzÞuðx; y; tÞ gðzÞ oy uðx; y; z; tÞ ¼ ½z gðzÞ/ðx; y; tÞ gðzÞ
ð2Þ
where g(z) is a function describing the nonlinear shear deformation of the sandwich panel gðzÞ ¼
N X n¼1
an
z2kn þ 1 2k ð2k n þ 1Þðh þ dÞ n
ou o/ o2 w ¼ ðz gk Þ gk 2 ; ox ox ox
ov ou o2 w ¼ ðz gk Þ gk 2 ; oy oy oy ou ov o/ ou o2 w þ ¼ ðz gk Þ þ ; cxy ¼ 2gk oy ox oy ox ox oy ow ou og ow cxz ¼ þ ¼ 1 k ; /þ ox oz ox oz ow ov ogk ow þ ¼ 1 cyz ¼ : /þ oy oz oy oz ey ¼
ð4Þ
2.1. Stain energy of the panel ð3Þ
and n = 1, 2, 3, . . . , are weight factors that satisfy a1 + a2 + a3 + = 1. The factor ‘‘kn’’ in the empirical shear function g(z) is selected in such a way that g(z) remains an odd function, which satisfies the following conditions: ex ðh þ dÞ ¼ ex ððh þ dÞÞ; ey ðh þ dÞ ¼ ey ððh þ dÞÞ: The factor kn is not necessary to be an integer—a fraction number such as 1/2, 3/2, 7/4, 5/2, etc., could also
The strain energy of the panel is Z Z Z 1 T U¼ ½r ½edV 2
ð5Þ
which is composed of six components, namely strain energy associated with tension/compression in x and y directions, and with shear strain in the xy plane for the facesheets and core, respectively. The strain energy induced by the tension/compression in the top and bottom facesheets in x direction can be expressed as
Fig. 2. (a) Behavior of shear function with respect to panel thickness. (b) Behavior of the derivative of shear function.
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
U fx
1 ¼ 2
Z
a 0
1 þ 2 ¼
Z
b
0
Z
a 0
Z
hþd
h
Z
b 0
Z
rfx efx dz dy dx
t12 Dfy2
Z
a
Z
rfx efx dz dy dx
þ t12 Dfy3
ðhþdÞ
ð6aÞ
Ef 1 Ef 1 K f 1 ; Dfx2 ¼ K f 2; 1 t12 t21 1 t12 t21 Ef 1 ¼ K f 3; 1 t12 t21 Ef 2 Ef 2 K f 1 ; Dfy2 ¼ K f 2; 1 t12 t21 1 t12 t21 Ef 2 ¼ K f 3; 1 t12 t21
Dfy1 ¼
Kf 1 ¼ 2
Z
hþd 2
ðz gk Þ dz; K f 2 ¼ 2
h
Kf 3 ¼ 2
Z
a
o/ o2 w ou o2 w þ dy dx ox oy 2 oy ox2 0 ! Z b 2 o w o2 w dy dx ox2 oy 2 0 Z
b
ð6bÞ
b
Dfx1 ¼
Dfy3
Z
0
where Df s are the stiffness of the facesheets with respect to the mid-plane of the panel
Dfx3
a
0
h
2 o/ dy dx ox 0 0 Z a Z b o/ o2 w 2Dfx2 dy dx ox ox2 0 0 Z a Z b 2 2 ow þ 2Dfx3 dy dx ox2 0 0 Z a Z b o/ ou þ t12 Dfy1 dy dx ox oy 0 0 Z a Z b o/ o2 w ou o2 w þ dy dx t12 Dfy2 ox oy 2 oy ox2 0 0 ! Z a Z b 2 o w o2 w þ t12 Dfy3 dy dx ; ox2 oy 2 0 0 1 Dfx1 2
Z
367
Z
and the three components of strain energy associated with shear deformation are Z hþd Z a Z b o/ ou þ U fxy ¼ Gfxy ðz gk Þ oy ox h 0 0 2 2 ow 2gk dxdy dz oxoy 2 Z a Z b 1 o/ ou Dfxy1 þ ¼ dxdy 2 oy ox 0 0 Z a Z b o/ ou o2 w þ dxdy 4Dfxy2 oy ox oxoy 0 0 ! Z a Z b 2 2 ow þ4Dfxy3 dxdy ; oxoy 0 0 2 Z hþd Z a Z b og ow U fxz ¼ Gfxz 1 k dxdy dz /þ ox oz h 0 0 2 Z a Z b 1 ow ¼ S fxz /þ dxdy; 2 ox 0 0 2 Z hþd Z a Z b og ow U fyz ¼ Gfyz 1 k dxdy dz uþ oy oz h 0 0 2 Z a Z b 1 ow ¼ S fyz uþ dxdy; 2 oy 0 0 ð6cÞ where
