Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations tài liệu, giáo án, bài giảng ,...

Journal of Thermal Stresses ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20 Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations Nguyen Dinh Duc, Dao Huy Bich & Pham Hong Cong To cite this article: Nguyen Dinh Duc, Dao Huy Bich & Pham Hong Cong (2016) Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations, Journal of Thermal Stresses, 39:3, 278-297 To link to this article: http://dx.doi.org/10.1080/01495739.2015.1125194 Published online: 15 Mar 2016 Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=uths20 Download by: [ECU Libraries] Date: 16 March 2016, At: 04:20 JOURNAL OF THERMAL STRESSES 2016, VOL 39, NO 3, 278–297 http://dx.doi.org/10.1080/01495739.2015.1125194 Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations Nguyen Dinh Duca , Dao Huy Bicha , and Pham Hong Conga,b Downloaded by [ECU Libraries] at 04:20 16 March 2016 a Advanced Materials and Structures Laboratory, University of Engineering and Technology, Vietnam National University, Hanoi, Vietnam; b Center for Informatics and Computing, Vietnam Academy of Sciences and Technology, Hanoi, Vietnam ABSTRACT ARTICLE HISTORY This paper investigates the nonlinear dynamic response of thick functionally graded materials (FGM) plates using the third-order shear deformation plate theory and stress function The FGM plate is assumed to rest on elastic foundations and subjected to thermal and damping loads Numerical results for dynamic response of the FGM plate are obtained by Runge–Kutta method The results show the influences of geometrical parameters, the material properties, the elastic foundations, and thermal loads on the nonlinear dynamic response of FGM plates Received April 2015 Accepted 17 May 2015 KEYWORDS FGM plate; nonlinear dynamic response; stress function; the third-order shear deformation plate theory; thermal loads Introduction Functionally graded materials (FGMs) are microscopically inhomogeneous composite materials in which the mechanical and thermal properties vary smoothly and continuously from one surface to the other Typically, these materials are made from a mixture of metal and ceramic or a combination of different metals by gradually varying the volume fraction of the constituent metals The properties of FGM plates and shells are assumed to vary through the thickness of the structures Due to the high temperature resistance, FGMs have many practical applications such as reactor vessels, aircrafts, space vehicles, defense industries, and other engineering structures As a result, in recent years, many investigations have been carried out on the dynamic and vibration of FGM plates and shells The vibration of functionally graded cylindrical shells has been investigated by Lam and Reddy [1] Lam and Hua considered the influence of boundary conditions on the frequency characteristics of a rotating truncated circular conical shell [2] In [3], Pradhan et al studied vibration characteristics of FGM cylindrical shells under various boundary conditions Yang and Shen [4] published the nonlinear analysis of FGM plates under transverse and in-plane loads Zhao et al [5] studied the free vibration of two-side simply supported laminated cylindrical panel through the mesh-free kp-Ritz method About vibration of FGM plates, Vel and Batra [6] gave a three-dimensional exact solution for the vibration of FGM rectangular plates Ferreira et al [7] received natural frequencies of FGM plates by meshless method Woo et al [8] investigated the nonlinear free vibration behavior of functionally graded plates Kadoli and Ganesan [9] studied the buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition Wu et al [10] published their results of nonlinear static and dynamic analysis of functionally graded plates Natural frequencies and buckling stresses of FGM plates were analyzed by Matsunaga [11] using two-dimensional higher-order deformation theory Shariyat obtained the dynamic thermal buckling of suddenly heated temperaturedependent FGM cylindrical shells under combined axial compression [12] and external pressure and dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical with temperature-dependent material properties under thermoelectromechanical loads [13] Zhao et al [14] studied free vibration CONTACT Nguyen Dinh Duc ducnd@vnu.edu.vn Advanced Materials and