# Buckling Analysis of Functionally Graded Material Plates Using Higher Order Shear Deformation Theory

Authors: B. Sidda Reddy; J. Suresh Kumar; C. Eswara Reddy; K. Vijaya Kumar Reddy
Journal: Journal of Composites
 (2013)

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## Abstract

The prime aim of the present study is to present analytical formulations and solutions for the buckling analysis of simply supported functionally graded plates (FGPs) using higher order shear deformation theory (HSDT) without enforcing zero transverse shear stresses on the top and bottom surfaces of the plate. It does not require shear correction factors and transverse shear stresses vary parabolically across the thickness. Material properties of the plate are assumed to vary in the thickness direction according to a power law distribution in terms of the volume fractions of the constituents. The equations of motion and boundary conditions are derived using the principle of virtual work. Solutions are obtained for FGPs in closed-form using Navier’s technique. Comparison studies are performed to verify the validity of the present results from which it can be concluded that the proposed theory is accurate and efficient in predicting the buckling behavior of functionally graded plates. The effect of side-to-thickness ratio, aspect ratio, modulus ratio, the volume fraction exponent, and the loading conditions on the critical buckling load of FGPs is also investigated and discussed.


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## Body

## 1. Introduction

Functionally graded materials (FGMs) are the new generation of novel composite materials in the family of engineering composites, whose properties are varied smoothly in the spatial direction microscopically to improve the overall structural performance. These materials offer great promise in high temperature environments, for example, wear-resistant linings for handling large heavy abrasive ore particles, rocket heat shields, heat exchanger tubes, thermo-electric generators, heat engine components, plasma facings for fusion reactors, and electrically insulating metal/ceramic joints and also these are widely used in many structural applications such as mechanics, civil engineering, optical, electronic, chemical, mechanical, biomedical, energy sources, nuclear, automotive fields, and ship building industries to minimize thermomechanical mismatch in metal-ceramic bonding. Most structures, irrespective of their use, will be subjected to dynamic loads during their operational life. Increased use of FGMs in various structural applications necessitates the development of accurate theoretical models to predict their response.In the past, a variety of plate theories have been proposed to study the buckling behavior of FGM plates. The classical plate theory (CPT) provides acceptable results only for the analysis of thin plates and neglects the transverse shear effects. Javaheri and Eslami [1], Abrate [2], Mohammadi et al. [3], Mahdavian [4], Feldman and Aboudi [5], Shariat et al. [6], and Tung and Duc [7] employed this theory to analyze buckling behavior of FG plates. However, for moderately thick plates CPT underpredicts deflections and overpredicts buckling loads and natural frequencies. The first-order shear deformation theories (FSDTs) are based on Reissner [8] and Mindlin [9] accounts for the transverse shear deformation effect by means of a linear variation of inplane displacements and stresses through the thickness of the plate, but requires a correction factor to satisfy the free transverse shear stress conditions on the top and bottom surfaces of the plate. Although the FSDT provides a sufficiently accurate description of response for thin to moderately thick plates, it is not convenient to use due to difficulty with determination of the correct value of shear correction factor [10]. The authors [11–16] used FSDT to analyze the buckling of FG plates. In order to overcome the limitations of FSDT many HSDTs were developed that involve higher order terms in Taylors expansions of the displacements in the thickness coordinate. Javaheri and Eslami [17], Najafizadeh and Heydari [18], Bodaghi and Saidi [19], Bagherizadeh et al. [20], and Mozafari and Ayob [21] used the HSDT to analyze the buckling behavior of FG plates. Ma and Wang [22] have investigated the axisymmetric large deflection bending and postbuckling behavior of a functionally graded circular plate under mechanical, thermal, and combined thermal-mechanical load based on classical nonlinear von Karman plate theory. They observed from their investigation that the power law index “n” has a significant effect on the midplane temperature, critical buckling temperature, and on the thermal post-buckling behavior of FGM plate. Hosseini-Hashemi et al. [23] have developed the closed-form solutions in analytical form to study the buckling behavior of in-plane loaded isotropic rectangular FG plates without any use of approximation for different boundary conditions using the Mindlin plate theory. Saidi et al. [24] employed the unconstrained third-order shear deformation theory to analyze the axisymmetric bending and buckling of FG solid circular plates in which the bending-stretching coupling exists. Oyekoya et al. [25] developed Mindlin type and Ressner type element for modeling of FG composite plate subjected to buckling and free vibration. Further, they studied the plate for the effect of different fiber distribution cases and the effects of fire distribution on buckling, and free vibration. Ghannadpour et al. [26] applied finite strip method to analyze the buckling behavior of rectangular FG plats under thermal load. The solution was obtained by the minimization of the total potential energy and solving the corresponding eigenvalue problem. Thai and Choi [27] presented a simple refined theory to analyze the buckling behavior of FG plates which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The governing equations were derived from the principle of minimum total potential energy. The effects of loading conditions and variations of power of functionally graded material, modulus ratio, aspect ratio, and thickness ratio were also investigated by these authors. Thai and Vo [10] have developed a new sinusoidal shear deformation theory to study the bending, buckling and vibration of FG plates accounting for sinusoidal distribution of transverse shear stress and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Uymaz and Aydogdu [28] analyzed the rectangular FG plates under different axial loadings for buckling based on small strain elasticity theory with different boundary conditions. They also investigated the effects of the different material composition and the plate geometry on the critical buckling loads and mode shapes.Lal et al. [29] have examined the second order statistics of postbuckling responses of FGM plate subjected to mechanical and thermal load with nonuniform temperature changes subjected to temperature independent and dependent material properties. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions, and types of loadings subjected to temperature independent and temperature dependent material properties were investigated through numerical examples.This paper aims to develop analytical formulations and solutions for the buckling analysis of functionally graded plates (FGPs) using higher order shear deformation theory (HSDT) without enforcing zero transverse shear stress on the top and bottom surfaces of the plate. This does not require shear correction factor. The plate material is graded through the thickness direction. The plate’s governing equations and its boundary conditions are derived by employing the principle of virtual work. Solutions are obtained for FGPs in closed-form using Navier’s technique and solving the eigenvalue equation. The present results are compared with the solutions of Thai and Choi [27] to verify the accuracy of the proposed theory in predicting the critical buckling loads of FG plates. The effect of side-to-thickness ratios, aspect ratios, and modulus ratios and the volume fraction exponent on the critical buckling loads are studied after establishing the accuracy of the present results for FG plates.


## 2. Theoretical Formulation

In formulating the higher order shear deformation theory, a rectangular plate of lengtha, width b, and thickness h is considered, which composed of functionally graded material through the thickness. Figure 1 shows the functionally graded material plate with the rectangular Cartesian coordinate system x, y, and z. The material properties are assumed to be varied in the thickness direction only and the bright and dark areas correspond to ceramic and metal particles, respectively. On the top surface (z=+h/2), the plate is composed of full ceramic and graded to the bottom surface (z=-h/2) which composed of full metal. The reference surface is the middle surface of the plate (z=0). The functionally graded material plate properties are assumed to be the function of the volume fraction of constituent materials. The functional relationship between the material property and the thickness coordinates is assumed to be
(1)P(z)=(Pt-Pb)(zh+12)n+Pb,
where P denotes the effective material property, Pt  and Pb denote the property on the top and bottom surface of the plate, respectively, and n is the material variation parameter that dictates the material variation profile through the thickness. The effective material properties of the plate, including Young’s modulus, E, density, ρ, and shear modulus, G, vary according to (1), and poisons ratio (υ) is assumed to be constant.Figure 1
Functionally graded plate and coordinates.


