DSpace at VNU: Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation

Journal of http://jvc.sagepub.com/ Vibration and Control Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation Nguyen Dinh Duc and Pham Hong Cong Journal of Vibration and Control published online July 2013 DOI: 10.1177/1077546313489717 The online version of this article can be found at: http://jvc.sagepub.com/content/early/2013/07/08/1077546313489717 Published by: http://www.sagepublications.com Additional services and information for Journal of Vibration and Control can be found at: Email Alerts: http://jvc.sagepub.com/cgi/alerts Subscriptions: http://jvc.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav >> OnlineFirst Version of Record - Jul 8, 2013 What is This? Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Article Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation Journal of Vibration and Control 0(0) 1–10 ! The Author(s) 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1077546313489717 jvc.sagepub.com Nguyen Dinh Duc and Pham Hong Cong Abstract This paper presents a first proposal to investigate the nonlinear dynamic response of imperfect symmetric thin sigmoidfunctionally graded material (S-FGM) plate resting on an elastic foundation and subjected to mechanical loads The formulations use classical plate theory taking into account geometrical nonlinearity, initial geometrical imperfection of the S-FGM plate and stress function The volume fractions of metal and ceramic are applied by sigmoid-law distribution (S-FGM) with metal-ceramic-metal layers The nonlinear equations are solved by the Runge-Kutta and Bubnov-Galerkin methods using stress function The obtained results show the effects of material, imperfection and elastic foundations on the dynamical response of S-FGM plate Keywords Classical plate theory (CPT), elastic foundation, imperfection, nonlinear dynamic response, thin S-FGM plate Introduction Functionally graded materials (FGMs), which microscopically are composites and made from a mixture of metal and ceramic constituents, have received considerable attention in recent years due to their high performance heat resistance capacity and excellent characteristics in comparison with conventional composites Therefore, FGMs have been chosen for use in temperature shielding structure components of aircraft, aerospace vehicles, nuclear plants and engineering structures in various industries As a result, many investigations have been carried out on the dynamics and vibration of FGM plates and shells One of the most popular FGM structures, which has been widely studied by using a simple power-law distribution (P-FGM) of the elastic modules varying with thickness, is a metal-ceramic composite structure An advantage of this is that the ceramic layer plays a similar role to the thermal resistance, whereas the metal layer will protect the mechanical deformation Recently, the static and dynamical properties of P-FGM have attracted the interest of the research community In this paper, we narrow our study on dynamical properties, and therefore just summarize the significant findings of the last few years Firstly, we should mention the findings of Zhao et al (2004), who studied the free vibration of a two-sided simply supported laminated cylindrical panel via the mesh-free kp-Ritz method This research has indeed provided us with the approximation calculation of the FGM panel In 2004, Vel and Batra also reported a three dimensional exact solution for the vibration of FGM rectangular plates Also in this year, Sofiyev and Schnack (2004) investigated the stability of functionally graded cylindrical shells under linearly Vietnam National University, Hanoi, Vietnam Received: 13 February 2013; accepted: 15 April 2013 Corresponding author: Nguyen Dinh Duc, Vietnam National University, 144 Xuan Thuy–Cau Giay, Hanoi, Vietnam Email: ducnd@vnu.edu.vn Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Journal