# A New Hyperbolic Shear Deformation Theory for Bending Analysis of Functionally Graded Plates

Authors: Tahar Hassaine Daouadji; Abdelaziz Hadj Henni; Abdelouahed Tounsi; Adda Bedia El Abbes
Journal: Modelling and Simulation in Engineering
 (2012)

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## Abstract

Theoretical formulation, Navier’s solutions of rectangular plates based on a new higher order shear deformation model are presented for the static response of functionally graded plates. This theory enforces traction-free boundary conditions at plate surfaces. Shear correction factors are not required because a correct representation of transverse shearing strain is given. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concern flexural behavior of FG plates with metal-ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fractions profiles, aspect ratios, and length to thickness ratios. Results are verified with available results in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static bending behavior of functionally graded plates.


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## Body

## 1. Introduction

The concept of functionally graded materials (FGMs) was first introduced in 1984 by a group of material scientists in Japan, as ultrahigh temperature-resistant materials for aircraft, space vehicles, and other engineering applications. Functionally graded materials (FGMs) are new composite materials in which the microstructural details are spatially varied through nonuniform distribution of the reinforcement phase. This is achieved by using reinforcement with different properties, sizes, and shapes, as well as by interchanging the role of reinforcement and matrix phase in a continuous manner. The result is a microstructure that produces continuous or smooth change on thermal and mechanical properties at the macroscopic or continuum level (Koizumi, 1993 [1]; Hirai and Chen, 1999 [2]). Now, FGMs are developed for general use as structural components in extremely high-temperature environments. Therefore, it is important to study the wave propagation of functionally graded materials structures in terms of nondestructive evaluation and material characterization.Several studies have been performed to analyze the mechanical or the thermal or the thermomechanical responses of FG plates and shells. A comprehensive review is done by Tanigawa (1995) [3]. Reddy (2000) [4] has analyzed the static behavior of functionally graded rectangular plates based on his third-order shear deformation plate theory. Cheng and Batra (2000) [5] have related the deflections of a simply supported FG polygonal plate given by the first-order shear deformation theory and third-order shear deformation theory to that of an equivalent homogeneous Kirchhoff plate. The static response of FG plate has been investigated by Zenkour (2006) [6] using a generalized shear deformation theory. In a recent study, Şimşek (2010) [7] has studied the dynamic deflections and the stresses of an FG simply supported beam subjected to a moving mass by using Euler Bernoulli, Timoshenko, and the parabolic shear deformation beam theory. Şimşek (2010) [8], Benchour et al. [9], and Abdelaziz et al. 2010 [10] studied the free vibration of FG beams having different boundary conditions using the classical, the first-order, and different higher-order shear deformation beam and plate theories. The nonlinear dynamic analysis of an FG beam with pinned-pinned supports due to a moving harmonic load has been examined by Şimşek (2010) [11] using Timoshenko beam theory.The primary objective of this paper is to present a general formulation for functionally graded plates (FGPs) using a new higher-order shear deformation plate theory with only four unknown functions. The present theory satisfies equilibrium conditions at the top and bottom faces of the plate without using shear correction factors. The hyperbolic function in terms of thickness coordinate is used in the displacement field to account for shear deformation. Governing equations are derived from the principle of minimum total potential energy. Navier solution is used to obtain the closed-form solutions for simply supported FG plates. To illustrate the accuracy of the present theory, the obtained results are compared with three-dimensional elasticity solutions and results of the first-order, and the other higher-order theories (Table1).Table 1
Displacement models.
Model
Theory
Unknownfunction
CPT
Classical plate theory [12]
3
ATDSP
Analytical tree dimensional solution for plate (3D) [13]
5
SSDPT
Sinusoidal shear deformation plate theory (Zenkour) [6]
5
PSDPT
Parabolic shear deformation plate theory (Reddy) [4]
5
NHPSDT
New hyperbolic shear deformation theory (present)
4In this study, a new displacement models for an analysis of simply supported FGM plates are proposed. The plates are made of an isotropic material with material properties varying in the thickness direction only. Analytical solutions for bending deflections of FGM plates are obtained. The governing equations are derived from the principle of minimum total potential energy. Numerical examples are presented to illustrate the accuracy and efficiency of the present theory by comparing the obtained results with those computed using various other theories.


## 2. Problem Formulation

Consider a plate of total thicknessh and composed of functionally graded material through the thickness. It is assumed that the material is isotropic, and grading is assumed to be only through the thickness. The xy  plane is taken to be the undeformed mid plane of the plate with the z-axis positive upward from the mid plane (Figure 1).Figure 1
Geometry of rectangular plate composed of FGM.


