# Bending Analysis of Functionally Graded Plates in the Context of Different Theories of Thermoelasticity

Authors: A. M. Zenkour; D. S. Mashat; K. A. Elsibai
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The quasistatic bending response is presented for a simply supported functionally graded rectangular plate subjected to a through-the-thickness temperature field under the effect of various theories of generalized thermoelasticity, namely, classical dynamical coupled theory, Lord and Shulman's theory with one relaxation time, and Green and Lindsay's theory with two relaxation times. The generalized shear deformation theory obtained by the first author is used. Material properties of the plate are assumed to be graded in the thickness direction according to a simple exponential law distribution in terms of the volume fractions of the constituents. The numerical illustrations concern quasistatic bending response of functionally graded square plates with two constituent materials are studied using the different theories of generalized thermoelasticity


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## Body

## 1. Introduction

In the recent years, functionally graded materials (FGMs) have gained considerable attention in many engineering applications. FGMs are considered as a potential structural material for future high-speed spacecraft and power generation industries. FGMs are new materials, microscopically inhomogeneous, in which the mechanical properties vary smoothly and continuously from one surface to the other [1–7].The effect of thermal loading on the displacement and stress fields for FGM plates and shells has been studied by a number of authors. For example, Wetherhold et al. [5] have considered the use of functionally graded materials to eliminate or control thermal deformation in beams and plates. Suresh and Mortensen [6] have discussed the large deformation of graded multilayered composites under mechanical and thermal loads. Praveen and Reddy [7] have investigated the response of functionally graded ceramic-metal plates using a finite element that accounts for the transverse shear strains, rotary inertia, and moderately large rotations in the von Karman sense.The theory of thermoelasticity that includes the effect of temperature change has been well established. According to this theory, the temperature field is coupled with the elastic strain field. This theory covers a wide range of extensions of classical dynamical coupled thermoelasticity. Lord and Shulman [8] and Green and Lindsay [9] have extended the coupled theory of thermoelasticity by introducing the thermal relaxation times in the constitutive equations. Additional thermoelasticity theories have been presented and investigated by other researchers [10–22].Lord and Shulman [8] have considered isotropic solids and introduced one relaxation time parameter into the Fourier heat conduction equation; that is, both the heat flux and its time derivative are considered in the heat conduction equation. The heat equation associated with this theory is thus hyperbolic. A direct consequence is that the paradox of infinite speed of propagation inherent in both the uncoupled and coupled theories of classical thermoelasticity is eliminated and the heat wave feature can be modeled by the generalized thermoelasticity. The Green and Lindsay's theory [9] does not violate the Fourier's law of heat conduction when the body under consideration has a center of symmetry. In this theory, both the equations of motion and heat conduction are hyperbolic but the equation of motion is modified and differs from that of classical coupled thermoelasticity theory.In this paper, a generalized nonclassical dynamic coupled thermoelasticity analysis is carried out on a functionally graded material plate. Through-the-thickness temperature distribution varying according to exponential law is considered and then temperature and stress behavior are presented for the mentioned plate. The governing equations of FGM plate for two-dimensional generalized thermoelastic problems are derived within the framework of the classical coupled theory, Lord and Shulman's theory, and Green and Lindsay's theory. The material properties of the functionally graded plate are assumed to vary continuously through the thickness according to an exponential law distribution of the volume fraction of the constituents. A generalized shear deformation theory is presented to obtain the governing equations. An exact solution for the coupled governing equations under simply supported boundary conditions is obtained. Numerical results are provided to show the influence of the material properties, and a temperature field on the displacement and stresses.


