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Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method
Finite Elements in Analysis and Design 47 (2011) 394–401
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Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method Mohammad Talha , B.N. Singh Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721 302, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 August 2009 Received in revised form 26 July 2010 Accepted 17 November 2010 Available online 6 January 2011
In this paper, large amplitude free flexural vibration analysis of shear deformable functionally graded material (FGM) plates are investigated. The material properties of the FGM plates are assumed to vary through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. The nonlinear finite element equations are obtained using higher order shear deformation theory with a special modification in the transverse displacement. The Green–Lagrange nonlinear strain–displacement relation with all higher order nonlinear strain terms is included in the formulation to account for the large deflection response of the plate. The fundamental equations are obtained using variational approach by employing traction free boundary conditions on the top and bottom faces of the plate. Results are obtained by employing an efficient C0 finite element with 13 degrees of freedom (DOFs) per node. Convergence tests and comparison studies have been carried out to establish the efficacy of the present model. The variation of nonlinear frequency ratio with the amplitude ratio is highlighted for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions. & 2010 Elsevier B.V. All rights reserved.
Keywords: Functionally graded material Higher order shear deformation theory Finite element method Large amplitude vibration Green–Lagrange
1. Introduction The accomplishment of functionally graded material is the realization of contemporary and distinct functions that cannot be achieved by the traditional composite materials. These are advanced composite materials with a microscopically inhomogeneous anatomy and are usually made from a mixtures of ceramic and metal using powder metallurgy techniques. Continuous changes in their microstructure distinguish FGM from other traditional composite materials. The material property of the FGM can be tailored to obtain the specific demand in different engineering applications in order to exploit the advantage of the properties of individual constituent. This is possible, because the material composition changes gradually in a preferred direction. The advantage of using this material is that it eliminates the interface problem due to smooth and continuous change of material properties from one surface to other [1,2]. Large amplitude free flexural vibration (LAFFV) behavior of a plate arises in many engineering applications, particularly in the panels of aircraft. When a structure is deflected substantially, i.e., half of its thickness, a considerable geometrical nonlinearity occurs, mostly due to the development of in-plane membrane
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stresses. These membrane stresses are tensile in nature that stiffens the plate. This stiffening effect results in the rise of resonance frequencies and change of mode shapes. Thus, the linear model is not being capable to determine the behavior of the structures completely. Therefore, in the recent years geometrically nonlinear flexural vibration of plates have received considerable attention compared to static large deflection behavior of plates. Since this area is fairly new, published literature on the nonlinear free and forced vibrations of FGM plate is limited in number and most of them are fascinated on linear problem. Reddy [3] presented theoretical formulation and finite element models (FEM) in the frame work of third order shear deformation theory for static and dynamic analyses of the FGM plates. Vel and Batra [4] presented a threedimensional analytical solution of simply supported rectangular FGM plates for free and forced vibrations. Suitable displacement functions which satisfy boundary conditions are used to solve governing equations by employing the power series method. Efraim and Eisenberger [5] obtained exact free vibration frequencies and modes of variable thickness thick annular FGM plates. Gunes and Reddy [6] investigated geometrically nonlinear analysis of circular FGM plates subjected to mechanical and thermal loads. They used Green–Lagrange strain tensor with all its terms in the analysis. Chen et al. [7] derived nonlinear partial differential equations for the vibration motion of initially stressed FGM plates. The formulation are derived for the nonlinear vibration motion of the FGM in a general state of arbitrary initial stresses, based on classical laminated plate theory (CLPT). Talha and Singh [8] studied free vibration and static analysis of
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
FGM plates using modified HSDT kinematics. The fundamental equations are obtained using variational principle by considering the stress free boundary conditions at the top and bottom faces of the plate. Yang and Shen [9] analyzed the free and forced vibration analyses for initially stressed FGM plates in thermal environment. Temperature dependent material properties are assumed for the analysis and the formulations are based on Reddy’s higher order shear deformation theory, which includes the thermal effects due to uniform temperature variation. Sundararajan et al. [10] developed nonlinear formulation based on von-Karman assumptions to study the free vibration characteristics of FGM plates in thermal environment. They obtained nonlinear governing equations using Lagrange’s equations of motion and solved using FEM, coupled with direct iterative technique. Chen [11] presented nonlinear vibration for FGM plate in the state of non-uniform initial stresses. Galerkin method have been used for transformation of the governing nonlinear partial differential equations into ordinary nonlinear differential equations, and the nonlinear and linear frequencies are obtained using the Runge– Kutta method. Ng et al. [12] presented the parametric resonance of FGM rectangular plates under harmonic in-plane loading. It is found that the parametric resonance of FGM rectangular plate varies by varying the power-law exponent, which controls the material distribution in the structures. Allahverdizadeh et al. [13] developed a semi-analytical approach for nonlinear free and forced axi-symmetric vibrations of a thin circular FGM plates. The formulation is based on the CLPT kinematics and the geometric nonlinearity is incorporated in vonKarman sense. Woo et al. [14] provided an analytical solution for the nonlinear free vibration behavior of FGM plates. The governing equations for thin rectangular FGM plates are obtained using the von-Karman theory for large transverse deflection, and mixed Fourier series analysis is used to get the solution. Huang and Shen [15] studied nonlinear vibration and dynamic response of FGM plates in thermal environments. The formulations are based on the HSDT kinematics and general von-Karman type equation, which includes thermal effects. Yang and Shen [16] investigated large deflection and postbuckling responses of FGM rectangular plates by using semi-analytical approach under transverse and in-plane loads. The formulations are based on the CLPT kinematics. Paraveen and Reddy [17] investigated the response of FGM plates using FEM that accounts for the transverse shear strains, rotary inertia and moderately large rotations in the von-Karman sense. The gradation of properties is assumed according to power-law throughout the thickness and comparisons have been made with homogeneous isotropic plates. The determination of accurate nonlinear behavior of the FGM fundamentally depends on the theory used to model the structure. The classical laminated plate theory which requires that normals to the mid-plane remain normal during plate deflections may be inappropriate for analysis of the FGM plates, in which volume fractions of two or more materials vary continuously as a function of position in a preferred direction. The inaccuracy occurs due to neglecting the effects of transverse shear and normal strains in the plate [18]. Due to continuous variation in material properties, the first order shear deformation theory and higher order shear deformation theory may be conveniently used in the analysis. It is noted that the first order shear deformation theory proposed by Mindlin [21] does not satisfy the parabolic variation of transverse shear strain in the thickness direction. Consequently, the solution accuracy of the FSDT depends on the shear correction factors. Therefore, it has to be incorporated to adjust the transverse shear stiffness. Generally, in the HSDT kinematics the in-plane displacements are assumed to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be constant. In the present study, the structural model kinematics assumes the
395
cubically varying in-plane displacement over the entire thickness, while the transverse displacement varies quadratically to achieve the accountability of normal strain and its derivative in calculation of transverse shear strains. Thus, the development of higher order shear deformation theory for describing the mechanical behavior of FGM structures has been of high importance to the researchers. In this regard the geometrically nonlinear free flexural vibrations of the FGM plates using HSDT kinematics by incorporating geometric nonlinearity in Green–Lagrange sense is appropriate to examine the responses of FGM structure accurately. It is accomplished from the literature that the nonlinear free vibration analysis of the FGM plates using HSDT kinematics with geometric nonlinearity in Green–Lagrange sense has not been reported in the literature to the best of the authors’ knowledge. The present research aims to develop a higher order shear deformation theory with a special modification in transverse displacement that provides an additional freedom to the displacements through the thickness, and consequently eliminates the over prediction. The geometric nonlinearity in the formulation has been incorporated by taking all higher order nonlinear strain terms associated with Green–Lagrange theory. The governing equations are formulated using the variational approach. A nonlinear C0 continuous isoparametric FEM is proposed to minimize the computational exercise required in the disposition of element matrices without compromising the solution credibility. The nonlinear fundamental frequencies are obtained for different thickness ratios, the aspect ratios, the amplitude ratios, and for different volume fraction indices and boundary conditions. The obtained results are compared with those available in the literature.
2. Theoretical formulation 2.1. Geometric configuration and material properties The FGM plate is regarded to be a single layer plate of uniform thickness. Here we ascertain the FGM plate of length a, width b and total thickness h made from an isotropic material of metal and ceramics, in which the composition varies from top to bottom surface as shown in Fig. 1. The top surface ðZ ¼ þ h=2Þ of the plate is ceramic rich, whereas the bottom surface ðZ ¼ h=2Þ is metal rich. The coordinates x, y are considered along the in-plane directions and z along the thickness direction. The formulations are restricted here with the assumption of a linear elastic material behavior for small strains and large displacements. The elastic material properties vary through the plate thickness according to the volume fractions of the constituents. Power-law distribution is commonly used to describe the variation of material a
x b Ceramic
+h/2 -h/2 Metal
y z
Fig. 1. Geometry and dimensions of the plate.
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where ax ¼ @cz =@x, bx ¼ @w=@x, yx ¼ @xz =@x, ay ¼ @cz =@y, by ¼ @w=@y, yy ¼ @xz =@y and xz ¼ az and f1 ðzÞ ¼ C1 zC2 z3 , f2 ðzÞ ¼ C3 z2 , f3 ðzÞ ¼ C4 z3 , f4 ðzÞ ¼ C5 z3 , f5 ¼ C1 z, f6 ¼ C1 z2 , C1 ¼ 1, C2 ¼ C4 ¼ 4=3h2 , C3 ¼ 1=2, C5 ¼ 1=3. The displacement vector for the model based on the modified displacement field as given in Eq. (4) is written as fLg ¼ ½u,v,w, cx , cy , cz , ax , ay , az , bx , by , yx , yy T
ð5Þ
2.3. Strain–displacement relation The nonlinear Green–Lagrange strain–displacement relation for FGM plate can be represented as [20] 9 8 9 > @u > 8 > > > > @x > > e xx > > > > > > > > @v > > > > > > > e yy > @y > > > > > > > > > > > > > > > > @w > > > > e > > > = < zz = < @z > fe g ¼ g yz ¼ @w þ @v @y @z > > > > > > > > > @u @w > >g > > > xz > > > > > þ @x > > > > > @z > > > > > > > > >g > > > xy > > > @v @u > > > > > ; > : @x þ @y > > > ; :
Fig. 2. Variation of the volume fractions Vc through the thickness.
properties (Fig. 2), which is expressed as [19] PðzÞ ¼ ðPc Pm ÞVc þPm
z 1 n þ Vc ðzÞ ¼ h 2
ð0 rn r1Þ
ð1Þ ð2Þ
where P denotes the effective material property, Pm and Pc represents the properties of the metal and ceramic, respectively, Vc is the volume fraction of the ceramic and n is the volume fraction exponent. The effective material properties of the plate including, Young’s modulus E, density r vary according to Eq. (1) and Poisson’s ratio n is assumed to be constant. The above power-law theory estimates a simple rule of mixtures which is used to find the effective properties of the ceramic–metal graded plate. This rule of mixtures applies in the thickness direction only.