### 2.1. Displacement Fields and Strains

The assumed displacement field is as follows:(1)u(x,y,z)=u0(x,y)-z∂wb∂x-f(z)∂ws∂x,v(x,y,z)=v0(x,y)-z∂wb∂y-f(z)∂ws∂y,w(x,y,z)=wb(x,y)+ws(x,y),
where u0 and v0 are the mid-plane displacements of the plate in the x and y directions, respectively; wb and ws are the bending and shear components of transverse displacement, respectively, while f(z) represents shape functions determining the distribution of the transverse shear strains and stresses along the thickness and is given as
(2)f(z)=z[1+3π2sech2(12)]-3π2htanh(zh).
It should be noted that unlike the first-order shear deformation theory, this theory does not require shear correction factors. The kinematic relations can be obtained as follows:
(3)εx=εx0+zkxb+f(z)kxs,εy=εy0+zkyb+f(z)kys,γxy=γxy0+zkxyb+f(z)kxys,γyz=g(z)γyzs,γxz=g(z)γxzs,εz=0,
where
(4)εx0=∂u0∂x,kxb=-∂2wb∂x2,kxs=-∂2ws∂x2,εy0=∂v0∂y,kyb=-∂2wb∂y2,kys=-∂2ws∂y2,γxy0=∂u0∂y+∂v0∂x,kxyb=-2∂2wb∂x∂y,kxys=-2∂2ws∂x∂y,γyzs=∂ws∂y,γxzs=∂ws∂x,g(z)=1-f′(z),f′(z)=df(z)dz.


### 2.2. Constitutive Relations

In FGM, material property gradation is considered through the thickness, and the expression given below represents the profile for the volume fraction(5)P(z)=(Pt-Pb)(zh+12)k+Pb,
where P  denotes a generic material property like modulus, Pt and Pb denote the property of the top and bottom faces of the plate, respectively, and k is a parameter that dictates material variation profile through the thickness. Here, it is assumed that modules E and G vary according to the equation (5), and ν is assumed to be a constant. The linear constitutive relations are
(6){σxσyτyzτxzτxy}=[Q11Q12000Q12Q1100000Q4400000Q5500000Q66]{εxεyγyzγxzγxy},
where
(7)Q11=E(z)1-ν2,Q12=νQ11,Q44=Q55=Q66=E(z)2(1+ν).


### 2.3. Governing Equations

The governing equations of equilibrium can be derived by using the principle of virtual displacements. The principle of virtual work in the present case yields(8)∫-h/2h/2∫Ω[σxδεx+σyδεy+τxyδγxy+τyzδγyz+τxzδγxz]dΩdz-∫ΩqδwdΩ=0,
where Ω is the top surface and q is the applied transverse load.Substituting and (6) into (8) and integrating through the thickness of the plate, (8) can be rewritten as
(9)∫Ω[Nxδεx0+Nyδεy0+Nxyδεxy0+Mxbδkxb+Mybδkyb+Mxybδkxyb+Mxsδkxs+Mysδkys+Mxysδkxys+Syzsδγyzs+Sxzsδγxzs]dΩ-∫ΩqδwdΩ=0,
where(10a){Nx,Ny,NxyMxb,Myb,MxybMxs,Mys,Mxys}=∫-h/2h/2(σx,σy,τxy){1zf(z)}dz,(10b)(Sxzs,Syzs)=∫-h/2h/2(τxz,τyz)g(z)dz,The governing equations of equilibrium can be derived from (9) by integrating the displacement gradients by parts and setting the coefficients δu0, δv0, δwb, and δws zero separately. Thus one can obtain the equilibrium equations associated with the present shear deformation theory as follows:
(11)δu:∂Nx∂x+∂Nxy∂y=0,δv:∂Nxy∂x+∂Ny∂y=0,δwb:∂2Mxb∂x2+2∂2Mxyb∂x∂y+∂2Myb∂y2+q=0,δws:∂2Mxs∂x2+2∂2Mxys∂x∂y+∂2Mys∂y2+∂Sxzs∂x+∂Syzs∂y+q=0.
Using (6) in (10a) and (10b) the stress resultants of a sandwich plate made up of three layers can be related to the total strains by
(12){NMbMs}=[ABBsADDsBsDsHs]{εkbks},S=Asγ,
where (13a)N={Nx,Ny,Nxy}t,Mb={Mxb,Myb,Mxyb}t,Ms={Mxs,Mys,Mxys}t,(13b)ε={εx0,εy0,γxy0}t,kb={kxb,kyb,kxyb}t,ks={kxs,kys,kxys}t,(13c)A=[A11A120A12A22000A66],B=[B11B120B12B22000B66],D=[D11D120D12D22000D66],(13d)Bs=[B11sB12s0B12sB22s000B66s],Ds=[D11sD12s0D12sD22s000D66s],Hs=[H11sH12s0H12sH22s000H66s],(13e)S={Sxzs,Syzs}t,γ={γxz,γyz}t,As=[A44s00A55s],where  Aij, Bij, and so forth are the plate stiffness, defined by(14a){A11B11D11B11sD11sH11sA12B12D12B12sD12sH12sA66B66D66B66sD66sH66s}=∫-h/2h/2Q11(1,z,z2,f(z),zf(z),f2(z)){1ν1-ν2}dz,(14b)(A22,B22,D22,B22s,D22s,H22s)=(A11,B11,D11,B11s,D11s,H11s),(14c)A44s=A55s=∫hn-1hnQ44[g(z)]2dz.Substituting from (12) into , we obtain the following equation:(15a)A11d11u0+A66d22u0+(A12+A66)d12v0-B11d111wb-(B12+2B66)d122wb-(B12s+2B66s)d122ws-B11sd111ws=0,(15b)A22d22v0+A66d11v0+(A12+A66)d12u0-B22d222wb-(B12+2B66)d112wb-(B12s+2B66s)d112ws-B22sd222ws=0,(15c)B11d111u0+(B12+2B66)d122u0+(B12+2B66)d112v0+B22d222v0-D11d1111wb-2(D12+2D66)d1122wb-D22d2222wb-D11sd1111ws-2(D12s+2D66s)d1122ws-D22sd2222ws=q,(15d)B11sd111u0+(B12s+2B66s)d122u0+(B12s+2B66s)d112v0+B22sd222v0-D11sd1111wb-2(D12s+2D66s)d1122wb-D22sd2222wb-H11sd1111ws-2(H12s+2H66s)d1122ws-H22sd2222ws+A55sd11ws+A44sd22ws=q,where  dij, dijl, and dijlm are the following differential operators:
(16)dij=∂2∂xi∂xj,dijl=∂3∂xi∂xj∂xl,dijlm=∂4∂xi∂xj∂xl∂xm,di=∂∂xi,(i,j,l,m=1,2).