hþd
gk ðz gk Þdz;
h
Dfxy1 ¼ Gfxy K f 1 ; Dfxy2 ¼ Gfxy K f 2 ; Dfxy3 ¼ Gfxy K f 3 ;
hþd
g2k dz:
h
Similarly, the strain energy of the facesheets associated with stress component in y direction is Z Z Z 1 a b hþd U fy ¼ rfy efy dz dy dx 2 0 0 h Z Z Z 1 a b h þ rfy efy dz dy dx 2 0 0 ðhþdÞ Z a Z b 2 ou ¼ Dfy1 dy dx oy 0 0 Z a Z b ou o2 w 2Dfy2 dy dx oy oy 2 0 0 Z a Z b 2 2 ow dy dx þ 2Dfy3 oy 2 0 0 Z a Z b o/ ou þ t12 Dfy1 dy dx ox oy 0 0
ogk ¼ 2Gfxz 1 dz; S fyz oz h Z hþd og ¼ 2Gfyz 1 k dz: oz h Z
S fxz
hþd
Following the same procedure, the strain energy components of the core are: Z Z Z 1 a b h U cx ¼ ðrcx ecx Þdz dy dx 2 0 0 h Z a Z b 2 1 o/ Dcx1 dy dx ¼ 2 ox 0 0 Z a Z b o/ o2 w 2Dcx2 dy dx ox ox2 0 0 Z a Z b 2 2 ow dy dx þ 2Dcx3 ox2 0 0 Z a Z b o/ ou þ t12 Dcy1 dy dx ox oy 0 0
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Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
t12 Dcy2 þ t12 Dcy3
U cy
1 ¼ 2
Z
a
Z
0
0
o/ o2 w ou o2 w þ dy dx ox oy 2 oy ox2 0 ! Z b 2 o w o2 w dy dx ; ox2 oy 2 0
Z
h
Z Z
b
a 0 a
Z
b
2 ou dy dx oy 0 0 Z a Z b ou o2 w 2Dcy2 dy dx oy oy 2 0 0 Z a Z b 2 2 ow dy dx þ 2Dcy3 oy 2 0 0 Z a Z b o/ ou þ t12 Dcy1 dy dx ox oy 0 0 Z a Z b o/ o2 w ou o2 w t12 Dcy2 þ dydx ox oy 2 oy ox2 0 0 ! Z a Z b 2 o w o2 w dy dx ; þ t12 Dcy3 ox2 oy 2 0 0
1 Dcy1 ¼ 2
U cxy ¼ Gcxy
Z
Z
h
a
Z
h
0 2
Z
a
ð7aÞ
b
S cxz ¼ Gcxz
Z 0
b
S cyz ¼ Gcyz
K c1 ¼ K c3 ¼
U cyz
ð7bÞ
2
ð7cÞ Ec1 Ec1 K c1 ; Dcx2 ¼ K c2 ; ð1 l12 l21 Þ ð1 l12 l21 Þ Ec1 K c3 ; ¼ ð1 l12 l21 Þ
Dcx3
h
og 1 k oz
2 dz;
2 og 1 k dz; oz h h
h Z h
2
ðz gk Þ dz; K c2 ¼
Z
h
gk ðz gk Þdz;
h
g2k dz:
þ U cxy þ U cxz þ U cyz :
ow dx dy dz ox oy 2 Z a Z b o/ ou þ dx dy ¼ Dcxy1 oy ox 0 0 Z a Z b o/ ou o2 w þ dx dy 4Dcxy2 oy ox ox oy 0 0 ! Z a Z b 2 2 ow þ 4Dcxy3 dx dy ; ox oy 0 0 2 Z h Z a Z b og ow ¼ Gcxz 1 k dx dy dz /þ ox oz h 0 0 2 Z a Z b 1 ow ¼ S cxz /þ dx dy; 2 ox 0 0 2 Z h Z a Z b og ow ¼ Gcyz 1 k dx dy dz uþ oy oz h 0 0 2 Z a Z b 1 ow ¼ S cyz uþ dx dy; 2 oy 0 0
Dcx1 ¼
Z
h
h
h
o/ ou þ ð z gk Þ oy ox
where
Z
Z
The total strain energy can thus be expressed as U ¼ U fx þ U fy þ U fxy þ U fxz þ U fyz þ U cx þ U cy
2gk
U cxz
Dcy3
Dcxy1 ¼ Gcxy K c1 ; Dcxy2 ¼ Gcxy K c2 ; Dcxy3 ¼ Gcxy K c3 ;
rcy ecy dz dy dx
h
0
Ec2 Ec2 K c1 ; Dcy2 ¼ K c2 ; ð1 l12 l21 Þ ð1 l12 l21 Þ Ec2 K c3 ; ¼ ð1 l12 l21 Þ
Dcy1 ¼
ð8Þ