## 2. Formulation of the Problem

We consider a solid rectangular plate of lengtha, width b, and thickness h, made of functionally graded materials. The material properties of the FGM plate are assumed to be function of the volume fraction of the constituent materials. Using the rectangular Cartesian coordinates (x,y,z) we take the functional graded between the physical properties and z for ceramic and metal FGM plate
(2.1)P(z)=PmeηP((2z+h)/2h),ηp=ln(PcPm),
where Pc and Pm are the corresponding properties of ceramic (top surface) and metal (bottom surface), respectively.The displacements of a material point located at(x,y,z) in the plate may be written as
(2.2)ux(x,y,z,t)=u-z∂xw+Ψφx,uy(x,y,z,t)=v-z∂yw+Ψφy,uz(x,y,z,t)=w,
where Ψ=ℏsin(πz/h),ℏ=h/π,and ∂x,∂y represent the differentiation with respect to x and y.(ux,uy,uz) are the displacements corresponding to the co-ordinate system and are functions of the spatial co-ordinates, (u,v,w) are the displacements along the axes x,y, and z, respectively, and φx and φy are the rotations about the y- and x-axes.The strain components will be(2.3)εx=εx0+zεx1+Ψεx2,εy=εy0+zεy1+Ψεy2,εxy=εxy0+zεxy1+Ψεxy2,εz=0,εyz=Ψ′φy,εxz=Ψ′φx,
where
(2.4)εx0=∂xu,εy0=∂yv,εxy0=∂xv+∂yu,εx1=-∂x2w,εy1=-∂y2w,εxy1=-2∂x∂yw,εx2=∂xφx,εy2=∂yφy,εxy2=∂xφy+∂yφx.


## 3. Theories of Thermoelasticity

In addition, the stress-strain-temperature relations for the linear thermoelastic materials are given, according to the generalized theories of thermoelasticity, by(3.1)σij=cijklεkl-βij(T+t1Ṫ-T0),
where T,T0,εij,βij,cijkl, and t1 are the absolute temperature, reference temperature, strain tensor components, components of tensor of stress-temperature moduli, components of tensor of elastic moduli, and the first relaxation time of Green and Lindsay's theory, respectively.In details, we can rewrite the stress components in the deferent theories of generalized thermoelasticity as follows:(3.2)σx=E1εx+E2εy-β(T+t1Ṫ-T0),σy=E1εy+E2εx-β(T+t1Ṫ-T0),σz=0,σyz=E3εyz,σxz=E3εxz,σxy=E3εxy,
where
(3.3)E1=E1-v2,E2=Ev1-v2,E3=E2(1+v),
and the material properties E and v are functions of z. In the absence of body forces and internal heat generation, the heat conduction equation will be in the form
(3.4)(κT,j),j-ρc(Ṫ+t2T̈)-βT0(u̇j+t3üj),j=0,
where t2 and t3 are additional relaxation times. A comma followed by index j denotes partial differentiation with respect to the position xj of a material particle. A superimposed dot indicates partial derivative with respect to time t. In addition to the elastic coefficients E, and v, the material properties κ,ρ,c, and β are also functions of z.It is clear that, by settingt1=0 in (3.2) and t2=t3=0 in (3.4), we get the field equations for the conventional coupled theory of thermoelasticity; whereas when t1=0 and t2=t3≠0, the equations reduce to the Lord and Shulman's theory and when t3=0 and t1 and t2 are nonvanishing, the equations reduce to the Green and Lindsay's theory.Note that the stress-temperature modulusβ is given in terms of Young's modulus E, Poisson's ratio v, and the thermal expansion coefficient α by the relation(3.5)β=αE1-2v.
Generally, this study assumes that Young’s modulus E, Poisson’s ratio v, material density ρ, thermal expansion coefficient α, specific heat capacity c, and thermal conductivity coefficient κ of the FGM change continuously through the thickness direction of the plate and obey the gradation relation given in (2.1).