### 2.4. Exact Solution for a Simply Supported FGM Plate

Rectangular plates are generally classified in accordance with the type of support used. We are here concerned with the exact solution of (15a)–(15d) for a simply supported FG plate. The following boundary conditions are imposed at the side edges:(17a)v0=wb=ws=∂ws∂y=Nx=Mxb=Mxs=0atx=-a2,a2,(17b)u0=wb=ws=∂ws∂x=Ny=Myb=Mys=0aty=-b2,b2,To solve this problem, Navier assumed the transverse mechanical and temperature loads, q, in the form of a double trigonometric series as
(18)q=q0sin(λx)sin(μy),
where  λ=π/a, μ=π/b, and q0 represents the intensity of the load at the plate center.Following the Navier solution procedure, we assume the following solution form foru0, v0, wb and ws that satisfies the following boundary conditions:
(19){u0v0wbws}={Ucos(λx)sin(μy)Vsin(λx)cos(μy)Wbsin(λx)sin(μy)Wssin(λx)sin(μy)},
where  U,V,Wb,  and Ws are arbitrary parameters to be determined subjected to the condition that the solution in (19) satisfies governing equations (15a)–(15d). One obtains the following operator equation:
(20)[C]{Δ}={P},
where {Δ}={U,V,Wb,Ws}t and [C] is the symmetric matrix given by
(21)[C]=[a11a12a13a14a12a22a23a24a13a23a33a34a14a24a34a44],
in which
(22)a11=A11λ2+A66μ2,a12=λμ(A12+A66),a13=-λ[B11λ2+(B12+2B66)μ2],a14=-λ[B11sλ2+(B12s+2B66s)μ2],a22=A66λ2+A22μ2,a23=-μ[(B12+2B66)λ2+B22μ2],a24=-μ[(B12s+2B66s)λ2+B22sμ2],a33=D11λ4+2(D12+2D66)λ2μ2+D22μ4,a34=D11sλ4+2(D12s+2D66s)λ2μ2+D22sμ4,a44=H11sλ4+2(H11s+2H66s)λ2μ2+H22sμ4+A55sλ2+A44sμ2,