Structures Laboratory, University of Engineering and Technology, Vietnam National University, 144 - Xuan Thuy Street, Hanoi, Vietnam © 2016 Taylor & Francis Downloaded by [ECU Libraries] at 04:20 16 March 2016 JOURNAL OF THERMAL STRESSES 279 analysis of functionally graded plates using the element-free kp-Ritz method Bich and Nguyen [15] investigated nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnel equations using classical shell theory Recently, Duc [16] published a valuable book “Nonlinear Static and Dynamic Stability of Functionally Graded Plates and Shells”, in which the results about nonlinear vibration of shear deformable FGM plates and shells are presented Mohammad and Singh [17] studied static response and free vibration of unsymmetrical FGM plates using first-order shear deformation theory with finite element method Duc and Cong [18] studied nonlinear dynamic response of imperfect symmetric thin S-FGM plate with metal–ceramic–metal layers on elastic foundation [18] Duc [19] has recently investigated the nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation Up to date, dynamic analysis of FGM plates and shells using the higher-order shear deformation theory has received great attention of the researchers Ungbhakorn and Wattanasakulpong [20] studied thermoelastic vibration analysis of third-order shear deformable functionally graded plates with distributed patch mass under thermal environment, the solutions are obtained by energy method Talha and Singh [21] studied static response and free vibration analysis of FGM plates using higher-order shear deformation theory with finite element models Huang and Shen [22] studied nonlinear vibration and dynamic response of FGM plates in thermal environments but volume fraction follows a simple power law for unsymmetrical FGM plate Swaminathan and Ragounadin [23] studied analytical solutions using a higher-order refined theory for the static analysis of antisymmetric angle-ply composite and sanwich plates Qian et al [24] studied static and dynamic deformations of thick functionally graded elastic plates using higher-order shear and normal deformable plate theory and meshless local Petrov– Galerkin method In all the metioned publications [20–24], the authors used finite element method and the displacement functions With some other publications [15, 18, 19], the authors have used the stress function and Volmir’s assumption with analytical approach to investigate the vibration and dynamic responses of FGM plates and shells applying classical plate and shell theories for thin structures For thick structures, we have to use the third-order shear deformation theory, but then Volmir’s assumption is useless There has not been any publication using the third-order shear deformation plate theory and stress function to study vibration and dynamic response for FGM plates Therefore, this paper investigates the nonlinear thermal dynamic response and nonlinear vibration of thick FGM plates using the third-order shear deformation plate theory and stress function (in two cases: uniform temperature rises and nonlinear temperature through the thickness of the plate) Numerical results for dynamic response of the FGM plate are obtained by Runge–Kutta method Theoretical formulation Consider a rectangular FGM plate on elastic foundations The plate is referred to a Cartesian coordinate system x, y, z, where xy is the midplane of the plate and z is the thickness coordinator, −h/2 ≤ z ≤ h/2 The length, width, and total thickness of the plate are a, b, and h, respectively (Figure 1) Figure Geometry of the FGM plate on elastic foundations 280 N D DUC ET AL By applying a simple power law distribution (P-FGM), the volume fractions of metal and ceramic, Vm and Vc , are assumed as [25] Vc (z) = N 2z + h 2h , Vm (z) = − Vc (z) (1) where the volume fraction index N is a non-negative number that defines the material distribution and can be chosen to optimize the structural response The effective properties Peff of the FGMs are determined by the modified mixed rules as follows [25]: Peff (z) = Pr Vc (z) + Pr Vm (z) c (2) m Downloaded by [ECU Libraries] at 04:20 16 March 2016 in which Pr is a symbol for the specific nature of the material such as elastic modulus E, thermal expansion coefficient α or density ρ, thermal conduction K, and subscripts m and c stand for the metal and ceramic constituents, respectively