Buckling and Vibration of Functionally Graded Non-uniform Circular Plates Resting on Winkler Foundation

An investigation on the effect of uniform tensile in-plane force on the radially symmetric vibratory characteristics of functionally graded circular plates of linearly varying thickness along radial direction and resting on a Winkler foundation has been carried out on the basis of classical plate theory. The non-homogeneous mechanical properties of the plate are assumed to be graded through the thickness and described by a power function of the thickness coordinate. The governing differential equation for such a plate model has been obtained using Hamilton’s principle. The differential transform method has been employed to obtain the frequency equations for simply supported and clamped boundary conditions. The effect of various parameters like volume fraction index, taper parameter, foundation parameter and the in-plane force parameter has been analysed on the first three natural frequencies of vibration. By allowing the frequency to approach zero, the critical buckling loads for both the plates have been computed. Threedimensional mode shapes for specified plates have been plotted. Comparison with existing results has been made.


INTRODUCTION
Now-a-days, technologists are able to tailor advanced materials by mixing two or more materials to get the desired mechanical properties along one/ more direction(s) due to their extensive demand in many fields of modern engineering applications.Functionally graded materials (FGMs) are the recent innovation of this class.In classic ceramic/metal FGMs, the ceramic phase offers thermal barrier effects and protects the metal from corrosion and oxidation and the FGM is toughened and strengthened by the metallic constituent.By nature, such materials are microscopically non-Latin American Journal of Solids and Structures 12 (2015) 2231-2258 homogeneous, in which the material properties vary continuously in a certain manner either along a line or in a plane or in space.The plate type structural elements of FGMs have their wide applications in solar energy generators, nuclear energy reactors, space shuttle etc. and particularly in defence -as penetration resistant materials used for armour plates and bullet-proof vests.This necessitates to study their dynamic behaviour with a fair amount of accuracy.
Since the introduction of FGMs by Japanese scientists in 1984 (Koizumi, 1993), a number of researches dealing with the vibration characteristics of FGM plates of various geometries has been made due to their practical importance.A critical review of the work upto 2012 has been given by Jha et al. (2013).In addition to this, natural frequencies of functionally graded anisotropic rectangular plates have been studied by Batra and Jin (2005) using first-order shear deformation theory coupled with the finite element method.The first and third-order shear deformation plate theories have been used by Ferreira et al. (2006) in analyzing the free vibrations of rectangular FGM plates using global collocation method by approximating the trial solution with multiquadric radial basis functions.The same method has been employed by Roque et al. (2007) to present the free vibration analysis of FGM rectangular plates using a refined theory.Zhao et al. (2009) used element-free kp-Ritz method for free vibration analysis of rectangular and skew plates with different boundary conditions taking four types of functionally graded materials on the basis of first-order shear deformation theory.Liu et al. (2010) have analysed the free vibration of FGM rectangular plates with inplane material inhomogeneity using Fourier series expansion and a particular integration technique on the basis of classical plate theory.Navier solution method has been used by Jha et al. (2012) to analyse the free vibration of FG rectangular plates employing higher order shear and normal deformation theory.The vibration behaviour of rectangular FG plates with non-ideal boundary conditions has been studied by Najafizadeh et al. (2012) using Levy method and Lindstedt-Poincare perturbation technique.The vibration solutions for FGM rectangular plate with in-plane material inhomogeneity have been obtained by Uymaz et al. (2012) using Ritz method and assuming the displacement functions in the form of Chebyshev polynomials on the basis of five-degree-of-freedom shear deformable plate theory.Thai et al. (2013) have developed an efficient shear deformation theory for vibration of rectangular FGM plates which accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors.Recently, Dozio (2014) has derived first-known exact solutions for free vibration of thick and moderately thick FGM rectangular plates with at least on pair of opposite edges simply supported on the basis of a family of two-dimensional shear and normal deformation theories with variable order.Very recently, the natural frequencies of FGM nanoplates are analyzed by Zare et al. (2015) for different combinations of boundary conditions by introducing a new exact solution method.