## 2.1. Displacement Fields and Strains

The assumed displacement field is as follows:(1)u(x,y,z)=u0(x,y)-z∂wb∂x-f(z)∂ws∂x,v(x,y,z)=v0(x,y)-z∂wb∂y-f(z)∂ws∂y,w(x,y,z)=wb(x,y)+ws(x,y),
where u0 and v0 are the mid-plane displacements of the plate in the x and y directions, respectively; wb and ws are the bending and shear components of transverse displacement, respectively, while f(z) represents shape functions determining the distribution of the transverse shear strains and stresses along the thickness and is given as
(2)f(z)=z[1+3π2sech2(12)]-3π2htanh(zh).
It should be noted that unlike the first-order shear deformation theory, this theory does not require shear correction factors. The kinematic relations can be obtained as follows:
(3)εx=εx0+zkxb+f(z)kxs,εy=εy0+zkyb+f(z)kys,γxy=γxy0+zkxyb+f(z)kxys,γyz=g(z)γyzs,γxz=g(z)γxzs,εz=0,
where
(4)εx0=∂u0∂x,kxb=-∂2wb∂x2,kxs=-∂2ws∂x2,εy0=∂v0∂y,kyb=-∂2wb∂y2,kys=-∂2ws∂y2,γxy0=∂u0∂y+∂v0∂x,kxyb=-2∂2wb∂x∂y,kxys=-2∂2ws∂x∂y,γyzs=∂ws∂y,γxzs=∂ws∂x,g(z)=1-f′(z),f′(z)=df(z)dz.


## 2.2. Constitutive Relations

In FGM, material property gradation is considered through the thickness, and the expression given below represents the profile for the volume fraction(5)P(z)=(Pt-Pb)(zh+12)k+Pb,
where P  denotes a generic material property like modulus, Pt and Pb denote the property of the top and bottom faces of the plate, respectively, and k is a parameter that dictates material variation profile through the thickness. Here, it is assumed that modules E and G vary according to the equation (5), and ν is assumed to be a constant. The linear constitutive relations are
(6){σxσyτyzτxzτxy}=[Q11Q12000Q12Q1100000Q4400000Q5500000Q66]{εxεyγyzγxzγxy},
where
(7)Q11=E(z)1-ν2,Q12=νQ11,Q44=Q55=Q66=E(z)2(1+ν).


## 2.3. Governing Equations

The governing equations of equilibrium can be derived by using the principle of virtual displacements. The principle of virtual work in the present case yields(8)∫-h/2h/2∫Ω[σxδεx+σyδεy+τxyδγxy+τyzδγyz+τxzδγxz]dΩdz-∫ΩqδwdΩ=0,
where Ω is the top surface and q is the applied transverse load.Substituting and (6) into (8) and integrating through the thickness of the plate, (8) can be rewritten as
(9)∫Ω[Nxδεx0+Nyδεy0+Nxyδεxy0+Mxbδkxb+Mybδkyb+Mxybδkxyb+Mxsδkxs+Mysδkys+Mxysδkxys+Syzsδγyzs+Sxzsδγxzs]dΩ-∫ΩqδwdΩ=0,
where(10a){Nx,Ny,NxyMxb,Myb,MxybMxs,Mys,Mxys}=∫-h/2h/2(σx,σy,τxy){1zf(z)}dz,(10b)(Sxzs,Syzs)=∫-h/2h/2(τxz,τyz)g(z)dz,The governing equations of equilibrium can be derived from (9) by integrating the displacement gradients by parts and setting the coefficients δu0, δv0, δwb, and δws zero separately. Thus one can obtain the equilibrium equations associated with the present shear deformation theory as follows:
(11)δu:∂Nx∂x+∂Nxy∂y=0,δv:∂Nxy∂x+∂Ny∂y=0,δwb:∂2Mxb∂x2+2∂2Mxyb∂x∂y+∂2Myb∂y2+q=0,δws:∂2Mxs∂x2+2∂2Mxys∂x∂y+∂2Mys∂y2+∂Sxzs∂x+∂Syzs∂y+q=0.
Using (6) in (10a) and (10b) the stress resultants of a sandwich plate made up of three layers can be related to the total strains by
(12){NMbMs}=[ABBsADDsBsDsHs]{εkbks},S=Asγ,
where (13a)N={Nx,Ny,Nxy}t,Mb={Mxb,Myb,Mxyb}t,Ms={Mxs,Mys,Mxys}t,(13b)ε={εx0,εy0,γxy0}t,kb={kxb,kyb,kxyb}t,ks={kxs,kys,kxys}t,(13c)A=[A11A120A12A22000A66],B=[B11B120B12B22000B66],D=[D11D120D12D22000D66],(13d)Bs=[B11sB12s0B12sB22s000B66s],Ds=[D11sD12s0D12sD22s000D66s],Hs=[H11sH12s0H12sH22s000H66s],(13e)S={Sxzs,Syzs}t,γ={γxz,γyz}t,As=[A44s00A55s],where  Aij, Bij, and so forth are the plate stiffness, defined by(14a){A11B11D11B11sD11sH11sA12B12D12B12sD12sH12sA66B66D66B66sD66sH66s}=∫-h/2h/2Q11(1,z,z2,f(z),zf(z),f2(z)){1ν1-ν2}dz,(14b)(A22,B22,D22,B22s,D22s,H22s)=(A11,B11,D11,B11s,D11s,H11s),(14c)A44s=A55s=∫hn-1hnQ44[g(z)]2dz.Substituting from (12) into , we obtain the following equation:(15a)A11d11u0+A66d22u0+(A12+A66)d12v0-B11d111wb-(B12+2B66)d122wb-(B12s+2B66s)d122ws-B11sd111ws=0,(15b)A22d22v0+A66d11v0+(A12+A66)d12u0-B22d222wb-(B12+2B66)d112wb-(B12s+2B66s)d112ws-B22sd222ws=0,(15c)B11d111u0+(B12+2B66)d122u0+(B12+2B66)d112v0+B22d222v0-D11d1111wb-2(D12+2D66)d1122wb-D22d2222wb-D11sd1111ws-2(D12s+2D66s)d1122ws-D22sd2222ws=q,(15d)B11sd111u0+(B12s+2B66s)d122u0+(B12s+2B66s)d112v0+B22sd222v0-D11sd1111wb-2(D12s+2D66s)d1122wb-D22sd2222wb-H11sd1111ws-2(H12s+2H66s)d1122ws-H22sd2222ws+A55sd11ws+A44sd22ws=q,where  dij, dijl, and dijlm are the following differential operators:
(16)dij=∂2∂xi∂xj,dijl=∂3∂xi∂xj∂xl,dijlm=∂4∂xi∂xj∂xl∂xm,di=∂∂xi,(i,j,l,m=1,2).