2.2. Kinetic energy of the panel The kinetic energy consists of five components, corresponding to the transverse motion of the panel and motion in x and y directions of the facesheets and core, respectively. Z Z 2 q a b ow Tz ¼ dy dx; 2 0 0 ot Z a Z b Z hþd 2 2qf ou T fx ¼ dz dy dx ot 2 0 0 h 2 2 Z Z 1 a b o/ o/ ow If1 ¼ 2I f 2 2 0 0 ot ot ox ot 2 2 ! ow þ If3 dy dx; ox ot Z a Z b Z hþd 2 2qf ov T fy ¼ dz dy dx ot 2 0 0 h 2 2 Z Z 1 a b ou ou ow If1 2I f 2 ¼ 2 0 0 ot ot oyot 2 2 ! ow þ If3 dy dx; oyot Z Z Z 2 qc a b h ou T cx ¼ dz dy dx 2 0 0 h ot 2 2 Z Z 1 a b o/ o/ ow I c1 ¼ 2I c2 2 0 0 ot ot ox ot 2 2 ! ow þ I c3 dy dx; ox ot
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
Z Z Z 2 qc a b h ov ¼ dz dy dx 2 0 0 h ot 2 2 Z Z 1 a b ou ou ow I c1 ¼ 2I c2 2 0 0 ot ot oyot 2 2 ! ow þ I c3 dy dx; oyot
T cy
where Dx1 ¼ Dfx1 þ Dcx1 ; Dx2 ¼ Dfx2 þ Dcx2 ; Dx3 ¼ Dfx3 þ Dcx3 ; Dy1 ¼ Dfy1 þ Dcy1 ; Dy2 ¼ Dfy2 þ Dcy2 ; Dy3 ¼ Dfy3 þ Dcy3 ; ð9Þ
where
Eqs. (12a)–(12c) are the governing equations of vibration for a rectangular sandwich panel. For a simply supported panel, these equations can be solved by assuming the following functions:
I f 3 ¼ qf K f 3 ; I c1 ¼ qc K c1 ; I c2 ¼ qc K c2 ; I c3 ¼ qc K c3 ; so the total kinetic energy can be written as T ¼ T z þ T fx þ T fy þ T cx þ T cy :
ð10Þ
Using HamiltonÕs variational principle, Z t2 d ðU T Þdt ¼ 0;
ð11Þ
t1
o3 w o3 w þ t12 Dfy2 þ l12 Dcy2 þ 2 Dfxy2 þ Dcxy2 3 ox ox oy 2
o2 u o2 / Dx1 2 t12 Dfy1 þ l12 Dcy1 þ Dfxy1 þ Dcxy1 ox ox oy 2 o / ow Dfxy1 þ Dcxy1 þ Gxz / þ oy 2 ox o2 / o3 w þ I ; 2 ot2 ox ot2
ðx; yÞ sinðxtÞ; w¼w yÞ sinðxtÞ; / ¼ /ðx;
ð13Þ
ðx; yÞ sinðxtÞ; u¼u where ðx; yÞ ¼ A sinðmpx=aÞ sinðnpy=bÞ; w
a system of differential equations can be obtained
¼ I 1
I 1 ¼ I f 1 þ I c1 ; I 2 ¼ I f 2 þ I c2 ; I 3 ¼ I f 3 þ I c3 ; Gxz ¼ S fxz þ S cxz ; Gyz ¼ S fyz þ S cyz :
q ¼ 2dqf þ 2hqc ; I f 1 ¼ qf K f 1 ; I f 2 ¼ qf K f 2 ;
Dx2
369
ð12aÞ
yÞ ¼ B cosðmpx=aÞ sinðnpy=bÞ; /ðx; ðx; yÞ ¼ C sinðmpx=aÞ cosðnpy=bÞ: u A system of homogeneous equations of constants A, B, and C can be obtained 2 32 3 A A11 þ A12 x2 A21 þ A22 x2 A00 6 7 6 7 6 B11 þ B12 x2 6 7 B00 B21 þ B22 x2 7 4 54 B 5 ¼ 0: C 11 þ C 12 x2
C 21 þ C 22 x2
C 31 þ C 32 x2
ð14Þ
o4 w o4 w Dx3 4 þ Dy3 4 þ 2t12 Dfy3 þ 2l12 Dcy3 ox oy o4 w o3 / o3 u D D þ4 Dfxy3 þ Dcxy3 x2 y2 ox2 oy 2 ox3 oy 3 3 o / o3 u t12 Dfy2 þ l12 Dcy2 þ 2 Dfxy2 þ Dcxy2 þ oxoy 2 ox2 oy o/ o2 w ou o2 w þ 2 Gyz þ 2 Gxz ox ox oy oy 3 4 2 3 ow o/ ou ow o4 w þ þ ¼ q 2 I 2 þ I3 ; ot oxot2 oy ot2 ox2 ot2 oy 2 ot2