### 2.1. Displacement Models

In order to approximate 3D plate problem to a 2D one, the displacement componentsu(x,y,z,t), v(x,y,z,t), and w(x,y,z,t) at any point in the plate are expanded in terms of the thickness coordinate. The elasticity solution indicates that the transverse shear stress varies parabolically through the plate thickness. This requires the use of a displacement field, in which the in-plane displacements are expanded as cubic functions of the thickness coordinate. In addition, the transverse normal strain may vary nonlinearly through the plate thickness. The displacement field which satisfies the above criteria may be assumed in the form:
(2)u(x,y,z)=uo(x,y)+zθx(x,y)+z2uo*(x,y)+z3θx*(x,y)v(x,y,z)=vo(x,y)+zθy(x,y)+z2vo*(x,y)+z3θy*(x,y)w(x,y,z)=wo(x,y),
where uo, vo, and wo  denote the displacements of a point (x,y) on the midplane.θ
x, θy are rotations of the normal to the mid plane about y and x-axes.u
0
*, v0*, θx*, and θy* are the higher order deformation terms defined at the mid plane.By substitution of displacement relations from (2) into the strain displacement equations of the classical theory of elasticity, the following relations are obtained:
(3)εx=εxo+zkx+z2εxo*+z3kx*,εy=εyo+zky+z2εyo*+z3ky*,εz=0,γxy=εxyo+zkxy+z2εxyo*+z3kxy*,γyz=φy+zεyzo+z2φy*,γxz=φx+zεxzo+z2φx*,
where
(4)εxo=∂uo∂x,εyo=∂vo∂y,εxyo=∂uo∂y+∂vo∂xkx=∂θx∂x,ky=∂θy∂y,kxy=∂θx∂y+∂θy∂xkx*=∂θx*∂x,ky*=∂θy*∂y,kxy*=∂θx*∂y+∂θy*∂xεxo*=∂uo*∂x,εyo*=∂vo*∂y,εxyo*=∂uo*∂y+∂vo*∂xφy=θy+∂wo∂y,φx=θx+∂wo∂xεyzo=2vo*,εxzo=2uo*,φy*=3θy*,φx*=3θx*.


### 2.2. Elastic Stress-Strain Relations

The elastic stress-strain relations depend on which assumption ofεz=0. In the case of functionally graded materials the constitutive equations can be written as
(5){σxσyτxyτyzτxz}=[Q11Q12000Q12Q2200000Q3300000Q4400000Q55][εxεyγxyγyzγxz],
where σ=(σx,σy,τxy,τyz,τxz)t are the stresses, ε=(εx,εy,γxy,γyz,γxz)t are the strains with respect to the axes, and Qij’s are the plane stress reduced elastic coefficients in the plate axes that vary through the plate thickness given by
(6)Q11=Q22=E(Z)1-υ2=(Ec-Em)((z/h)+(1/2))n+Em1-υ2,Q12=Q21=υQ11,Q33=Q44=Q55=(1-υ2)2(1+υ)Q11,
where Ec is the modulus of elasticity of the ceramic material and Em is the modulus of elasticity of the metal.


### 2.3. Governing Equations of Motion

The work done by the actual forces in moving through virtual displacements, which are consistent with the geometric constraints of a body, is set to zero to obtain the equation of motion and this is known as energy principle. It is useful in (a) deriving governing equations and the boundary conditions and (b) obtaining approximate solutions by virtual methods.Energy principles provide alternative means to obtain the governing equations and their solutions. In the present study, the principle of virtual work is used to derive the equations of motion of functionally graded plates.The governing equations of displacement model in (2) will be derived using the dynamic version of the principle of virtual displacements; that is,
(7)∫0T‍(δU+δV-δK)dt=0,
where δU= virtual strain energy, δV= virtual work done by applied forces, δK= virtual kinetic energy, δU+δV= total potential energy.The virtual strain energy, work done, and kinetic energy is given by(8)δU=∫A{∫-h/2h/2[σxδ∈x+σyδ∈y+τxyδγxy+τxzδγxz+τyzδγyz]dz∫-h2h2[σxδ∈x+σyδ∈y+τxyδγxy}dxdy,δV=-∫(qw0+N-x∂wo∂xδ∂wo∂x+N-xy∂wo∂yδ∂wo∂x+N-yx∂wo∂xδ∂wo∂y+N-y∂wo∂yδ∂wo∂y)dxdy,δK=∫A{∫-h/2h/2‍ρ0[(u˙0+zθ˙x+z2u˙0*+z3θ˙x*)×(δu˙0+zδθ˙x+z2δu˙0*+z3δθ˙x*)+(v˙0+zθ˙y+z2v˙0*+z3θ˙y*)×(δv˙0+zδθ˙y+z2δv˙0*+z3δθ˙y*)+w˙0δw˙0(u˙0+zθ˙x+z2u˙0*+z3θ˙x*)]dz∫-h/2h/2‍}dxdy,
where q = distributed load over the surface of the plate.N
-
x and N-y the inplane loads perpendicular to the edges x=0 and y=0, respectively, N-xyN-yx the distributed shear forces parallel to the edges x=0 and y=0, respectively, ρ0=density  of  plate  material,u˙0=∂u0/∂t,v˙0=∂v0/∂t,  and so forth, indicate the time derivatives.Substituting forδU, δV, and δK in the virtual work statement in (7) and integrating through the thickness, integrating by parts, and collecting the coefficients of δuo, δvo, δwo, δθx, δθy, δuo*, δvo*, δθx*, δθy*, the following equations of motion are obtained:
(9)δu0:∂Nx∂x+∂Nxy∂y=I1u¨0+I2θ¨x+I3u¨0*+I4θ¨x*,δv0:∂Ny∂y+∂Nxy∂x=I1v¨0+I2θ¨y+I3v¨0*+I4θ¨y*,δw0:∂Qx∂x+∂Qy∂y+q+N~=I1w¨0,
where
(10)N~=N-x∂2wo∂x2+N-xy∂2wo∂y∂x+N-yx∂2wo∂x∂y+N-y∂2wo∂y2δθx:∂Mx∂x+∂Mxy∂y-Qx=I2u¨0+I3θ¨x+I4u¨0*+I5θ¨x*,δθy:∂My∂y+∂Mxy∂x-Qy=I2v¨0+I3θ¨y+I4v¨0*+I5θ¨y*,δu0*:∂Nx*∂x+∂Nxy*∂y-2Sx=I3u¨0+I4θ¨x+I5u¨0*+I6θ¨x*,δv0*:∂Ny*∂y+∂Nxy*∂x-2Sy=I3v¨0+I4θ¨y+I5v¨0*+I6θ¨y*,δθx*:∂Mx*∂x+∂Mxy*∂y-3Qx*=I4u¨0+I5θ¨x+I6u¨0*+I7θ¨x*,δθy*:∂My*∂y+∂Mxy*∂x-3Qy=I4v¨0+I5θ¨y+I6v¨0*+I7θ¨y*,
where the force and moment resultants are defined as
(11){Nx∣Nx*Ny∣Ny*Nxy∣Nxy*}=∑L=1n‍∫-h/2h/2‍{σxσyτxy}[1∣z2]dz,{Mx∣Mx*My∣My*Mxy∣Mxy*}=∑L=1n‍∫-h/2h/2{σxσyτxy}[z∣z3]dz.
And the transverse force resultants and the inertias are given by
(12){Qx∣Sx∣Qx*Qy∣Sy∣Qy*}=∑L=1n‍∫-h/2h/2{τxzτyz}[1∣z∣z2]dz,(13)I1,I2,I3,I4,I5,I6,I7=∫-h/2h/2[(ρc-ρm)(2z-h2h)n+ρm]×(1,z,z2,z3,z4,z5,z6)dz.
The resultants in (11)-(12) can be related to the total strains in (5) by the following matrix:
(14):{NN*—MM*—QQ*}=[A∣B∣0Bt-∣Db-∣0-0-∣0-∣Ds-]{ε0ε0*—KsK*—φφ*},
where
(15)N=[NxNyNxy]t;N*=[Nx*Ny*Nxy*]t.N,N* are called the inplane force resultants
(16)M=[MxMyMxy]t;M*=[Mx*My*Mxy*]t.M,M* are called as moment resultants
(17)Q=[QxQy]t;Q*=[SxSyQx*Qy*].tQ,Q* denote the transverse force result
(18)ε0=[εx0εy0εxy0]t;ε0*=[εx0*εy0*εxy0*]tKs=[KxKyKxy]t;K*=[Kx*Ky*Kxy*]tφ=[φxφy]t;φ*=[εxz0εyz0φx*φy*]t.
The matrices [A], [B], [D], and [Ds] are the plate stiffness whose elements can be calculated using (5) and (11)-(12).