@u @x
2 @u @y
2
þ þ
2 @v @x
2 @v @y
þ þ
2 @w @x
2 @w @y
9 > > > > > > > > > > > > > > > > > > > =
2 2 þ @v þ @w @z @z þ 2> h i > > > > > @u @u @v @v @w > > > > 2 @y @z þ @y @z þ @w > > @y @z > > > > h i > > > > > > @u @u @v @v @w @w > > 2 @x @z þ @x @z þ @x > > @z > > > > > > h i > > > > @u @u @v @v @w @w > > ; : 2 @x @y þ @x @y þ @x @y
fe g ¼ fe l g þfe nl g
In the present study, system of governing equations for FGM plate is derived by using variational approach. The origin of the material coordinates is at the middle of the plate as shown in Fig. 1. For accurate analysis of transverse shear effects in the mathematical formulation the HSDT kinematics has been used, with a special modification in the transverse displacement. The in-plane displacements u , v and the transverse displacement w for the plate is assumed as uðx,y,z,tÞ ¼ uðx,y,tÞ þzcx ðx,y,tÞ þ z2 xx ðx,y,tÞ þ z3 rx ðx,y,tÞ vðx,y,z,tÞ ¼ vðx,y,tÞ þ zcy ðx,y,tÞ þ z2 xy ðx,y,tÞ þ z3 ry ðx,y,tÞ ð3Þ
where t is the time, u, v, and w are the corresponding displacements of a point on the mid-plane. cx and cy are the rotations of normal to the mid-plane about the y-axis and x-axis, respectively. The functions xx , xy , rx and ry are the higher order terms in the Taylor series expansion defined in the mid-plane of the plate. The higher order terms are determined by diminishing the transverse shear stresses txz ¼ t4 and tyz ¼ t5 on the top and bottom surfaces of the plate, and by applying this boundary condition the displacement field is modified as u ¼ u þ f1 ðzÞcx þf2 ðzÞax þ f3 ðzÞbx þ f4 ðzÞyx v ¼ v þ f1 ðzÞcy þf2 ðzÞay þ f3 ðzÞby þ f4 ðzÞyy w ¼ w þ f5 ðzÞcz þ f6 ðzÞaz
2
@u @z
ð6Þ
or, in compact form Eq. (6) may be written as
2.2. Displacement field and strains
wðx,y,z,tÞ ¼ wðx,y,tÞ þzcz ðx,y,tÞ þ z2 xz ðx,y,tÞ
8 > > > > > > > > > > > > > > > > > > > 1<
ð4Þ
ð7Þ
where fe l g and fe nl g are the linear and nonlinear strain vectors, respectively. Substituting Eq. (6) in Eq. (3) the strain displacement of the FGM plate is expressed as 8 nl0 9 28 9 8 nl1 93 8 09 e1 > k1 > > > k11 > e1 > > > > > > > > > > > > > > > > > 6> >7 > 0> > 1> > nl0 > > nl1 > > > > > 6 > > > > > > > > e k e k > > > >7 > > > > 2 2 2 2 6 > > > > > > > > > > 6> >7 > 0> > 1> > nl0 > > nl1 > > > > > = 6< k = 1 < k =7 nl1 >7 6 e k > > > > > > 2 2 2e 2k > > > > > > > 6 >7 > 4> > 4> > 4 > > 4 > > > > > > > > > 6 7 > > > > > > > > 6> >7 > > k15 > > > e05 > 2enl0 2knl1 > > > > > 5 > 5 > > > > > > > > > 4> 5 > > > > > > > > : e0 ; : k1 ; > > ; ; : 2enl0 > : 2knl1 > 6 6 6 6 8 nl2 93 28 9 k1 > > k21 > > > > > > > 6> >7 > > nl2 > > 2> > > 6> 7 > > > k k > > > > 2 2> > > 6> 7 > > > > > > > 6> nl2 =7 < 0 = 1< k 6 7 3 26 7 þz 6 þ 2 nl2 >7 k > > > 2 2k > 6> 7 4> 4 > > > > > > > > 6> 7 > > > > > > k25 > 6> 7 2knl2 > > > 5 > > > > 4> >5 > > > 2; > > : : 2knl2 ; k6 6 8 nl3 93 8 nl4 9 28 9 3 k k > > > > > > > >7 > > > 1 > > 1 > > k1 > 6> > > > > nl3 nl4 > > > > > > 3 6> 7 > > > > > > k k k > > > > > > 2 2 6> >7 > > > > > 2> > > > > > > > 6> 7 nl3 > nl4 > = = = < < < k k 0 6 7 1 1 3 3 36 4 7 þz þ þz 6 nl3 nl4 7 2> 2k > 2k > >7 > > 2> > > >0> 6> > > >7 > > > 4nl3 > > 4nl4 > > > 6> > > > > > > > > > 6> 7 0> 2k 2k5 > > > > > > > 5 > > > > > 4> 5 > > > > > > 3; > > > : ; : 2knl3 ; : 2knl4 > k6 6 6
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
9 8 8 nl6 9 > > knl5 k1 > > > > 1 > > > > > > > > > > > nl5 nl6 > > > > > k > > > > k 2 > > > > 2 > > > > > > > > = = < < 0 0 51 61 þz þz nl5 2k > > > 2> 2 0 > > > 4 > > > > > > > > > > > 2knl5 > > > > > > 0 > > > 5 > > > > > > > > > ; > nl5 ; : 2knl6 > : 2k6 6
397
R h=2 R h=2 R h=2 where ½D1 ¼ h=2 ½TTl ½Q ½Tl dz, ½D2 ¼ h=2 ½TTl ½Q ½Tnl dz, ½D3 ¼ h=2 R h=2 ½TTnl ½Q ½Tl dz, ½D4 ¼ h=2 ½TTnl ½Q ½Tnl dz and [Q] is the stiffness ð8Þ
coefficients. 2.6. Kinetic energy
The above strain–displacement relation (Eq. (8)) can be expressed in terms of mid-plane strain vector by classifying the linear strain vector fel g and nonlinear strains vector fenl g separately, as
The kinetic energy of the FGM plate can be expressed as Z ð17Þ T ¼ 12 rfu_ gT fu_ gdV V
where
where r and fug are the density and global displacement vector of the plate. The global displacement field model as given by Eq. (4) may be represented as
fe l g ¼ fe01 e02 e03 e04 e05 e06 k11 k12 k13 k14 k15 k16 k21 k22 k24 k25 k26 k31 k32 k36 g
fug ¼ ½NfLg
feg ¼ fel g þ fenl g ¼ ½Tl fe l g þ 12½Tnl fe nl g
ð9Þ
ð10Þ fe nl g ¼
9 8 nl0 nl0 nl0 nl0 nl0 nl0 nl1 nl1 nl1 nl1 nl1 nl1 nl2 nl2 nl2 nl2 > > > e1 e2 e3 e4 e5 e6 k1 k2 k3 k4 k5 k6 k1 k2 k3 k4 > 1 < nl2 nl2 nl3 nl3 nl3 nl3 nl3 nl3 nl4 nl4 nl4 nl4 nl4 nl4 nl5 nl5 = k5 k6 k1 k2 k3 k4 k5 k6 k1 k2 k3 k4 k5 k6 k1 k2 > 2> > > ; : 0 knl5 knl5 knl5 knl6 knl6 0 0 0 knl6 5 4 6 1 2 6
ð11Þ The linear strain vector in terms of mid-plane strain vector fel g can be further expressed as fel g201 ¼ ½w2013 fLg131
2.4. Constitutive relations The constitutive relation describes how the stresses and strains are related within the plate and is expressed as 9 2 9 8 38 exx > sxx > Q11 Q12 Q13 0 0 0 > > > > > > > 6 > > > 7 > > > > > eyy > syy > 0 0 0 7> > > > > 6 Q12 Q22 Q23 > > > > > 6 > > > 7> > > > < szz = 6 Q13 Q23 Q33 < 7 e 0 0 0 7 zz = 6 ¼6 ð13Þ 7 g s > 6 0 > > 0 0 7> 0 0 Q44 > > 6 > > yz > > yz > 7> > > > > > > gxz > s > > 6 > > 0 7 0 0 0 0 Q55 > > > 5> > 4 > > xz > > > > ; ; : sxy > :g > 0 0 0 0 0 Q 2
3
where Q11 ¼ Q22 ¼ Q33 ¼ EðzÞð1n Þ=ð13n 2n Þ, Q12 ¼ Q13 ¼ Q23 ¼ EðzÞnð1 þ nÞ=ð13n2 2n3 Þ, Q44 ¼ Q55 ¼ Q66 ¼ EðzÞ=2ð1þ nÞ. The elastic modulus E, and the coefficients Qij vary in the thickness direction of the plate according to Eqs. (1) and (2). 2.5. Strain energy of FGM plate The strain energy of the FGM plate is given by Z 1 U¼ fegT fsg dV 2 v
Eq. (15) in expanded form is written as U¼
1 2
V
f l gTi ½D1 f l gi þ 12 f l gTi ½D2 f nl gi þ 12 f nl gTi ½D3 f l gi þ 14 f nl gTi ½D4 f nl gi
e
e
e
A
Z
A
e
e
e
3. Finite element implementation A nine-noded quadrilateral C0 isoparametric element is used in the present study. The domain is discretized into a set of finite elements. Over each of the elements, the displacement vector and element geometry of the model is expressed by fLg ¼
NN X i¼1
Ni fLgi ; x ¼
NN X i¼1
Ni xi ; y ¼
NN X
Ni yi
ð21Þ
i¼1
where Ni is the interpolation function (shape function) for the ith node, fLgi is the vector of unknown displacements for the ith node, NN is the number of nodes per element and xi and yi are Cartesian coordinate of the ith node. 3.1. Governing equation
ð22Þ for i ¼ 1,2, . . . where T is the kinetic energy, U is the strain energy, fqi g and fq_ i g are the generalized coordinates and generalized velocities, respectively. The equilibrium equation for nonlinear free vibration analysis with large deformation (i.e., small strains and large displacements relation) can be represented as € þð½K þ gKc þ 12 ½Knl1 þ ½Knl2 þ 12½Knl3 Þfqg ¼ 0 ½Mfqg
! dA
The governing equation for nonlinear free vibration problem of the FGM plate can be derived using variational principle, which is the generalization of the principle of virtual displacement. Lagrange equation for a conservative system can be written as 8 2 3 9 cz 2 cz 2 > > > > ax @@x þ bx @@x > > > 6 7 > > > > > 6 7 Z = < 2 2 6 7 d @T @U @ g 6 @cz @xz 7dv ¼ 0 þ y þ a þ þ x y @x @y 6 7 > dt @fq_ i g @fqi g @fqi g > 2 > > 6 > 2 2 7 > > 4 5 > > > xz > > ; : þ by @w þ yy @@y @y
ð14Þ
By substituting the strains and stresses from Eqs. (9) and (13), the above equation becomes Z 1 U¼ fe þ enl gTi ½Q fel þ enl gi dV ð15Þ 2 V l
e
Substituting Eq. (17) into Eq. (18), the kinetic energy becomes Z Z Z rfL_ gT ½NT ½NfL_ g dz dA ¼ 12 fL_ gT ½mfL_ g dA ð20Þ T ¼ 12
xy
66
2
e
where fLg is as defined in Eq. (5), and the function of thickness coordinate ½N may be represented as 2 3 1 0 0 f1 ðzÞ 0 0 f2 ðzÞ 0 0 f3 ðzÞ 0 f4 ðzÞ 0 6 7 ð19Þ ½N ¼ 4 0 1 0 0 f1 ðzÞ 0 0 f2 ðzÞ 0 0 f3 ðzÞ 0 f4 ðzÞ 5 0 0 1 0 0 f5 ðzÞ 0 0 f6 ðzÞ 0 0 0 0
ð12Þ
where ½w and fLg are a matrix of differential operator and displacement field vector, respectively. The superscripts 0, 1, 2–3 in Eq. (10) and nl0, nl1, and nl2–6 in Eq. (11) represents the membrane, bending and higher order terms, respectively.