For a non-trivial solution, the frequencies of vibration can be determined from the characteristic equation 2 3 A11 þ A12 x2 A21 þ A22 x2 A00 6 7 2 det 6 B00 B21 þ B22 x2 7 4 B11 þ B12 x 5 ¼ 0; C 11 þ C 12 x2 C 21 þ C 22 x2 C 31 þ C 32 x2 ð15Þ where A11 ¼ a2 b2 mpGxz b2 m3 p3 Dx2 a2 mn2 p3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ;
ð12bÞ A12 ¼ a2 b2 mpI 2 ; 3
C
3
o w ow þ t12 Dfy2 þ l12 Dcy2 þ 2 Dfxy2 þ Dcxy2 3 oy ox2 oy 2 o2 / ou D21 2 t12 Dfy1 þ l12 Dcy1 þ Dfxy1 þ Dcxy1 oy oxoy o2 u ow o2 u o3 w Dfxy1 þ Dcxy1 þ G u þ þ I ; ¼ I yz 1 2 ox2 oy ot2 oy ot2
Dy2
ð12cÞ
A21 ¼ a3 b2 Gxz þ ab2 m2 p2 Dx1 þ a3 n2 p2 Dxy1 ; A22 ¼ a3 b2 I 1 ; A00 ¼ a2 bmnp2 ðDxy1 þ t12 Dfy1 þ l12 Dcy1 Þ; B11 ¼ a2 b2 npGyz a2 n3 p3 Dy2 b2 m2 np3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ;
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B12 ¼ pa2 b2 nI 2 ; B00 ¼ ab2 mnp2 ðDxy1 þ t12 Dfy1 þ l12 Dcy1 Þ; B21 ¼ a2 b3 Gyz þ a2 bn2 p2 Dy1 þ b3 m2 p2 Dxy1 ; B22 ¼ a2 b3 I 1 ; C 11 ¼ a2 b4 m2 p2 Gxz þ a4 b2 n2 p2 Gyz þ b4 m4 p4 Dx3 þ a4 n4 p4 Dy3 þ a2 b2 m2 n2 p4 ð4Dxy3 þ 2t12 Dfy3 þ 2l12 Dcy3 Þ; C 12 ¼ a2 b2 ðI 3 p2 ða2 n2 þ b2 m2 Þ þ a2 b2 qÞ; C 21 ¼ a3 b4 mpGxz ab4 m3 p3 Dx2 a3 b2 mn2 p3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ; C 22 ¼ a3 b4 mpI 2 ; C 31 ¼ npa4 b3 Gyz a4 bn3 p3 Dy2 a2 b3 m2 np3 ð2Dxy2 þ t12 Dfy2 þ l12 Dcy2 Þ; C 32 ¼ a4 b3 npI 2 :
3. Results and discussions The nonlinear transverse shear deformation of a sandwich core is described by the proposed displacement function, Eqs. (2) and (3). For demonstration purpose, the results presented thereafter are corresponding to the case in which n = 1, a1 = 1 and therefore, gk ¼
1 ð2k þ 1Þðh þ dÞ2k
z2kþ1 :
ð16Þ
For the cases illustrated in the following, the panel size is a = b = 200 mm. The influence of different values of k is shown in Fig. 3 in which h = 5 mm and d = 0.5 mm. The material properties includes density 2.8 · 103 kg/m3, elastic modulus 70 GPa, PoisonÕs ratio 0.33 for face-
Fig. 3. Effect of index k on frequency prediction.
sheets and density 40 kg/m3, elastic modulus (in applicable directions) 10 MPa, PoisonÕs ratio 0.3 for core if not explicitly stated otherwise. The results from the classical thin plate theory as well as low order (or linear) shear deformation theory are also presented. The result from thin plate theory was used as the reference for comparison. It is seen that both thin plate theory and linear shear deformation theory overestimate the mode frequency of a sandwich panel, particularly when the transverse shear modulus of core (Gxz and Gyz) is low. In comparison, the proposed refined model offers the