## 2.1. Displacement Models

In order to approximate 3D plate problem to a 2D one, the displacement componentsu(x,y,z,t), v(x,y,z,t), and w(x,y,z,t) at any point in the plate are expanded in terms of the thickness coordinate. The elasticity solution indicates that the transverse shear stress varies parabolically through the plate thickness. This requires the use of a displacement field, in which the in-plane displacements are expanded as cubic functions of the thickness coordinate. In addition, the transverse normal strain may vary nonlinearly through the plate thickness. The displacement field which satisfies the above criteria may be assumed in the form:
(2)u(x,y,z)=uo(x,y)+zθx(x,y)+z2uo*(x,y)+z3θx*(x,y)v(x,y,z)=vo(x,y)+zθy(x,y)+z2vo*(x,y)+z3θy*(x,y)w(x,y,z)=wo(x,y),
where uo, vo, and wo  denote the displacements of a point (x,y) on the midplane.θ
x, θy are rotations of the normal to the mid plane about y and x-axes.u
0
*, v0*, θx*, and θy* are the higher order deformation terms defined at the mid plane.By substitution of displacement relations from (2) into the strain displacement equations of the classical theory of elasticity, the following relations are obtained:
(3)εx=εxo+zkx+z2εxo*+z3kx*,εy=εyo+zky+z2εyo*+z3ky*,εz=0,γxy=εxyo+zkxy+z2εxyo*+z3kxy*,γyz=φy+zεyzo+z2φy*,γxz=φx+zεxzo+z2φx*,
where
(4)εxo=∂uo∂x,εyo=∂vo∂y,εxyo=∂uo∂y+∂vo∂xkx=∂θx∂x,ky=∂θy∂y,kxy=∂θx∂y+∂θy∂xkx*=∂θx*∂x,ky*=∂θy*∂y,kxy*=∂θx*∂y+∂θy*∂xεxo*=∂uo*∂x,εyo*=∂vo*∂y,εxyo*=∂uo*∂y+∂vo*∂xφy=θy+∂wo∂y,φx=θx+∂wo∂xεyzo=2vo*,εxzo=2uo*,φy*=3θy*,φx*=3θx*.


## 2.2. Elastic Stress-Strain Relations

The elastic stress-strain relations depend on which assumption ofεz=0. In the case of functionally graded materials the constitutive equations can be written as
(5){σxσyτxyτyzτxz}=[Q11Q12000Q12Q2200000Q3300000Q4400000Q55][εxεyγxyγyzγxz],
where σ=(σx,σy,τxy,τyz,τxz)t are the stresses, ε=(εx,εy,γxy,γyz,γxz)t are the strains with respect to the axes, and Qij’s are the plane stress reduced elastic coefficients in the plate axes that vary through the plate thickness given by
(6)Q11=Q22=E(Z)1-υ2=(Ec-Em)((z/h)+(1/2))n+Em1-υ2,Q12=Q21=υQ11,Q33=Q44=Q55=(1-υ2)2(1+υ)Q11,
where Ec is the modulus of elasticity of the ceramic material and Em is the modulus of elasticity of the metal.


## 2.3. Governing Equations of Motion

The work done by the actual forces in moving through virtual displacements, which are consistent with the geometric constraints of a body, is set to zero to obtain the equation of motion and this is known as energy principle. It is useful in (a) deriving governing equations and the boundary conditions and (b) obtaining approximate solutions by virtual methods.Energy principles provide alternative means to obtain the governing equations and their solutions. In the present study, the principle of virtual work is used to derive the equations of motion of functionally graded plates.The governing equations of displacement model in (2) will be derived using the dynamic version of the principle of virtual displacements; that is,
(7)∫0T‍(δU+δV-δK)dt=0,
where δU= virtual strain energy, δV= virtual work done by applied forces, δK= virtual kinetic energy, δU+δV= total potential energy.The virtual strain energy, work done, and kinetic energy is given by(8)δU=∫A{∫-h/2h/2[σxδ∈x+σyδ∈y+τxyδγxy+τxzδγxz+τyzδγyz]dz∫-h2h2[σxδ∈x+σyδ∈y+τxyδγxy}dxdy,δV=-∫(qw0+N-x∂wo∂xδ∂wo∂x+N-xy∂wo∂yδ∂wo∂x+N-yx∂wo∂xδ∂wo∂y+N-y∂wo∂yδ∂wo∂y)dxdy,δK=∫A{∫-h/2h/2‍ρ0[(u˙0+zθ˙x+z2u˙0*+z3θ˙x*)×(δu˙0+zδθ˙x+z2δu˙0*+z3δθ˙x*)+(v˙0+zθ˙y+z2v˙0*+z3θ˙y*)×(δv˙0+zδθ˙y+z2δv˙0*+z3δθ˙y*)+w˙0δw˙0(u˙0+zθ˙x+z2u˙0*+z3θ˙x*)]dz∫-h/2h/2‍}dxdy,
where q = distributed load over the surface of the plate.N
-
x and N-y the inplane loads perpendicular to the edges x=0 and y=0, respectively, N-xyN-yx the distributed shear forces parallel to the edges x=0 and y=0, respectively, ρ0=density  of  plate  material,u˙0=∂u0/∂t,v˙0=∂v0/∂t,  and so forth, indicate the time derivatives.Substituting forδU, δV, and δK in the virtual work statement in (7) and integrating through the thickness, integrating by parts, and collecting the coefficients of δuo, δvo, δwo, δθx, δθy, δuo*, δvo*, δθx*, δθy*, the following equations of motion are obtained:
(9)δu0:∂Nx∂x+∂Nxy∂y=I1u¨0+I2θ¨x+I3u¨0*+I4θ¨x*,δv0:∂Ny∂y+∂Nxy∂x=I1v¨0+I2θ¨y+I3v¨0*+I4θ¨y*,δw0:∂Qx∂x+∂Qy∂y+q+N~=I1w¨0,
where
(10)N~=N-x∂2wo∂x2+N-xy∂2wo∂y∂x+N-yx∂2wo∂x∂y+N-y∂2wo∂y2δθx:∂Mx∂x+∂Mxy∂y-Qx=I2u¨0+I3θ¨x+I4u¨0*+I5θ¨x*,δθy:∂My∂y+∂Mxy∂x-Qy=I2v¨0+I3θ¨y+I4v¨0*+I5θ¨y*,δu0*:∂Nx*∂x+∂Nxy*∂y-2Sx=I3u¨0+I4θ¨x+I5u¨0*+I6θ¨x*,δv0*:∂Ny*∂y+∂Nxy*∂x-2Sy=I3v¨0+I4θ¨y+I5v¨0*+I6θ¨y*,δθx*:∂Mx*∂x+∂Mxy*∂y-3Qx*=I4u¨0+I5θ¨x+I6u¨0*+I7θ¨x*,δθy*:∂My*∂y+∂Mxy*∂x-3Qy=I4v¨0+I5θ¨y+I6v¨0*+I7θ¨y*,
where the force and moment resultants are defined as
(11){Nx∣Nx*Ny∣Ny*Nxy∣Nxy*}=∑L=1n‍∫-h/2h/2‍{σxσyτxy}[1∣z2]dz,{Mx∣Mx*My∣My*Mxy∣Mxy*}=∑L=1n‍∫-h/2h/2{σxσyτxy}[z∣z3]dz.
And the transverse force resultants and the inertias are given by
(12){Qx∣Sx∣Qx*Qy∣Sy∣Qy*}=∑L=1n‍∫-h/2h/2{τxzτyz}[1∣z∣z2]dz,(13)I1,I2,I3,I4,I5,I6,I7=∫-h/2h/2[(ρc-ρm)(2z-h2h)n+ρm]×(1,z,z2,z3,z4,z5,z6)dz.
The resultants in (11)-(12) can be related to the total strains in (5) by the following matrix:
(14):{NN*—MM*—QQ*}=[A∣B∣0Bt-∣Db-∣0-0-∣0-∣Ds-]{ε0ε0*—KsK*—φφ*},
where
(15)N=[NxNyNxy]t;N*=[Nx*Ny*Nxy*]t.N,N* are called the inplane force resultants
(16)M=[MxMyMxy]t;M*=[Mx*My*Mxy*]t.M,M* are called as moment resultants
(17)Q=[QxQy]t;Q*=[SxSyQx*Qy*].tQ,Q* denote the transverse force result
(18)ε0=[εx0εy0εxy0]t;ε0*=[εx0*εy0*εxy0*]tKs=[KxKyKxy]t;K*=[Kx*Ky*Kxy*]tφ=[φxφy]t;φ*=[εxz0εyz0φx*φy*]t.
The matrices [A], [B], [D], and [Ds] are the plate stiffness whose elements can be calculated using (5) and (11)-(12).