## 2.4. Exact Solution for a Simply Supported FGM Plate

Rectangular plates are generally classified in accordance with the type of support used. We are here concerned with the exact solution of (15a)–(15d) for a simply supported FG plate. The following boundary conditions are imposed at the side edges:(17a)v0=wb=ws=∂ws∂y=Nx=Mxb=Mxs=0atx=-a2,a2,(17b)u0=wb=ws=∂ws∂x=Ny=Myb=Mys=0aty=-b2,b2,To solve this problem, Navier assumed the transverse mechanical and temperature loads, q, in the form of a double trigonometric series as
(18)q=q0sin(λx)sin(μy),
where  λ=π/a, μ=π/b, and q0 represents the intensity of the load at the plate center.Following the Navier solution procedure, we assume the following solution form foru0, v0, wb and ws that satisfies the following boundary conditions:
(19){u0v0wbws}={Ucos(λx)sin(μy)Vsin(λx)cos(μy)Wbsin(λx)sin(μy)Wssin(λx)sin(μy)},
where  U,V,Wb,  and Ws are arbitrary parameters to be determined subjected to the condition that the solution in (19) satisfies governing equations (15a)–(15d). One obtains the following operator equation:
(20)[C]{Δ}={P},
where {Δ}={U,V,Wb,Ws}t and [C] is the symmetric matrix given by
(21)[C]=[a11a12a13a14a12a22a23a24a13a23a33a34a14a24a34a44],
in which
(22)a11=A11λ2+A66μ2,a12=λμ(A12+A66),a13=-λ[B11λ2+(B12+2B66)μ2],a14=-λ[B11sλ2+(B12s+2B66s)μ2],a22=A66λ2+A22μ2,a23=-μ[(B12+2B66)λ2+B22μ2],a24=-μ[(B12s+2B66s)λ2+B22sμ2],a33=D11λ4+2(D12+2D66)λ2μ2+D22μ4,a34=D11sλ4+2(D12s+2D66s)λ2μ2+D22sμ4,a44=H11sλ4+2(H11s+2H66s)λ2μ2+H22sμ4+A55sλ2+A44sμ2,