From Eq (1) and Eq (2), the effective properties of the FGM plate can be written as follows: [E(z), α(z), ρ(z), K (z)] = [Em , αm , ρm , Km ] + [Ecm , αcm , ρcm , Kcm ] 2z + h 2h N (3) where Ecm = Ec − Em , αcm = αc − αm , ρcm = ρc − ρm , Kcm = Kc − Km (4) and the Poisson ratio v(z) is assumed to be constant v(z) = v Suppose that the FGM plate is subjected to a transverse load and compressive axial loads In this study, the Reddy’s third-order shear deformation theory is used to obtain the motion, compatibility equations At the same time, the stress function is applied for determining the nonlinear dynamic response and vibration of the FGM plate The strain–displacement relations taking into account the Von Karman nonlinear terms and the third-order shear deformation plate theory are [26–28]  0  1  3   εx κx κx εx γxz κxz  0  1  3 γxz  εy  =  εy  + z  κy  + z  κy  , = +z (5)       γyz γyz0 κyz γxy γ κ κ xy xy xy with  εx0   0  εy  =   γxy  κx3   3  κy  =   κxy γxz γyz0 =  ∂w ∂x    ∂v ∂w   , +   ∂y ∂y ∂u ∂v ∂w ∂w ∂y + ∂x + ∂x ∂y  ∂u ∂x +  κx1   1   κy  =      ∂φx κxy ∂y +  ∂φx ∂2w ∂x + ∂x2   ∂φy   −c1  + ∂∂yw2  ∂y   ∂φy ∂φx ∂2w + + ∂y ∂x ∂x∂y ∂w φx + ∂x κxz , = −3c1 κyz φy + ∂w ∂y  ∂φx ∂x ∂φy ∂y  φx + φy + ∂w ∂x ∂w ∂y  ∂φy ∂x   ,  (6) in which c1 = 4/3h2 , εx , εy are normal strains, γxy is the in-plane shear strain, and γxz , γyz are the transverse shear deformations Also u, v, ω are the displacement components along the x, y, z directions, respectively, and φx , φy are the slope rotations in the (x, z) and (y, z) planes, respectively JOURNAL OF THERMAL STRESSES 281 The strains are related in the compatibility equation [27, 28] ∂ εy0 ∂ γxy ∂ εx0 + − = ∂y2 ∂x2 ∂x∂y ∂ 2w ∂x∂y − ∂ 2w ∂ 2w ∂x2 ∂y2 (7) Hooke’s law for a plate which included the effect of temperature is defined as follows: σx , σy = E [(εx , εy ) + ν εy , εx − (1 + ν)α T(1, 1)] − ν2 (8) E = γxy , γxz , γyz 2(1 + ν) σxy , σxz , σyz where T is temperature rise from stress-free initial state or temperature difference between two surfaces of the FGM plate The force and moment resultants of the plate can be expressed in terms of stress components across the thickness of the plate as Downloaded by [ECU Libraries] at 04:20 16 March 2016 h/2 (Ni , Mi , Pi ) = σi 1, z, z3 dz; i = x, y, xy −h/2 (9) h/2 (Qi , Ri ) = σj (1, z )dz; i = x, y; j = xz, yz −h/2 Inserting Eqs (3), (5), and (8) into Eq (9) gives the constitutive relations as  0 1  (E1 , E2 , E4 ) εx + νεy + (E2 , E3 , E5 ) kx + νky (Nx , Mx , Px ) = − ν2 + (E4 , E5 , E7 ) k3x + νk3y − (1 + ν) ( a , b ,  0 1  (E1 , E2 , E4 ) εy + νεx + (E2 , E3 , E5 ) ky + νkx Ny , My , Py = − ν2 + (E4 , E5 , E7 ) k3y + νk3x − (1 + ν) ( a , b , c)  c)  + (E2 , E3 , E5 ) k1xy + (E4 , E5 , E7 ) k3xy (E1 , E2 , E4 ) γxy 2(1 + ν) + (E3 , E5 ) k2xz (E1 , E3 ) γxz (Qx , Rx ) = 2(1 + ν) Qy , Ry = (E1 , E3 ) γyz0 + (E3 , E5 ) k2yz 2(1 + ν) Nxy , Mxy , Pxy =   (10) where h/2 (E1 , E2 , E3 , E4 , E5 , E7 ) = 1, z, z2 , z3 , z4 , z6 E(z)dz −h/2 h/2 ( a, b, c) (11) (1, z, z )E(z)α(z) T(z)dz = −h/2 and specific expressions of coefficients Ei (i = 1/5, 7) are given in Appendix For using late, the reverse relations are obtained from Eq (10) 1 Nx − νNy − E2 k1x − E4 k3x + a , εy0 = Ny − νNx − E2 k1y − E4 k3y + E1 E1 = 2(1 + ν)Nxy − E2 k1xy − E4 k3xy E1 εx0 = γxy a (12) 282 N D DUC ET AL According to higher-order shear deformation theory, the equations of motion are [26] ∂Nxy ∂Nx ∂ 2u ∂ φx ∂ 3w + = I0 + J1 − c1 I3 ∂x ∂y ∂t ∂t ∂x∂t 2 ∂Nxy ∂Ny ∂ φy ∂ v ∂ 3w + = I0 + J1 − c1 I3 ∂x ∂y ∂t ∂t ∂y∂t ∂Qy ∂Qx + − c2 ∂x ∂y ∂ 2w ∂ 2w + ∂x2 ∂y − k1 w + k2 Downloaded by [ECU Libraries] at 04:20 16 March 2016 + c1 I3 ∂Ry ∂Rx + ∂x ∂y + Nx + q = I0 ∂ 3v ∂ 3u + ∂x∂t ∂y∂t ∂ 2w ∂ 2w ∂ 2w + 2N + N + c1 xy y ∂x2 ∂x∂y ∂y2 ∂ 2w ∂w + 2εI0 − c12 I6 ∂t ∂t ∂ Pxy ∂ Py ∂ Px + + ∂x2 ∂x∂y ∂y2 ∂ 4w ∂ 4w + ∂x2 ∂t ∂y2 ∂t ∂ φy ∂ φx + ∂x∂t ∂y∂t + J4 ∂ 2u ∂ φx ∂ 3w + K2 − c1 J4 ∂t ∂t ∂x∂t ∂ φy ∂ 2v ∂ 3w − Qy − c2 Ry = J1 + K2 − c1 J4 ∂t ∂t ∂y∂t ∂Mxy ∂Pxy ∂Px ∂Mx + − c1 + ∂x ∂y ∂x ∂y ∂Mxy ∂My ∂Py ∂Pxy + − c1 + ∂x ∂y ∂x ∂y − Qx − c2 Rx = J1 (13) where h/2 ρ (z) zi dz, Ii = (i = 0, 1, 2, 3, 4, 6) −h/2 K2 = I2 − 2c1 I4 + c12 I6 , Ji = Ii − c1 Ii+2 , c2 = 3c1 (14) and specific expressions of coefficients Ii (i = ÷ 4, 6) are given in Appendix, and k1 is Winkler foundation modulus, k2 is the shear layer foundation stiffness of Pasternak model, q is an external pressure uniformly