## 4. Solution of the Problem

To solve the problem, we obtain the stress and moment resultants for the FGM plate by integrating the stress components given in (3.2) over the thickness and written as(4.1)Nx=A11εx0+A12εy0+A21εx1+A22εy1+A31εx2+A32εy2-B1(T+t1Ṫ-To),Ny=A11εy0+A12εx0+A21εy1+A22εx1+A31εy2+A32εx2-B1(T+t1Ṫ-To),Nxy=A13εxy0+A23εxy1+A33εxy2,Mx=A21εx0+A22εy0+A41εx1+A42εy1+A51εx2+A52εy2-B2(T+t1Ṫ-To),My=A21εy0+A22εx0+A41εy1+A42εx1+A51εy2+A52εx2-B2(T+t1Ṫ-To),Mxy=A23εxy0+A43εxy1+A53εxy2,Sx=A31εx0+A32εy0+A51εx1+A52εy1+A61εx2+A62εy2-B3(T+t1Ṫ-To),Sy=A31εy0+A32εx0+A51εy1+A52εx1+A61εy2+A62εx2-B3(T+t1Ṫ-To),Sxy=A33εxy0+A53εxy1+A63εxy2,Qxz=A73φx,Qyz=A73φy.
Here, the coefficients Ai1,Ai2(i=1,…,6),Aj3(j=1,…,7), and Bk(k=1,2,3) are defined by(4.2){A11,A21,A31,A41,A51,A61}=∫-h/2+h/2E1{1,z,Ψ,z2,zΨ,Ψ2}dz,{A12,A22,A32,A42,A52,A62}=∫-h/2+h/2E2{1,z,Ψ,z2,zΨ,Ψ2}dz,{A13,A23,A33,A43,A53,A63,A73}=∫-h/2+h/2E3{1,z,Ψ,z2,zΨ,Ψ2,Ψ′}dz,{B1,B2,B3}=∫-h/2+h/2β{1,z,Ψ}dz.By using Hamilton's principle, the governing equations can be obtained in the form(4.3)∂xNx+∂yNxy=0,∂xNxy+∂yNy=0,∂x2Mx+2∂x∂yMxy+∂y2My=0,∂xSx+∂ySxy-Qxz=0,∂xSxy+∂ySy-Qyz=0.
The edges of the plate are assumed to be simply supported and maintained at the reference temperature. That is,
(4.4)w=v=φy=Nx=Mx=Sx=T=0,atx=0,a,w=u=φx=Ny=My=Sy=T=0,aty=0,b,
For the present problem, the solution for the change in temperature is sought in the form
(4.5)T(x,y,z,t)=eIωtτ(z)sin(λx)sin(μy),
where λ=π/a,μ=π/b, and I=-1. This temperature identically satisfies the boundary conditions given in (4.4) at the edges of the plate. The function τ(z) will be obtained from the solution of the heat equation (3.4). In addition, we assume the following solution form for (u,v,w,φx,φy) that satisfies the boundary conditions:(4.6){uvwφxφy}={Ucos(λx)sin(μy)Vsin(λx)cos(μy)Wsin(λx)sin(μy)Xcos(λx)sin(μy)Ysin(λx)cos(μy)}eIωt,
where U,V,W,X, and Y are arbitrary parameters. Finally, we get the stress components
(4.7)σx=-eIωt{E1[λU-zλ2W+ΨλX]+E2[μV-zμ2W+ΨμY]+βτ}sin(λx)sin(μy)+βTo,σy=-eIωt{E2[λU-zλ2W+ΨλX]+E1[μV-zμ2W+ΨμY]+βτ}sin(λx)sin(μy)+βTo,σyz=E3eIωtΨ′Ysin(λx)cos(μy),σxz=E3eIωtΨ′Xcos(λx)sin(μy),σxy=E3eIωt[μU+λV-2zλμW+Ψ(μX+λY)]cos(λx)cos(μy).
Substitution for T from (4.5) into (3.4), with aid of (4.6), gives the following partial differential equation with variable coefficients:
(4.8)d2τdz2+1κdκdzdτdz-[λ2+μ2+Iρcωκ(1-t2ω)]τ+IωβT0(1+It3ω)κ[λU+μV-z(λ2+μ2)W+Ψ(λX+μY)]=0.
Inserting into the above ordinary differential equation the material properties κ,ρ,c, and β from (2.1), we obtain
(4.9)d2τdz2+ηκhdτdz-[λ2+μ2+Iρmcmωκm(1-ωt2)e(ηρc-ηκ)((2z+h)/2h)]τ+IωβmT0(1+It3ω)κme(ηβ-ηκ)((2z+h)/2h)[λU+μV-z(λ2+μ2)W+hπsin(πzh)(λX+μY)]=0,
where
(4.10)ηκ=ln(κcκm),ηρc=ln(ρcccρmcm),ηβ=ln(βcβm).The exact closed form solution of (4.9) is given by
(4.11)τ(z)=e-ηκz/2h[BI(-ξ,ξ)C1+BK(ξ,ξ)C2+ωT0βmF(z)],
where C1 and C2 are the integration constants, and the functions BI and BK are the modified Bessel's functions of the first and second kinds, respectively, in which(4.12)ξ=ηκ2+h2(λ2+μ2)|ηρc-ηκ|,ζ=2Iγω(t2ω-I)ηκ-ηρce(ηρc-ηκ)((2z+h)/4h).
Note that the term
(4.13)γ=ρmcmh2κm
represents the characteristic time of heat conduction through length h. The function F(z) is given by
(4.14)F(z)=BK(ξ,ζ)∫BI(-ξ,ζ)ΘΔdz-BI(-ξ,ζ)∫BK(ξ,ζ)ΘΔdz,
where
(4.15)Δ=κm{BK(ξ,ζ)BI(1-ξ,ζ)γω(t2ω-I)+BI(-ξ,ζ)[γω(t2ω-I)BK(1+ξ,ζ)-Iξ(ηρc-ηκ)e(ηκ-ηρc)((2z+h)/4h)BK(ξ,ζ)]},Θ=-he(2z(2ηβ-ηρc)+h(2ηβ-ηκ-ηρc))/4h[λU+μV-z(λ2+μ2)W+ℏsin(πzh)(λX+μY)](1+It3ω).
Note that the constants C1and C2 are given from the temperature boundary conditions at the lower and upper surfaces of the plate
(4.16)τ(z)={τ0,atz=-h2,τ1,atz=+h2.