Z
ð18Þ
ð16Þ
ð23Þ
where ½M, g,½Kc ,½K,½Knl1 ,½Knl2 ,½Knl3 , and fqg, are the global mass matrix, penalty parameter that enforces constraints, the global
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linear stiffness matrix arising due to constrains, the global linear stiffness matrix, the global nonlinear stiffness matrices and global displacement vector, respectively. Assuming the system vibrating in principal mode with natural frequency o, Eq. (23) can be conveniently reduced to nonlinear generalized eigenvalue problem as ½Kfqg ¼ l½Mfqg
ð24Þ
2
with l ¼ o , where o is defined as frequency of natural vibration. This eigenvalue problem is solved by employing direct iterative procedure. Initially, the linear response with normalized first mode is calculated by solving Eq. (24) with all nonlinear [Knl1], [Knl2], and [Knl3] terms fixed to zero. This normalized vector is scaled up in such a way that the maximum deflection is equal to the desired amplitude w/h (w is the maximum transverse displacement and h is the thickness of the plate). After that nonlinear stiffness matrices [Knl1], [Knl2], and [Knl3] are obtained with the first mode. Then using updated stiffness matrices, eigenvalue and its eigenvector are obtained. This attained eigenvector is used to compute the nonlinear stiffness matrices for the next iteration. This exercise continues till two eigenvalues from the two subsequent iterations reached the tolerance limit r 103 . The convergence is supposed to reach at said tolerance limit, and the corresponding o is the nonlinear frequency (onl ) of the vibrated FGM plate.
The following sets of boundary conditions are considered in the present study: Simply supported: (SSSS) u0 ¼ w0 ¼ cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0
at x ¼ 0 and a
v0 ¼ w0 ¼ cx ¼ ay ¼ az ¼ bx ¼ yy ¼ 0
at y ¼ 0 and b
Clamped: (CCCC) u0 ¼ v0 ¼ w0 ¼ cx ¼ cy ¼ cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0 at x ¼ 0,a and y ¼ 0,b Simply supported and clamped: (SCSC) u0 ¼ w0 ¼ cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0
at x ¼ 0 and y ¼ 0
u0 ¼ v0 ¼ w0 ¼ cx ¼ cy ¼ cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0 at x ¼ a, y ¼ b Simply supported and clamped: (SSCC) u0 ¼ w0 ¼ cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0
at x ¼ 0 and a
u0 ¼ v0 ¼ w0 ¼ cx ¼ cy ¼ cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0 at y ¼ 0,b 4.1. Convergence and validation study
4. Numerical results In the present work, large amplitude free flexural vibration behavior of the FGM plates has been addressed. The LAFFV responses are computed using the proposed mathematical model in conjunction with nonlinear FEM. The geometric nonlinearity is incorporated in the formulation applying Green–Lagrange theory by taking all higher order terms. A computer program has been developed in MATLAB 7.5.0 (R2007b) environment. The validation and efficacy of the proposed algorithm is examined by comparing the results with those available in the literature. A nine-noded Lagrange isoparametric element, with 117 degrees of freedom per element for the present HSDT kinematics has been used for discretizing the plate. Full integration schemes (3 3) and selective integration schemes (2 2) are used for thick and thin plates, respectively, to compute the results. In the present analysis simply supported boundary conditions are used to check the efficacy of the model. However, the formulation and code does not impose any limitations. The properties of the FGM constituents at room temperature (300 K) are shown in Table 1, which have been used for the computation of the results throughout the study, unless specified otherwise. The features of volume fraction of the ceramic phase through the dimensionless thickness is depicted in Fig. 2. In the analysis, it is assumed that the materials are perfectly elastic throughout the deformation.
Table 1 Properties of the FGM components. Material
Aluminium (Al) Alumina (Al2O3) Zirconia (ZrO2) Silicon nitride (Si3N4) Titanium alloy (Ti–6Al–4V) Stainless steel(SUS304)
The accuracy of the present finite element formulation is validated by comparing the results with those available in the literature [15] which is based on the HSDT and general von-Karman type of nonlinearity. A convergence study is also presented. A FGM square plate, simply supported at all four edges is analyzed. In this example, the analysis is performed with volume fraction index n¼2, aspect ratio a=b ¼ 1 with sides of the plate a ¼ b ¼ 0:2 m, for different amplitude ratios (wmax =h). wmax is the maximum deflection at the center of the plate. The FGM plate comprised titanium alloy, Ti–6Al–4V (metal) and zirconia, ZrO2 (ceramic). The results have been carried out in terms of frequency ratio onl =ol with various mesh divisions as shown in Table 2. This clearly shows that Table 2 Comparison of fundamental nonlinear frequency ratio ðonl =ol Þ for the various values of amplitude ratios of a square SSSS (Ti–6AL–4V/ZrO2) FGM plate with different mess sizes. Mesh size
Amplitude ratio ðwmax =hÞ 0.2
0.4
0.6
0.8
1.0
Present (2 2) Present (3 3) Present (4 4) Present (5 5) Huang and Shen [15]
1.0493 1.0476 1.0467 1.0455 1.021
1.1597 1.1468 1.1405 1.1409 1.082
1.2951 1.2857 1.2786 1.2765 1.176
1.4864 1.4492 1.4300 1.4244 1.296
% Difference
2.343
5.162
7.873
9.014
1.6333 1.6094 1.6064 1.6055 1.436 10.557
Table 3 Comparison of fundamental nonlinear frequency ratio ðonl =ol Þ for the various values of amplitude ratios of a square SSSS (Si3N4/SUS304) FGM plate (a/h ¼10). Properties ðwmax =hÞ
E (N/m2)
n
r ðkg=m3 Þ
70 109 380 109 151 109 427 109 105.7 109 207.78 109
0.30 0.30 0.30 0.28 0.298 0.28
2707 3800 3000 3210 4429 8166
0.2 0.4 0.6 0.8 1.0
FEM Ref. [10]
Present
% Difference
1.0063 1.0654 1.1707 1.3155 1.4789
1.0198 1.1164 1.2377 1.4214 1.6390
1.323 3.712 5.413 7.452 9.768
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
the solution accuracy and the rate of convergence with mesh refinement are good for frequency ratio onl =ol for the different values of amplitude ratios (wmax =h). Based on the convergence, it is concluded that ð5 5Þ mesh is sufficient for LAFFV analysis. Table 3 shows the comparison of the frequency ratio onl =ol for Si3N4/ SUS304 simply supported square FGM plate. The side to thickness ratio (a/h) is taken as 10. The top surface of the plate is ceramic rich (Si3N4), and bottom surface is metal rich (SUS304). It can be observed that the solutions from this study agrees well with those presented by Sundararajan et al. [10]. It is accomplished from Tables 2 and 3, as the amplitude ratio (wmax =h) increases, the % difference in frequency ratio onl =ol increases. This is due to the fact that Refs. [15,10], had incorporated the geometric nonlinearity in von-Karman sense. Whereas, in the present study geometric nonlinearity is included in Green–Lagrange sense with all higher order nonlinear strain terms. However, it has been observed from the detailed analysis that, the von-Karman type of geometric nonlinearity gives fairly good results up to ðwmax =hÞ ¼ 0:6, for the sakeof conciseness it is not shown here. The difference in the result suggests the requirement of a better mathematical model for the structures which undergo small strain having large deformation and/or large rotation.