distributed on the surface of the plate, ε is damping coefficient The stress function f (x, y, t) is introduced as Nx = ∂ 2f , ∂y2 Ny = ∂ 2f , ∂x2 Nxy = − ∂ 2f ∂x∂y (15) Replacing Eq (15) into the first two Eqs (13) yields ∂ 2u J1 ∂ φx c1 I3 ∂ w =− + 2 ∂t I0 ∂t I0 ∂x∂t (16a) J1 ∂ φy c1 I3 ∂ w ∂ 2v = − + ∂t I0 ∂t I0 ∂y∂t (16b) By substituting Eq (16a) and Eq (16b) into the last three Eqs (13), we obtain the last three Eqs (13) rewritten as follows: ∂Qy ∂Qx + − c2 ∂x ∂y − k1 w + k2 + J4 c1 − ∂Ry ∂Rx + ∂x ∂y ∂ 2w ∂ 2w ∂ 2w + 2N + N + c1 y xy ∂x2 ∂x∂y ∂y2 ∂ 2w ∂w + + 2εI0 ∂t ∂t ∂ φx J1 I3 c1 ∂ φy + J c − ∂x∂t I0 ∂y∂t ∂ 2w ∂ 2w + ∂x2 ∂y J1 I3 c1 I0 + Nx + q = I0 c12 I32 − c12 I6 I0 ∂ Pxy ∂ Py ∂ Px + + ∂x2 ∂x∂y ∂y2 ∂ 4w ∂ 4w + ∂x2 ∂t ∂y2 ∂t (17a) JOURNAL OF THERMAL STRESSES 283 ∂Mxy ∂Mx + − c1 ∂x ∂y ∂Pxy ∂Px + ∂x ∂y − Qx − c2 Rx = K2 − J12 I0 ∂ φx + ∂t c1 I3 J1 − c1 J4 I0 ∂ 3w ∂x∂t (17b) ∂Mxy ∂My + − c1 ∂x ∂y ∂Pxy ∂Py + ∂x ∂y − Qy − c2 Ry = K2 − J12 I0 ∂ φy + ∂t c1 I3 J1 − c1 J4 I0 ∂ 3w ∂y∂t (17c) By setting Eq (12) and Eq (15) into the deformation compatibility Eq (7), we obtain E1 ∂ 4f ∂ 4f ∂ 4f +2 2 + 4 ∂x ∂x ∂y ∂y = ∂ 2w ∂x∂y − ∂ 2w ∂ 2w ∂x2 ∂y2 (18) By substituting Eq (6) into Eq (10) and then into Eq (17), the system of motion Eq (17) is rewritten as follows: Downloaded by [ECU Libraries] at 04:20 16 March 2016 L11 (w) + L12 (φx ) + L13 φy + P w, f + q = I0 ∂ 2w ∂w + + 2εI0 ∂t ∂t + J4 c1 − J1 I3 c1 I0 c12 I32 − c12 I6 I0 ∂ 4w ∂ 4w + 2 2 ∂x ∂t ∂y ∂t 3 ∂ φx J1 I3 c1 ∂ φy + J4 c1 − ∂x∂t I0 ∂y∂t (19a) L21 (w) + L22 (φx ) + L23 φy = K2 − J12 I0 ∂ φx + ∂t c1 I3 J1 − c1 J4 I0 ∂ 3w ∂x∂t (19b) L31 (w) + L32 (φx ) + L33 φy = K2 − J12 I0 ∂ φy + ∂t c1 I3 J1 − c1 J4 I0 ∂ 3w ∂y∂t (19c) and the linear operators Lij i = − 3, j = − and the nonlinear operator P are given in Appendix The system of four Eqs (18) and (19) combined with boundary conditions and initial conditions could be used for nonlinear dynamical analysis of FGM plates using the higher-order shear deformation theory in the next section Nonlinear dynamic analysis Four edges of the plate are simply supported and immovable (IM) In this case, boundary conditions are [27, 28] w = u = φy = Mx = Px = 0, Nx = Nx0 at x = 0, a w = v = φx = My = Px = 0, Ny = Ny0 at y = 0, b (20) in which Nx0 , Ny0 are the forces are the jets when the edges are IM in the plane of the plate The approximate solutions of the system of Eq (18) and Eq (19) satisfying the boundary condition [Eq (20)] can be written as [26]: w x, y, t = W (t) sin αx sin βy φx x, y, t = φy x, y, t = x (t) cos αx sin βy (21) y (t) sin αxcosβy nπ where α = mπ a , β = b and m, n = 1, 2, are the natural numbers of half waves in the corresponding direction x, y, and W, x , y are the amplitudes which are functions dependent on time 284 N D DUC ET AL By substituting displacement functions [Eq (21)] into compatibility Eq (18), we define the stress function as 1 f x, y, t = A1 (t) cos 2αx + A2 (t) cos 2βy + Nx0 y2 + Ny0 x2 (22a) 2 in which E1 β 2 W , 32α A1 = E1 α 2 W 32β A2 = (22b) Downloaded by [ECU Libraries] at 04:20 16 March 2016 By replacing Eq (21) and Eq (22) into the equations of motion [Eq (19)] and then applying Galerkin method, we obtain 16 l11 − Nx0 α +Ny0 β W + l12 x + l13 y + n1 W + q mnπ ∂W mπ ∂ x nπ ∂ y ∂ 2W + ρ2 + ρ2 (23a) = n2 + 2εI0 ∂t ∂t a ∂t b ∂t l21 W + l22 x + l23 l31 W + l32 x + l33 y = ρ1 mπ ∂ W ∂2 x + ρ ∂t a ∂t (23b) y = ρ1 ∂2 y nπ ∂ W + ρ ∂t b ∂t (23c) J2 where m, n are odd numbers, ρ1 = K2 − I10 , ρ2 = c1II03 J1 − c1 J4 Specific expressions of coefficients lij i = ÷ 3, j = ÷ , and n1 , n2 are given in Appendix Equation (23) is used to analyze the nonlinear dynamic response of thick FGM plates on the elastic foundation applying the higher-order shear deformation theory in thermal environment Consider the FGM plate with all edges which are simply supported and IM (all edges IM) under thermal load The condition expressing the immovability on the edges, u = (on x = 0, a) and v = (on y = 0, b), is satisfied in an average sense as [27, 28] b a ∂u dxdy = 0, ∂x a b ∂v dxdy = ∂y (24) From Eq (6) and Eq (12) of which mention relations [Eq (15)], we obtain the following expressions E2 c1 E ∂u = f,yy − vf,xx − φx,x + φx,x + w,xx − ∂x E1 E1 E1 ∂w ∂x ∂v E2 c1 E = f,xx − vf,yy − φy,y + φy,y + w,yy − ∂y E1 E1 E1 ∂w ∂y + + a E1 (25) a E1 By placing Eqs (21) and (22) into Eq (25) then substituting the obtained result into Eq (24), we