Thermal Vibration of Magnetostrictive Material in Laminated Plates by the GDQ Method


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1 The Open echanics Journal, 007, 1, Thermal Vibration of agnetostrictive aterial in Laminated Plates by the GDQ ethod C.C. Hong * Department of echanical Engineering, Hsiuping Institute of Technology, Taichung, 41 Taiwan, ROC Abstract: The study of magnetostrictive material in laminated plate under thermal vibration is calculated by using the generalized differential quadrature (GDQ) method. The YS shear deformation effect is included in the time dependent of displacement field. In the thermoelastic stressstrain relations that containing the linear temperature rise and the magnetostrictive coupling terms with velocity feedback control. We use the GDQ method to normalize and discrete the dynamic differential equations in terms of displacements and shear rotations into the form of dynamic discretized equations. Four edges of rectangular laminate with simply supported boundary conditions are considered. The time responses of thermal stresses and center displacement with and without velocity control are obtained. Keywords: agnetostrictive material, thermal vibration, generalized differential quadrature method, GDQ, shear deformation, velocity feedback control. 1. ITRODUCTIO One of the new trends of material in the mechanical engineering is the functionally gradient materials (FG). In each type of FG, there are some functional layers e.g. piezoelectric, magnetostrictive, electrostrictive, shape memory alloys, in their sections can make a controlled progressive change of smart structures [1,]. agnetostrictive materials have the agnetoelectric coupling property under the action of magnetism and mechanism. TerfenolD is one of available agnetostrictive materials, can be used usually in the sensors and actuators [3]. In 005, Lee and Reddy presented the finite element method to analyze the nonlinear response of laminated plate of agnetostrictive material under thermomechanical loading [3]. They found the temperature can decrease the amplitude and period of deflection. In 005, Pradhan presented the analytical solutions for the FG shells of embedded agnetostrictive layers under vibration [1]. It was found that the agnetostrictive layers should be placed away from the neutral plane and made thinner to get better attenuation effects. In 006, Ramirez, Heyliger and Pan present the Ritz approach to get an approximate solution for the free vibration problem of twodimensional magnetoelectroelastic laminates [4]. They obtained the elastic displacements, electric potential, and magnetic potential numerical solutions. In 004, Buchanan presented the candidate materials of multiphase magnetoelectroelastic composites [5]. They found the multiphase material properties vary depending upon the ratio of fiber material to matrix material. In 003, Hong and Jane presented the GDQ method to study the shear deformation in the thermal vibration and bending of laminated plates [6,7]. They found that the maximum deflection and interlaminar stresses at center posi *Address correspondence to this author at the Department of echanical Engineering, Hsiuping Institute of Technology, Taichung, 41 Taiwan, ROC; tion of laminate increasing with the sidetothickness ratio value decreasing. In 005, Hong et al. presented the GDQ method to study the thermal vibration of a thermal sleeve [8]. They found that the computational GDQ solutions of the natural frequency, displacement and thermal stress. In 006, Hong et al. used the GDQ method to study the piezoelectric material under mechanical and electric loads [9]. They found that the numerical GDQ solutions of stress and electric potential function for the analyses of local symmetric pressure with no overshoot and no Gibbs effect and under tractionfree boundary condition of the strip. ow it is interesting to study and find the natural frequency, displacement and thermal stress of the agnetostrictive material in laminated plates with the GDQ method.. FORULATIO.1. Displacement Field The time dependent of displacements fields are assumed in the following YS (YangorrisStavsky, 1966) form [6], which is the firstorder shear deformation theory in the linear equation: u = u 0 (x, y, + z x (x, y, v = v 0 (x, y, + z y (x, y, w = w(x, y, where u 0 and v 0 are tangential displacements, w is transverse displacement of the middleplane, x and y are the shear rotations, t is time... GDQ ethod The GDQ method was presented in 1990 by Shu and Richards [11]. The GDQ method approximates the derivative of function would be used [6] and can be restated that: the derivative of a smooth function at a discrete point in a domain can be discretized by using an approximated weighting linear sum of the function values at all the discrete points in the direction [10,11]. For example, the firstorder and the / Bentham Science Publishers Ltd.