5. Parametric study Based on the established approach and analysis of aforementioned sections, the simply supported (SSSS), clamped (CCCC), simply supported–simply supported and clamped–clamped (SSCC) and simply supported and clamped (SCSC) boundary conditions have been employed in the analysis. It is also observed that ð5 5Þ mesh gives good convergence as mentioned earlier, and have been used for accomplishing the results, unless otherwise stated. Table 4 shows the effect of volume fraction index n and thickness ratio a=h ¼ 10 on the frequency ratio (onl =ol ) of a SSSS square titanium alloy (Ti–6AL–4V) and zirconium oxide (ZrO2) FGM plate. The properties of the constituents are provided in Table 1 as mentioned earlier. The top surface of the plate is ceramic rich whereas the bottom surface is metal rich. The non-dimensional linear frequency is assumed as o ¼ oða2 =hÞ½rm ð1n2 Þ=Em 1=2 . It is observed from the table that the frequency ratio decreases with the increase in the volume fraction index n. This decrease in frequency ratio is on expected lines, because increase in the volume fraction index n means that a plate has a smaller ceramic component, and thus its stiffness is reduced and consequently its non-dimensional fundamental linear frequency is decayed. It can be additionally viewed that the frequency ratio enhances with the rise in amplitude ratio. Table 5 shows the effect of variation of thickness ratios (a=h ¼ 52100Þ, ranging from thick to thin plates, with the volume fraction index n. Here, FGM plates consisting of (Ti–6AL–4V/ZrO2) with all simply supported edges. The aspect ratio is taken as a=b ¼ 1 with volume fraction index n ¼ 0 and 1. Choosing n index as zero
399
corresponds to the fully ceramic plate and 1 complies linear variation of ceramic and metal constituent throughout the thickness of the plate. It is observed that the frequency ratio (onl =ol ) decreases with the increase of thickness ratio and it shows more significant differences at lower thickness ratio than larger thickness ratio. This implies that the higher order theory is more suitable for thick plates and higher order terms must be considered for the thick plate condition. It is also revealed that the nonlinear frequency ratio is slightly higher for fully ceramic plate i.e., (n ¼ 0) considered here, than (n ¼ 1). It is also accomplished that the frequency ratio increases with the increase of amplitude ratio. However, mixed types of behavior on the frequency ratio have also been observed due to the presence of severe nonlinearity. Therefore, it can be concluded that there is a great need to analyze the structures with geometric nonlinearity in Green–Lagrange sense to closely monitor the response of structures. Table 6 shows the variation of frequency ratio (onl =ol ) with amplitude ratio (wmax =h) for SCSC square stainless steel (SUS304) and silicon nitride (Si3N4) FGM plate. The fundamental linear frequency is non-dimensionalized as o ¼ oða2 =hÞðrc =Ec Þ1=2 . It is observed from the table that the frequency ratio decreases with the increase in the volume fraction index n up to a certain value, say n ¼ 2, and then the frequency ratio (onl =ol ) increases with additional increase in volume fraction index n. It is realized that the value of linear frequency decreases for higher value of index n and accordingly the nonlinear frequency decreases. It is also observed that the degradation in linear frequency at higher volume fraction index n is more than that of the nonlinear frequency. Hence, the overall trend of frequency ratio (onl =ol ) increases. The frequency ratio is on higher side for a=h ¼ 10 compared to a=h ¼ 20. Table 7 describes the frequency ratio (onl =ol ) for aluminum (Al) zirconia (ZrO2) square FGM plate for different volume fraction index n and thickness ratios a=h ¼ 10,20. Here it is ascertained that the SSCC boundary conditions and the fundamental linear frequency are non-dimensionalized as o ¼ oða2 =hÞðrc =Ec Þ1=2 . The
Table 5 Effect of variation of thickness ratios (a/h ¼ 5–100) with the volume fraction index n for (Ti–6AL–4V/ZrO2) FGM plates consisting with SSSS boundary condition. a/b
n
a/h
Amplitude ratio ðwmax =hÞ 0.5
1.0
1.2
1.4
1.6
1.8
2.0
1
0
5 15 25 50 100
2.1878 1.5223 1.1636 1.1662 1.5491
2.3648 1.9828 1.5159 1.5809 1.5291
2.5774 2.5139 1.7680 1.7383 1.7330
2.5253 2.0053 1.9931 1.8765 1.7827
2.8965 2.1878 2.0526 1.9418 1.8225
2.9843 2.3212 2.2484 2.0165 2.1086
2.9779 2.4036 2.2014 2.1394 1.8534
1
1
5 15 25 50 100
1.9459 1.4192 1.3077 1.2114 1.2080
2.3176 1.6088 1.6066 1.5622 1.4951
2.4829 1.7816 1.7874 1.7250 1.6407
2.3710 1.8540 1.7932 1.8279 1.7625
2.7433 2.1428 1.8241 1.5265 1.8041
2.5920 2.2791 2.1910 2.1771 1.8915
2.7165 2.3801 2.1139 2.1221 1.9076
Table 4 Effect of volume fraction index n and thickness ratio ða=h ¼ 10Þ on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square SSSS (Ti–6AL–4V/ZrO2) FGM plate. n
0 0.2 0.4 0.6 0.8 1.0
Amplitude ratio ðwmax =hÞ 0.5
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
1.2144 1.1989 1.1613 1.1406 1.0626 1.1250
1.6953 1.6266 1.5997 1.5757 1.5438 1.5328
1.7564 1.7070 1.6052 1.6156 1.8075 1.8160
1.9675 1.8837 1.8285 1.7887 1.7566 1.9892
2.0297 1.8896 1.7956 1.7815 1.8318 1.7559
2.1317 1.9528 1.7913 1.7658 1.8659 1.8536
2.2563 2.1609 2.1533 2.0857 2.0083 1.9366
2.3442 2.2769 2.0778 1.9838 1.7678 1.8902
3.2223 2.4834 2.2024 2.2071 2.1853 2.0427
400
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
Table 6 Effect of volume fraction index n and thickness ratio (a=h ¼ 10,20) on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square SCSC (SUS304/Si3N4) FGM plate. a/h
n
Amplitude ratio ðwmax =hÞ 0.4
0.8
1.2
1.6
2.0
2.5
3.0
10
0 0.5 1.0 2.0 5.0 10 20
1.8021 1.5874 1.4142 1.0170 1.3258 1.6972 1.7078
2.3109 1.0008 1.2675 1.2644 1.2858 1.4295 2.1414
2.6300 1.5458 1.5129 1.5040 1.4767 1.5295 1.5363
2.8015 1.6314 1.6463 1.6929 1.5057 1.6933 1.7776
3.0557 2.9435 1.8700 1.7843 1.8317 1.9354 2.0201
3.4309 3.13089 2.0262 2.1517 2.1517 2.1743 2.3155
3.7179 3.5417 2.1559 2.2543 2.2543 3.3274 3.4180
20
0 0.5 1.0 2.0 5.0 10 20
1.1703 1.1439 1.1851 1.0328 1.3726 1.3783 1.6682
1.2342 1.1953 1.2575 1.2820 1.2919 1.2919 1.4623
1.4923 1.4240 1.4339 1.3666 1.4855 1.4841 1.5517
1.7982 1.7515 1.8003 1.7797 1.4880 1.6734 1.7034
1.9464 1.8907 1.8464 1.7966 1.8993 1.9651 2.0167
2.1040 1.9308 1.9821 1.6951 1.9368 2.0987 2.1705
2.6612 2.0971 1.9718 1.8074 2.0235 2.0094 2.3860
Table 7 Effect of volume fraction index n and thickness ratio ða=h ¼ 10,20Þ on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square SSCC (Al/ZrO2) FGM plate. a/h
n
Amplitude ratio ðwmax =hÞ 0.4
0.8
1.2
1.6
2.0
2.5
3.0
10
0 0.5 1.0 2.0 5.0 10
1.2879 1.2263 1.0790 1.0697 1.6017 1.7078
1.6377 1.2516 1.2415 1.2372 1.3289 1.5407
1.8887 1.4220 1.4197 1.4792 1.5387 1.7285
1.6682 1.6014 1.6472 1.5358 1.6404 1.7400
1.7069 1.6600 1.5719 1.5892 1.7025 1.8018
1.8243 1.6789 1.4982 1.6344 1.8584 2.0114
1.9261 1.8934 1.7899 1.7471 1.9465 2.0812
20
0 0.5 1.0 2.0 5.0 10
1.0567 1.1672 1.0625 1.0711 1.0820 1.1137
1.0921 1.2404 1.2209 1.2118 1.2751 1.3337
1.4404 1.4658 1.3774 1.3295 1.4282 1.4963
1.6349 1.6680 1.6434 1.5458 1.5680 1.7200
1.8816 1.7973 1.7420 1.7123 1.7458 1.8116
1.8294 1.7620 1.6145 1.6254 1.7045 1.7907
1.9375 2.1963 2.0012 1.7120 1.7294 1.8068
increasing frequency trend at a particular amplitude ratio and then steadily increases with further increase in amplitude ratio showing hardening type of character. This behavior is most likely due to the change in stiffness values, and apparently the redistribution of mode shapes at certain level of amplitude of vibrations. It is also revealed that frequency ratio (onl =ol ) first decreases with the increase of volume fraction index n and again increases with the increase of index value n, say ðn ¼ 10Þ. Table 8 shows the effect of volume fraction index n with the frequency ratio (onl =ol ) for (Al/ZrO2) square FGM plate for different volume fraction index n and thickness ratios a=h ¼ 10,20. The plate is clamped at all its edges. It is seen that the frequency ratio decreases with the increase of thickness ratio (a/h). A mixed type of behavior in the frequency ratio (onl =ol ) is observed.
6. Conclusions The nonlinear free flexural vibration analysis of the FGM plate is analyzed using higher order shear deformation theory with a special amendment in the transverse displacement associated with nonlinear FEM in the capacity of Green–Lagrange theory. To implement this nonlinear model a nine-noded C0 continuous isoparametric Lagrangian element with 13 DOFs per node is developed and applied to accomplish the frequency ratio by utilizing direct iterative method. The governing equations are derived using the variational approach. Convergence and validation studies have been carried out to ascertain the accuracy of the present formulation. Numerical results for different aspect ratios, the thickness ratios, the volume fraction indices and different combinations of the boundary conditions have been presented. The results show the necessity and importance of the higher order nonlinear terms.
Acknowledgments The authors gratefully acknowledge the financial support by the All India Council for Technical Education (AICTE), New Delhi (F. no. 1-10/RID/NDF-PG(19)/2008-09, dated 13th March 2009) a statutory body of the Government of India. References
Table 8 Effect of volume fraction index n and thickness ratio ða=h ¼ 10,20Þ on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square CCCC (Al/ZrO2) FGM plate. a/h
n
Amplitude ratio ðwmax =hÞ 0.4
0.8
1.2
1.6
2.0
2.5
3.0
10
0 0.5 1.0 2.0 5.0 10
1.9684 1.6091 1.4210 1.1243 1.2078 1.0433
2.0164 2.0060 1.5484 1.3296 1.6388 1.7939
2.1732 1.8838 1.3118 1.2678 1.3404 1.3167
2.2922 2.1279 2.0159 1.6241 1.3452 1.7074
2.3682 1.8076 1.6685 1.6463 1.3237 1.4038
2.4913 2.0541 2.0171 1.9837 1.5808 1.7607
2.6415 2.3469 2.3221 2.1529 1.6261 1.8508
20
0 0.5 1.0 2.0 5.0 10
1.3890 1.2011 1.1774 1.0718 1.1739 1.2796
1.4091 1.3644 1.2441 1.1538 1.1704 1.2395
1.3681 1.3396 1.3141 1.2878 1.3160 1.4699
1.4796 1.5293 1.4864 1.2739 1.4655 1.4930
1.7224 1.7078 1.5947 1.3288 1.5160 1.6376
1.9074 1.8938 1.6689 1.5686 1.7768 1.7981
1.9950 1.9192 1.7569 1.6802 1.7035 2.0432
value of volume fraction index n varies from 0 to 10. It is observed that the frequency ratio increases with the rise of amplitude ratio (wmax =h). However, it is ascertained that there is sudden drop in the
[1] M. Koizumi, FGM activities in Japan, Composites Part B 28B (1997) 1–4. [2] M. Koizumi, The concept of FGM, Proceedings of the Second International Symposium on FGM, vol. 34, 1993, pp. 3–10. [3] J.N. Reddy, Analysis of functionally graded plates, Int. J. Numer. Methods Eng. 47 (2000) 663–684. [4] S.S. Vel, R.C. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, J. Sound Vib. 272 (2003) 703–730. [5] E. Efraim, M. Eisenberger, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, J. Sound Vib. 299 (2007) 720–738. [6] R. Gunes, J.N. Reddy, Nonlinear analysis of functionally graded circular plates under different load and boundary conditions, Int. J. Struct. Stab. Dyn. 8 (2008) 131–159. [7] C.S. Chen, T.J. Chen, R.D. Chien, Nonlinear vibration of initially stressed functionally graded plates, Thin-Walled Struct. 44 (2006) 844–851. [8] M. Talha, B.N. Singh, Static response and free vibration analysis of FGM plates using higher order shear deformation theory, Appl. Math. Modell. 34 (2010) 3991–4011. [9] J. Yang, H.S. Shen, Vibration characteristics and transient response of sheardeformable functionally graded plates in thermal environments, J. Sound Vib. 255 (2001) 579–602. [10] N. Sundararajan, T. Prakash, M. Ganapathi, Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environment, Finite Element Anal. Des. 42 (2005) 152–168. [11] C.S. Chen, Nonlinear vibration of a shear deformable functionally graded plate, Compos. Struct. 68 (2005) 295–302. [12] T.Y. Ng, K.Y. Lam, K.M. Liew, Effects of FGM materials on the parametric resonance of plate structures, Comput. Methods Appl. Mech. Eng. 190 (2000) 953–962.