have Nx0 = − a − mnπ 1−v E1 + − v2 − v2 (E2 − c1 E4 ) α + vβ y − c1 E4 α + vβ W α + vβ W a − (E2 − c1 E4 ) vα − v mnπ − v2 E1 + vα + β W − v2 Ny0 = − x (26a) x +β y − c1 E4 vα + β W (26b) JOURNAL OF THERMAL STRESSES 285 By replacing Eqs (26a) and (26b) into the equations of motion [Eq (23)], we have l11 + + mπ a nπ b + a x + l13 y + l14 xW + l15 yW + l16 W + l17 W 16 ∂ 2W ∂W mπ ∂ x nπ ∂ y q = n + 2εI + ρ + ρ 2 mnπ ∂t ∂t a ∂t b ∂t l21 W + l22 x + l23 y = ρ1 (27a) mπ ∂ W ∂2 x + ρ ∂t a ∂t (27b) ∂2 y nπ ∂ W + ρ (27c) ∂t b ∂t and specific expressions of coefficients l1i (i = − 7) are given in Appendix The effects of temperature in Eq (27) appear in a a are identified upon the uniform increasing temperature or nonlinear temperature transferring through the thickness of the plate Taking linear parts of the set of Eq (27) and placing q = 0, the natural frequencies (ω) of the plate can be determined directly by solving determinant l31 W + l32 Downloaded by [ECU Libraries] at 04:20 16 March 2016 W + l12 1−v l11 + mπ a x + l33 nπ a b 1−v l21 + ρ2 mπ a ω l31 + ρ2 nπ b ω + y = ρ1 l + ρ nπ ω2 + n2 ω2 l12 + ρ2 mπ 13 b a ω l22 + ρ1 ω2 l23 l32 l33 + ρ1 ω2 =0 (28) Solving Eq (28) yields three frequencies of the FGM plates, and the smallest one is being considered Uniform temperature rises The plate is placed in the environment which temperature is steadily increased from the beginning value Ti to the ending value Tf , the temperature difference T = Tf − Ti is a constant Where a is determined from Eq (11) and a = P T, P = h Em αm + Em αcm + Ecm αm Ecm αcm + N+1 2N + (29) Through the thickness temperature gradient In many applications of FGM plates, the temperature distribution in the plate is uneven Usually much higher temperature in rich ceramic surface than the surface of the rich metal sheet of FGM plate In this case, the temperature through thickness of plate is described by the Fourier temperature equation as follows [28]: d dT K (z) = 0, dz dz T (z = h/2) = Tc , T (z = −h/2) = Tm (30) in which Tc and Tm is corresponding temperature on the ceramic and metal surfaces, respectively The solution of Eq (30) can be obtained by polynomial series Taking the first seven terms of the series, the solution for temperature distribution across the thickness of the plate becomes [28] κ T (z) = Tm + T −κ N Kcm /Km p=0 pN+1 (−Kcm /Km )p p=0 pN+1 p (31) in which κ = (2z + h)/2h and T = Tc − Tm are identified as the temperature differences between the ceramic and metal surfaces of the plate 286 N D DUC ET AL By switching Eq (31) into Eq (11), we obtain a a (32) =H T and specific expression of coefficients H are given in Appendix Numerical results and discussion Consider a FGM plates acted on by an uniformly distributed transverse load q = Q sin t (Q is the amplitude of uniformly excited load, is the frequency of the load) The fourth-order Runge–Kutta method is used to solve Eq (27) in which a is identified through Eq (29) in the case of the uniform increasing temperature and in the case of the nonlinear temperature transferring through the thickness of the structure, a is taken from Eq (32) To illustrate the present approach, we consider a ceramic– metal FGM plate that consists of aluminum (metal) and alumina (ceramic) with the following properties [19, 27, 28] Downloaded by [ECU Libraries] at 04:20 16 March 2016 Ec = 380 × 109 N/m2 , Em = 70 × 10 N/m , v = 0.3 ρc = 3800 kg/m3 , αc = 7.4 × 10−6 ◦ C−1 , −6 ◦ −1 ρm = 2702 kg/m , αm = 23 × 10 C , Kc = 10.4 W/mK Km = 204 W/mK and the ratio of geometric parameters of the selected plate a/b = 1, a/h = 20 Table presents a comparison of the fundamental frequency parameter established in this paper with the result of Ungbhakorn and Wattanasakulpong [20] They used the energy function and choosing results under the type of moving position function for study From Table 1, it can be seen that the present values are not significantly different from the result in [20] Table showing the effects of the group numbers of (m, n) and the volume coefficient ratio N on natural frequencies We can see that when the volume coefficient ratio N is increased while the natural frequencies decrease, and the value of the (m, n) is increased while the natural frequencies increases Table also shows that the lowest natural frequency corresponds mode (m, n) = (1, 1) The effects of uniform increasing temperature to natural frequencies are