of Vibration and Control 0(0) increasing dynamic tensional loading and obtained the result for the stability of functionally graded truncated conical shells subjected to a periodic impulsive loading Sofiyev et al (2005) also published the result of the stability of P-FGM ceramic-metal cylindrical shells under a periodic axial impulsive loading Ferreira et al (2006) received natural frequencies of FGM plates with a meshless method Zhao et al (2006) has also developed the element-free kp-Ritz method (which was applied successfully to an FGM panel in 2004) to calculate the free vibration analysis of the complicated FGM structures, i.e conical shell panels It is not only the mechanical effects that have been investigated, the studies of the dynamical properties of P-FGM structures under the thermal and thermomechanical loads have also been a particular interest of many authors Liew et al (2006a, 2006b) studied the nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient and dynamic stability of rotating cylindrical shells subjected to periodic axial loads Woo et al (2006) investigated the nonlinear free vibration behavior of functionally graded plates Kadoli and Ganesan (2006) studied the buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition Wu et al (2006) also published their results on nonlinear static and dynamic analysis of functionally graded plates Sofiyev (2007) has considered the buckling of functionally graded truncated conical shells under dynamic axial loading Prakash et al (2007) studied the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps In 2008, Darabi et al obtained the nonlinear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading Matsunaga (2008) analyzed natural frequencies and buckling stresses of FGM plates using 2-D higherorder deformation theory Moreover, the P-FGM plate and shell structures under the complicated dynamic loads have recently been a hot topic in the research community Shariyat (2008a, 2008b) also obtained the dynamic thermal buckling for suddenly heated temperature-dependent FGM cylindrical shells under combined axial compression and external pressure as well as under thermoelectro-mechanical loads Allahverdizadeh et al (2008) studied nonlinear free vibration and nonlinear forced vibration for thin circular functionally graded plates Sofiyev (2009, 2011) investigated the vibration of freely supported FGM conical shells subjected to external pressure and clamped FGM conical shells under external loads Shen (2009) published a valuable book, Functionally Graded Materials, Nonlinear Analysis of Plates and Shells, in which the results about nonlinear vibration of shear deformable FGM plates are presented Zhang and Li (2010) studied the dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load Ibrahim and Tawfik (2010) investigated limit-cycle oscillations of FGM plates subjected to aerodynamic and thermal loads Mohammad and Singh (2010) studied the dynamic response of P-FGM plates using first order shear deformation theory by finite element method Fakhari and Ohadi (2011) also investigated nonlinear vibration control of P-FGM plates with piezoelectric layers in a thermal environment using the finite element method Unlike the others, they have assumed that the material properties of P-FGM are temperaturedependent Mollarazi et al (2012) presented analysis of free vibration for FGM cylinders using a meshless method The dynamic stability of FGM skew thin plate subjected to a uniformly distributed tangential follower force has been investigated by Miao Ruan et al (2012) Najafov et al (2012) studied the vibration of axially compressed truncated conical shells with a functionally graded middle layer surrounded by elastic medium In order to increase the loading ability, a very good choice is often the stiffener Therefore, the research on the dynamics of stiffened FGM plates and shells has also been of interest Recently, Bich et al (2012) investigated nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels using the classical shell theory Duc (2013) studied the nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on an elastic foundation We have witnessed that dynamic analysis of P-FGM plates and shells has comparatively received a lot of attention over the last two decades However, it is not only the metal-ceramic and ceramic-metal of P-FGM mentioned above that displays high thermal resistance – in modern engineering and technology, there are also many structures that share similar properties In order to increase the adapting ability to a high temperature, structures with the top and bottom surfaces are made of ceramic and the core of the structure is made of metal (Duc and Cong, 2012) Moreover, FGM can be used with the top and bottom metallic surfaces together with the ceramic core to increase the strength and thermal resistance The sigmoid-FGM (S-FGM) plate considered in this paper with metal-ceramic-metal layers is an example of these structures This paper presents a first proposal to investigate the nonlinear dynamic response of imperfect symmetrical, thin S-FGM plates with metal-ceramic-metal layers on an elastic foundation using classical plate theory Numerical results for the dynamic response of the S-FGM plate are obtained by Bubnov-Galerkin and Rugge-Kutta methods and using stress function Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d [1–10] [PREPRINTER stage] (JVC) Duc and Cong Nonlinear dynamics of imperfect S-FGM plate where Consider a thin rectangular S-FGM plate that consists of functionally graded materials and rests on an elastic foundation The outer surface layers of the plate are metal-rich, but the midplane layer is purely ceramic (Figure 1) Please note that the dynamic response of S-FGM plate with ceramic-metal-ceramic layers was considered by Duc and Cong (2012) 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, Àh=2 z h=2 The length, width and total thickness of the plate are a, b and h, respectively (Figure 1) By applying a simple Sigmoid-law distribution, the volume fractions of metal and ceramic, Vm and Vc , are assumed as: 8  2z ỵ h N > > > , h=2 z > < h Vc zị ẳ   > > 1ị 2z ỵ h N > > : , z h=2 h and the Poisson ratio  is assumed constant zị ẳ : Suppose that the symmetrical S-FGM plate is subjected to a transverse load of intensity q0 In the present study, the classical theory of thin plates is used to obtain the motion and compatibility equations, as well as expression for determining the dynamic response of the S-FGM plate The train-displacement relations taking into account the von Karman nonlinear terms are (Brush and Almroth, 1975) 0 1 "x "x x @ "y A ẳ @ "0y A ỵ z@ y A 4ị xy 2xy xy and Ecm ẳ Ec Em , Vm zị ẳ Vc zị: 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 From equation (1) the effective properties of the S-FGM plate can be written as follows (Duc and Cong, 2012): ð3Þ with 0 1 1 "x u,x ỵ w2,x =2 x w,xx @ "0y A ẳ @ v,y ỵ w2 =2 A, @ y A ¼ À@ w,yy A: ,y xy w,xy xy u,y ỵ v,x ỵ w,x w,y 5ị where "0x and "0y are the normal strains, xy is the shear strain on the midplane of the plate; u, v, and w are the midplane displacement components along the x, y, and z axes ð, Þ indicates a partial derivative The strains are related in the compatibility equation  2 2 @2 "0x @2 "0y @2 xy @ w @ w@ w ẳ ỵ 2: 2 @x@y @x @y @y @x @x@y ð6Þ Hooke law for an FGM plate is defined as ½E, Š ¼ ½Em , m Š 8  > 2z ỵ h N > > , < h ỵ ẵEcm , cm   > 2z ỵ h N > > , : h cm ¼ c À m Àh=2 z ð2Þ z h=2 à E  "x , "y ị ỵ "y , "x ị 2 E xy : ẳ 21 ỵ ị Á x , y ¼ xy Figure S-FGM plate on elastic foundation Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 ð7Þ XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Journal of Vibration and Control 0(0) The force and moment resultants of the plate can be expressed in terms of stress components across the plate thickness as R h=2 where Ji ẳ h=2 zi zịdz i ẳ 0, 1, 2ị and k1 