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[13] A. Allahverdizadeh, M.H. Naei, M. Nikkhah Bahrami, Nonlinear free and forced vibration analysis of thin circular functionally graded plates, J. Sound Vib. 310 (2008) 966–984. [14] J. Woo, S.A. Meguid, L.S. Ong, Nonlinear free vibration of functionally graded plates, J. Sound Vib. 289 (2006) 595–611. [15] X.L. Huang, H.S. Shen, Nonlinear vibration and dynamic response of functionally graded plates in thermal environments, Int. J. Solids Struct. 41 (2004) 2403–2427. [16] J. Yang, H.S. Shen, Non-linear analysis of functionally graded plates under transverse and in-plane loads, Int. J. Non-linear Mech. 38 (2003) 467–482.
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[17] G.N. Paraveen, J.N. Reddy, Nonlinear transient thermoelastic analysis of functionally graded ceramic–metal plates, Int. J. Solids Struct. 35 (1998) 4457–4476. [18] H. Matasunaga, Analysis of functionally graded plates, Compos. Struct. 82 (2008) 499–512. [19] J.S. Park, J.H. Kim, Thermal postbuckling and vibration analyses of functionally graded plates, J. Sound Vib. 289 (2006) 77–93. [20] R.D. Cook, D.S. Malkus, M.E. Plesha, Concept and Application of Finite Element Analysis, third ed., John Wiley and Sons, 1989. [21] R.D. Mindlin, Influence of rotatory inertia and shear on flexural vibrations of isotropic elastic plates, J. Appl. Mech. 73 (1951) 31–38.
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Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel
Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method Mohammad Talha , B.N. Singh Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur 721 302, India
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 August 2009 Received in revised form 26 July 2010 Accepted 17 November 2010 Available online 6 January 2011
In this paper, large amplitude free flexural vibration analysis of shear deformable functionally graded material (FGM) plates are investigated. The material properties of the FGM plates are assumed to vary through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. The nonlinear finite element equations are obtained using higher order shear deformation theory with a special modification in the transverse displacement. The Green–Lagrange nonlinear strain–displacement relation with all higher order nonlinear strain terms is included in the formulation to account for the large deflection response of the plate. The fundamental equations are obtained using variational approach by employing traction free boundary conditions on the top and bottom faces of the plate. Results are obtained by employing an efficient C0 finite element with 13 degrees of freedom (DOFs) per node. Convergence tests and comparison studies have been carried out to establish the efficacy of the present model. The variation of nonlinear frequency ratio with the amplitude ratio is highlighted for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions. & 2010 Elsevier B.V. All rights reserved.
Keywords: Functionally graded material Higher order shear deformation theory Finite element method Large amplitude vibration Green–Lagrange
1. Introduction The accomplishment of functionally graded material is the realization of contemporary and distinct functions that cannot be achieved by the traditional composite materials. These are advanced composite materials with a microscopically inhomogeneous anatomy and are usually made from a mixtures of ceramic and metal using powder metallurgy techniques. Continuous changes in their microstructure distinguish FGM from other traditional composite materials. The material property of the FGM can be tailored to obtain the specific demand in different engineering applications in order to exploit the advantage of the properties of individual constituent. This is possible, because the material composition changes gradually in a preferred direction. The advantage of using this material is that it eliminates the interface problem due to smooth and continuous change of material properties from one surface to other [1,2]. Large amplitude free flexural vibration (LAFFV) behavior of a plate arises in many engineering applications, particularly in the panels of aircraft. When a structure is deflected substantially, i.e., half of its thickness, a considerable geometrical nonlinearity occurs, mostly due to the development of in-plane membrane
Corresponding author. Tel.: +91 3222 283026; fax: +91 3222 255303.
E-mail addresses: [email protected], [email protected] (M. Talha). 0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2010.11.006
stresses. These membrane stresses are tensile in nature that stiffens the plate. This stiffening effect results in the rise of resonance frequencies and change of mode shapes. Thus, the linear model is not being capable to determine the behavior of the structures completely. Therefore, in the recent years geometrically nonlinear flexural vibration of plates have received considerable attention compared to static large deflection behavior of plates. Since this area is fairly new, published literature on the nonlinear free and forced vibrations of FGM plate is limited in number and most of them are fascinated on linear problem. Reddy [3] presented theoretical formulation and finite element models (FEM) in the frame work of third order shear deformation theory for static and dynamic analyses of the FGM plates. Vel and Batra [4] presented a threedimensional analytical solution of simply supported rectangular FGM plates for free and forced vibrations. Suitable displacement functions which satisfy boundary conditions are used to solve governing equations by employing the power series method. Efraim and Eisenberger [5] obtained exact free vibration frequencies and modes of variable thickness thick annular FGM plates. Gunes and Reddy [6] investigated geometrically nonlinear analysis of circular FGM plates subjected to mechanical and thermal loads. They used Green–Lagrange strain tensor with all its terms in the analysis. Chen et al. [7] derived nonlinear partial differential equations for the vibration motion of initially stressed FGM plates. The formulation are derived for the nonlinear vibration motion of the FGM in a general state of arbitrary initial stresses, based on classical laminated plate theory (CLPT). Talha and Singh [8] studied free vibration and static analysis of
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
FGM plates using modified HSDT kinematics. The fundamental equations are obtained using variational principle by considering the stress free boundary conditions at the top and bottom faces of the plate. Yang and Shen [9] analyzed the free and forced vibration analyses for initially stressed FGM plates in thermal environment. Temperature dependent material properties are assumed for the analysis and the formulations are based on Reddy’s higher order shear deformation theory, which includes the thermal effects due to uniform temperature variation. Sundararajan et al. [10] developed nonlinear formulation based on von-Karman assumptions to study the free vibration characteristics of FGM plates in thermal environment. They obtained nonlinear governing equations using Lagrange’s equations of motion and solved using FEM, coupled with direct iterative technique. Chen [11] presented nonlinear vibration for FGM plate in the state of non-uniform initial stresses. Galerkin method have been used for transformation of the governing nonlinear partial differential equations into ordinary nonlinear differential equations, and the nonlinear and linear frequencies are obtained using the Runge– Kutta method. Ng et al. [12] presented the parametric resonance of FGM rectangular plates under harmonic in-plane loading. It is found that the parametric resonance of FGM rectangular plate varies by varying the power-law exponent, which controls the material distribution in the structures. Allahverdizadeh et al. [13] developed a semi-analytical approach for nonlinear free and forced axi-symmetric vibrations of a thin circular FGM plates. The formulation is based on the CLPT kinematics and the geometric nonlinearity is incorporated in vonKarman sense. Woo et al. [14] provided an analytical solution for the nonlinear free vibration behavior of FGM plates. The governing equations for thin rectangular FGM plates are obtained using the von-Karman theory for large transverse deflection, and mixed Fourier series analysis is used to get the solution. Huang and Shen [15] studied nonlinear vibration and dynamic response of FGM plates in thermal environments. The formulations are based on the HSDT kinematics and general von-Karman type equation, which includes thermal effects. Yang and Shen [16] investigated large deflection and postbuckling responses of FGM rectangular plates by using semi-analytical approach under transverse and in-plane loads. The formulations are based on the CLPT kinematics. Paraveen and Reddy [17] investigated the response of FGM plates using FEM that accounts for the transverse shear strains, rotary inertia and moderately large rotations in the von-Karman sense. The gradation of properties is assumed according to power-law throughout the thickness and comparisons have been made with homogeneous isotropic plates. The determination of accurate nonlinear behavior of the FGM fundamentally depends on the theory used to model the structure. The classical laminated plate theory which requires that normals to the mid-plane remain normal during plate deflections may be inappropriate for analysis of the FGM plates, in which volume fractions of two or more materials vary continuously as a function of position in a preferred direction. The inaccuracy occurs due to neglecting the effects of transverse shear and normal strains in the plate [18]. Due to continuous variation in material properties, the first order shear deformation theory and higher order shear deformation theory may be conveniently used in the analysis. It is noted that the first order shear deformation theory proposed by Mindlin [21] does not satisfy the parabolic variation of transverse shear strain in the thickness direction. Consequently, the solution accuracy of the FSDT depends on the shear correction factors. Therefore, it has to be incorporated to adjust the transverse shear stiffness. Generally, in the HSDT kinematics the in-plane displacements are assumed to be a cubic expression of the thickness coordinate and the out-of-plane displacement to be constant. In the present study, the structural model kinematics assumes the
395
cubically varying in-plane displacement over the entire thickness, while the transverse displacement varies quadratically to achieve the accountability of normal strain and its derivative in calculation of transverse shear strains. Thus, the development of higher order shear deformation theory for describing the mechanical behavior of FGM structures has been of high importance to the researchers. In this regard the geometrically nonlinear free flexural vibrations of the FGM plates using HSDT kinematics by incorporating geometric nonlinearity in Green–Lagrange sense is appropriate to examine the responses of FGM structure accurately. It is accomplished from the literature that the nonlinear free vibration analysis of the FGM plates using HSDT kinematics with geometric nonlinearity in Green–Lagrange sense has not been reported in the literature to the best of the authors’ knowledge. The present research aims to develop a higher order shear deformation theory with a special modification in transverse displacement that provides an additional freedom to the displacements through the thickness, and consequently eliminates the over prediction. The geometric nonlinearity in the formulation has been incorporated by taking all higher order nonlinear strain terms associated with Green–Lagrange theory. The governing equations are formulated using the variational approach. A nonlinear C0 continuous isoparametric FEM is proposed to minimize the computational exercise required in the disposition of element matrices without compromising the solution credibility. The nonlinear fundamental frequencies are obtained for different thickness ratios, the aspect ratios, the amplitude ratios, and for different volume fraction indices and boundary conditions. The obtained results are compared with those available in the literature.
2. Theoretical formulation 2.1. Geometric configuration and material properties The FGM plate is regarded to be a single layer plate of uniform thickness. Here we ascertain the FGM plate of length a, width b and total thickness h made from an isotropic material of metal and ceramics, in which the composition varies from top to bottom surface as shown in Fig. 1. The top surface ðZ ¼ þ h=2Þ of the plate is ceramic rich, whereas the bottom surface ðZ ¼ h=2Þ is metal rich. The coordinates x, y are considered along the in-plane directions and z along the thickness direction. The formulations are restricted here with the assumption of a linear elastic material behavior for small strains and large displacements. The elastic material properties vary through the plate thickness according to the volume fractions of the constituents. Power-law distribution is commonly used to describe the variation of material a
x b Ceramic
+h/2 -h/2 Metal
y z
Fig. 1. Geometry and dimensions of the plate.