## 3. Numerical Results and Discussions

The study has been focused on the static behavior of functionally graded plate based on the present new higher-order shear deformation model. Here, some representative results of the Navier solution obtained for a simply supported rectangular plate are presented.A functionally graded material consisting of aluminum-alumina is considered. The following material properties are used in computing the numerical values (Bouazza et al. [14]).(i)
Metal (aluminum, Al):EM=70 GPa;  ν=0.3.(ii)
Ceramic (alumina, Al2O3): EC=380 GPa;  ν=0.3.Now, a functionally graded material consisting of aluminum and alumina is considered. Young's modulus for aluminum is 70 GPa while for alumina is 380 GPa. Note that, Poisson's ratio is selected constant for both and equal to 0.3. The various nondimensional parameters used are
(23)w-=10h3Ea4q0w(a2,b2),u-x=100h3Ea4q0ux(a2,b2,-h4),u-y=100h3Ea4q0uy(a2,b2,-h6),σ-x=haq0σx(a2,b2,h2),σ-y=haq0σy(a2,b2,h3),τ-xy=haq0τxy(0,0,-h3),τ-yz=haq0τyz(a2,0,h6),τ-xz=haq0τxz(0,b2,0).It is clear that the deflection increases as the side-to-thickness ratio decreases. The same results were obtained in most literatures. In addition, the correlation between the present new higher-order shear deformation theory and different higher-order and first-order shear deformation theories is established by the author in his recent papers. It is found that this theory predicts the deflections and stresses more accurately when compared to the first- and third-order theories.For the sake of completeness, results of the present theory are compared with those obtained using a new Navier-type three-dimensionally exact solution for small deflections in bending of linear elastic isotropic homogeneous rectangular plates. The center deflectionw and the distribution across the plate thickness of in-plane longitudinal stress σxand longitudinal tangential stress τxy are compared with the results of the 3D solution and are shown in Tables 2 and 3. The present solution is realized for a quadratic plate, with the following fixed data: a=1,b=1,Em=Ec=E=1,q0=1,ν=0.3 and three values for the plate thickness: h=0.01,h=0.03, and  h=0.1. It is to be noted that the present results compare very well with the 3D solution. All deflections again compare well with the 3D solution and show good convergence with the average 3D solution.Table 2
Center deflections of isotropic homogenous plates (k=0, E  and  a/b=1).
h
/
a
CPT [12]
3D [13]  z=0
SSDPT [6]
Present  theory:NHPSDT
Reddy [4]
0.01
44360.9
44384.7
44383.84
44383.86
44383.87
0.03
1643.00
1650.94
1650.646
1650.652
1650.657
0.1
44.3609
46.7443
46.6548
46.65655
46.65836Table 3
Distribution of stresses across the thickness of isotropic homogenous plates (E; a/b=1 and  k=0).
h
/
a
z
σ
x(a/2, b/2, z)
τ
x
y(0, 0,-z)
3D
SSDPT
NHPSDT
Reddy
3D
SSDPT
NHPSDT
Reddy
0.01
0.005
2873.3
2873.39
2873.422
2873.41
1949.6
1949.36
1949.086
1949.061
0.004
2298.6
2298.57
2298.597
2298.593
1559.2
1559.04
1558.854
1558.843
0.003
1723.9
1723.84
1723.861
1723.865
1169.1
1168.99
1168.883
1168.895
0.002
1149.2
1149.18
1149.197
1149.205
779.3
779.18
779.127
779.151
0.001
574.6
574.58
574.585
574.591
389.6
389.55
389.523
389.541
0.000
0.000
0.000
0.00000
0.000
0.000
0.000
0.000
0.000
0.03
0.015
319.4
319.445
319.445
319.437
217.11
217.156
217.082
217.058
0.012
255.41
255.415
255.416
255.413
173.26
173.282
173.255
173.244
0.009
191.49
191.472
191.475
191.48
129.75
129.682
129.686
129.698
0.006
127.63
127.603
127.607
127.615
86.41
86.313
86.330
86.354
0.003
63.8
63.788
63.790
63.796
43.18
43.72
43.126
43.143
0.000
0.000
0.000
0.0000
0.000
0.000
0.000
0.000
0.000
0.10
0.05
28.89
28.9307
28.928
28.92
19.92
20.0476
20.021
20.003
0.04
22.998
23.0055
23.004
23.000
15.606
15.6459
15.638
15.629
0.03
17.182
17.166
17.167
17.171
11.558
11.4859
11.494
11.504
0.02
11.423
11.3994
11.402
11.410
7.642
7.5315
7.546
7.565
0.01
5.702
5.6858
5.687
5.693
3.803
3.7265
3.7369
3.751
0.00