2 30 The Open echanics Journal, 007, Volume 1 C.C. Hong secondorder derivatives of function f * (x, y) at coordinates (x i, y j ) of grid point (i, j) can be discretized by: f * A x f * i, j i,l, f * B l, j y f * i, j j,m, i,m f * A () x i, j i,l f * l, j, f * B () y i, j j,m f * i,m, f * xy i, j (m) where A i, j B f * j,m l,m (m) and B i, j () denote the weighting coefficients for the m th order derivative of the function f * (x, y) with respect to the x and y directions..3. Thermoelastic StressStrain Relations with agnetostrictive Effect We consider a homogeneous, orthotropic, rectangular laminated host plate that mounted with magnetostrictive layers under uniformly distributed loading and thermal effect. Typical threelayer laminate with upper surface magnetostrictive layer which is shown in Fig., where a and b is the length in the x, y direction of the plate, h * is the plate thickness, p 1 and p are the inplane distributed forces, q is the applied pressure load, T = T 0 (x, y, + z h T * 1 (x, y, is the temperature difference between the laminate and curing area. For simplicity, the heat conduction equation for the host laminate would be used [6]. For the plane stress in a laminated material, the inplane stresses constitute the membrance stresses, bending stresses and thermal stresses with magnetostrictive effect for the k th layer are in the following equations [3]: x y xy 0 0 e e e 36 Q 11 Q 1 Q 16 = Q 1 Q Q 6 Q 16 Q 6 Q H z x x T y y T xy xy T and the interlaminar shear stresses are calculated from the following equation: yz xz = Q 44 Q 45 Q 45 Q 55 e 14 e 4 0 e 15 e H z yz xz where x and y are the coefficients of thermal expansion, xy is the coefficient of thermal shear. Q ij is the so called transformed reduced stiffness can be in terms of the elastic stiffness of materials and can be explained more detail by Whitney [1]. x, y, xy are inplane strains and yz, xz are shear strains in terms of displacement components and shear rotation, respectively. e ij is the transformed magnetostrictive coupling moduli. H z is the magnetic field intensity, expressed in the following equation. H z (x, y, = k c I (x, y, (3) (4) Fig.. Typical threelayer laminate with magnetostrictive layers.
3 Thermal Vibration of agnetostrictive aterial in Laminated Plates by the GDQ ethod The Open echanics Journal, 007, Volume 1 31 with velocity feedback control I w (x, y, = c( where k c t is the coil constant, I (x, y, is the coil current, c( is the control gain..4. Dynamic Equilibrium Differential Equations The dynamic equilibrium differential equations in terms of displacements and shear rotations included the magnetostrictive loads are expressed in the following matrix forms [3,6,1]: A 11 A 16 A 66 A 16 A 1 + A 66 A A 16 A 1 + A 66 A 6 A 66 A 6 A A 55 A 45 A 44 B 11 B 16 B 66 B 16 B 1 B B 16 B 1 B 6 B 66 B 6 B u 0 u 0 x xy u 0 v 0 v 0 y x xy v 0 y w x B 11 B 16 