shown in Table and Figure We can see that the increasing of temperature makes the decreasing of frequencies and with the same temperature variation, the uniform increasing temperature has the smaller natural frequencies than the nonlinear temperature transferring through the thickness of the plate Table also shows the effect of the elastic foundations: when κ1 , κ2 increase, they lead to the increase of natural frequencies Figures and illustrate the effect of geometric factors of the FGM plate on nonlinear dynamic response in the case of all FM edges with N = 1, (m, n) = (1, 1), ε = 0.1, T = and without elastic Table Comparison of fundamental frequency parameter γ = ωh ρE c for Al/Al2 O3 square plates without elastic foundations and c [a/b = 1, (m, n) = (1, 1), and T = 0] h/a = 0.1 h/a = 0.2 Source N = 0.5 N = 1.0 N = 10.0 N = 0.5 N = 1.0 N = 10.0 Present [20] 0.0492 0.0490 0.0443 0.0442 0.0364 0.0364 0.1819 0.1807 0.1640 0.1631 0.1306 0.1301 Table Effect of power law index N on natural frequencies (s−1 ) of FGM plates with a/b = 1, a/h = 20, k1 = 0, k2 = 0, and T = N ω1 (m = 1, n = 1) ω2 (m = 1, n = 3) ω3 (m = 1, n = 5) ω4 (m = 3, n = 5) ω5 (m = 5, n = 5) 0.5 2542.00 2161.47 1941.73 1631.57 1293.84 12414.12 10576.59 9509.09 7966.74 6318.62 30917.59 26429.64 23795.80 19836.97 15736.64 39638.47 33933.64 30571.07 25430.42 20175.45 56189.06 48224.60 43494.50 36044.94 28599.48 JOURNAL OF THERMAL STRESSES 287 Table Effect of temperature field and elastic foundation on natural frequencies (s−1 ) of FGM plates with a/b = 1, (m, n) = (1, 1), a/h = 20, and N = (k1 ; k2 ) (GPa/m; GPa.m) T (◦ C) (0; 0) 50 105 50 105 50 105 (0.25; 0) Downloaded by [ECU Libraries] at 04:20 16 March 2016 (0.25; 0.02) Uniform temperature rise 1941.73 1570.25 1015.19 2303.00 1999.79 1601.28 2779.46 2533.92 2232.86 Natural frequency (s−1 ) Through the thickness temperature gradient 1941.73 1808.09 1648.62 2303.00 2191.50 2061.90 2779.46 2687.79 2583.21 Figure Variation of natural frequency ω(s−1 ) with volume fraction index N and uniform temperature rise Figure Effect of ratio a/b on nonlinear dynamic response of the FGM plate Downloaded by [ECU Libraries] at 04:20 16 March 2016 288 N D DUC ET AL Figure Effect of ratio a/h on nonlinear dynamic response of the FGM plate foundation From Figure 3, we see that the amplitude of the FGM plate increases when increasing the ratio a/b And Figure describes the effect of the ratio a/h on the nonlinear dynamic response of FGM plates The plate fluctuates stronger when the ratio a/h increases Figure shows the effect of the power law index N on the nonlinear dynamic response of FGM plates with a/b = 1, a/h = 20, T = 0, ε = 0.1 and without elastic foundation when the frequency of external force q is far away from the natural frequency of the FGM plate with N = 0, 1, It can be seen Figure Effect of power law index N on nonlinear dynamic response of the FGM plates JOURNAL OF THERMAL STRESSES 289 Downloaded by [ECU Libraries] at 04:20 16 March 2016 that the amplitude of the nonlinear dynamic response of FGM plate increases when increasing the power law index N Figures and show the effect of elastic foundations on the nonlinear dynamic response of the FGM plate with a/b = 1, a/h = 20, N = 1, T = Figure shows the effect of the Winkler foundation It is clear that the amplitude of the plate decreases when the module k1 of Winkler foundation increases The Pasternak foundation with parameter k2 also has a similar behavior in Figure The graphs in Figures and show the beneficial effects of elastic foundations on the nonlinear dynamic response of FGM Figure Effect of the linear Winkler foundation on nonlinear dynamic response of the FGM plate Figure Effect of the Pasternak foundation on nonlinear dynamic response of the FGM plate 290 N D DUC ET AL Downloaded by [ECU Libraries] at 04:20 16 March 2016 plates, namely the amplitude of the plate decreases when it is resting on elastic foundations, and the beneficial effect of the Pasternak foundation is better than the Winkler one Figure indicates the effect of excited force amplitude on nonlinear dynamic response in the case of Q = (1200, 2500, 4000 N/m2 ) and T = The FGM plate’s amplitude increases when the excited force amplitude increases Figures and 10, respectively, shows the effect of temperature on the nonlinear dynamic response of FGM plates in both cases the uniform temperature rises