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000In Table4, the effect of volume fraction exponent on the dimensionless stresses and displacements of an FGM square plate (a/h=10) is given. This table shows comparison between results for plates subjected to uniform or sinusoidal distributed loads, respectively. As it is well known, the uniform load distribution always overpredicts the displacements and stresses magnitude. As the plate becomes more and more metallic, the difference increases for deflection w and in-plane longitudinal stress σx while it decreases for in-plane normal stress σy. It is important to observe that the stresses for a fully ceramic plate are the same as that for a fully metal plate. This is because the plate for these two cases is fully homogeneous, and the stresses do not depend on the modulus of elasticity. Results in Table 4 should serve as benchmark results for future comparisons.Table 4
Effects of volume fraction exponent and loading on the dimensionless stresses and displacements of a FGM square plate (a/h=10).
k
Theory
w
σ
x
σ
y
τ
y
z
τ
x
z
τ
x
y
0
NHPSDT
0.4665
2.8928
1.9104
0.4424
0.5072
1.2851
ceramic
SSDPT
0.4665
2.8932
1.9103
0.4429
0.5114
1.2850
Reddy
0.4665
2.8920
1.9106
0.4411
0.4963
1.2855
NHPSDT
0.9421
4.2607
2.2569
0.54404
0.50721
1.1573
1
SSDPT
0.9287
4.4745
2.1692
0.5446
0.5114
1.1143
Reddy
0.94214
4.25982
2.25693
0.54246
0.49630
1.15725
NHPSDT
1.2228
4.8890
2.1663
0.5719
0.4651
1.0448
2
SSDPT
1.1940
5.2296
2.0338
0.5734
0.4700
0.9907
Reddy
1.22275
4.88814
2.16630
0.56859
0.45384
1.04486
NHPSDT
1.3533
5.2064
1.9922
0.56078
0.4316
1.0632
3
SSDPT
1.3200
5.6108
1.8593
0.5629
0.4367
1.0047
Reddy
1.3530
5.20552
1.99218
0.55573
0.41981
1.06319
NHPSDT
1.4653
5.7074
1.7143
0.50075
0.4128
1.1016
5
SSDPT
1.4356
6.1504
1.6104
0.5031
0.4177
1.0451
Reddy
1.46467
5.70653
1.71444
0.49495
0.40039
1.10162
NHPSDT
1.6057
6.9547
1.3346
0.4215
0.4512
1.1118
10
SSDPT
1.5876
7.3689
1.2820
0.4227
0.4552
1.0694
Reddy
1.60541
6.95396
1.33495
0.41802
0.43915
1.1119
∞
NHPSD
2.5327
2.8928
1.9104
0.4424
0.5072
1.2851
métal
SSDPT
2.5327
2.8932
1.9103
0.4429
0.5114
1.2850
Reddy
2.5328
2.8920
1.9106
0.4411
0.4963
1.2855Tables5 and 6 compare the deflections and stresses of different types of the FGM square plate (a/b=1,k=0) and FGM rectangular plate (b=3a,k=2).The deflections decrease as the aspect ratio a/bincreases and this irrespective of the type of the FGM plate. All theories (SSDPT, PSDPT, and NHPSDT) give the same axial stress  σx  and  σy for a fully ceramic plate (k=0). In general, the axial stress increases with the volume fraction exponent  k. The transverse shear stress for a FGM plate subjected to a distributed load. The results show that the transverse shear stresses may be indistinguishable. As the volume fraction exponent increases for FGM plates, the shear stress will increase, and the fully ceramic plates give the smallest shear stresses.Table 5
Comparison of normalized displacements and stresses of a FGM square plate (a/b=1) and k=0.
a
/
h
Theory
w
σ
x
σ
y
τ
y
z
τ
x
z
τ
x
y
NHPSDT
0.5866
1.1979
0.7536
0.4307
0.4937
0.4908
4
SSDPT
0.5865
1.1988
0.7534
0.4307
0.4973
0.4906
Reddy
0.5868
1.1959
0.7541
0.4304
0.4842
0.4913
NHPSDT
0.4665
2.8928
1.9104
0.4424
0.5072
1.2851
10
SSDPT
0.4665
2.8932
1.9103
0.4429
0.5114
1.2850
Reddy
0.4666
2.8920
1.9106
0.4411
0.4963
1.2855
NHPSDT
0.4438
28.7342
19.1543
0.4466
0.5119
12.9884
100
SSDPT
0.4438
28.7342
19.1543
0.4472
0.5164
13.0125
Reddy
0.4438
28.7341
19.1543
0.4448
0.5004
12.9885Table 6
Comparison of normalized displacements and stresses of a FGM rectangular plate (b=3a) and k=2.
a
/
h
Theory
w
σ
x
σ
y
τ
y
z
τ
x
z
τ
x
y
NHPSDT
4.0569
5.2804
0.6644
0.6084
0.6699
0.5900
4
SSDPT
3.99
5.3144
0.6810
0.6096
0.6796
0.5646
Reddy
4.0529
5.2759
0.6652
0.6058
0.6545
0.5898
NHPSDT
3.5543
12.9252
1.6938
0.61959
0.6841
1.4898
10
SSDPT
3.5235
12.9374
1.7292
0.6211
0.6910
1.4500
Reddy
3.5537
12.9234
1.6941
0.6155
0.6672
1.4898
NHPSDT
3.4824
25.7712
3.3971
0.6214
0.6878
2.9844
20
SSDPT
3.4567
25.7748
3.4662
0.6232
0.6947
2.9126
Reddy
3.48225
25.7703
3.3972
0.6171
0.6704
2.9844
NHPSDT
3.4593
128.728
17.0009
0.6220
0.6894
14.9303
100
SSDPT
3.4353
128.713
17.3437
0.6238
0.6963
14.584
Reddy
3.45937
128.7283
17.0009
0.6177
0.67176
14.9303Figures2 and 3  show the variation of the center deflection with the aspect and side-to-thickness ratios, respectively. The deflection is maximum for the metallic plate and minimum for the ceramic plate. The difference increases as the aspect ratio increases while it may be unchanged with the increase of side-to-thickness ratio. One of the main inferences from the analysis is that the response of FGM plates is intermediate to that of the ceramic and metal homogeneous plates (see also Table 4). It is to be noted that, in the case of thermal or combined loads and under certain conditions, the above response is not intermediate.Figure 2