Under normal working conditions, plate type structures may be subjected to in-plane stressing arising from hydrostatic, centrifugal and thermal stresses ( Brayan (1890-91), Leissa (1982), Wang et al. (2004)), which may induce buckling, a phenomenon which is highly undesirable.Numerous studies dealing with the effect of uniaxial/biaxial in-plane forces on the vibrational behaviour of FGM plates are available in the literature and reported by Feldman and Aboudi (1997), Najafizadeh and Heydari (2008), Mahdavian (2009), Baferani et al. (2012), Javaheri and Eslami (2002), Prakash and Ganapati (2006), Zhao (2009), Zhang et al. (2014).Among these, Feldman and Aboudi (1997) analysed the elastic-bifurcational buckling of FG rectangular plates under in-plane compressive loading employing a combination of micromechanical and structural approach.The closed form solutions for the axisymmetric buckling of FG circular plates under uniform radial compression have been obtained by Najafizadeh and Heydari (2008) on the basis of higher order shear deformation plate theory.Baferani et al. (2012) have used Bessel functions to analyze the symmetric and asymmetric buckling modes of functionally graded annular plates under mechanical and thermal loads.Thermal buckling of FGM rectangular plates on the basis of classical plate theory for four different types of thermal loading has been presented by Javaheri and Eslami (2002) using Galerkin's method.Finite element method has been applied by Prakash and Ganapati (2006) to analyse the asymmetric thermal buckling and vibration characteristics of FGM circular plates.Recently, Zhang et al. (2014) have used local Kringing meshless method for the mechanical and thermal buckling analysis of rectangular FGM plates on the basis of first-order shear deformation plate theory.
Plates with tapered thickness are broadly used in many engineering structural elements such as turbine disks, aircraft wings and clutches etc.With appropriate variation of thickness, these plates can have significantly greater efficiency for vibration as compared to the plate of uniform thickness and also provide the advantage of reduction in weight and size.Several researches have been made to study the vibrational behaviour of functionally graded plates of varying thickness.Exact element method has been used by Efraim and Eigenberger (2007) for the vibration analysis of thick annular isotropic and FGM plates of three forms of thickness variation: linear, quadratically concave and quadratically convex.Naei et al. (2007) have presented the buckling analysis of radially loaded FGM circular plate of linearly varying thickness using finite-element method.Free vibration analysis of functionally graded thick annular plates with linear and quadratic thickness variation along the radial direction is investigated by Tajeddini and Ohadi (2011) using the polynomial-Ritz method.Recently, the free vibrations of FGM circular plates of linearly varying thickness under axisymmetric condition have been analysed by Shamekhi (2013) using a meshless method in which point interpolation approach is employed for constructing the shape functions for Galerkin weak form formulation.
The problem of plates resting on an elastic foundation has achieved great importance in modern technology and foundation engineering.Airport runways, submerged floating tunnels, bridge decks, building footings, reinforced-concrete pavements of highways, railway tracks, buried pipelines and foundation of storage tanks etc. are some direct applications of the foundations.From the viewpoint of practical utility, the commonly used elastic foundation is Winkler's model.In this model, the foundation is virtually replaced by mutually independent, closely spaced linear elastic springs providing the resistance and reaction at every point which is taken to be proportional to the deflection at that point.This consideration leads to satisfactory results in a variety of problems.In this regard, numerous studies analyzing the effect of Winkler foundation on the dynamic behaviour of non-FGM plates are available in the literature and the recent ones are reported by Gupta et al. (2006), Hsu (2010), Li and Yuan (2011), Kägo and Lellep (2013), Zhong et al. (2014), Ghaheri et al. (2014), to mention a few.However, regarding FGM plates, a very limited work is available and done by Amini et al. (2009), Kumar and Lal (2013), Kiani and Eslami (2013), Joodaky (2013), Fallah (2013), Chakraverty and Pradhan (2014).Of these, Amini et al. (2009) Kumar and Lal (2013) predicted the natural frequencies for axisymmetric vibrations of two-directional FG annular plates resting on Winkler foundation using differential quadrature method and Chebyshev collocation technique.
In the present study, a semi analytical approach: differential transform method proposed by Zhou (1986), has been employed to study the effect of in-plane force on the axisymmetric vibrations of functionally graded circular plate of linearly varying thickness and resting on a Winkler foundation.According to this method, the governing differential equation of motion for the present model of the plate gets reduced to a recurrence relation.Use of this recurrence relation in the transformed boundary conditions together with the regularity condition, one obtains a set of two algebraic equations.These resulting equations have been solved using MATLAB to get the frequencies.The material properties i.e.Young's modulus and density are assumed to be graded in the thickness direction and these properties vary according to a power-law in terms of volume fractions of the constituents.The natural frequencies are obtained for clamped and simply supported boundary conditions with different values of volume fraction indexn g , in-plane force parameter N , taper parameter γ and foundation parameter f K .Critical buckling loads for varying values of plate parameter have been computed.Three-dimensional mode shapes for the first three modes of vibration have been presented for the specified plates.A comparison of results has been given.1) is given by (Leissa, (1969)):