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where ax ¼ @cz =@x, bx ¼ @w=@x, yx ¼ @xz =@x, ay ¼ @cz =@y, by ¼ @w=@y, yy ¼ @xz =@y and xz ¼ az and f1 ðzÞ ¼ C1 zC2 z3 , f2 ðzÞ ¼ C3 z2 , f3 ðzÞ ¼ C4 z3 , f4 ðzÞ ¼ C5 z3 , f5 ¼ C1 z, f6 ¼ C1 z2 , C1 ¼ 1, C2 ¼ C4 ¼ 4=3h2 , C3 ¼ 1=2, C5 ¼ 1=3. The displacement vector for the model based on the modified displacement field as given in Eq. (4) is written as fLg ¼ ½u,v,w, cx , cy , cz , ax , ay , az , bx , by , yx , yy T
ð5Þ
2.3. Strain–displacement relation The nonlinear Green–Lagrange strain–displacement relation for FGM plate can be represented as [20] 9 8 9 > @u > 8 > > > > @x > > e xx > > > > > > > > @v > > > > > > > e yy > @y > > > > > > > > > > > > > > > > @w > > > > e > > > = < zz = < @z > fe g ¼ g yz ¼ @w þ @v @y @z > > > > > > > > > @u @w > >g > > > xz > > > > > þ @x > > > > > @z > > > > > > > > >g > > > xy > > > @v @u > > > > > ; > : @x þ @y > > > ; :
Fig. 2. Variation of the volume fractions Vc through the thickness.
properties (Fig. 2), which is expressed as [19] PðzÞ ¼ ðPc Pm ÞVc þPm
z 1 n þ Vc ðzÞ ¼ h 2
ð0 rn r1Þ
ð1Þ ð2Þ
where P denotes the effective material property, Pm and Pc represents the properties of the metal and ceramic, respectively, Vc is the volume fraction of the ceramic and n is the volume fraction exponent. The effective material properties of the plate including, Young’s modulus E, density r vary according to Eq. (1) and Poisson’s ratio n is assumed to be constant. The above power-law theory estimates a simple rule of mixtures which is used to find the effective properties of the ceramic–metal graded plate. This rule of mixtures applies in the thickness direction only.
@u @x
2 @u @y
2
þ þ
2 @v @x
2 @v @y
þ þ
2 @w @x
2 @w @y
9 > > > > > > > > > > > > > > > > > > > =
2 2 þ @v þ @w @z @z þ 2> h i > > > > > @u @u @v @v @w > > > > 2 @y @z þ @y @z þ @w > > @y @z > > > > h i > > > > > > @u @u @v @v @w @w > > 2 @x @z þ @x @z þ @x > > @z > > > > > > h i > > > > @u @u @v @v @w @w > > ; : 2 @x @y þ @x @y þ @x @y
fe g ¼ fe l g þfe nl g
In the present study, system of governing equations for FGM plate is derived by using variational approach. The origin of the material coordinates is at the middle of the plate as shown in Fig. 1. For accurate analysis of transverse shear effects in the mathematical formulation the HSDT kinematics has been used, with a special modification in the transverse displacement. The in-plane displacements u , v and the transverse displacement w for the plate is assumed as uðx,y,z,tÞ ¼ uðx,y,tÞ þzcx ðx,y,tÞ þ z2 xx ðx,y,tÞ þ z3 rx ðx,y,tÞ vðx,y,z,tÞ ¼ vðx,y,tÞ þ zcy ðx,y,tÞ þ z2 xy ðx,y,tÞ þ z3 ry ðx,y,tÞ ð3Þ
where t is the time, u, v, and w are the corresponding displacements of a point on the mid-plane. cx and cy are the rotations of normal to the mid-plane about the y-axis and x-axis, respectively. The functions xx , xy , rx and ry are the higher order terms in the Taylor series expansion defined in the mid-plane of the plate. The higher order terms are determined by diminishing the transverse shear stresses txz ¼ t4 and tyz ¼ t5 on the top and bottom surfaces of the plate, and by applying this boundary condition the displacement field is modified as u ¼ u þ f1 ðzÞcx þf2 ðzÞax þ f3 ðzÞbx þ f4 ðzÞyx v ¼ v þ f1 ðzÞcy þf2 ðzÞay þ f3 ðzÞby þ f4 ðzÞyy w ¼ w þ f5 ðzÞcz þ f6 ðzÞaz
2
@u @z
ð6Þ
or, in compact form Eq. (6) may be written as
2.2. Displacement field and strains
wðx,y,z,tÞ ¼ wðx,y,tÞ þzcz ðx,y,tÞ þ z2 xz ðx,y,tÞ
8 > > > > > > > > > > > > > > > > > > > 1<
ð4Þ
ð7Þ
where fe l g and fe nl g are the linear and nonlinear strain vectors, respectively. Substituting Eq. (6) in Eq. (3) the strain displacement of the FGM plate is expressed as 8 nl0 9 28 9 8 nl1 93 8 09 e1 > k1 > > > k11 > e1 > > > > > > > > > > > > > > > > > 6> >7 > 0> > 1> > nl0 > > nl1 > > > > > 6 > > > > > > > > e k e k > > > >7 > > > > 2 2 2 2 6 > > > > > > > > > > 6> >7 > 0> > 1> > nl0 > > nl1 > > > > > = 6< k = 1 < k =7
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
9 8 8 nl6 9 > > knl5 k1 > > > > 1 > > > > > > > > > > > nl5 nl6 > > > > > k > > > > k 2 > > > > 2 > > > > > > > > = = < < 0 0 51 61 þz þz nl5 2k > > > 2> 2 0 > > > 4 > > > > > > > > > > > 2knl5 > > > > > > 0 > > > 5 > > > > > > > > > ; > nl5 ; : 2knl6 > : 2k6 6
397
R h=2 R h=2 R h=2 where ½D1 ¼ h=2 ½TTl ½Q ½Tl dz, ½D2 ¼ h=2 ½TTl ½Q ½Tnl dz, ½D3 ¼ h=2 R h=2 ½TTnl ½Q ½Tl dz, ½D4 ¼ h=2 ½TTnl ½Q ½Tnl dz and [Q] is the stiffness ð8Þ
coefficients. 2.6. Kinetic energy
The above strain–displacement relation (Eq. (8)) can be expressed in terms of mid-plane strain vector by classifying the linear strain vector fel g and nonlinear strains vector fenl g separately, as
The kinetic energy of the FGM plate can be expressed as Z ð17Þ T ¼ 12 rfu_ gT fu_ gdV V
where
where r and fug are the density and global displacement vector of the plate. The global displacement field model as given by Eq. (4) may be represented as
fe l g ¼ fe01 e02 e03 e04 e05 e06 k11 k12 k13 k14 k15 k16 k21 k22 k24 k25 k26 k31 k32 k36 g
fug ¼ ½NfLg
feg ¼ fel g þ fenl g ¼ ½Tl fe l g þ 12½Tnl fe nl g
ð9Þ
ð10Þ fe nl g ¼
9 8 nl0 nl0 nl0 nl0 nl0 nl0 nl1 nl1 nl1 nl1 nl1 nl1 nl2 nl2 nl2 nl2 > > > e1 e2 e3 e4 e5 e6 k1 k2 k3 k4 k5 k6 k1 k2 k3 k4 > 1 < nl2 nl2 nl3 nl3 nl3 nl3 nl3 nl3 nl4 nl4 nl4 nl4 nl4 nl4 nl5 nl5 = k5 k6 k1 k2 k3 k4 k5 k6 k1 k2 k3 k4 k5 k6 k1 k2 > 2> > > ; : 0 knl5 knl5 knl5 knl6 knl6 0 0 0 knl6 5 4 6 1 2 6
ð11Þ The linear strain vector in terms of mid-plane strain vector fel g can be further expressed as fel g201 ¼ ½w2013 fLg131
2.4. Constitutive relations The constitutive relation describes how the stresses and strains are related within the plate and is expressed as 9 2 9 8 38 exx > sxx > Q11 Q12 Q13 0 0 0 > > > > > > > 6 > > > 7 > > > > > eyy > syy > 0 0 0 7> > > > > 6 Q12 Q22 Q23 > > > > > 6 > > > 7> > > > < szz = 6 Q13 Q23 Q33 < 7 e 0 0 0 7 zz = 6 ¼6 ð13Þ 7 g s > 6 0 > > 0 0 7> 0 0 Q44 > > 6 > > yz > > yz > 7> > > > > > > gxz > s > > 6 > > 0 7 0 0 0 0 Q55 > > > 5> > 4 > > xz > > > > ; ; : sxy > :g > 0 0 0 0 0 Q 2
3
where Q11 ¼ Q22 ¼ Q33 ¼ EðzÞð1n Þ=ð13n 2n Þ, Q12 ¼ Q13 ¼ Q23 ¼ EðzÞnð1 þ nÞ=ð13n2 2n3 Þ, Q44 ¼ Q55 ¼ Q66 ¼ EðzÞ=2ð1þ nÞ. The elastic modulus E, and the coefficients Qij vary in the thickness direction of the plate according to Eqs. (1) and (2). 2.5. Strain energy of FGM plate The strain energy of the FGM plate is given by Z 1 U¼ fegT fsg dV 2 v
Eq. (15) in expanded form is written as U¼
1 2
V
f l gTi ½D1 f l gi þ 12 f l gTi ½D2 f nl gi þ 12 f nl gTi ½D3 f l gi þ 14 f nl gTi ½D4 f nl gi
e
e
e
A
Z
A
e
e
e
3. Finite element implementation A nine-noded quadrilateral C0 isoparametric element is used in the present study. The domain is discretized into a set of finite elements. Over each of the elements, the displacement vector and element geometry of the model is expressed by fLg ¼
NN X i¼1
Ni fLgi ; x ¼
NN X i¼1
Ni xi ; y ¼
NN X
Ni yi
ð21Þ
i¼1
where Ni is the interpolation function (shape function) for the ith node, fLgi is the vector of unknown displacements for the ith node, NN is the number of nodes per element and xi and yi are Cartesian coordinate of the ith node. 3.1. Governing equation
ð22Þ for i ¼ 1,2, . . . where T is the kinetic energy, U is the strain energy, fqi g and fq_ i g are the generalized coordinates and generalized velocities, respectively. The equilibrium equation for nonlinear free vibration analysis with large deformation (i.e., small strains and large displacements relation) can be represented as € þð½K þ gKc þ 12 ½Knl1 þ ½Knl2 þ 12½Knl3 Þfqg ¼ 0 ½Mfqg
! dA
The governing equation for nonlinear free vibration problem of the FGM plate can be derived using variational principle, which is the generalization of the principle of virtual displacement. Lagrange equation for a conservative system can be written as 8 2 3 9 cz 2 cz 2 > > > > ax @@x þ bx @@x > > > 6 7 > > > > > 6 7 Z = < 2 2 6 7 d @T @U @ g 6 @cz @xz 7dv ¼ 0 þ y þ a þ þ x y @x @y 6 7 > dt @fq_ i g @fqi g @fqi g > 2 > > 6 > 2 2 7 > > 4 5 > > > xz > > ; : þ by @w þ yy @@y @y
ð14Þ
By substituting the strains and stresses from Eqs. (9) and (13), the above equation becomes Z 1 U¼ fe þ enl gTi ½Q fel þ enl gi dV ð15Þ 2 V l
e
Substituting Eq. (17) into Eq. (18), the kinetic energy becomes Z Z Z rfL_ gT ½NT ½NfL_ g dz dA ¼ 12 fL_ gT ½mfL_ g dA ð20Þ T ¼ 12
xy
66
2
e
where fLg is as defined in Eq. (5), and the function of thickness coordinate ½N may be represented as 2 3 1 0 0 f1 ðzÞ 0 0 f2 ðzÞ 0 0 f3 ðzÞ 0 f4 ðzÞ 0 6 7 ð19Þ ½N ¼ 4 0 1 0 0 f1 ðzÞ 0 0 f2 ðzÞ 0 0 f3 ðzÞ 0 f4 ðzÞ 5 0 0 1 0 0 f5 ðzÞ 0 0 f6 ðzÞ 0 0 0 0
ð12Þ
where ½w and fLg are a matrix of differential operator and displacement field vector, respectively. The superscripts 0, 1, 2–3 in Eq. (10) and nl0, nl1, and nl2–6 in Eq. (11) represents the membrane, bending and higher order terms, respectively.