## 3. Analytical Solution for the Simply Supported Plate

Leta be simply supported rectangular plate with length a and width b which is subjected to in-plane loading in two directions (N-x=-λ1Ncr,N-y=-λ2Ncr,N-xy=0). The following expressions of displacements are chosen based on Navier’s approach to automatically satisfy the simply supported boundary conditions of the plate:(19a)u0(x,y,t)=∑m=1∞‍∑n=1∞‍Umncosαxsinβy,(19b)v0(x,y,t)=∑m=1∞‍∑n=1∞‍Vmnsinαxcosβy,(19c)w0(x,y,t)=∑m=1∞‍∑n=1∞‍Wmnsinαxsinβy,(19d)θx(x,y,t)=∑m=1∞‍∑n=1∞‍Xmncosαxsinβy,(19e)θy(x,y,t)=∑m=1∞‍∑n=1∞‍Ymnsinαxcosβy,(19f)u0*(x,y,t)=∑m=1∞‍∑n=1∞‍Umn*cosαxsinβy,(19g)Vo*(x,y,t)=∑m=1∞‍∑n=1∞‍Vmn*sinαxcosβy,(19h)θx*(x,y,t)=∑m=1∞‍∑n=1∞‍Xmn*cosαxsinβy,(19i)θy*(x,y,t)=∑m=1∞‍∑n=1∞‍Ymn*sinαxcosβy.The eigenproblem related to governing equations is defined as(20)([S]9×9-λ[ς]9×9)X=0,
where [S] collects all stiffness terms and [ς] collects all terms related to the in-plane forces. In (20) X are the modes of buckling associated with the buckling loads defined as λ. For each value of m and n, there is a unique value of Ncr. The critical buckling load is the smallest value of Ncr (m,n).