is Winkler foundation modulus and k2 is the shear layer foundation stiffness of Pasternak model With Zh=2 Ni , Mi ị ẳ i 1, zịdz, i ẳ x, y, xy 8ị J0 ẳ m h ỵ h=2 Inserting equations (4) and (7) into equation (8) gives the constitutive relations as   h 0 Nx , Mx ị ẳ E , E ị " ỵ " x y 2 i ỵ E2 , E3 ị x ỵ y   h E1 , E2 ị "0y ỵ "0x N y , My ẳ 1 9ị ỵE2 , E3 ị y ỵ x h i E1 , E2 ị xy Nxy , Mxy ẳ 21 ỵ ị ỵ E2 , E3 ịxy : 1ỵ where: E1 ẳ Em h ỵ E3 ẳ Ecm h , Nỵ1 E2 ẳ Em h3 Ecm h3 ỵ : 12 2N ỵ 1ịN ỵ 2ịN ỵ 3ị 10ị The equations of motion for a thin FGM plate on the elastic foundation based on the classical plate yheory (CPT) and can be written as (Nayfeh and Pai, 2004) @Nx @Nxy @2 u @3 w ỵ ẳ J0 J1 @t @x@t2 @x @y @Nxy @Ny @2 v @3 w ỵ ẳ J0 J1 @t @y@t2 @x @y 2 @ w @ w ỵ Ny k1 w ỵ k2 r2 w ỵ q0 @x@y @y   @ w @ u @3 v ẳ J0 ỵ J1 ỵ @t @x@t2 @y@t2   @ w @4 w J2 ỵ : @x2 @t2 @y2 @t2 ỵ 2Nxy 12ị J1 ẳ m h3 cm h3 ỵ : 12 2N ỵ 1ịN ỵ 2ịN ỵ 3ị 13ị The substitution of equation (9) into equation (12) leads to: @Nx @Nxy @2 u @Nxy @Ny @2 v ỵ ẳ J0 , ỵ ¼ J0 : ð14Þ @t @t @x @y @x @y   @2 w @2 w @2 w 2 D w Nx ỵ 2Nxy ỵ Ny k1 w ỵ k2 r w @x @x@y @y   4 @ w @ w @ w ỵ J0 J2 ẳ q0 : þ @t @x2 @t2 @y2 @t2 ð15Þ E3 where Á ẳ @2 =@x2 ỵ @2 =@y2 and D ẳ 1v For solving equations (14) and (15) we introduce stress function ẳ x, yị so that Nx ẳ For using late, the reverse relations are obtained from equation (9)   Á À Áà ÂÀ "0x , "0y ¼ Nx , Ny À  Ny , Nx E1 11ị ỵ  ị xy ẳ Nxy : E1 @2 Mxy @2 My @2 Mx @2 w þ þ þ N x @x2 @x2 @x@y @y2 J2 ẳ cm h ; Nỵ1 @2 ; @y2 Ny ¼ @2 ’ ; @x2 Nxy ¼ À @2 ’ @x@y ð16Þ Volmir’s assumption can be used in the dynamical analysis (Volmir, 1972; Bich et al., 2012; Duc, 2013) By 2 taking the inertia J0 @@t2u ! and J0 @@t2v ! into consideration because u ( w, v ( w, equations (14) are satisfied Inserting equation (16) into the equation (15) for perfect plate leads to DÁ2 w @2 ’ @2 w @2 ’ @2 w B @y2 @x2 @x@y @x@y ỵ C @2 w B C B 2 C ỵ J0 @@ ’@ w A @t À k1 w þ k2 r2 w @x2 @y2   @ w @4 w J2 ỵ ẳ q0 : @x2 @t2 @y2 @t2 ð17Þ Equation (17) includes two dependent unknowns w and ’ To obtain a second equation, relating the unknowns, the geometrical compatibility for an imperfect plate can be used: "0x,yy ỵ "0y,xx xy,xy ẳ w2,xy w,xx w,yy !  2 À wÃ,xy ÀwÃ,xx wÃ,yy : Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 ð18Þ XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Duc and Cong in which wà is a known function representing an initial small imperfection of the S-FGM plate Setting equations (11) and (16) into equation (18) gives the compatibility equation of an imperfect FGM plate as   " 2 2 # @4 ’ @4 @4 @ w @ w@ w ỵ2 2ỵ ẳ 2 E1 @x4 @x @y @y @x@y @x @y " # 2 à à @2 wà @ w @ w À ð19Þ @x@y @x @y2 For an imperfect FGM plate, equation (17) is modified into form as (Bich et al., 2012; Duc, 2013): DÁ2 ðw À wÃ Þ 2 @ ’@ w @2 ’ @2 w B @y2 @x2 À @x@y @x@y C @2 w B C ÀB C ỵ J0 2 @ @ @ w A @t ỵ 2 k1 w ỵ k2 r2 w @x @y   @ w @4 w J2 ỵ ẳ q0 @x2 @t2 @y2 @t2 Mx ¼ 0, My ¼ 0, Nx ¼ 0, Ny ¼ 0, Nxy ¼ 0; Nxy ¼ 0; f€ ðtÞ þ m1 f ðtÞ þ m2 f3 ðtÞ þ m3 f0 ẳ m4 q0 : 25ị 4B a J0 B2h ỵ J2 m2 B2a ỵ n2 2 4 Dm2 B2a ỵn2 ị 128m2 n2 B3a E1 f20 À 2 2 27 4Ba B2h 9Bh m Ba ỵn ị K DB3 K DB 2 m2 B2 ỵn2 ị a a a ỵ 4B2 ỵ 4B2 m1 ¼  ð20Þ h at x ¼ 0, a at y ẳ 0, b: 21ị The mentioned conditions (21) can be satisfied if the deflectionwand the stress function ’ are represented by: w ẳ f tị sin m x sin n y ẳ gtị sin m x sin n y m, n - odd numbers Equations (23) and (24) can be simplified as follows: in which: Equations (19) and (20) are the basic relations used to investigate the dynamic response of imperfect S-FGM plate on elastic foundations They are nonlinear in the dependent unknowns w and ’ Suppose that the S-FGM plate is simply supported at its edges and subjected to q transverse loads q0 ðtÞ The boundary conditions can be expressed as w ¼ 0, w ¼ 0, ab Df tị f0 ị 4m ỵ 22m 2n ỵ 4n ! 