Dimensionless center deflection(w) as function of the aspect ratio (a/b) of an FGM plate.Figure 3
Dimensionless center deflection(w) as a function of the side-to-thickness ratio (a/h) of an FGM square plate.Figures7 and 8 depict the through-the-thickness distributions of the shear stresses τyz  and  τxz,  the in-plane longitudinal and normal stresses σx and σy, and the longitudinal tangential stress τxy in the FGM plate under the uniform load. The volume fraction exponent of the FGM plate is taken as k=2 in these figures. Distinction between the curves in Figures 8 and 9 is obvious. As strain gradients increase, the inhomogeneities play a greater role in stress distribution calculations. The through-the-thickness distributions of the shear stresses τyz and τxz are not parabolic, and the stresses increase as the aspect ratio decreases. It is to be noted that the maximum value occurs at z≅0.2, not at the plate center as in the homogeneous case.As exhibited in Figures5 and 6, the in-plane longitudinal and normal stresses, σx and σy, are compressive throughout the plate up to z≅0.155 and then they become tensile. The maximum compressive stresses occur at a point on the bottom surface and the maximum tensile stresses occur, of course, at a point on the top surface of the FGM plate. However, the tensile and compressive values of the longitudinal tangential stress, τxy (cf. Figure 7), are maximum at a point on the bottom and top surfaces of the FGM plate, respectively. It is clear that the minimum value of zero for all in-plane stresses σx, σy and τxy occurs at z≅0.153 and this is irrespective of the aspect and side-to-thickness ratios.Finally, the exact maximum deflections of simply supported FGM square plate are compared in Figure4 for various ratios of module, Em/Ec (for a given thickness, a/h=10). This means that the deflections are computed for plates with different ceramic-metal mixtures. It is clear that the deflections decrease smoothly as the volume fraction exponent decreases and as the ratio of metal-to-ceramic modules increases.Figure 4
The effect of anisotropy on the dimensionless maximum deflection(w) of an FGM plate for different values of k.Figure 5
Variation of in-plane longitudinal stress(σxx) through-the thickness of an FGM plate for different values of the side-to-thickness ratio.Figure 6
Variation of in-plane normal stress(σyy) through-the thickness of an FGM plate for different values of the aspect ratio.Figure 7
Variation of longitudinal tangential stress(τxy) through-the thickness of an FGM plate for different values of the aspect ratio.Figure 8
Variation of transversal shear stress(τyz) through-the thickness of an FGM plate for different values of the aspect ratio.Figure 9
Variation of transversal shear stress(τxz) through the thickness of an FGM plate for different values of the aspect ratio.


## 4. Conclusion

In this study, a new higher-order shear deformation model is proposed to analyze the static behavior of functionally graded plates. Unlike any other theory, the theory presented gives rise to only four governing equations resulting in considerably lower computational effort when compared with the other higher-order theories reported in the literature having more number of governing equations. Bending and stress analysis under transverse load were analyzed, and results were compared with previous other shear deformation theories. The developed theories give parabolic distribution of the transverse shear strains and satisfy the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The accuracy and efficiency of the present theories have been demonstrated for static behavior of functionally graded plates. All comparison studies demonstrated that the deflections and stresses obtained using the present new higher-order shear deformation theories (with four unknowns) and other higher shear deformation theories such as PSDPT and SSDPT (with five unknowns) are almost identical. The extension of the present theory is also envisaged for general boundary conditions and plates of a more general shape. In conclusion, it can be said that the proposed theory NHPSDT is accurate and simple in solving the static behaviors of FGM plates.



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*Source: 159806-2012-10-10.xml*