B 66 B 16 B 1 B 6 B 16 B 1 B 6 B 66 B 6 B D 11 D 16 D 66 D 16 D 1 + D 66 D 6 D 16 D 1 + D 66 D 6 D 66 D 6 D x x x xy x y y x y xy y y t w xy A 55 A 45 A 45 A 44 A 55 A A 45 A t w w x x y y x y x y x y A 55 A 45 A 45 A 44 = f 1 f f 3 f 4 f I x t y t u 0 t v 0 t w t H t w y x y u 0 t v 0 t x t y t (5) where f 1,, f 5 are the expressions of thermal loads (, ), mechanical loads (p 1, p, q) and magnetostrictive loads (, ). f 1 = x x + xy y + p 1 + x x + xy y f = xy x + y y + p + xy x + y y f 3 = q f 4 = x x + xy y + x x + xy y f 5 = xy x + y y + xy x + y y h * ( y, y ) = (Q 1 h * x + Q y + Q 6 xy )(T 0, z z h T 1)dz * h * ( xy, xy ) = (Q 16 h * x + Q 6 y + Q 66 xy )(T 0, z z h T 1)dz * h * h * ( x, x ) = e 31 H z (1, z )dz ( y, y ) = e 3 H z (1, z )dz h * h * ( xy, xy ) = e 36 H z (1, z )dz h * h * h * h * (A ij, B ij, D ij ) = Q ij (1, z, z )dz, (i, j = 1,, 6) h * h * Ai * j * = k k Qi * j * dz, (i *, j * = 4,5; = 6 i *, = 6 j * ) h * h * (, H, I) = 0 (1, z, z )dz in which k, k are the shear correction coefficients, 0 is the density of ply..5. Dynamic Discretized Equations We apply the weighting coefficients of discretized equations () in the twodimensional generalized differential qradrature (GDQ) method to discrete the differential equations (5) under the vibration of time sinusoidal displacement and temperature: u = [u 0 (x, y) + z x (x, y)]sin( mn, v = [v 0 (x, y) + z y (x, y)]sin( mn w = w(x, y)sin( mn, T = [T 0 (x, y) + z h * T 1(x, y)]sin(
4 3 The Open echanics Journal, 007, Volume 1 C.C. Hong where mn is natural frequency of the plate, is frequency of applied heat flux. And the following nondimensional parameters are introduced: X = x / a, Y = y / b, U = u 0 / a, V = v 0 / b, under the vibration of time sinusoidal displacement and temperature. W = 10h * w /( x T 1 a ) We obtain the following dynamic discretized equations in matrix notation: [ A ]{ SUVW }+ [ B ]{ SSIXY }+ [ KE] { SWSI} (6) + [ FQ] { UVWSI}= { F} where { SUVW }= { A () i,l U l, j V l,m () V i,m A () i,l W l, j { SSIXY }= { A () i,l x l, j () y l, j { SWSI}= { A i,l W l, j B j,m x i,m U l,m y l,m W i,m y l, j () U i,m W l,m A () i,l V l, j () W i,m x l,m y i,m () y i,m A i,l x l, j () x i,m { UVWSI}= {U i, j V i, j W i, j x i, j y i, j { F}= {F 1 F F 3 F 4 F 5 The elements of 5 9 matrix [ A ], 5 6 matrix [ B ], 5 6 matrix [ KE] and 5 5 matrix [ FQ] are as follows: A 11 = (A 11 / a)sin( mn A 1 = (A 16 / b)sin( mn A 13 = (A 66 a / b )sin( mn A 14 = (A 16 b / a )sin( mn A 15 = [(A 1 + A 66 )/a]sin( mn A 16 = (A 6 / b)sin( mn A 17 = A 18 = A 19 A 1 = (A 16 / a)sin( mn A = [(A 1 + A 66 )/b]sin( mn A 3 = (A 6 a / b )sin( mn A 4 = (A 66 b / a )sin( mn A 5 = (A 6 / a)sin( mn A 6 = (A / b)sin( mn A 7 = A 8 = A 9 A 31 = A 3 = A 33 = A 34 = A 35 = A 36 A 37 = xt 1 10h * A 55 sin( mn A 38 = x T 1 10h * (A 45 a / b)sin( mn A 39 = xt 1 10h * (A 44 a / b )sin( mn A 41 = (B 11 / a)sin( mn