## 4. Results and Discussion

### 4.1. Comparative Studies

To validate the accuracy of the present higher order theory in predicting the critical buckling load of FG plates subjected to different in-plane loading conditions (uniaxial compression:λ1=-1,λ2=0; biaxial compression: λ1=-1,λ2=-1; biaxial compression and tension λ1=-1,λ2=1), four numerical examples are presented and discussed. The material properties adopted here are as follows.Aluminium Young’s modulus (Em): 70 GPa, density ρm = 2702 kg/m3, and Poisson’s ratio (υ): 0.3.Alumina Young’s modulus (Ec): 380 GPa, density ρc = 3800 kg/m3, and Poisson’s ratio (υ): 0.3.For convenience, the critical buckling load is presented in nondimensionalized form as(21)N-=Ncra2Emh3.Example 1.
The first comparison is carried out for simply supported FG plate subjected to uniaxial compression along thex-axis (λ1=-1,λ2=0). The comparisons of nondimensionalized critical buckling loads (N-) obtained by the present theory and those given by Thai and Choi [27] are presented in Table 1. It can be observed that the predicted results are almost identical with that of Thai and Choi [27]. It can also be seen that the critical buckling load decreases with the increase of power-law index value while it increases with the increase of side-to-thickness ratio and aspect ratio. Furthermore, increasing of thickness ratio and aspect ratio not only increases the critical buckling load values, but also causes the changes in critical buckling modes. This can be observed when the plate is subjected to inplane compression along the x-axis with aspect ratio value of 2. The critical buckling mode varies from (3,1) to (2,1) as the side-to-thickness ratio increases from 5 to 10. The maximum percentage error between present HSDT and Thai and Choi [27] is 3.9 at a/h=5, a/b=2, and n=2.Table 1
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to uniaxial compression along the x-axis (λ1=-1,λ2=0).
Power-law index,n
a
/
b
a
/
h
Source
0
0.5
1
2
5
10
20
100
0.5
5
[27]
6.7203
4.4235
3.4164
2.6451
2.1484
1.9213
1.7115
1.3737
Present
6.714
4.409
3.39
2.61
2.124
1.90
1.705
1.371
10
[27]
7.405
4.82
3.71
2.88
2.41
2.18
1.93
1.52
Present
7.397
4.81
3.70
2.87
2.40
2.18
1.93
1.5231
20
[27]
7.599
4.93
3.79
2.95
2.49
2.26
2.00
1.56
Present
7.590
4.924
3.78
2.95
2.48
2.26
2.00
1.56
50
[27]
7.65
4.96
3.81
2.97
2.51
2.29
2.025
1.58
Present
7.64
4.95
3.81
2.973
2.51
2.28
2.02
1.57
100
[27]
7.66
4.968
3.82
2.98
2.52
2.29
2.028
1.58
Present
7.65
4.96
3.81
2.97
2.51
2.292
2.02
1.58
1
5
[27]
16.02
10.62
8.22
6.34
5.05
4.48
4.00
3.25
Present
16.00
10.57
8.146
6.23
4.97
4.44
3.98
3.25
10
[27]
18.57
12.12
9.33
7.26
6.03
5.45
4.83
3.81
Present
18.54
12.08
9.299
7.21
5.99
5.42
4.82
3.81
20
[27]
19.35
12.56
9.66
7.53
6.34
5.76
5.09
3.99
Present
19.31
12.53
9.649
7.51
6.32
5.75
5.08
3.98
50
[27]
19.58
12.69
9.763
7.61
6.43
5.8
5.17
4.04
Present
19.54
12.67
9.743
7.601
6.42
5.84
5.16
4.03
100
[27]
19.61
12.71
9.77
7.62
6.45
5.87
5.18
4.05
Present
19.57
12.69
9.75
7.61
6.43
5.86
5.17
4.04
1.5
5
[27]
28.19a
19.25a
15.03a
7.61a
8.47a
7.29a
6.61a
5.63a
Present
28.15a
19.09a
14.76a
11.06
8.25a
7.20a
6.56a
5.612a
10
[27]
40.74a
26.90a
20.80a
16.07a
12.95a
11.53a
10.29a
8.31a
Present
40.58a
26.72a
20.57a
15.81a
12.74a
11.42a
10.22a
8.27a
20
[27]
45.89a
29.90a
23.02a
17.92a
14.94a
13.52a
11.98a
9.44a
Present
45.64a
29.71a
22.85a
17.75a
14.81a
13.425a
11.90a
9.39a
50
[27]
47.57a
30.86a
23.74a
18.51a
15.628a
14.21a
12.56a
9.82a
Present
47.29a
30.67a
23.59a
18.39a
15.51a
14.12a
12.48a
9.76a
100
[27]
47.82a
31.01a
23.84a
18.60a
15.72a
14.31a
12.65a
9.87a
Present
47.53a
30.82a
23.69a
18.48a
15.62a
14.23a
12.57a
9.81a
2
5
[27]
37.74b
26.36b
20.74b
15.58b
10.95b
9.15c
8.39c
7.44b
Present
37.67b
26.11b
20.29b
14.99b
10.65b
9.04c
8.317c
7.40b
10
[27]
64.08a
42.50a
32.89a
25.37a
20.21a
17.92a
16.02a
13.03a
Present
63.78a
42.14a
32.46a
24.86a
19.84a
17.72a
15.90a
12.96a
20
[27]
74.3a
48.49a
37.35a
29.05a
24.14a
21.81a
19.33a
15.27a
Present
73.80a
48.10a
37.00a
28.71a
23.86a
21.61a
19.18a
15.17a
50
[27]
77.80a
50.48a
38.83a
30.28a
25.53a
23.27a
20.53a
16.05a
Present
77.20a
50.09a
38.51a
30.02a
25.32a
23.04a
20.36a
15.93a
100
[27]
78.32a
50.78a
39.05a
30.47a
25.74a
23.45a
20.71a
16.17a
Present
77.71a
50.38a
38.74a
30.22a
25.54a
23.26a
20.55a
16.04a
aMode for plate is (m,n) = (2, 1).
bMode for plate is (m,n) = (3, 1).
cMode for plate is (m,n) = (4, 1).Example 2.
The next comparison is performed for the simply supported FG plate subjected to inplane biaxial compression (λ1=-1,λ2=-1). The results of critical buckling loads obtained by the present theory and those reported by Thai and Choi [27] are presented in Table 2 and observed the close agreement between the results. It can be seen that, in this loading condition also, the nondimensionalized critical buckling load decreases with the increase of power-law index, while it increases with the increase of aspect ratio and side-to-thickness ratio, but only one critical buckling mode exists in any case of the aspect ratio, thickness ratio, modulus ratio, and power-law index. The maximum percentage error between present HSDT and Thai and Choi [27] is 2.90 at a/h=5, a/b=2, and n=2.Table 2
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to biaxial compression (λ1=-1,λ2=-1).
Power-law index,n
a
/
b
a
/
h
Source
0
0.5
1
2
5
10
20
100
0.5
5
[27]
5.376
3.539
2.733
2.116
1.719
1.537
1.369
1.099
Present
5.371
3.527
2.715
2.092
1.700
1.527
1.364
1.097
10
[27]
5.926
3.857
2.969
2.312
1.933
1.752
1.551
1.220
Present
5.918
3.850
2.961
2.302
1.925
1.747
1.548
1.218
20
[27]
6.079
3.857
3.034
2.367
1.996
1.815
1.604
1.255
Present
6.072
3.940
3.029
2.362
1.991
1.812
1.602
1.253
50
[27]
6.124
3.971
3.053
2.382
2.014
1.834
1.620
1.265
Present
6.117
3.966
3.049
2.379
2.011
1.831
1.618
1.263
100
[27]
6.131
3.974
3.056
2.385
2.016
1.837
1.622
1.266
Present
6.123
3.970
3.052
2.382
2.014
1.834
1.620
1.265
1
5
[27]
8.011
5.313
4.112
3.172
2.527
2.240
2.004
1.629
Present
8.001
5.288
4.073
3.120
2.487
2.221
1.994
1.626
10
[27]
9.289
6.062
4.670
3.632
3.018
2.726
2.417
1.910
Present
9.273
6.045
4.650
3.608
2.998
2.715
2.410
1.906
20
[27]
9.676
6.283
4.834
3.769
3.172
2.883
2.549
1.996
Present
9.658
6.270
4.821
3.757
3.162
2.876
2.544
1.992
50
[27]
9.791
6.349
4.882
3.809
3.219
2.931
2.589
2.217
Present
9.772
6.336
4.872
3.801
3.212
2.925
2.584
2.018
100
[27]
9.807
6.358
4.889
3.815
3.225
2.938
2.595
2.025
Present
9.788
6.345
4.879
3.807
3.219
2.932
2.590
2.021
1.5
5
[27]
11.682
7.830
6.080
4.664
3.618
3.172
2.851
2.360
Present
11.665
7.782
6.000
4.559
3.544
3.138
2.833
2.354
10
[27]
14.608
9.569
7.379
5.728
4.712
4.238
3.766
2.996
Present
14.571
9.528
7.331
5.671
4.666
4.212
3.749
2.987
20
[27]
15.589
10.133
7.798
6.076
5.101
4.630
4.096
3.214
Present
15.542
10.098
7.766
6.046
5.075
4.611
4.082
3.203
50
[27]
15.888
10.336
7.924
6.182
5.221
4.753
4.200
3.280
Present
15.837
10.270
7.897
6.160
5.203
4.737
4.186
3.270
100
[27]
15.931
10.328
7.942
6.197
5.239
4.771
4.215
3.290
Present
15.880
10.295
7.916
6.177
5.222
4.756
4.201
3.280
2
5
[27]
15.724
10.662
8.309
6.335
4.775
4.138
3.739
3.153
Present
15.698
10.581
8.172
6.156
4.661
4.088
3.712
3.143
10
[27]
21.505
14.155
10.932
8.464
6.875
6.148
5.477
4.396
Present
21.429
14.071
10.830
8.345
6.782
6.095
5.444
4.378
20
[27]
23.697
15.426
11.875
9.247
7.737
7.007
6.204
4.888
Present
23.590
15.346
11.802
9.177
7.674
6.964
6.171
4.858
50
[27]
24.394
15.824
12.170
9.493
8.013
7.293
6.444
5.036
Present
24.276
15.746
12.108
9.442
7.970
7.255
6.412
5.011
100
[27]
24.497
15.883
12.213
9.529
8.055
7.335
6.480
5.059
Present
24.378
15.805
12.153
9.482
8.015
7.299
6.448
5.034Example 3.
The third comparison is carried out for the simply supported FG plates under inplane biaxial compression and tension(λ1=-1,λ2=1). The results predicted by present theory are compared with Thai and Chois [27] results and good agreement between the results can be observed. The results are presented in Table 3.  It can also be seen that, under biaxial compression and tension, the critical buckling load decreases with the increase of power-law index value while it increases with the increase of side-to-thickness ratio and aspect ratio, same as in uniaxial and biaxial compression. Also, increasing of thickness ratio and aspect ratio not only increases the critical buckling load values, but also causes changes in critical buckling modes. This can be observed when the aspect ratio value is 1.5. The critical buckling mode varies from (2,1) to (1,2) for the values and the side-to-thickness ratio increases from 5 to 10 and 20 to 50. The maximum percentage error between present HSDT and Thai and Choi [27] is 2.90 at a/h=5, a/b=2, and n=2.Table 3
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to biaxial compression and tension (λ1=-1,λ2=-1).
Power-law index,n
a
/
b
a
/
h
Source
0
0.5
1
2
5
10
20
100
0.5
5
[27]
8.960
5.898
4.555
3.527
2.865
2.562
2.282
1.832
Present
8.953
5.879
4.525
3.487
2.833
2.545
2.274
1.829
10
[27]
9.874
6.428
4.948
3.853
3.222
2.200
2.585
2.033
Present
9.863
6.416
4.934
3.837
3.208
2.911
2.580
2.031
20
[27]
10.132
6.575
5.057
3.944
3.326
3.025
2.674
2.091
Present
10.120
6.566
5.049
3.936
3.319
3.020
2.670
2.089
50
[27]
10.207
6.618
5.089
3.971
3.356
3.056
2.700
2.108
Present
10.195
6.610
5.082
3.965
3.352
3.052
2.697
2.105
100
[27]
10.218
6.624
5.093
3.974
3.361
3.061
2.704
2.110
Present
10.206
6.616
5.087
3.969
3.356
3.057
2.700
2.108
1
5
[27]
26.20a
17.77a
13.84a
10.55a
7.95a
6.89a
6.23a
5.25a
Present
26.16a
17.63a
13.62a
10.26a
7.76a
6.81a
6.18a
5.23a
10
[27]
35.84a
23.59b
18.22a
14.10a
11.45a
10.24a
9.12a
7.32a
Present
35.71a
23.45b
18.04a
13.90a
11.30a
10.15a
9.07a
7.297
20
[27]
39.49a
25.71a
19.79a
15.41a
12.88a
11.67a
10.34a
8.13a
Present
39.31a
25.57a
19.67a
15.29a
12.79a
11.60a
10.28a
8.09a
50
[27]
40.65a
26.37a
20.283
15.82a
13.35a
12.15a
10.74a
8.39b
Present
40.46a
26.24a
20.179
15.73a
13.28a
12.09a
10.68a
8.35b
100
[27]
40.82a
26.47a
20.35a
15.88a
13.42a
12.22a
10.79a
8.43b
Present
40.62a
26.34a
20.25a
15.80a
13.35a
12.16a
10.74a
8.39b
1.5
5
[27]
29.02b
20.11b
15.78b
11.90b
8.52b
7.24b
6.60b
5.74b
Present
28.97b
19.92b
15.45b
11.47b
8.29b
7.15b
6.54b
5.72b
10
[27]
37.982
24.878
19.186
14.893
12.252
11.020
9.791
7.789
Present
37.884
24.773
19.060
14.745
12.133
10.950
9.748
7.767
20
[27]
40.531
26.346
20.244
15.798
13.262
12.038
10.650
8.355
Present
40.408
26.255
20.190
15.718
13.194
11.988
10.612
8.329
50
[27]
41.308
26.789
20.601
16.072
13.575
12.358
10.919
8.529
Present
41.177
26.702
20.532
16.016
13.528
12.317
10.883
8.502
100
[27]
41.421
26.854
20.649
16.112
13.624
12.405
10.958
8.554
Present
41.289
26.768
20.582
16.059
13.577
12.365
10.923
8.527
2
5
[27]
26.206
17.770
13.849
10.559
7.959
6.897
6.232
5.256
Present
26.164
17.636
13.620
10.261
7.768
6.814
6.187
5.239
10
[27]
35.842
23.592
18.221
14.107
11.458
10.247
9.128
7.326
Present
35.715
23.451
18.050
13.909
11.303
10.159
9.073
7.297
20
[27]
39.495
25.710
19.793
15.412
12.888
11.678
10.340
8.134
Present
39.317
25.576
19.670
15.295
12.791
11.607
10.286
8.096
50
[27]
40.657
26.374
20.283
15.822
13.355
12.154
10.740
8.393
Present
40.460
26.243
20.179
15.737
13.284
12.092
10.687
8.352
100
[27]
40.829
26.472
20.355
15.882
13.425
12.226
10.800
8.432
Present
40.629
26.341
20.254
15.803
13.358
12.165
10.747
8.390
aMode for plate is (m,n) = (2, 1).
bMode for plate is (m,n) = (1, 2).Example 4.
The last comparison is carried out for the simply supported FG plates under inplane uniaxial compression (λ1=-1,λ2=0). The results predicted by present theory are compared with the classical plate theory (CPT) [16] and first-order shear deformation theory (FSDT) [16] are presented in Table 4. The maximum percentage error between present HSDT and CPT and HSDT and FSDT for  n=0.1, 1, and 10, respectively are 5.633954, 0.19667, 5.195466, and 0.42725, 8.294579, 4.39547 at a/h=10. It is also seen that, the CPT and FSDT overpredicts the critical buckling load for all thickness ratios.Table 4
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to uniaxial compression (λ1=-1,λ2=0).
b
/
h
n
=
0.1
n
=
1
n
=
10
CPT [16]
FSDT [16]
HSDT
CPT [16]
FSDT [16]
HSDT
CPT [16]
FSDT [16]
HSDT
10
17.68
16.76
16.74
9.78
9.33
9.29
5.87
5.66
5.42
20
17.68
17.4
17.41
9.78
9.66
9.64
5.87
5.78
5.75
40
17.68
17.62
17.59
9.78
9.75
9.73
5.87
5.85
5.83
50
17.68
17.64
17.61
9.78
9.76
9.74
5.87
5.86
5.84
100
17.68
17.67
17.64
9.78
9.77
9.75
5.87
5.87
5.86