MATHEMATICAL FORMULATION
where w is the transverse deflection, D the flexural rigidity, ν the Poisson's ratio and a comma followed by a suffix denotes the partial derivative with respect to that variable.

N
For a harmonic solution, the deflection w can be expressed as where ω is the radian frequency and can be expressed as . The Eq. ( 1) reduces to (2 ) Latin American Journal of Solids and Structures 12 (2015) 2231-2258 Assuming that the top and bottom surfaces of the plate are ceramic and metal-rich, respectively, for which the variations of the Young's modulus ( ) E z and the density ( ) z ρ in the thickness direc- tion are taken as follows (Dong, (2008)): where c E , c ρ and m E , m ρ denote the Young's modulus and the density of ceramic and metal constituents, respectively and ( ) V z is the volume fraction of ceramic as follows: where g is used for the volume fraction index of the material.
The flexural rigidity and mass density of the plate material are given by Using Eqs.(4-6) into Eq.( 7) and Eq. ( 8), we obtain Assuming linear variation in the thickness i.e.
( ) , γ being the taper parameter and 0 h is the non-dimensional thickness of the plate at the center.Substituting the values of D and ρ from Eq. ( 9) and Eq. ( 10) into Eq.( 11), we get Where Eq. ( 12) is a fourth-order differential equation with variable coefficients whose exact solution is not possible.The approximate solution with appropriate boundary and regularity conditions has been obtained employing differential transform method.

Boundary and regularity conditions
Following Zhou (1986), the relations which should be satisfied for simply supported/clamped plate at the boundary and regularity condition at the centre are given as follows: (i) Simply-supported edge (iii) Regularity conditions at the centre ( 0 r = ) of the plate where r M is the radial bending moment and r Q the radial shear force.
Latin American Journal of Solids and Structures 12 (2015) 2231-2258 3 METHOD OF SOLUTION

Description of the method
According to differential transform method Zhou (1986), an analytic function (r) f in a domain 0 r r a − ≤ is expressed as a power series about the point 0 r .The differential transform of its The inverse transformation of the function k F is defined as Combining the above two expressions, we have, In actual applications, the function (r) f is expressed by a finite series.So, Eq. ( 18) may be written as: The convergence of the natural frequencies decides the value of n .Some basic theorems which are frequently used in the practical problems are given in Table 1.
Original Functions Transformed Functions

Transformation of the governing differential equation
Applying the transformation rules given in Table 1, the transformed form of the governing differential equation ( 12) around 0 0 r = can be written as