A 4 = (B 16 / b)sin( mn A 43 = (B 66 a / b )sin( mn A 44 = (B 16 b / a )sin( mn A 45 = (B 1 )(1 / a)sin( mn A 46 = (B 6 / b)sin( mn A 47 = A 48 = A 49 A 51 = (B 16 / a)sin( mn A 5 = (B 1 )(1 / b)sin( mn A 53 = (B 6 a / b )sin( mn A 54 = (B 66 b / a )sin( mn A 55 = (B 6 / a)sin( mn A 56 = (B / b)sin( mn A 57 = A 58 = A 59 B 11 = (B 11 / a )sin( mn B 1 = [B 16 /(ab)]sin( mn B 13 = (B 66 / b )sin( mn B 14 = (B 16 / a )sin( mn B 15 = [(B 1 )/(ab)]sin( mn B 16 = (B 6 / b )sin( mn B 1 = (B 16 / a )sin( mn B = [(B 1 )/(ab)]sin( mn B 3 = (B 6 / b )sin( mn B 4 = (B 66 / a )sin( mn B 5 = [B 6 /(ab)]sin( mn B 6 = (B / b )sin( mn B 31 = B 3 = B 33 = B 34 = B 35 = B 36 B 41 = (D 11 / a )sin( mn B 4 = [D 16 /(ab)]sin( mn B 43 = (D 66 / b )sin( mn B 44 = (D 16 / a )sin( mn B 45 = [(D 1 + D 66 )/(ab)]sin( mn B 46 = (D 6 / b )sin( mn B 51 = (D 16 / a )sin( mn B 5 = [(D 1 + D 66 )/(ab)]sin( mn B 53 = (D 6 / b )sin( mn B 54 = (D 66 / a )sin( mn B 55 = [D 6 /(ab)]sin( mn B 56 = (D / b )sin( mn k KE 11 = T x 1 a [ 1 10h * a k c( e c 31(z k z k1 )] mn cos( mn k KE 1 = T x 1 a [ 1 10h * b k c( e c 36(z k z k1 )] mn cos( mn KE 13 = KE 14 = KE 15 = KE 16
5 Thermal Vibration of agnetostrictive aterial in Laminated Plates by the GDQ ethod The Open echanics Journal, 007, Volume 1 33 k KE 1 = T x 1 a [ 1 10h * a k c( e c 36(z k z k1 )] mn cos( mn k KE = T x 1 a [ 1 10h * b k c( e c 3(z k z k1 )] mn cos( mn KE 3 = KE 4 = KE 5 = KE 6 KE 31 = KE 3 KE 33 = (A 55 / a)sin( mn KE 34 = (A 45 / b)sin( mn KE 35 = (A 45 / a)sin( mn KE 36 = (A 44 / b)sin( mn KE 41 = xt 1 10h * A 55 a sin( mn T x 1 a [ 1 10h * a k c( 1 c 3 k e 31 KE 4 = xt 1 10h * (A 45 a / b)sin( mn T x 1 a [ 1 10h * b k c( 1 c 3 k e 36 KE 43 = KE 44 = KE 45 = KE 46 KE 51 = xt 1 10h * A 45 a sin( mn T x 1 a [ 1 10h * a k c( 1 c 3 k e 36 KE 5 = x T 1 10h * (A 44 a / b)sin( mn T x 1 a [ 1 10h * b k c( 1 c 3 k e 3 KE 53 = KE 54 = KE 55 = KE 56 z 3 3 k z k1 ] h * mn cos( mn z 3 3 k z k1 ] h * mn cos( mn z 3 3 k z k1 ] h * mn cos( mn z 3 3 k z k1 ] h * mn cos( mn FQ 11 = mn a sin( mn FQ 1 = FQ 13 = FQ 15 FQ 14 = H mn sin( mn FQ = mn b sin( mn FQ 5 = H mn sin( mn FQ 1 = FQ 3 = FQ 4 FQ 31 = FQ 3 = FQ 34 = FQ 35 FQ 33 = mn [ x T 1 a /(10h * )]sin( mn FQ 41 = H mn a sin( mn FQ 4 = FQ 43 FQ 44 = (A 55 +I mn )sin( mn FQ 45 = A 45 sin( mn FQ 51 = FQ 53 FQ 5 = H mn b sin( mn FQ 54 = A 45 sin( mn FQ 55 = (A 44 + I mn )sin( mn in which F 1,..., F 5 are represented in the following discretized equation: F 1 = ( 1 a F = ( 1 a F 3 = q i, j F 4 = ( 1 a xl, j + 1 b xyi,m )sin( + p 1i, j xyl, j + 1 yi,m + p i, b j )sin( xl, j + 1 xyi,m )sin( b F 5 = ( 1 xyl, a j + 1 yi,m )sin( b in which p 1 and p are the inplane distributed forces, q is the applied pressure load. The force resultants x, xy, y and moment resultants x, xy, y are expressed as follows: T T T T T x = x, xy = xy, y = y, x = x, xy = xy, T y = y in which { T } is the thermal force resultant, { T } is the thermal moment resultant. 