### 4.2. Parametric Study

The effect of side-to-thickness ratio, aspect ratio, and the modulus ratio on nondimensionalized critical buckling load for simply supported FG plate made of Al/Al2O3 with εz=0 is investigated. Figures 2–4 represent the variation of nondimensionalized critical buckling load with side-to-thickness ratio, aspect ratio, and modulus ratio, respectively, under uniaxial compression. From Figures 2–4, it is important to observe that the nondimensionalized critical buckling loads are higher for ceramic rich plates and lower for metal rich plates. The critical buckling loads of FGM plates are intermediate to that of ceramic and metal. From Figure 2 it is seen that the effect of shear deformation is significant for a side-to-thickness ratio less than 10 and diminishes with increase of side-to-thickness ratio. The increase of aspect ratio increases the critical buckling load due to increase of stiffness of the plate as shown in Figure 3.Figure 2
Effect of side-to-thickness ratios (a/h) on nondimensionalized critical buckling load (N-) under uniaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 3
Effect of aspect ratios (a/b) on nondimensionalized critical buckling load (N-) under uniaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 4
Effect of modulus ratio (Em/Ec) on nondimensionalized critical buckling load (N-) under uniaxial compression for a simply supported FG plate for various material variation parameters (n).From Figure4, it can be seen that the nondimensional critical buckling load decreases as the metal-to-ceramic modulus ratio increases and decreases as the power law index increases. This is due to the fact that higher values of metal-to-ceramic modulus ratio correspond to high portion of metal.The effect of side-to-thickness ratio, aspect ratio, modulus ratio, and power law index values on nondiensionalized critical buckling load for a simply supported FG plate under biaxial compression is shown in Figures5, 6, and 7. The same can be observed as in the case of uniaxial compression. It is important to observe that the critical buckling loads are larger in uniaxial compression and smaller in biaxial compression. Figure 8 shows the variation of nondimensionalized critical buckling load of different modulus ratios and power law index values under inplane compression and tension. It can be observed that critical buckling load decreases with the increase of modulus ratio and power-law index values.Figure 5
Effect of side to thickness ratios (a/h) on nondimensionalized critical buckling load (N-) under biaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 6
Effect of aspect ratios (a/b) on nondimensionalized critical buckling load (N-) under biaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 7
Effect of modulus ratios (Em/Ec) on nondimensionalized critical buckling load (N-) under biaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 8
Effect of modulus ratios (Em/Ec) on nondimensionalized critical buckling load          (N-) under biaxial compression and tension for a simply supported FG plate for various material variation parameters (n) (mode for the plate (m,n) = (1,2)).