flexibility of using selected high order terms by adjusting weight distribution and, therefore, has the versatility to handle the constitutive parameters in a relatively wide range. As illustrated in Fig. 2 of the shear function g(z), it becomes clear now that a smaller value for parameter k may be employed for low modulus core and a larger value for high modulus core. Combination of the terms with various k values and appropriate weights may be used for cores with intermediate shear modulus. In practical applications, the terms and the associate parameters may be selected based on preliminary experimental investigation. A comparison between the refined shear deformation model (for k = 1) and the low order (linear) shear deformation theory as well as the classic thin plate theory is provided in Fig. 4(a)–(d). The transverse shear modulus of the core is assumed to be Gcxz = Gcyz with four different values used: Gcxz = Gcyz = 50, 200, 500 and 1000 MPa. Results from finite element analysis are also plotted in Fig. 4(b). The FEA software code used was ABAQUS. The facesheests are represented by one-layer of solid hexahedral brick elements at the top and bottom respectively. The soft core is also modeled as equivalent solid brick elements that have same size as the facesheet elements in panelÕs length and width directions. A few scenarios were established where the core has 2, 5 and 10 layers elements of the same size (small cubic). The results were quite similar and the frequency from smaller element case was slight lower than in the bigger element case. The results presented were from the scenario where the core has 10 layers of elements (1 mm by 1 mm by 1 mm) and the facesheets have one-layer of elements (1 mm by 1 mm by 0.5 mm). It can be seen from the four figures that the differences among the three models increase as the core thickness increases. It can also be found that the transverse shear modulus of the core has noticeable effect. The lower the shear modulus, the greater the difference is. When Gcxz = Gcyz = 1000 MPa, the differences among the three models are insignificant; as the shear modulus of the core increases, the difference becomes negligible. For the facesheet with 0.5 mm thickness (4.5% of total thickness of the panel), the prediction from linear shear deformation theory is very close to that from thin plate theory; as the facesheet thickness increases, the
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Fig. 4. Comparison for effects of core thickness on frequency prediction.