and the nonlinear temperature transferring Figure Nonlinear dynamic response of the FGM plate with different loads Figure Effect of uniform temperature rise on nonlinear dynamic response of the FGM plate Downloaded by [ECU Libraries] at 04:20 16 March 2016 JOURNAL OF THERMAL STRESSES 291 Figure 10 Effect of through the thickness temperature gradient on nonlinear dynamic response of the FGM plate Figure 11 Effect of temperature on nonlinear dynamic response of the FGM plate through the thickness of the plates Figure shows the effects of uniform temperatures rise with T = (0, 50, 105 ◦ C) on the nonlinear dynamic response of FGM plates The dynamic response amplitude increases when the temperature T increases Impact of nonlinear temperature transferring through the thickness of the plates is shown in Figure 10 In this case, the temperature at the metal-rich outer surface is kept in Tm = 27◦ C (room temperature) and temperature transferring from the ceramic surface through the thickness of plate Tc = (27; 77; 132◦ C) The figure shows the amplitude of FGM plates increased with increasing temperature values in the ceramic-rich surface 292 N D DUC ET AL Downloaded by [ECU Libraries] at 04:20 16 March 2016 Comparison of nonlinear dynamic response of FGM plates in case of the uniform temperature rises and nonlinear temperature transferring through the thickness of the plate is shown in Figure 11 The temperature variations T = 100◦ C From the figure, we see that the amplitude in the case of the uniform increasing temperature is lager than one in the case of the nonlinear temperature transferring through the thickness of the FGM plate Figure 12 represents the nonlinear dynamic response of the plates in uniform increasing temperature ( T = 50◦ C) when the frequency of the external force is near to the natural frequency of the plate The nonlinear dynamic responses with volume ratio N = 1, natural frequency of the plate ω = 1570.25 (s−1 ), Figure 12 Nonlinear dynamic response of the thick FGM plate Figure 13 Effect of damping ε on nonlinear responses of the thick FGM plates JOURNAL OF THERMAL STRESSES 293 frequency of the external load = 1500 (s−1 ) We can easily see that the natural frequencies of the FGM plate are clearly dependent on the constituent volume fraction N, and when the frequencies of the external forces close to the natural frequencies of the plate, the interesting phenomenon is observed like the harmony beat phenomenon of a linear vibration Effect of damping on nonlinear responses is presented in Figure 13 with three values of damping coefficient (ε = 0.1, 30.0, 800.0) The damping influences very small to the amplitude–time curves of the nonlinear dynamic response of the thick FGM plates Downloaded by [ECU Libraries] at 04:20 16 March 2016 Concluding remarks The nonlinear thermal dynamic responses of FGM plates have been investigated in this paper We summarize the following main findings: • Motionally differential equations for analyzing nonlinear dynamic response of FGM plates on elastic foundations using the third-order shear deformation plate theory and stress function are obtained • The nonlinear thermal dynamic responses of FGM plates are analyzed in two cases: uniform temperature rises and temparature varies through the thickness of the plates • The effects of the elastic foundations on nonlinear thermal dynamic response of thick FGM plate are considered Interestingly, the elastic foundations impact positively on dynamic behavior of the plate, and the beneficial effect of the Pasternak foundation is better than the Winkler one • Damping lighly influences to the nonlinear response of the thick FGM plates • The volume ratio N and geometrical parameters (a/b, a/h ratios) effected extremely on the nonlinear thermal dynamic responses of the FGM plates Thus, it is obvious that vibration and thermal dynamic response of the FGM plates depend significantly on many factors volume ratio N, elastic foundations, temperature, and geometrical parameters of the plate Therefore, when we change these parameters, we can actively control the dynamic responses and vibration of the FGM plate Appendix Ecm Nh2 Ecm h ; E2 = ; N+1 2(N + 1)(N + 2) Em h 1 + Ecm h3 − 12 4(N + 1) (N + 2)(N + 3) Ecm h4 3 − + N + 4(N + 2) (N + 3)(N + 4) Ecm h5 1 Em h 12 + − + − 80 N + 16 2(N + 2) (N + 2)(N + 3) (N + 2)(N + 4)(N + 5)   30 − 32(N+2) + 16(N+2)(N+3) 64  