3. SOE UERICAL RESULTS AD DISCUS SIOS We consider the crossply laminates including shear deformation under four sides simply supported. The elastic modules of the typical host material and TerfenolD magnetostrictive material are listed in the following Table 1 [3,6]. The value of shear correction coefficient is K K = 5/6. For the simplification the typical host material conductivity Btu /(sin F), specific heat C v.16 Btu /(lbm F) are used to calculate K = /(C v ) value, and find some of the lowest frequency of applied heat flux and vibration frequency 11 of plate, Table 1. aterial Properties of Typical Host and TerfenolD aterial E 1 E G 1 E G 3 E G 1 E 1 (lb / in 3 ) x (1 / F) y (1 / F) Typical Host TerfenolD
6 34 The Open echanics Journal, 007, Volume 1 C.C. Hong at time t =0.003s, 1s, s,, 9s, for a / b = 1, threelayer (0 m /90 /0 ) laminate as shown in Fig. (). vibration of sinusoidal temperature only ( T 0, T 1 = 1.0 F, p 1 = p = q ) at time 6sec, m = n = 1 mode shape, with k c c( = 10 8, aspect ratio a / b.5, 1.0, and.0, sidetothickness ratio a / h * =100, 50, 0, 10 and 5. Fig. (3) shows that W (a /,b /) in the grid point = 9 9, and of GDQ method for the upper surface magnetostrictive layer of the threelayer (0 m /90 /0 ) We find that W (a /,b /) is in good convergence in grid point = Fig. (a). The lowest frequency of threelayer (0 m /90 /0 ) laminate. Fig. (3a). Convergence for a / b.5, threelayer (0 m /90 /0 ) Fig. (b). The lowest vibration frequency 11 of threelayer (0 m /90 /0 ) laminate. The magnetostrictive coupling moduli is e 31 = e 3 = E m d m and E m = 6.5GPa, d m = ma 1 for TerfenolD. We use the following coordinates for the grid points: x i.5[1 cos( i 1 )]a,i = 1,,..., 1 y j.5[1 cos( j 1 )]b, j = 1,,..., (7) 1 Firstly, we make the studies of dynamic convergence of center deflection amplitude W (a /,b /) in the thermal Fig. (3b). Convergence for a / b = 1.0, threelayer (0 m /90 /0 ) Fig. (4) shows that W (a /,b /) in the grid point = 9 9, and of GDQ method for the upper and lower surface magnetostrictive layers of the ten
7 Thermal Vibration of agnetostrictive aterial in Laminated Plates by the GDQ ethod The Open echanics Journal, 007, Volume 1 35 layer (0 m /90 /0 /90 /0 ) s symmetrical The superscript of m denotes magnetostrictive layer, the subscript s denotes the symmetric of laminate. We find that the 1313 grid point have the convergence result and used further in the GDQ analyses of time responses for deflection and stress. We used the k c c( values to control and reduce the amplitude of center deflection W (a /,b /). Fig. (4c). Convergence for a / b =.0, (0 m /90 /0 /90 /0 ) s Figs. (5,6) show that the k c c( values with time for thick laminated plate ah * = 10 and thin laminated ah * = 100, in threelayer and tenlayer magnetostrictive layer of