Z
ð18Þ
ð16Þ
ð23Þ
where ½M, g,½Kc ,½K,½Knl1 ,½Knl2 ,½Knl3 , and fqg, are the global mass matrix, penalty parameter that enforces constraints, the global
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linear stiffness matrix arising due to constrains, the global linear stiffness matrix, the global nonlinear stiffness matrices and global displacement vector, respectively. Assuming the system vibrating in principal mode with natural frequency o, Eq. (23) can be conveniently reduced to nonlinear generalized eigenvalue problem as ½Kfqg ¼ l½Mfqg
ð24Þ
2
with l ¼ o , where o is defined as frequency of natural vibration. This eigenvalue problem is solved by employing direct iterative procedure. Initially, the linear response with normalized first mode is calculated by solving Eq. (24) with all nonlinear [Knl1], [Knl2], and [Knl3] terms fixed to zero. This normalized vector is scaled up in such a way that the maximum deflection is equal to the desired amplitude w/h (w is the maximum transverse displacement and h is the thickness of the plate). After that nonlinear stiffness matrices [Knl1], [Knl2], and [Knl3] are obtained with the first mode. Then using updated stiffness matrices, eigenvalue and its eigenvector are obtained. This attained eigenvector is used to compute the nonlinear stiffness matrices for the next iteration. This exercise continues till two eigenvalues from the two subsequent iterations reached the tolerance limit r 103 . The convergence is supposed to reach at said tolerance limit, and the corresponding o is the nonlinear frequency (onl ) of the vibrated FGM plate.
The following sets of boundary conditions are considered in the present study: Simply supported: (SSSS) u0 ¼ w0 ¼ cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0
at x ¼ 0 and a
v0 ¼ w0 ¼ cx ¼ ay ¼ az ¼ bx ¼ yy ¼ 0
at y ¼ 0 and b
Clamped: (CCCC) u0 ¼ v0 ¼ w0 ¼ cx ¼ cy ¼ cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0 at x ¼ 0,a and y ¼ 0,b Simply supported and clamped: (SCSC) u0 ¼ w0 ¼ cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0
at x ¼ 0 and y ¼ 0
u0 ¼ v0 ¼ w0 ¼ cx ¼ cy ¼ cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0 at x ¼ a, y ¼ b Simply supported and clamped: (SSCC) u0 ¼ w0 ¼ cy ¼ ax ¼ az ¼ by ¼ yx ¼ 0
at x ¼ 0 and a
u0 ¼ v0 ¼ w0 ¼ cx ¼ cy ¼ cz ¼ ax ¼ ay ¼ az ¼ bx ¼ by ¼ yx ¼ yy ¼ 0 at y ¼ 0,b 4.1. Convergence and validation study
4. Numerical results In the present work, large amplitude free flexural vibration behavior of the FGM plates has been addressed. The LAFFV responses are computed using the proposed mathematical model in conjunction with nonlinear FEM. The geometric nonlinearity is incorporated in the formulation applying Green–Lagrange theory by taking all higher order terms. A computer program has been developed in MATLAB 7.5.0 (R2007b) environment. The validation and efficacy of the proposed algorithm is examined by comparing the results with those available in the literature. A nine-noded Lagrange isoparametric element, with 117 degrees of freedom per element for the present HSDT kinematics has been used for discretizing the plate. Full integration schemes (3 3) and selective integration schemes (2 2) are used for thick and thin plates, respectively, to compute the results. In the present analysis simply supported boundary conditions are used to check the efficacy of the model. However, the formulation and code does not impose any limitations. The properties of the FGM constituents at room temperature (300 K) are shown in Table 1, which have been used for the computation of the results throughout the study, unless specified otherwise. The features of volume fraction of the ceramic phase through the dimensionless thickness is depicted in Fig. 2. In the analysis, it is assumed that the materials are perfectly elastic throughout the deformation.
Table 1 Properties of the FGM components. Material
Aluminium (Al) Alumina (Al2O3) Zirconia (ZrO2) Silicon nitride (Si3N4) Titanium alloy (Ti–6Al–4V) Stainless steel(SUS304)
The accuracy of the present finite element formulation is validated by comparing the results with those available in the literature [15] which is based on the HSDT and general von-Karman type of nonlinearity. A convergence study is also presented. A FGM square plate, simply supported at all four edges is analyzed. In this example, the analysis is performed with volume fraction index n¼2, aspect ratio a=b ¼ 1 with sides of the plate a ¼ b ¼ 0:2 m, for different amplitude ratios (wmax =h). wmax is the maximum deflection at the center of the plate. The FGM plate comprised titanium alloy, Ti–6Al–4V (metal) and zirconia, ZrO2 (ceramic). The results have been carried out in terms of frequency ratio onl =ol with various mesh divisions as shown in Table 2. This clearly shows that Table 2 Comparison of fundamental nonlinear frequency ratio ðonl =ol Þ for the various values of amplitude ratios of a square SSSS (Ti–6AL–4V/ZrO2) FGM plate with different mess sizes. Mesh size
Amplitude ratio ðwmax =hÞ 0.2
0.4
0.6
0.8
1.0
Present (2 2) Present (3 3) Present (4 4) Present (5 5) Huang and Shen [15]
1.0493 1.0476 1.0467 1.0455 1.021
1.1597 1.1468 1.1405 1.1409 1.082
1.2951 1.2857 1.2786 1.2765 1.176
1.4864 1.4492 1.4300 1.4244 1.296
% Difference
2.343
5.162
7.873
9.014
1.6333 1.6094 1.6064 1.6055 1.436 10.557
Table 3 Comparison of fundamental nonlinear frequency ratio ðonl =ol Þ for the various values of amplitude ratios of a square SSSS (Si3N4/SUS304) FGM plate (a/h ¼10). Properties ðwmax =hÞ
E (N/m2)
n
r ðkg=m3 Þ
70 109 380 109 151 109 427 109 105.7 109 207.78 109
0.30 0.30 0.30 0.28 0.298 0.28
2707 3800 3000 3210 4429 8166
0.2 0.4 0.6 0.8 1.0
FEM Ref. [10]
Present
% Difference
1.0063 1.0654 1.1707 1.3155 1.4789
1.0198 1.1164 1.2377 1.4214 1.6390
1.323 3.712 5.413 7.452 9.768
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
the solution accuracy and the rate of convergence with mesh refinement are good for frequency ratio onl =ol for the different values of amplitude ratios (wmax =h). Based on the convergence, it is concluded that ð5 5Þ mesh is sufficient for LAFFV analysis. Table 3 shows the comparison of the frequency ratio onl =ol for Si3N4/ SUS304 simply supported square FGM plate. The side to thickness ratio (a/h) is taken as 10. The top surface of the plate is ceramic rich (Si3N4), and bottom surface is metal rich (SUS304). It can be observed that the solutions from this study agrees well with those presented by Sundararajan et al. [10]. It is accomplished from Tables 2 and 3, as the amplitude ratio (wmax =h) increases, the % difference in frequency ratio onl =ol increases. This is due to the fact that Refs. [15,10], had incorporated the geometric nonlinearity in von-Karman sense. Whereas, in the present study geometric nonlinearity is included in Green–Lagrange sense with all higher order nonlinear strain terms. However, it has been observed from the detailed analysis that, the von-Karman type of geometric nonlinearity gives fairly good results up to ðwmax =hÞ ¼ 0:6, for the sakeof conciseness it is not shown here. The difference in the result suggests the requirement of a better mathematical model for the structures which undergo small strain having large deformation and/or large rotation.