8m n ab ab À f ðtÞ gðtÞ À k1 f tị k2 f tị m ỵ n 4 ab ab ỵ J0 f tị ỵ J2 2m ỵ 2n f tị ẳ q0 : 4 m n ð24Þ ð22Þ Ã w ðx, yị ẳ f0 sin m x sin n y n where m ¼ m a , n ¼ b in which m, n ¼ 1, 2, , are natural numbers representing the number of half waves in the x and y directions respectively; fðtÞ is the deflection amplitude; f0 ¼ const, varying between and 1, represents the size of the imperfections The introduction of equation (22) into equations (19) and (20) and applying the Galerkin method gives À Á ab À Á 4m n ¼ ÀE1 f2 tị f20 : gtị 4m ỵ 22m 2n þ 4n ð23Þ h 4Ba 128m2 n2 B3a E1 À Á à m2 ¼  À Á J0 B2h ỵ J2 m2 B2a ỵ n2 2 9B2 m2 B2a ỵ n2 h 4 D m2 B2a ỵ n2 m3 ẳ J0 B2h ỵ J2 m2 B2a ỵ n2 2 B2h m4 ẳ 16b2 J0 B2h ỵ J2 m2 B2a ỵ n2 2 mn2 b b a4 Ba ¼ , Bh ¼ , K1 ¼ k1 , a h D D D ¼ , J0 ¼ h J0 , J2 ¼ J2 , h K2 ¼ k2 E1 ¼ a2 , D E1 h2 Equation (25), for obtaining the nonlinear dynamic response, the initial conditions are assumed as f 0ị ẳ f0 , f0ị ¼ The applied loads varying as a function of time The nonlinear equation (25) can be solved by the Newmark’s numerical integration method or by the Runge-Kutta method Numerical results and discussion The imperfect symmetrical S-FGM plate considered here a square plate: a ¼ b ¼ m, h ¼ 0:01 m The plates are simply supported at all edges The combination of materials consists of aluminum (Em ¼ 70:109 N=m2 , m ¼ 2702 kg=m3 ) and alumina (Ec ¼ 380:109 N=m2 , c ¼ 3800 kg=m3 ) The Poisson ration  is chosen to be 0:3 for simplicity The plate subjected to an uniformly distributed excited transverse load q0 tị ẳ p sin t Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Journal of Vibration and Control 0(0) The nonlinear dynamic response of the S-FGM plate acted on by the harmonic uniformly excited transverse load q0 tị ẳ p sin t is obtained by solving equation (25) combined with the initial conditions and by use of the Runge-Kutta method Figure shows the graph of maximum deflection of 38 periods; Figure shows the nonlinear response of the FGM plate of long period with different intensity of loads: p ¼ 1500 N=m2 and p ¼ 2500 N=m2 ;  ¼ 500, N ¼ As yet, there have been no reports on the static and dynamic buckling for symmetric S-FGM plate with metal-ceramic-metal layers We are therefore limited in our comparisons The relation of maximum deflection and the velocity of maximum deflection when ðN ¼ 1Þ and q0 ðtÞ ¼ 1500 sinð500tÞ are presented in Figure Figure shows the influence of power law indices on nonlinear dynamic responses of the S-FGM plate (N ẳ 1, 2, 3; q0 tị ẳ 1500 sin500tị) Figure shows the effect of the imperfection (f0 ¼ 0:001, 0:003) on nonlinear dynamic responses of the S-FGM plate Figure is chosen with Figure Dynamic response of the S-FGM plate (N ẳ 1, q0 tị ẳ 1500 sinð500tÞ) Figure Deflection–velocity relation Figure Dynamic response with different intensity of loads Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Duc and Cong Figure Nonlinear response of S-FGM plate with variety of volume fraction index N Figure Influence of imperfection on nonlinear dynamic response of the S-FGM plate Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Journal of Vibration and Control 0(0) Figure Influence of elastic foundations on nonlinear dynamic responses of the S-FGM plate Figure Effect of dimension ratio b=a on nonlinear dynamic response of the S-FGM q0 ðtÞ ¼ 75000 sinð500tÞ and N ¼ The increase in imperfection will lead to the increase of the amplitude of maximum deflection Figure shows the influence of elastic foundations K1 , K2 on nonlinear dynamic responses of the S-FGM plate with q0 tị ẳ 1500 sin500tị and N ẳ From Figure we conclude that these elastic foundations have a strong effect on the nonlinear dynamic response of the S-FGM plate Compared to the case corresponding to the coefficient K1 , the Pasternak-type elastic foundation with the coefficient K2 has a stronger effect Figures and show the effect of the geometrical parameters on nonlinear dynamic response of the SFGM plate Figure shows the effect of dimension ratio b=a on the nonlinear dynamic response of the S-FGM