## 5. Numerical Results and Discussion

We present exact results for a simply supported FG square plate subjected to a transient thermal load. Since it is common in high-temperature applications to employ a ceramic top layer as a thermal barrier to a metallic structure, we choose the constituent materials of the FG plate to be Aluminum (Al) and Silicon (SiC) having the following material properties:(5.1)Al:Em=70GPa,vm=0.3,αm=23.4×10-6/K,κm=233W/mk,cm=896J/kgK,ρm=2707kg/m3,SiC:Ec=427GPa,vc=0.17,αc=4.3×10-6/K,κc=65W/mK,cc=670J/kgK,ρc=3100kg/m3.
The dimensions of the simply supported FG plate are a=b=0.25 m. In addition, the values of different parameters are used as follows:
(5.2)T0=1,τ0=0,τ1=1,ω=5×10-41/s.
Numerical results are presented in terms of the nondimensional variables defined as
(5.3)t=t103γ,T=1τ1T(a2,b2,Z,t),w=1hαmτ1u3(a2,b2,Z,t),σ1=1Emαmτ1σx(a2,b2,Z,t),σ5=ahEmαmτ1σxz(0,b2,Z,t),σ6=1Emαmτ1σxy(0,0,Z,t),Z=zh,
The results of the classical coupled theory (C-T), Lord and Shulman's theory (L-S), and Green and Lindsay's theory (G-L) for various values of time parameter t are listed in Table 1. The relaxation times for these theories are chosen to be
(5.4)C-T:t1=t2=t3=0,L-S:t1=0,t2=t3=0.02s,G-L:t1=0.01s,t2=2t1,t3=0.
The dimensionless temperature T, displacement w, longitudinal stress σ1, transverse shear stress σ5, and in-plane shear stress σ6 are given at z=0,0,1/2,1/4, and 1/2, respectively. It can be found from Table 1 that the results obtained by the L-S model agree well with those obtained by C-T whereas the G-L model gives an accurate prediction of the results that slightly differ from the above two models. The temperature given in the context of all theories may be unchanged.Table 1
Dimensionless results for FGM square plates according to various theories of thermoelasticity with different time parameters.
tTheoryTwσ1σ5σ60.05C-T0.3331221020.3839669850.5696398320.393610131-0.404103471L-S0.3331221050.3839669880.5696398320.393610133-0.404103471G-L0.3331221050.3839666750.5696393650.393609811-0.4041031390.10C-T0.3199152200.3687443180.5469836550.377986612-0.388031140L-S0.3199152220.3687443210.5469836550.377986614-0.388031140G-L0.3199152220.3687437010.5469827320.377985978-0.3880304850.20C-T0.2687381090.3097559740.4593334910.317481666-0.325851963L-S0.2687381110.3097559770.4593334910.317481667-0.325851963G-L0.2687381110.3097548000.4593317430.317480460-0.3258507230.50C-T-0.018542705-0.021372904-0.032152421-0.0220235210.022808982L-S-0.018542705-0.021372904-0.032152421-0.0220235210.022808982G-L-0.018542705-0.021374846-0.032155303-0.0220255120.0228110270.80C-T-0.289521187-0.333711203-0.495370643-0.3421661200.351416781L-S-0.289521189-0.333711205-0.495370643-0.3421661220.351416781G-L-0.289521189-0.333712205-0.495372125-0.3421671460.3514178321.00C-T-0.335469789-0.386673003-0.573677254-0.3963899680.406967624L-S-0.335469792-0.386673006-0.573677254-0.3963899700.406967624G-L-0.335469792-0.386672791-0.573676933-0.3963897490.406967397It is well known that G-L theory is accurate to predict temperature, displacement, and stresses, so some results have been plotted in Figures1–8. The through-the-thickness variation of the temperature and stresses is plotted in Figures 1, 2, 3, and 4 for different values of the time parameter t. The through-the-thickness variation of the longitudinal stress σ1, transverse shear stress