laminate, respectively. Fig. (3c). Convergence for a / b =.0, threelayer (0 m /90 /0 ) Fig. (5). k c c( values vs time t (sec) in (0 m /90 /0 ) laminated plate. Fig. (4a). Convergence for a / b.5, (0 m /90 /0 /90 /0 ) s Fig. (6). k c c( values vs time t (sec) in (0 m /90 /0 /90 /0 ) s Fig. (4b). Convergence for a / b = 1.0, (0 m /90 /0 /90 /0 ) s Fig. (7) shows that time response of the nondimensional transverse center deflection amplitude W (a /,b /) with and without k c c( values at ah * = 10 and ah * = 100, respectively, for threelayer (0 m /90 /0 ) laminated plate, ab= 1, = 13 13, m = 1, n = 1, q, T 1 = 1, p 1 = p under shear effect. We find that the values of the center deflection amplitude W (a /,b /) with k c c(
8 36 The Open echanics Journal, 007, Volume 1 C.C. Hong are smaller than the values of W (a /,b /) without k c c(. The amplitude W (a /,b /) can be controlled and adjusted to a desired smaller value by using a suitable k c c( value, especially at time 0.003s, 1.s, 5.9s and 8.8s in Fig. (7a) with thick plate ah * = 10 of the GDQ method. But in Fig. (7b) with thin plate ah * = 100 of the GDQ method, the variations are small, except at time 0.003s and 0.8s. Fig. (9) shows that time response of the nondimensional transverse center deflection amplitude W (a /,b /) with and without k c c( values at ah * = 10 and ah * = 100, respectively, for tenlayer (0 m /90 /0 /90 /0 ) s symmetric laminate, ab= 1, = 13 13, m = 1, n = 1, q, T 1 = 1, p 1 = p under shear effect. We find that the values of the center deflection amplitude W (a /,b /) with k c c( are smaller than the values of W (a /,b /) without k c c(. The amplitude W (a /,b /) can be controlled and adjusted to a desired smaller value by using a suitable k c c( value, especially at time 0.003s,.6s, 5.s and 7.8s in Fig. (9a) with thick plate ah * = 10 of the GDQ method. But in Fig. (9b) with thin plate ah * = 100 of the GDQ method, the variations are small, except at time 5.1s. Fig. (7a). Thick, threelayer laminated magnetostrictive plate. Fig. (8a). Thick, threelayer laminated magnetostrictive plate. Fig. (7b). Thin, threelayer laminated magnetostrictive plate. Fig. (8). shows that time response of the dominated nondimensional stress x = x h * / ( x T 1 ae ) at center position of lower surface Z.5h * with and without k c c( values as the analyses of deflection W (a /,b /) at ah * = 10 and ah * = 100, respectively, for threelayer (0 m /90 /0 ) laminate, ab= 1, = 13 13, m = 1, n = 1, q, T 1 = 1, p 1 = p under shear effect. We find that the amplitudes of the stress x do not become the smaller values under the controlled W (a /,b /) condition in Fig. (8a) with plate ah * = 10 of GDQ method. But in Fig. (8b) with thin plate ah * = 100 of the GDQ method, the variations are small. Fig. (8b). Thin, threelayer laminated magnetostrictive plate. Fig. (9a). Thick, tenlayer laminated magnetostrictive plate.