plate with q0 tị ẳ 1500 sin200tị, N ẳ Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] Duc and Cong Figure Effect of dimension ratio a=h on nonlinear dynamic response of the S-FGM Figure shows the effect of dimension ratio a=h on nonlinear dynamic response of the S-FGM plate with q0 tị ẳ 1500 sin500tị, N ẳ Figures 79 show us that the mechanical loading ability in S-FGM plate is better than P-FGM plate under the same conditions, i.e the size and the external force Conclusions This paper presents the first proposal to investigate the nonlinear dynamic response of imperfect symmetric thin S-FGM plate with metal-ceramic-metal layers resting on an elastic foundation Numerical results for the dynamic response of the S-FGM plate are obtained by Rugge-Kutta method and stress function The obtained results show the effects of material, imperfection, elastic foundations and geometrical parameters on the dynamical response of S-FGM plates Thus it is obvious that the dynamic response of the considered S-FGM plate depends on many factors significantly: volume ratio N, elastic foundation, imperfection and geometrical parameters of the FGM plate Therefore, when we change these parameters, we can actively control the dynamic response of the S-FGM plate Funding This work was supported by Project QGDA.12.03 of the Foundation for Science and Technology Development of Vietnam National University, Hanoi The authors are grateful for this support References Allahverdizadeh A, Naei MH and Bahrami MN (2008) Nonlinear free and forced vibration analysis of thin circular functionally graded plates Journal of Sound and Vibration 310: 966–984 Bich DH, Dung DV and Nam VH (2012) Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels Composite Structures 94: 2465–2473 Brush DD and Almroth BO (1975) Buckling of Bars, Plates and Shells New York: McGraw-Hill Darabi M, Darvizeh M and Darvizeh A (2008) Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading Composite Structures 83: 201–211 Duc ND (2013) Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation Journal of Composite Structures 99: 88–96 Duc ND and Cong PH (2012) Nonlinear dynamic response of FGM plate In Proceeding of the Second International Conference on Engineering Mechanics and Automation ICEMA2, Hanoi, Vietnam, August 16–17, 2012, pp.271–278 Fakhari V and Ohadi A (2011) Nonlinear vibration control of functionally graded plate with piezoelectric layers in thermal environment Journal of Vibration and Control 17(3): 449–469 Ferreira AJM, Batra RC and Roque CMC (2006) Natural frequencies of FGM plates by meshless method Journal of Composite Structures 75: 593–600 Ibrahim HH and Tawfik M (2010) Limit-cycle oscillations of functionally graded material plates subject to aerodynamic and thermal loads Journal of Vibration and Control 16(14): 2147–2166 Kadoli R and Ganesan N (2006) Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition Journal of Sound and Vibration 289: 450–480 Liew KM, Hu GY, Ng TY and Zhao X (2006a) Dynamic stability of rotating cylindrical shells subjected to periodic axial loads International Journal of Solids and Structures 43: 7553–7570 Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 XML Template (2013) [22.6.2013–2:19pm] //blrnas3/cenpro/ApplicationFiles/Journals/SAGE/3B2/JVCJ/Vol00000/130419/APPFile/SG-JVCJ130419.3d (JVC) [1–10] [PREPRINTER stage] 10 Journal of Vibration and Control 0(0) Liew KM, Yang J and Wu YF (2006b) Nonlinear vibration of a coating-FGM-substrate cylindrical panel subjected to a temperature gradient Computer Methods in Applied Mechanics and Engineering 195: 1007–1026 Matsunaga H (2008) Free vibration and stability of FGM plates according to a 2-D high order deformation theory Journal of Composite Structures 82: 499–512 Miao Ruan, Zhong-Min Wang and Yan Wang (2012) Dynamic stability of functionally graded materials skew plates subjected to uniformly distributed tangential follower forces Journal of