FREQUENCY EQUATIONS
Since, the subscripts of the F − terms should be non-negative, so in Eq. ( 20), the subscript k should starts with 3. Starting with 3 k = in Eq. ( 20), we get a recursive relation i.e. 4 F is deter- mined in terms of 0 F and 2 F in terms of 2 F and 4 F and so on.Therefore, all the F terms can be expressed in terms of 0 F and 2 F .Now, applying the boundary condition (21) on the re- sulted k F expressions, we get the following equations: .Eq. ( 24) can be expressed in matrix form as follows: For a non-trivial solution of Eq. ( 25), the frequency determinant must vanish and hence Similarly, for clamped edge condition, the corresponding frequency determinant is given by

NUMERICAL RESULTS AND DISCUSSION
The frequency Eqs. ( 26) and ( 27) provide the values of the frequency parameter Ω .The lowest three roots of these equations have been obtained using MATLAB to investigate the influence of in-plane force parameter N , taper parameter γ , foundation parameter f K and volume fraction index g on the frequency parameter Ω for both the boundary conditions.In the present analysis, the values of Young's modulus and density for aluminium as metal and alumina as ceramic constituents are taken from Dong (2008), as follows: The variation in the values of Poisson's ratio is assumed to be negligible all over the plate and its value is taken as ν = 0.3.From the literature, the values of various parameters are taken as: Volume fraction index g = 0, 1, 3, 5; In-plane force parameter N = -30, -20, -10, 0, 10, 20, 30; Taper parameter γ = -0.5, -0.3, -0.1, 0, 0.1, 0.3, 0.5; and Foundation parameter f K = 0, 10, 50, 100.In order to choose an appropriate value of k , a computer program developed to evaluate frequency parameter Ω was run for different sets of the values of plate parameters for both the boundary conditions taking k = 48, 49,…, 56.Then, the difference between the values of the frequency parameter Ω for two successive values of k was checked continually with varying values of k for various values of plate parameters till the accuracy of four decimals is attained i.e.
( ) ( ) 1 0.00005 for all the three modes i = 1, 2, 3.For the present study, the number of terms k has been taken as 55 as there was no further improvement in the values of Ω for the first three modes for both the boundary conditions, considered here.The relative error 56 / 1 rel j E = Ω Ω − , j = 48 (1) 56 for a specified plate N = 30, g = 5, γ = 0.5, f K = 100 has been shown in Figure 2 for the first three modes of vibration, as maximum deviations were obtained for this data.The numerical results have been given in Tables 2-6 and presented in Figures 3-8.It is observed that the values of the frequency parameter Ω for simply supported plate are less than those for a clamped plate for the same set of the values of other parameters.The values of critical buckling loads for a clamped plate are higher than that for the corresponding simply supported plate.In Figures 3(a, b, c), the behaviour of volume fraction index g on the frequency parameter Ω for two values of in-plane force parameter N = -5, 10 and taper parameter γ = -0.5, 0.5 for a fixed value of foundation parameter f K = 50 for both the plates has been presented.It has been observed that the value of frequency parameter Ω decreases with the increasing value of g for both tensile as well as compressive in-plane forces for both the plates except for the fundamental mode of vibration.The corresponding rate of decrease is higher for smaller values of g (< 2) as compared to the higher values of g (> 3).Further, it increases with the increase in the number of modes.In case of fundamental mode (Figure 3(a)), the frequency parameter Ω increases as g increases for both the plates when plate becomes thinner and thinner towards the boundary (i.e.