σ5, and in-plane shear stress σ6 changes significantly as a function of time. For example, at t=0 the magnitude of the temperature and longitudinal stress is maximum at a point on the top surface of the plate. However at the maximum values of the transverse shear stress occur at z = 0.298 for lower values of t. The magnitude of the in-plane stress is maximum at a point on the top surface of the plate and it increases as t increases. Since material properties and the temperature change vary through the thickness, then the through-the-thickness variation of the longitudinal stress is nonlinear and the maximum values of the transverse shear stress dose not occur at the center of the plate.Figure 1
Through-the-thickness variation of dimensionless temperatureT of FGM square plate for different values of the time parameter t.Figure 2
Through-the-thickness variation of dimensionless longitudinal stressσ1 of FGM square plate for different values of the time parameter t.Figure 3
Through-the-thickness variation of dimensionless transverse shear stressσ5 of FGM square plate for different values of the time parameter t.Figure 4
Through-the-thickness variation of dimensionless in-plane shear stressσ6 of FGM square plate for different values of the time parameter t.Figure 5
Dimensionless displacementw versus time parameter t of FGM square plate for different values of relaxation time t*.Figure 6
Dimensionless longitudinal stressσ1 versus time parameter t of FGM square plate for different values of relaxation time t*.Figure 7
Dimensionless transverse shear stressσ5 versus time parameter t of FGM square plate for different values of relaxation time t*.Figure 8
Dimensionless in-plane shear stressσ6 versus time parameter t of FGM square plate for different values of relaxation time t*.Now, we discuss the effect of relaxation times in G-L model. Let the first relaxation timet1=t* and the second relaxation time be double of it. Figures 5, 6, 7, and 8 present the displacement and stresses versus the time parameter using different values of the relaxation time t*. It is clear that all results are very sensitive to the variation of the relaxation time. The results may be inaccurate for higher values of t*.


## 6. Conclusion

In this paper, the numerical illustrations concern quasistatic bending response of FG square plates are studied in the context of the generalized thermoelasticity theories. A refined shear deformation theory is used for this purpose. Material properties of the plate are assumed to be graded in the thickness direction according to a simple exponential law distribution in terms of the volume fractions of the constituents. An exact solution for the present problem is obtained. Numerical results are provided to show the influence of the material properties, and a temperature field on the displacement and stresses.From these results, we can conclude that(1)
the results of G-L model give an accurate prediction comparing with those obtained by the other two models;(2)
the results of L-S model agree well with those obtained by C-T model;(3)
the temperature may be independent of the parameters used in the different theories;(4)
for higher values of the relaxation times, L-S and G-L models may be failed to get accurate solution comparing with the C-T model.



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