9 Thermal Vibration of agnetostrictive aterial in Laminated Plates by the GDQ ethod The Open echanics Journal, 007, Volume 1 37 Fig. (9b). Thin, tenlayer laminated magnetostrictive plate. Fig. (10) shows that time response of the dominated nondimensional stress y = y h * / ( x TaE ) at center position of lower surface Z.5h * with and without k c c( values as the analyses of deflection W (a /,b /) at ah * = 10 and ah * = 100, respectively, for tenlayer (0 m /90 /0 /90 /0 ) s symmetric laminate, ab= 1, = 13 13, m = 1, n = 1, q, T 1 = 1, p 1 = p under shear effect. We find that the amplitudes of the stress y are almost the same value in controlled and uncontrolled W (a /,b /) condition of thick plate ah * = 10 and thin plate ah * = 100 by the GDQ method. Fig. (10a). thick, tenlayer laminated magnetostrictive plate. Fig. (10b). Thin, tenlayer laminated magnetostrictive plate. 4. COCLUSIOS (a) The numerical GDQ calculations provides an efficient method to compute the deflection and stress in the cross ply laminated plate with magnetostrictive layer subjected to thermal vibration of sinusoidal temperature including shear deformation. (b) The amplitude of transverse center deflection W (a /,b /) can be controlled to a smaller desirable value with suitable velocity feedback gain k c c( value, especially in the thick plate ah * = 10 by using the GDQ method. (c) The amplitude of the dominated stress x is not become a smaller values under the corresponding controlled W (a /,b /) condition for threelayer (0 m /90 /0 ) laminate in the thick plate ah * = 10 of the GDQ method. But the amplitudes of the dominated stress y are almost the same value in the corresponding controlled and uncontrolled W (a /,b /) condition for tenlayer (0 m /90 /0 /90 /0 ) s symmetric laminate of the GDQ method. (d) The GDQ method provide the less grid points ( = 13 13) and less computational time to get the numerical solutions of deflection and stress for magnetostrictive layer laminates. ACKOWLEDGEETS The completion of this study was made possible by a grant SC951E from ational Science Council, Taiwan, ROC. REFERECES [1] Pradhan SC. Vibration suppression of FG shells using embedded magnetostrictive layers. Int J Solids Struct 005; 4: [] Wojciechowski S. ew trends in the development of mechanical engineering materials. J ater Process Tech 000; 106: [3] Lee SJ, Reddy J. onlinear response of laminated composite plates under thermomechanical loading. Int J onlinear ech 005; 40: [4] Ramirez F, Heyliger PR, Pan E. Free vibration response of twodimensional magnetoelectroelastic laminated plates. J Sound Vibrat 006; 9: [5] Buchanan GR. Layered versus multiphase magnetoelectroelastic composites. Composites Part B. Eng 004; 35: [6] Hong CC, Jane KC, Shear deformation in thermal bending analysis of laminated plates by the GDQ method. ech Res Commun 003; 30: [7] Hong CC, Liao HW, Lee LT, Ke JB, Jane KC, Thermally induced vibration of a thermal sleeve with the GDQ method. Int J ech Sci 005; 47: [8] Hong CC, Liao HW, Lee LT, Ke JB, Jane KC, Thermally induced vibration of a thermal sleeve with the GDQ method. Int J ech Sci 005; 47: [9] Hong CC, Liao HW, Jane KC. umerical analyses of piezoelectric materials with the GDQ method. Comput at Sci 006; 35: [10] Bert CW, Jang SK, Striz AG. onlinear bending analysis of orthotropic rectangular plates by the method of differential quadrature. Comput ech 1989; 5: [11] Shu C, Du H. Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analyses of beams and plates. Int J Solds Struct 1997; 34: [1] Whitney J. Structural analysis of laminated anisotropic plates. Lancaster, PA, USA: Technomic Publishing Company, Inc., Received: July 7, 007 Revised: August 3, 007 Accepted: August 9, 007
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