Em h Ecm h7  15 90 −  + (N+2)(N+3)(N+4) + (N+2)(N+3)(N+4)(N+5)   448 N+1 360 − (N+2)(N+3)(N+4)(N+6)(N+7) E1 = Em h + E3 = E4 = E5 = E7 = 294 N D DUC ET AL I0 = ρm h + ρcm h ρcm Nh2 ; I1 = ; N+1 2(N + 1)(N + 2) I2 = ρm h3 + ρcm h3 − 12 4(N + 1) (N + 2)(N + 3) I3 = ρcm h4 3 − + N + 4(N + 2) (N + 3)(N + 4) ρcm h5 1 12 ρm h5 + − + − 80 N + 16 2(N + 2) (N + 2)(N + 3) (N + 2)(N + 4)(N + 5)   30 − 32(N+2) + 16(N+2)(N+3) 64  ρm h7 ρcm h7  15 90  −  I6 = + (N+2)(N+3)(N+4) + (N+2)(N+3)(N+4)(N+5)   448 N+1 360 − (N+2)(N+3)(N+4)(N+6)(N+7) Downloaded by [ECU Libraries] at 04:20 16 March 2016 I4 = L11 (w) = ∂ 2w ∂ 2w + [E1 − 3c1 E3 − c2 (E3 − 3c1 E5 )] 2(1 + ν) ∂x2 ∂y − L12 (φx ) = c12 − ν2 E7 − E42 E1 − k1 w + k2 ∂ 2w ∂ 2w + ∂x2 ∂y ∂φx c1 E2 E4 E42 + E − − c E − [E1 − 3c1 E3 − c2 (E3 − 3c1 E5 )] 2(1 + ν) ∂x − ν2 E1 E1 + c1 L13 φy ∂ 4w ∂ 4w ∂ 4w + 2 2+ 4 ∂x ∂x ∂y ∂y ν + 1+ν − v2 E5 − E2 E4 E1 − c1 ν + − ν2 1+ν E7 − E42 E1 ∂ φx ∂x∂y2 ∂ φy ∂y3 ∂φy c1 E2 E4 E42 + E − − c E − [E1 − 3c1 E3 − c2 (E3 − 3c1 E5 )] 2(1 + ν) ∂y − ν2 E1 E1 + c1 P x, f = v + 1+ν − ν2 E5 − E2 E4 E1 − c1 v + − ν2 1+ν E7 − E42 E1 ∂ φy ∂x2 ∂y ∂ 2f ∂ 2w ∂ 2f ∂ 2w ∂ 2f ∂ 2w − + ∂y2 ∂x2 ∂x∂y ∂x∂y ∂x2 ∂y2 L21 (w) = c1 − ν2 − L22 (φx ) = L23 φy = c1 E − c1 E42 E2 E4 − E5 + E1 E1 ∂ 3w ∂ 3w + ∂x∂y ∂x ∂w [(E1 − 3c1 E3 ) + c2 (E3 − 3c1 E5 )] 2(1 + ν) ∂x ∂ φx ∂ φx E22 E2 E4 E − − 2c1 E5 − + 2 − ν ∂x 2(1 + ν) ∂y E1 E1 − [(E1 − 3c1 E3 ) + c2 (E3 − 3c1 E5 )] φx 2(1 + ν) E2 E2 E4 E3 − − 2c1 E5 − (1 − ν) E1 E1 + c12 E7 − E42 E1 ∂ φy ∂x∂y + c12 E7 − ∂ φx ∂x3 E42 E1 JOURNAL OF THERMAL STRESSES L31 (w) = c1 E2 E4 E42 c E − c − E5 + 1 − ν2 E1 E1 − L32 (φx ) = Downloaded by [ECU Libraries] at 04:20 16 March 2016 l11 = − + c12 E7 − E3 − c12 − ν2 mπ a + nπ b 2 E7 − mπ a E42 E1 ν + 1+ν − v2 E5 − E2 E4 E1 − c1 v + 1+ν − ν2 E5 − n1 = − E1 16 E2 E4 E1 mπ a n2 = I0 + c12 I6 − − c1 + nπ b + + c12 E7 − E42 E1 mπ a E5 − ν + − ν2 1+ν nπ b I32 I0 + E5 − nπ b E2 E4 E2 − c1 E − E1 E1 E7 − v + 1−ν 1+ν nπ b mπ a − k1 − k2 nπ c1 [E1 − 3c1 E3 − c2 (E3 − 3c1 E5 )] + 2(1 + ν) b − ν2 + c1 ∂ φx ∂x∂y [E1 − 3c1 E3 + c2 (E3 − 3c1 E5 )] φy 2(1 + ν) mπ c1 [E1 − 3c1 E3 − c2 (E3 − 3c1 E5 )] + 2(1 + ν) a − ν2 + c1 E42 E1 E22 E2 E4 − 2c1 E5 − E1 E1 [E1 − 3c1 E3 − c2 (E3 − 3c1 E5 )] 2(1 + ν) − l13 = − ∂w [E1 − 3c1 E3 + c2 (E3 − 3c1 E5 )] 2(1 + ν) ∂y ∂ φy ∂ φy + − ν ∂y2 2(1 + ν) ∂x2 − l12 = − ∂ 3w ∂ 3w + ∂y3 ∂x2 ∂y E2 E2 E4 E3 − − 2c1 E5 − (1 − v) E1 E1 L33 φy = 295 E42 E1 mπ a nπ b E2 E4 E2 − c1 E − E1 E1 E7 − E42 E1 mπ a nπ b mπ a + nπ b c1 mπ c1 E42 E2 E4 mπ nπ + c1 E − − E5 + 1−ν a a b E1 E1 mπ − [(E1 − 3c1 E3 ) + c2 (E3 − 3c1 E5 )] 2(1 + ν) a l21 = − mπ nπ E22 E2 E4 + E − − 2c1 E5 − 1−ν a 2(1 + ν) b E1 E1 − [(E1 − 3c1 E3 ) + c2 (E3 − 3c1 E5 )] 2(1 + ν) l22 = − l23 = − mnπ E2 E2 E4 E3 − − 2c1 E5 − (1 − ν) ab E1 E1 + c12 E7 − E42 E1 + c12 E7 − E42 E1 296 N D DUC ET AL nπ c1 nπ E2 E4 E42 mπ + c E − c − E + − ν2 b E1 E1 a b nπ − [E1 − 3c1 E3 + c2 (E3 − 3c1 E5 )] 2(1 + ν) b mnπ E2 E2 E4 E2 E3 − − 2c1 E5 − + c12 E7 − =− (1 − v) ab E1 E1 E1 nπ mπ E22 E2 E4 =− + E3 − − 2c1 E5 − 1−ν b 2(1 + ν) a E1 E1 − [E1 − 3c1 E3 + c2 (E3 − 3c1 E5 )] 2(1 + ν) l31 = − l32 l33 (E2 − c1 E4 ) Downloaded by [ECU Libraries] at 04:20 16 March 2016 l14 = nπ b +v mnπ − v2 (E2 − c1 E4 ) l15 = l16 = − l17 mπ a mnπ 2 mπ v + nπ a b − v2 + c12 E7 − E42 E1 mπ a nπ b 4c1 E4 α + 2vα β + β mnπ − v2 E1 mπ = n1 − a 1−v H=h (−Kcm /Km )p p=0 pN+1 + 2v Em αm pN+2 + mπ a nπ b Em αcm +Ecm αm (p+1)N+2 + + nπ b Ecm αcm (p+2)N+2 (−Kcm /Km )p p=0 pN+1 Funding This work was supported by the Grant in Mechanics coded 107.02-2015.03 of the National Foundation for Science and Technology 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Postbuckling of Higher Order Shear Deformable Functionally Graded Plates on Elastic Foundations, Compos Struct., vol 93, pp 2874–2881, 2011  ... beneficial effects of elastic foundations on the nonlinear dynamic response of FGM Figure Effect of the linear Winkler foundation on nonlinear dynamic response of the FGM plate Figure Effect of the Pasternak... gradient on nonlinear dynamic response of the FGM plate Figure 11 Effect of temperature on nonlinear dynamic response of the FGM plate through the thickness of the plates Figure shows the effects of. .. curves of the nonlinear dynamic response of the thick FGM plates Downloaded by [ECU Libraries] at 04:20 16 March 2016 Concluding remarks The nonlinear thermal dynamic responses of FGM plates