difference between the linear shear deformation theory and the thin plate theory increases. Apparently, the classic thin plate theory overestimates the natural frequency. The low order linear shear deformation theory only slightly improves the estimation of natural frequency. The prediction from the refined shear model is considerably different from that from either low order shear theory or thin plate theory, especially when the transverse
shear modulus of core is low. The proposed refined (high order) shear deformation model provides a more agreeable prediction of the vibration frequency of the sandwich panel than the other models considered. The effect of transverse shear modulus of the core on natural frequency is presented in Fig. 5(a) and the effect of transverse shear modulus of facesheets in Fig. 5(b). In both cases the thickness of the core is fixed at h = 5 mm,
Fig. 5. (a) Effect of transverse shear modulus of core (Gcxz = Gcyz). (b) Effect of transverse shear modulus of facesheets (Gfxz = Gfyz).
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and the parameter k is the ratio of facesheet thickness to core height. Clearly, the larger the shear modulus, the higher the vibration frequency will be. This result is quite expected because of the correlation between shear modulus and panel stiffness. This effect is also related to the relative thickness of the facesheets with respect to the core thickness. Fig. 6 shows the effect of core orthotropy for k = 0.1 (thin facesheet) and 0.4 (thick facesheet). With Gcxz specified, the natural frequency increases as Gcyz increases and the relation is nonlinear. Fig. 7 shows the effect of facesheet orthotropy with k = 0.1 and 0.4. With Gfxz specified, the natural frequency increases as Gfyz increases. The relation is almost linear for the case of thin facesheets, but is nonlinear for the thick facesheets. A special case is when the sandwich panel becomes a solid plate, i.e., the thickness of core is set at zero. The results are shown in Fig. 8. When the thickness of the plate is small, all three models provide the identical results (this provides another validation of the proposed
Fig. 8. Comparison of mode frequency from various models for isotropic solid plate.
refined shear deformation model). As the thickness of the plate increases, the predictions from both low order and high order theories deviate gradually from that of
Fig. 6. Effect of orthotropy of transverse shear modulus of core in refined model: (a) k = 0.1 (thin facesheet); (b) k = 0.4 (thick facesheet).
Fig. 7. Effect of orthotropy of transverse shear modulus of facesheet: (a) k = 0.1 (thin facesheet); (b) k = 0.4 (thick facesheet).
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thin plate theory. The results also show that the difference between the low order and high order theories is negligible when they are used for a solid plate. This suggests that a linear transverse theory provides sufficient accuracy in studying a solid plate with various thicknesses. Although high order shear deformation model partially corrects the overestimation of shear deformation effect so that the natural frequency predicted is lower than those by classical thin plate theory and low order theory, the natural frequency is still higher than those obtained from FEA, in which the modulus of both the facesheets and core in thickness direction are taken into account. Further computation verifies that the higher the modulus in thickness direction, the smaller the difference between analytical results and FEA results. Either low order or high order theories, though correct the effect of shear deformation to certain extent, still overestimate the effect of modulus in thickness direction. One of the assumptions made in this model is that the deformation in thickness direction is small so that strain energy associated with this deformation can be neglected. This implies that the modulus of the panel in the thickness direction is assumed to be infinite. However, a theoretical model with the deformation in thickness direction considered would be extremely difficult and was therefore not attempted in this paper. If the shear modulus of facesheets is much higher than that of the core, and the thickness is much smaller, only the shear modulus of the core may be taken into account—in this case, predictions by both low order and high order models are very close to those from FEA. The two analytical solutions are useful to parametrically study the effect of all the structural and materials parameters in sandwich panel, except the modulus of facesheet and core in thickness direction.
4. Conclusions A refined shear deformation model was proposed to investigate the natural frequency of a thick rectangular sandwich panel with orthotropic facesheets and deformable soft core. The high order model is based on a nonlinear function with multiple and adjustable weights to describe the transverse shear deformation. A noticeable improvement over the linear shear theory can be achieved using this model, especially for the sandwich core with a low transverse shear modulus. The predictions on the modal frequency of a thick sandwich panel from this model are in better agreement with results from finite element analysis. This model provides an analytical approach to quantitatively investigate the effect of the constitutive parameters of a sandwich panel on its modal frequency.