## 4.1. Comparative Studies

To validate the accuracy of the present higher order theory in predicting the critical buckling load of FG plates subjected to different in-plane loading conditions (uniaxial compression:λ1=-1,λ2=0; biaxial compression: λ1=-1,λ2=-1; biaxial compression and tension λ1=-1,λ2=1), four numerical examples are presented and discussed. The material properties adopted here are as follows.Aluminium Young’s modulus (Em): 70 GPa, density ρm = 2702 kg/m3, and Poisson’s ratio (υ): 0.3.Alumina Young’s modulus (Ec): 380 GPa, density ρc = 3800 kg/m3, and Poisson’s ratio (υ): 0.3.For convenience, the critical buckling load is presented in nondimensionalized form as(21)N-=Ncra2Emh3.Example 1.
The first comparison is carried out for simply supported FG plate subjected to uniaxial compression along thex-axis (λ1=-1,λ2=0). The comparisons of nondimensionalized critical buckling loads (N-) obtained by the present theory and those given by Thai and Choi [27] are presented in Table 1. It can be observed that the predicted results are almost identical with that of Thai and Choi [27]. It can also be seen that the critical buckling load decreases with the increase of power-law index value while it increases with the increase of side-to-thickness ratio and aspect ratio. Furthermore, increasing of thickness ratio and aspect ratio not only increases the critical buckling load values, but also causes the changes in critical buckling modes. This can be observed when the plate is subjected to inplane compression along the x-axis with aspect ratio value of 2. The critical buckling mode varies from (3,1) to (2,1) as the side-to-thickness ratio increases from 5 to 10. The maximum percentage error between present HSDT and Thai and Choi [27] is 3.9 at a/h=5, a/b=2, and n=2.Table 1
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to uniaxial compression along the x-axis (λ1=-1,λ2=0).
Power-law index,n
a
/
b
a
/
h
Source
0
0.5
1
2
5
10
20
100
0.5
5
[27]
6.7203
4.4235
3.4164
2.6451
2.1484
1.9213
1.7115
1.3737
Present
6.714
4.409
3.39
2.61
2.124
1.90
1.705
1.371
10
[27]
7.405
4.82
3.71
2.88
2.41
2.18
1.93
1.52
Present
7.397
4.81
3.70
2.87
2.40
2.18
1.93
1.5231
20
[27]
7.599
4.93
3.79
2.95
2.49
2.26
2.00
1.56
Present
7.590
4.924
3.78
2.95
2.48
2.26
2.00
1.56
50
[27]
7.65
4.96
3.81
2.97
2.51
2.29
2.025
1.58
Present
7.64
4.95
3.81
2.973
2.51
2.28
2.02
1.57
100
[27]
7.66
4.968
3.82
2.98
2.52
2.29
2.028
1.58
Present
7.65
4.96
3.81
2.97
2.51
2.292
2.02
1.58
1
5
[27]
16.02
10.62
8.22
6.34
5.05
4.48
4.00
3.25
Present
16.00
10.57
8.146
6.23
4.97
4.44
3.98
3.25
10
[27]
18.57
12.12
9.33
7.26
6.03
5.45
4.83
3.81
Present
18.54
12.08
9.299
7.21
5.99
5.42
4.82
3.81
20
[27]
19.35
12.56
9.66
7.53
6.34
5.76
5.09
3.99
Present
19.31
12.53
9.649
7.51
6.32
5.75
5.08
3.98
50
[27]
19.58
12.69
9.763
7.61
6.43
5.8
5.17
4.04
Present
19.54
12.67
9.743
7.601
6.42
5.84
5.16
4.03
100
[27]
19.61
12.71
9.77
7.62
6.45
5.87
5.18
4.05
Present
19.57
12.69
9.75
7.61
6.43
5.86
5.17
4.04
1.5
5
[27]
28.19a
19.25a
15.03a
7.61a
8.47a
7.29a
6.61a
5.63a
Present
28.15a
19.09a
14.76a
11.06
8.25a
7.20a
6.56a
5.612a
10
[27]
40.74a
26.90a
20.80a
16.07a
12.95a
11.53a
10.29a
8.31a
Present
40.58a
26.72a
20.57a
15.81a
12.74a
11.42a
10.22a
8.27a
20
[27]
45.89a
29.90a
23.02a
17.92a
14.94a
13.52a
11.98a
9.44a
Present
45.64a
29.71a
22.85a
17.75a
14.81a
13.425a
11.90a
9.39a
50
[27]
47.57a
30.86a
23.74a
18.51a
15.628a
14.21a
12.56a
9.82a
Present
47.29a
30.67a
23.59a
18.39a
15.51a
14.12a
12.48a
9.76a
100
[27]
47.82a
31.01a
23.84a
18.60a
15.72a
14.31a
12.65a
9.87a
Present
47.53a
30.82a
23.69a
18.48a
15.62a
14.23a
12.57a
9.81a
2
5
[27]
37.74b
26.36b
20.74b
15.58b
10.95b
9.15c
8.39c
7.44b
Present
37.67b
26.11b
20.29b
14.99b
10.65b
9.04c
8.317c
7.40b
10
[27]
64.08a
42.50a
32.89a
25.37a
20.21a
17.92a
16.02a
13.03a
Present
63.78a
42.14a
32.46a
24.86a
19.84a
17.72a
15.90a
12.96a
20
[27]
74.3a
48.49a
37.35a
29.05a
24.14a
21.81a
19.33a
15.27a
Present
73.80a
48.10a
37.00a
28.71a
23.86a
21.61a
19.18a
15.17a
50
[27]
77.80a
50.48a
38.83a
30.28a
25.53a
23.27a
20.53a
16.05a
Present
77.20a
50.09a
38.51a
30.02a
25.32a
23.04a
20.36a
15.93a
100
[27]
78.32a
50.78a
39.05a
30.47a
25.74a
23.45a
20.71a
16.17a
Present
77.71a
50.38a
38.74a
30.22a
25.54a
23.26a
20.55a
16.04a
aMode for plate is (m,n) = (2, 1).
bMode for plate is (m,n) = (3, 1).
cMode for plate is (m,n) = (4, 1).Example 2.
The next comparison is performed for the simply supported FG plate subjected to inplane biaxial compression (λ1=-1,λ2=-1). The results of critical buckling loads obtained by the present theory and those reported by Thai and Choi [27] are presented in Table 2 and observed the close agreement between the results. It can be seen that, in this loading condition also, the nondimensionalized critical buckling load decreases with the increase of power-law index, while it increases with the increase of aspect ratio and side-to-thickness ratio, but only one critical buckling mode exists in any case of the aspect ratio, thickness ratio, modulus ratio, and power-law index. The maximum percentage error between present HSDT and Thai and Choi [27] is 2.90 at a/h=5, a/b=2, and n=2.Table 2
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to biaxial compression (λ1=-1,λ2=-1).
Power-law index,n
a
/
b
a
/
h
Source
0
0.5
1
2
5
10
20
100
0.5
5
[27]
5.376
3.539
2.733
2.116
1.719
1.537
1.369
1.099
Present
5.371
3.527
2.715
2.092
1.700
1.527
1.364
1.097
10
[27]
5.926
3.857
2.969
2.312
1.933
1.752
1.551
1.220
Present
5.918
3.850
2.961
2.302
1.925
1.747
1.548
1.218
20
[27]
6.079
3.857
3.034
2.367
1.996
1.815
1.604
1.255
Present
6.072
3.940
3.029
2.362
1.991
1.812
1.602
1.253
50
[27]
6.124
3.971
3.053
2.382
2.014
1.834
1.620
1.265
Present
6.117
3.966
3.049
2.379
2.011
1.831
1.618
1.263
100
[27]
6.131
3.974
3.056
2.385
2.016
1.837
1.622
1.266
Present
6.123
3.970
3.052
2.382
2.014
1.834
1.620
1.265
1
5
[27]
8.011
5.313
4.112
3.172
2.527
2.240
2.004
1.629
Present
8.001
5.288
4.073
3.120
2.487
2.221
1.994
1.626
10
[27]
9.289
6.062
4.670
3.632
3.018
2.726
2.417
1.910
Present
9.273
6.045
4.650
3.608
2.998
2.715
2.410
1.906
20
[27]
9.676
6.283
4.834
3.769
3.172
2.883
2.549
1.996
Present
9.658
6.270
4.821
3.757
3.162
2.876
2.544
1.992
50
[27]
9.791
6.349
4.882
3.809
3.219
2.931
2.589
2.217
Present
9.772
6.336
4.872
3.801
3.212
2.925
2.584
2.018
100
[27]
9.807
6.358
4.889
3.815
3.225
2.938
2.595
2.025
Present
9.788
6.345
4.879
3.807
3.219
2.932
2.590
2.021
1.5
5
[27]
11.682
7.830
6.080
4.664
3.618
3.172
2.851
2.360
Present
11.665
7.782
6.000
4.559
3.544
3.138
2.833
2.354
10
[27]
14.608
9.569
7.379
5.728
4.712
4.238
3.766
2.996
Present
14.571
9.528
7.331
5.671
4.666
4.212
3.749
2.987
20
[27]
15.589
10.133
7.798
6.076
5.101
4.630
4.096
3.214
Present
15.542
10.098
7.766
6.046
5.075
4.611
4.082
3.203
50
[27]
15.888
10.336
7.924
6.182
5.221
4.753
4.200
3.280
Present
15.837
10.270
7.897
6.160
5.203
4.737
4.186
3.270
100
[27]
15.931
10.328
7.942
6.197
5.239
4.771
4.215
3.290
Present
15.880
10.295
7.916
6.177
5.222
4.756
4.201
3.280
2
5
[27]
15.724
10.662
8.309
6.335
4.775
4.138
3.739
3.153
Present
15.698
10.581
8.172
6.156
4.661
4.088
3.712
3.143
10
[27]
21.505
14.155
10.932
8.464
6.875
6.148
5.477
4.396
Present
21.429
14.071
10.830
8.345
6.782
6.095
5.444
4.378
20
[27]
23.697
15.426
11.875
9.247
7.737
7.007
6.204
4.888
Present
23.590
15.346
11.802
9.177
7.674
6.964
6.171
4.858
50
[27]
24.394
15.824
12.170
9.493
8.013
7.293
6.444
5.036
Present
24.276
15.746
12.108
9.442
7.970
7.255
6.412
5.011
100
[27]
24.497
15.883
12.213
9.529
8.055
7.335
6.480
5.059
Present
24.378
15.805
12.153
9.482
8.015
7.299
6.448
5.034Example 3.
The third comparison is carried out for the simply supported FG plates under inplane biaxial compression and tension(λ1=-1,λ2=1). The results predicted by present theory are compared with Thai and Chois [27] results and good agreement between the results can be observed. The results are presented in Table 3.  It can also be seen that, under biaxial compression and tension, the critical buckling load decreases with the increase of power-law index value while it increases with the increase of side-to-thickness ratio and aspect ratio, same as in uniaxial and biaxial compression. Also, increasing of thickness ratio and aspect ratio not only increases the critical buckling load values, but also causes changes in critical buckling modes. This can be observed when the aspect ratio value is 1.5. The critical buckling mode varies from (2,1) to (1,2) for the values and the side-to-thickness ratio increases from 5 to 10 and 20 to 50. The maximum percentage error between present HSDT and Thai and Choi [27] is 2.90 at a/h=5, a/b=2, and n=2.Table 3
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to biaxial compression and tension (λ1=-1,λ2=-1).
Power-law index,n
a
/
b
a
/
h
Source
0
0.5
1
2
5
10
20
100
0.5
5
[27]
8.960
5.898
4.555
3.527
2.865
2.562
2.282
1.832
Present
8.953
5.879
4.525
3.487
2.833
2.545
2.274
1.829
10
[27]
9.874
6.428
4.948
3.853
3.222
2.200
2.585
2.033
Present
9.863
6.416
4.934
3.837
3.208
2.911
2.580
2.031
20
[27]
10.132
6.575
5.057
3.944
3.326
3.025
2.674
2.091
Present
10.120
6.566
5.049
3.936
3.319
3.020
2.670
2.089
50
[27]
10.207
6.618
5.089
3.971
3.356
3.056
2.700
2.108
Present
10.195
6.610
5.082
3.965
3.352
3.052
2.697
2.105
100
[27]
10.218
6.624
5.093
3.974
3.361
3.061
2.704
2.110
Present
10.206
6.616
5.087
3.969
3.356
3.057
2.700
2.108
1
5
[27]
26.20a
17.77a
13.84a
10.55a
7.95a
6.89a
6.23a
5.25a
Present
26.16a
17.63a
13.62a
10.26a
7.76a
6.81a
6.18a
5.23a
10
[27]
35.84a
23.59b
18.22a
14.10a
11.45a
10.24a
9.12a
7.32a
Present
35.71a
23.45b
18.04a
13.90a
11.30a
10.15a
9.07a
7.297
20
[27]
39.49a
25.71a
19.79a
15.41a
12.88a
11.67a
10.34a
8.13a
Present
39.31a
25.57a
19.67a
15.29a
12.79a
11.60a
10.28a
8.09a
50
[27]
40.65a
26.37a
20.283
15.82a
13.35a
12.15a
10.74a
8.39b
Present
40.46a
26.24a
20.179
15.73a
13.28a
12.09a
10.68a
8.35b
100
[27]
40.82a
26.47a
20.35a
15.88a
13.42a
12.22a
10.79a
8.43b
Present
40.62a
26.34a
20.25a
15.80a
13.35a
12.16a
10.74a
8.39b
1.5
5
[27]
29.02b
20.11b
15.78b
11.90b
8.52b
7.24b
6.60b
5.74b
Present
28.97b
19.92b
15.45b
11.47b
8.29b
7.15b
6.54b
5.72b
10
[27]
37.982
24.878
19.186
14.893
12.252
11.020
9.791
7.789
Present
37.884
24.773
19.060
14.745
12.133
10.950
9.748
7.767
20
[27]
40.531
26.346
20.244
15.798
13.262
12.038
10.650
8.355
Present
40.408
26.255
20.190
15.718
13.194
11.988
10.612
8.329
50
[27]
41.308
26.789
20.601
16.072
13.575
12.358
10.919
8.529
Present
41.177
26.702
20.532
16.016
13.528
12.317
10.883
8.502
100
[27]
41.421
26.854
20.649
16.112
13.624
12.405
10.958
8.554
Present
41.289
26.768
20.582
16.059
13.577
12.365
10.923
8.527
2
5
[27]
26.206
17.770
13.849
10.559
7.959
6.897
6.232
5.256
Present
26.164
17.636
13.620
10.261
7.768
6.814
6.187
5.239
10
[27]
35.842
23.592
18.221
14.107
11.458
10.247
9.128
7.326
Present
35.715
23.451
18.050
13.909
11.303
10.159
9.073
7.297
20
[27]
39.495
25.710
19.793
15.412
12.888
11.678
10.340
8.134
Present
39.317
25.576
19.670
15.295
12.791
11.607
10.286
8.096
50
[27]
40.657
26.374
20.283
15.822
13.355
12.154
10.740
8.393
Present
40.460
26.243
20.179
15.737
13.284
12.092
10.687
8.352
100
[27]
40.829
26.472
20.355
15.882
13.425
12.226
10.800
8.432
Present
40.629
26.341
20.254
15.803
13.358
12.165
10.747
8.390
aMode for plate is (m,n) = (2, 1).
bMode for plate is (m,n) = (1, 2).Example 4.
The last comparison is carried out for the simply supported FG plates under inplane uniaxial compression (λ1=-1,λ2=0). The results predicted by present theory are compared with the classical plate theory (CPT) [16] and first-order shear deformation theory (FSDT) [16] are presented in Table 4. The maximum percentage error between present HSDT and CPT and HSDT and FSDT for  n=0.1, 1, and 10, respectively are 5.633954, 0.19667, 5.195466, and 0.42725, 8.294579, 4.39547 at a/h=10. It is also seen that, the CPT and FSDT overpredicts the critical buckling load for all thickness ratios.Table 4
Comparison of nondimensionalized critical buckling load (N-) of simply supported Al/Al2O3 plate subjected to uniaxial compression (λ1=-1,λ2=0).
b
/
h
n
=
0.1
n
=
1
n
=
10
CPT [16]
FSDT [16]
HSDT
CPT [16]
FSDT [16]
HSDT
CPT [16]
FSDT [16]
HSDT
10
17.68
16.76
16.74
9.78
9.33
9.29
5.87
5.66
5.42
20
17.68
17.4
17.41
9.78
9.66
9.64
5.87
5.78
5.75
40
17.68
17.62
17.59
9.78
9.75
9.73
5.87
5.85
5.83
50
17.68
17.64
17.61
9.78
9.76
9.74
5.87
5.86
5.84
100
17.68
17.67
17.64
9.78
9.77
9.75
5.87
5.87
5.86