γ = 0.5) in presence of tensile in-plane force (i.e.N =10).However, for the simply supported plate vibrating in the fundamental mode, the frequency parameter is found monotonically increasing with increasing values of g irrespective of the values of γ ( ± 0.5) as well as in-plane force parameter N (-5, 10).The effect of in-plane force parameter N on the frequency parameter Ω for three different values of taper parameter γ = -0.3,0, 0.3 taking g = 5 and f K =10 for both the plates vibrating in the first three modes has been shown in Figures 4(a, b, c).It has been noticed that the value of the frequency parameter increases with the increasing value of N whatever be the values of other plate parameters.This effect increases with the increasing number of modes.The rate of increase in the values of Ω with N increases as the plate becomes thicker and thicker towards the outer edge i.e. γ changes from positive values to negative values.The rate of increase is higher for simply supported plate as compared to clamped plate.Figures 5(a, b, c) show the graphs for foundation parameter f K verses frequency parameter Ω for two different values of taper parameter γ = -0.5, 0.5 and in-plane force parameter N = -5, 10 for a fixed value of volume fraction index g = 5 for all the three modes.It can be seen that the frequency parameter Ω increases with the increasing values of foundation parameter f K .This effect is more pronounced for tensile in-plane forces (i.e.N = 10) as compared to compressive inplane forces (i.e.N = -5) with respect to the change in the values of taper parameter γ from 0.5 to -0.5 keeping other plate parameters fixed and also increases with the increase in the number of modes for both the plates.
To study the effect of taper parameter γ on the frequency parameter Ω , the graphs have been plotted for fixed value of foundation parameter f K = 50 for two values of volume fraction index g = 0, 5 and three values of in-plane force parameter N = -5, 0, 10 and given in Figures 6(a,   b, c) for all the three modes and for both the plates.It has been observed that for a clamped plate vibrating in the fundamental mode of vibration, the values of the frequency parameter Ω decrease monotonically with the increasing values of taper parameter γ from -0.5 to 0.5 whatever be the values of g as well as N .In case of simply supported plate, for g = 0, the values of frequency parameter has a point of minima having a tendency of shifting from -0.1 towards 0.3 as N changes from compressive to tensile i.e. takes the values as -5, 0, 10.However, for g = 5, the values of Ω increase continuously for N = -5, 0 while there is a point of minima in the vicinity of γ = 0 for N = 10.For the second mode of vibration, the values of frequency parameter Ω monotonically decrease with the increasing values of g and N for both the plates except for simply supported plate for N = -5, g = 5.In this case, there is a point of maxima in the vicinity of γ = -0.1.
The rate of decrease in the values of Ω with increasing values of taper parameter γ is higher in case of clamped plate as compared to simply supported plate keeping other parameters fixed.The effect of in-plane force parameter is more pronounced for γ = 0.5 for both the plates whatever be the value of g .In case of the third mode of vibration, the behaviour of the frequency parameter Ω with the taper parameter γ is almost similar to that of the second mode for both the plates except that the rate of decrease is much higher than the second mode.By allowing the frequency to approach zero, the values of the critical buckling load parameter cr N in compression for different values of volume fraction index g = 1, 3; taper parameter γ = -0.3,0.3 and foundation parameter f K = 0, 10, 100 for both the plates have been computed in Table 4.For selected values of volume fraction index g = 0, 5; taper parameter γ = -0.5, -0.1, 0, 0.1, 0.5 for f K = 50, the plots for the critical buckling load parameter cr N for both the plates vibrating in the fundamental mode of vibration have been given in Figures 7(a, b).From the results, it has been observed that the values of the critical buckling load parameter cr N for a clamped plate are higher than that for the corresponding simply supported plate.Also, the value  Vol'mir (1966) exact solutions, Pardoen (1978) obtained by using finite element method have been presented in Table 5.An excellent agreement among the results has been noticed.Three-dimensional mode shapes for the first three modes of vibrations for a specified plate g = 5, N = 30, γ = -0.5, f K = 100 are shown in Figure 8 for both the plates.
The results for the frequency parameter Ω with those obtained by Rayleigh-Ritz method (Singh and Saxena, 1995), exact element method (Eisenberger and Jabareen, 2001), generalized differential quadrature rule (Wu and Liu, 2001) has been given in Table 6.A close agreement of the results for both the boundary conditions shows the versatility of the present technique.Table 7: Percentage variation in the values of frequency parameter with respect to γ = 0 when γ changes from -0.5 to 0 and 0 to 0.5 for N = 0, 30 and g = 0, 5 taking f K = 50.