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References [1] Maheri MR, Adams RD. Steady-state flexural vibration damping of honeycomb sandwich beams. Compos Sci Technol (UK) 1994; 52(3):333–47. [2] Chen YH, Sheu JT. Dynamic characteristics of layered beam with flexible core. J Vib Accoust 1994;116(3):350–6. [3] Sakiyama T, Matsuda H, Morita C. Free vibration analysis of continuous sandwich beams with elastic or viscoelastic cores by applying the discrete Green function. J Sound Vib 1996;198(4): 439–54. [4] Marur SR, Kant T. Transient dynamics of laminated Beams: An evaluation with a high-order refined theory. Compos Struct 1998; 11:1–11. [5] Nilsson E, Nilsson AC. Prediction and measurement of some dynamic properties of sandwich structures with honeycomb and foam cores. J Sound Vib 2002;251(3):409–30. [6] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 1951;18:31–8. [7] Dawe DJ, Roufaeil OL. Rayleigh–Rits vibration analysis of Mindlin plates. J Sound Vib 1980;69(3):345–59. [8] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51(4):745–52. [9] Kant T, Arora CP, Varaiya JH. Finite element transient analysis of composite and sandwich plates based on a refined theory and a mode superposition method. Compos Struct 1992;22(2):109–20. [10] Kant T, Kommineni JR. Large amplitude free vibration analysis of cross-Ply composite and sandwich laminates with a refined theory and C0 finite elements. Comput Struct 1994;50(1):123–34. [11] Lee LJ, Fan YJ. Bending and vibration analysis of composite sandwich plates. Comput Struct 1996;60(1):103–12. [12] Bardell NS, Dunsdon JM, Langley RS. Free vibration analysis of coplanar sandwich panels. Compos Struct 1997;38(1–4):463–75. [13] Lu C-H, Libove C. Beamlike harmonic vibration of variablethickness sandwich plates. AIAA J 1991;29(2):299–305. [14] Wang CM. Deducing thick plate solutions from classical thin plate solutions. Struct Eng Mech 2001;11(1):89–104. [15] Kant T, Swaminathan K. Free vibration of isotropic, orthotropic, and multilayer plates based on higher order refined theories. J Sound Vib 2001;241(2):319–27. [16] Pai PF, Palazotto AN. A higher-order sandwich plate theory accounting for 3-D stress. Int J Solids Struct 2001;38(21–32): 5045–62. [17] Meunier M, Shenoi RA. Dynamic analysis of composite sandwich plates with damping modelled using high-order shear deformation theory. Compos Struct 2001;54(2):243–54. [18] Makhecha DP, Ganapathi M, Patel BP. Vibration and damping analysis of laminated/sandwich composite plates using higherorder theory. J Reinf Plast Compos 2002;21(6):559–75. [19] Noor AK, Burton WS, Bert CW. Computational models for sandwich panels and shells. Appl Mech Rev 1996;49(3):155–99. [20] Allen HG. Analysis of sandwich panels: The significance of shear deformation. Composites 1970;1(6):215–20. [21] Mead DJ. A comparison of some equations for the flexural vibration of damped sandwich beams. J Sound Vib 1982;83(3): 363–77. [22] Pandya BN, Kant T. Higher-order shear deformable theories for flexure of sandwich plates—finite element evaluation. Int J Solids Struct 1988;24(12):1267–86. [23] Noor AK, Burton WS. Assessment of shear deformation theories for multilayered composite plates. Appl Mech Rev 1989;42(1): 1–13. [24] Ha KH. Finite element analysis of sandwich plates: An overview. Comput Struct 1990;37(4):397–403. [25] Burton WS, Noor AK. Assessment of computational models for sandwich panels and shells. Comput Methods Appl Mech Eng 1995;124(1–2):125–51.
374
Q. Liu, Y. Zhao / Composite Structures 72 (2006) 364–374
[26] Liu Q, Zhao Y. Prediction of natural frequencies of vibration for a sandwich panel using thick plate theory. J Sandwich Struct Mater 2001;3(4):289–309.
[27] Liu Q, Zhao Y. A parametric study on natural frequency of sandwich panel using refined shear model, ETCE 2002, Houston, TX, 4–5 February 2002, vol. 2. p. 535–40.