## 4.2. Parametric Study

The effect of side-to-thickness ratio, aspect ratio, and the modulus ratio on nondimensionalized critical buckling load for simply supported FG plate made of Al/Al2O3 with εz=0 is investigated. Figures 2–4 represent the variation of nondimensionalized critical buckling load with side-to-thickness ratio, aspect ratio, and modulus ratio, respectively, under uniaxial compression. From Figures 2–4, it is important to observe that the nondimensionalized critical buckling loads are higher for ceramic rich plates and lower for metal rich plates. The critical buckling loads of FGM plates are intermediate to that of ceramic and metal. From Figure 2 it is seen that the effect of shear deformation is significant for a side-to-thickness ratio less than 10 and diminishes with increase of side-to-thickness ratio. The increase of aspect ratio increases the critical buckling load due to increase of stiffness of the plate as shown in Figure 3.Figure 2
Effect of side-to-thickness ratios (a/h) on nondimensionalized critical buckling load (N-) under uniaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 3
Effect of aspect ratios (a/b) on nondimensionalized critical buckling load (N-) under uniaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 4
Effect of modulus ratio (Em/Ec) on nondimensionalized critical buckling load (N-) under uniaxial compression for a simply supported FG plate for various material variation parameters (n).From Figure4, it can be seen that the nondimensional critical buckling load decreases as the metal-to-ceramic modulus ratio increases and decreases as the power law index increases. This is due to the fact that higher values of metal-to-ceramic modulus ratio correspond to high portion of metal.The effect of side-to-thickness ratio, aspect ratio, modulus ratio, and power law index values on nondiensionalized critical buckling load for a simply supported FG plate under biaxial compression is shown in Figures5, 6, and 7. The same can be observed as in the case of uniaxial compression. It is important to observe that the critical buckling loads are larger in uniaxial compression and smaller in biaxial compression. Figure 8 shows the variation of nondimensionalized critical buckling load of different modulus ratios and power law index values under inplane compression and tension. It can be observed that critical buckling load decreases with the increase of modulus ratio and power-law index values.Figure 5
Effect of side to thickness ratios (a/h) on nondimensionalized critical buckling load (N-) under biaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 6
Effect of aspect ratios (a/b) on nondimensionalized critical buckling load (N-) under biaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 7
Effect of modulus ratios (Em/Ec) on nondimensionalized critical buckling load (N-) under biaxial compression for a simply supported FG plate for various material variation parameters (n).Figure 8
Effect of modulus ratios (Em/Ec) on nondimensionalized critical buckling load          (N-) under biaxial compression and tension for a simply supported FG plate for various material variation parameters (n) (mode for the plate (m,n) = (1,2)).


## 5. Conclusions

A higher order shear deformation theory was successfully developed and applied to study the buckling behavior of functionally graded plates without enforcing zero transverse shear stresses on the top and bottom surfaces of the plate. The present formulation was compared with the refined theory developed by Thai and Choi [27] and proved very accurate for the buckling problem. This eliminated the need of using shear correction factors. It can be concluded that the present theory is accurate and efficient in predicting the critical buckling load of simply supported FG plates. Hence, the present findings will be useful benchmark for evaluating other future plate theories and numerical methods such as the finite element and meshless methods.



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