CONCLUSIONS
The effect of thickness variation and elastic foundation on the buckling and free axisymmetric vibration of functionally graded circular plate has been analysed employing differential transform method.The numerical results show that: 1. Values of frequency parameter Ω for an isotropic plate are higher than those for the corresponding FGM plate i.e. frequency parameter decreases as the contribution of metal constituent increases.
2. The values of frequency parameter Ω for a clamped plate are higher than that for the corresponding simply supported plate whatever be the values of other plate parameters.
3. As the plate becomes thicker and thicker towards the outer edge i.e. the value of the taper parameter γ changes from negative to positive, the values of the frequency parameter Ω decreases continuously and the rate of decrease is higher for the clamped plate as compared to the simply supported plate.
4. With the increase in the value of volume fraction index g , the frequency parameter Ω decreases irrespective to the values of the other plate parameters.The rate of decrease increases with increase in the number of modes for both the plates.
5. The values of frequency parameter also increase with the increase in the values of foundation parameter f K whatever be the values of other plate parameters.
6.The values of critical buckling loads for an isotropic plate ( g = 0) are higher than the corresponding FGM plate ( g > 0) i.e. in order to obtain the highest critical buckling loads, the ceramic isotropic plate is much suitable than FGM plate.
7. The clamped plate pursue more critical buckling load than the corresponding simply supported plate.
8. The values of the critical buckling load parameter cr N decreases with the increasing values of volume fraction index g keeping other plate parameters fixed.9.As the foundation stiffness increases, the values of critical buckling load parameter cr N increases for both the plates.
10.The percentage variations in the values of the frequency parameter Ω with varying values of in-plane force parameter N = 0, 30 and volume fraction index g = 0, 5 for both the plates for two cases (i) when the plate becomes thicker and thicker toward the outer edge i.e. γ changes from 0 to -0.5 (ii) when the plate becomes thinner and thinner towards the outer edge i.e. γ change from 0 to 0.5 for all the three modes have been computed and given in Table 7.It has been noticed that for both the plates, in absence of in-plane force when the plate changes its nature from isotropic to FGM, the percentage variation decreases for all the modes and for both the cases.This behaviour remains same in the presence of tensile in-plane force, when the plate is vibrating in the fundamental mode of vibration while percentage variation increases for the second and third modes of vibration.
11.The differential transform method, being straightforward and easy to apply for such type of problems, gives highly accurate results with less computational efforts as compared to the other conventional methods like differential quadrature and finite element methods etc.The results presented in this paper can serve as benchmark solutions for future investigations.

unctionally
Graded Non-uniform Circular Plates Resting on Winkler Foundation 2235 Latin American Journal of Solids and Structures 12 (2015) 2231-2258 section of tapered FGM circular plate 0 and resting on Winkler foundation.
Ω of degree m .

*
denotes that the frequency parameter does not exist because of buckling

have provided the Latin American Journal of Solids and Structures 12 (2015) 2231-2258 exact
three-dimensional vibration results for rectangular FGM plates resting on Winkler foundation by employing Chebyshev polynomials and Ritz's method.Recently,

Table 1 :
Transformation Rules for one-dimensional DTM.

Table 2 :
Values of frequency parameter for simply supported plate.
Latin American Journal ofSolids and Structures 12 (2015) 2231-2258* denotes that the frequency parameter does not exist because of buckling

Table 3 :
Values of frequency parameter for clamped plate.

Table 4 :
Critical buckling load in compression.

Latin American Journal of Solids and Structures 12 (2015) 2231-2258 of
cr N decreases with the increasing values of the volume fraction index g for both the boundary conditions whereas the value of cr N increases with the increasing values of f K for all the values of g and γ for both the plates.A comparative study for critical buckling load parameter cr N for an isotropic plate with Gupta and Ansari (1998) obtained by using Ritz method,

Table 5 :
Wu and Liu (2001)abareen (2001)by exact element method cWu and Liu (2001)by generalized differential quadrature rule Comparison of frequency parameter Ω for linear varying thickness taking Singh and Saxena (1995))by Rayleigh-Ritz method b f K = .

Table 6 :
Comparison of critical buckling load parameter cr