5. Parametric study Based on the established approach and analysis of aforementioned sections, the simply supported (SSSS), clamped (CCCC), simply supported–simply supported and clamped–clamped (SSCC) and simply supported and clamped (SCSC) boundary conditions have been employed in the analysis. It is also observed that ð5 5Þ mesh gives good convergence as mentioned earlier, and have been used for accomplishing the results, unless otherwise stated. Table 4 shows the effect of volume fraction index n and thickness ratio a=h ¼ 10 on the frequency ratio (onl =ol ) of a SSSS square titanium alloy (Ti–6AL–4V) and zirconium oxide (ZrO2) FGM plate. The properties of the constituents are provided in Table 1 as mentioned earlier. The top surface of the plate is ceramic rich whereas the bottom surface is metal rich. The non-dimensional linear frequency is assumed as o ¼ oða2 =hÞ½rm ð1n2 Þ=Em 1=2 . It is observed from the table that the frequency ratio decreases with the increase in the volume fraction index n. This decrease in frequency ratio is on expected lines, because increase in the volume fraction index n means that a plate has a smaller ceramic component, and thus its stiffness is reduced and consequently its non-dimensional fundamental linear frequency is decayed. It can be additionally viewed that the frequency ratio enhances with the rise in amplitude ratio. Table 5 shows the effect of variation of thickness ratios (a=h ¼ 52100Þ, ranging from thick to thin plates, with the volume fraction index n. Here, FGM plates consisting of (Ti–6AL–4V/ZrO2) with all simply supported edges. The aspect ratio is taken as a=b ¼ 1 with volume fraction index n ¼ 0 and 1. Choosing n index as zero
399
corresponds to the fully ceramic plate and 1 complies linear variation of ceramic and metal constituent throughout the thickness of the plate. It is observed that the frequency ratio (onl =ol ) decreases with the increase of thickness ratio and it shows more significant differences at lower thickness ratio than larger thickness ratio. This implies that the higher order theory is more suitable for thick plates and higher order terms must be considered for the thick plate condition. It is also revealed that the nonlinear frequency ratio is slightly higher for fully ceramic plate i.e., (n ¼ 0) considered here, than (n ¼ 1). It is also accomplished that the frequency ratio increases with the increase of amplitude ratio. However, mixed types of behavior on the frequency ratio have also been observed due to the presence of severe nonlinearity. Therefore, it can be concluded that there is a great need to analyze the structures with geometric nonlinearity in Green–Lagrange sense to closely monitor the response of structures. Table 6 shows the variation of frequency ratio (onl =ol ) with amplitude ratio (wmax =h) for SCSC square stainless steel (SUS304) and silicon nitride (Si3N4) FGM plate. The fundamental linear frequency is non-dimensionalized as o ¼ oða2 =hÞðrc =Ec Þ1=2 . It is observed from the table that the frequency ratio decreases with the increase in the volume fraction index n up to a certain value, say n ¼ 2, and then the frequency ratio (onl =ol ) increases with additional increase in volume fraction index n. It is realized that the value of linear frequency decreases for higher value of index n and accordingly the nonlinear frequency decreases. It is also observed that the degradation in linear frequency at higher volume fraction index n is more than that of the nonlinear frequency. Hence, the overall trend of frequency ratio (onl =ol ) increases. The frequency ratio is on higher side for a=h ¼ 10 compared to a=h ¼ 20. Table 7 describes the frequency ratio (onl =ol ) for aluminum (Al) zirconia (ZrO2) square FGM plate for different volume fraction index n and thickness ratios a=h ¼ 10,20. Here it is ascertained that the SSCC boundary conditions and the fundamental linear frequency are non-dimensionalized as o ¼ oða2 =hÞðrc =Ec Þ1=2 . The
Table 5 Effect of variation of thickness ratios (a/h ¼ 5–100) with the volume fraction index n for (Ti–6AL–4V/ZrO2) FGM plates consisting with SSSS boundary condition. a/b
n
a/h
Amplitude ratio ðwmax =hÞ 0.5
1.0
1.2
1.4
1.6
1.8
2.0
1
0
5 15 25 50 100
2.1878 1.5223 1.1636 1.1662 1.5491
2.3648 1.9828 1.5159 1.5809 1.5291
2.5774 2.5139 1.7680 1.7383 1.7330
2.5253 2.0053 1.9931 1.8765 1.7827
2.8965 2.1878 2.0526 1.9418 1.8225
2.9843 2.3212 2.2484 2.0165 2.1086
2.9779 2.4036 2.2014 2.1394 1.8534
1
1
5 15 25 50 100
1.9459 1.4192 1.3077 1.2114 1.2080
2.3176 1.6088 1.6066 1.5622 1.4951
2.4829 1.7816 1.7874 1.7250 1.6407
2.3710 1.8540 1.7932 1.8279 1.7625
2.7433 2.1428 1.8241 1.5265 1.8041
2.5920 2.2791 2.1910 2.1771 1.8915
2.7165 2.3801 2.1139 2.1221 1.9076
Table 4 Effect of volume fraction index n and thickness ratio ða=h ¼ 10Þ on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square SSSS (Ti–6AL–4V/ZrO2) FGM plate. n
0 0.2 0.4 0.6 0.8 1.0
Amplitude ratio ðwmax =hÞ 0.5
1.0
1.2
1.4
1.6
1.8
2.0
2.5
3.0
1.2144 1.1989 1.1613 1.1406 1.0626 1.1250
1.6953 1.6266 1.5997 1.5757 1.5438 1.5328
1.7564 1.7070 1.6052 1.6156 1.8075 1.8160
1.9675 1.8837 1.8285 1.7887 1.7566 1.9892
2.0297 1.8896 1.7956 1.7815 1.8318 1.7559
2.1317 1.9528 1.7913 1.7658 1.8659 1.8536
2.2563 2.1609 2.1533 2.0857 2.0083 1.9366
2.3442 2.2769 2.0778 1.9838 1.7678 1.8902
3.2223 2.4834 2.2024 2.2071 2.1853 2.0427
400
M. Talha, B.N. Singh / Finite Elements in Analysis and Design 47 (2011) 394–401
Table 6 Effect of volume fraction index n and thickness ratio (a=h ¼ 10,20) on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square SCSC (SUS304/Si3N4) FGM plate. a/h
n
Amplitude ratio ðwmax =hÞ 0.4
0.8
1.2
1.6
2.0
2.5
3.0
10
0 0.5 1.0 2.0 5.0 10 20
1.8021 1.5874 1.4142 1.0170 1.3258 1.6972 1.7078
2.3109 1.0008 1.2675 1.2644 1.2858 1.4295 2.1414
2.6300 1.5458 1.5129 1.5040 1.4767 1.5295 1.5363
2.8015 1.6314 1.6463 1.6929 1.5057 1.6933 1.7776
3.0557 2.9435 1.8700 1.7843 1.8317 1.9354 2.0201
3.4309 3.13089 2.0262 2.1517 2.1517 2.1743 2.3155
3.7179 3.5417 2.1559 2.2543 2.2543 3.3274 3.4180
20
0 0.5 1.0 2.0 5.0 10 20
1.1703 1.1439 1.1851 1.0328 1.3726 1.3783 1.6682
1.2342 1.1953 1.2575 1.2820 1.2919 1.2919 1.4623
1.4923 1.4240 1.4339 1.3666 1.4855 1.4841 1.5517
1.7982 1.7515 1.8003 1.7797 1.4880 1.6734 1.7034
1.9464 1.8907 1.8464 1.7966 1.8993 1.9651 2.0167
2.1040 1.9308 1.9821 1.6951 1.9368 2.0987 2.1705
2.6612 2.0971 1.9718 1.8074 2.0235 2.0094 2.3860
Table 7 Effect of volume fraction index n and thickness ratio ða=h ¼ 10,20Þ on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square SSCC (Al/ZrO2) FGM plate. a/h
n
Amplitude ratio ðwmax =hÞ 0.4
0.8
1.2
1.6
2.0
2.5
3.0
10
0 0.5 1.0 2.0 5.0 10
1.2879 1.2263 1.0790 1.0697 1.6017 1.7078
1.6377 1.2516 1.2415 1.2372 1.3289 1.5407
1.8887 1.4220 1.4197 1.4792 1.5387 1.7285
1.6682 1.6014 1.6472 1.5358 1.6404 1.7400
1.7069 1.6600 1.5719 1.5892 1.7025 1.8018
1.8243 1.6789 1.4982 1.6344 1.8584 2.0114
1.9261 1.8934 1.7899 1.7471 1.9465 2.0812
20
0 0.5 1.0 2.0 5.0 10
1.0567 1.1672 1.0625 1.0711 1.0820 1.1137
1.0921 1.2404 1.2209 1.2118 1.2751 1.3337
1.4404 1.4658 1.3774 1.3295 1.4282 1.4963
1.6349 1.6680 1.6434 1.5458 1.5680 1.7200
1.8816 1.7973 1.7420 1.7123 1.7458 1.8116
1.8294 1.7620 1.6145 1.6254 1.7045 1.7907
1.9375 2.1963 2.0012 1.7120 1.7294 1.8068
increasing frequency trend at a particular amplitude ratio and then steadily increases with further increase in amplitude ratio showing hardening type of character. This behavior is most likely due to the change in stiffness values, and apparently the redistribution of mode shapes at certain level of amplitude of vibrations. It is also revealed that frequency ratio (onl =ol ) first decreases with the increase of volume fraction index n and again increases with the increase of index value n, say ðn ¼ 10Þ. Table 8 shows the effect of volume fraction index n with the frequency ratio (onl =ol ) for (Al/ZrO2) square FGM plate for different volume fraction index n and thickness ratios a=h ¼ 10,20. The plate is clamped at all its edges. It is seen that the frequency ratio decreases with the increase of thickness ratio (a/h). A mixed type of behavior in the frequency ratio (onl =ol ) is observed.
6. Conclusions The nonlinear free flexural vibration analysis of the FGM plate is analyzed using higher order shear deformation theory with a special amendment in the transverse displacement associated with nonlinear FEM in the capacity of Green–Lagrange theory. To implement this nonlinear model a nine-noded C0 continuous isoparametric Lagrangian element with 13 DOFs per node is developed and applied to accomplish the frequency ratio by utilizing direct iterative method. The governing equations are derived using the variational approach. Convergence and validation studies have been carried out to ascertain the accuracy of the present formulation. Numerical results for different aspect ratios, the thickness ratios, the volume fraction indices and different combinations of the boundary conditions have been presented. The results show the necessity and importance of the higher order nonlinear terms.
Acknowledgments The authors gratefully acknowledge the financial support by the All India Council for Technical Education (AICTE), New Delhi (F. no. 1-10/RID/NDF-PG(19)/2008-09, dated 13th March 2009) a statutory body of the Government of India. References
Table 8 Effect of volume fraction index n and thickness ratio ða=h ¼ 10,20Þ on the fundamental nonlinear frequency ratio ðonl =ol Þ of a square CCCC (Al/ZrO2) FGM plate. a/h
n
Amplitude ratio ðwmax =hÞ 0.4
0.8
1.2
1.6
2.0
2.5
3.0
10
0 0.5 1.0 2.0 5.0 10
1.9684 1.6091 1.4210 1.1243 1.2078 1.0433
2.0164 2.0060 1.5484 1.3296 1.6388 1.7939
2.1732 1.8838 1.3118 1.2678 1.3404 1.3167
2.2922 2.1279 2.0159 1.6241 1.3452 1.7074
2.3682 1.8076 1.6685 1.6463 1.3237 1.4038
2.4913 2.0541 2.0171 1.9837 1.5808 1.7607
2.6415 2.3469 2.3221 2.1529 1.6261 1.8508
20
0 0.5 1.0 2.0 5.0 10
1.3890 1.2011 1.1774 1.0718 1.1739 1.2796
1.4091 1.3644 1.2441 1.1538 1.1704 1.2395
1.3681 1.3396 1.3141 1.2878 1.3160 1.4699
1.4796 1.5293 1.4864 1.2739 1.4655 1.4930
1.7224 1.7078 1.5947 1.3288 1.5160 1.6376
1.9074 1.8938 1.6689 1.5686 1.7768 1.7981
1.9950 1.9192 1.7569 1.6802 1.7035 2.0432
value of volume fraction index n varies from 0 to 10. It is observed that the frequency ratio increases with the rise of amplitude ratio (wmax =h). However, it is ascertained that there is sudden drop in the
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