Vibration and Control 18(7): 913–923 Mohammad T and Singh BN (2010) Static response and free vibration analysis of FGM plates using higher order shear deformation theory Applied Mathematical Modeling 34(12): 3991–4011 Mollarazi HR, Foroutan M and Moradi-Dastjerdi R (2012) Analysis of free vibration of functionally graded material cylinders by a meshless method Journal of Composite Materials 46(5): 507–515 Najafov A, Sofiyev A, Ozyigit P and Yucel KT (2012) Vibration and stability of axially compressed truncated conical shells with functionally graded middle layer surrounded by elastic medium Journal of Vibration and Control DOI: 1077546312461025, first published on October 25, 2012 Nayfeh AH and Pai PF (2004) Linear and Nonlinear Structural Mechanics New Jersey: Wiley-Interscience Prakash T, Sundararajan N and Ganapathi M (2007) On the nonlinear axisymmetric dynamic buckling behavior of clamped functionally graded spherical caps Journal of Sound and Vibration 299: 36–43 Shariyat M (2008a) Dynamic thermal buckling of suddenly heated temperature-dependent FGM cylindrical shells under combined axial compression and external pressure International Journal of Solids and Structures 45: 2598–2612 Shariyat M (2008b) Dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical with temperaturedependent material properties under thermo-electromechanical loads International Journal of Mechanical Sciences 50: 1561–1571 Shen Hui-Shen (2009) Functionally Graded materials, Nonlinear Analysis of plates and shells London, New York: CRC Press, Taylor & Francis Group Sofiyev AH (2004) The stability of functionally graded truncated conical shells subjected to a periodic impulsive loading International Journal of Solids and Structures 41: 3411–3424 Sofiyev AH (2007) The buckling of functionally graded truncated conical shells under dynamic axial loading Journal of Sound and Vibration 305: 808–826 Sofiyev AH (2009) The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure Composite Structures 89: 356–366 Sofiyev AH (2011) On the vibration and stability of clamped FGM conical shells under external loads Journal of Composite Materials 45(7): 771–788 Sofiyev AH and Schnack E (2004) The stability of functionally graded cylindrical shells under linearly increasing dynamic tensional loading Engineering Structures 26: 1321–1331 Vel SS and Batra RC (2004) Three dimensional exact solution for the vibration of FGM rectangular plates Journal of Sound and Vibration 272(3): 703–730 Volmir AS (1972) Non-linear Dynamics of Plates and Shells Science edn Moscow: Scientific Totals Woo J, Meguid SA and Ong LS (2006) Nonlinear free vibration behavior of functionally graded plates Journal of Sound and Vibration 289: 595–611 Wu Tsung-Lin, Shukla KK and Huang HJ (2006) Nonlinear static and dynamic analysis of functionally graded plates International Journal of Applied Mechanics and Engineering 11: 679–698 Zhang J and Li S (2010) Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load Composite Structures 92: 2979–2983 Zhao X, Li Q, Liew KM and Ng TY (2006) The elementfree kp-Ritz method for free vibration analysis of conical shell panels Journal of Sound and Vibration 295: 906–922 Zhao X, Ng TY and Liew KM (2004) Free vibration of twosided simply-supported laminated cylindrical panel via the mesh-free kp-Ritz method International Journal of Mechanical Sciences 46: 123–142 Downloaded from jvc.sagepub.com at CARLETON UNIV on June 9, 2014 ... imperfection and elastic foundations on the dynamical response of S-FGM plate Keywords Classical plate theory (CPT), elastic foundation, imperfection, nonlinear dynamic response, thin S-FGM plate. .. Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation Journal of Vibration and Control 0(0) 1–10 !... dynamics of imperfect S-FGM plate where Consider a thin rectangular S-FGM plate that consists of functionally graded materials and rests on an elastic foundation The outer surface layers of the plate