ANewHyperbolic Shear Deformation Theory for Bending Analysis of Functionally Graded Plates

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.


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, S ¸ims ¸ek (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.S ¸ims ¸ek (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 pinnedpinned supports due to a moving harmonic load has been examined by S ¸ims ¸ek (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 (Table 1).
In 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.

Problem Formulation
Consider a plate of total thickness h 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).

Displacement Fields and Strains.
The assumed displacement field is as follows: where u 0 and v 0 are the mid-plane displacements of the plate in the x and y directions, respectively; w b and w s 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 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: where (4)

Constitutive Relations.
In FGM, material property gradation is considered through the thickness, and the expression given below represents the profile for the volume fraction where P denotes a generic material property like modulus, P t and P b 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

Modelling and Simulation in Engineering
ν is assumed to be a constant.The linear constitutive relations are where 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 where Ω is the top surface and q is the applied transverse load.Substituting (3) and ( 6) into (8) and integrating through the thickness of the plate, (8) can be rewritten as where The governing equations of equilibrium can be derived from ( 9) by integrating the displacement gradients by parts and setting the coefficients δu 0 , δv 0 , δw b , and δw s zero separately.
Thus one can obtain the equilibrium equations associated with the present shear deformation theory as follows: 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 where where A i j , B i j , and so forth are the plate stiffness, defined by Substituting from ( 12) into (11), we obtain the following equation: where d i j , d i jl , and d i jlm are the following differential operators: 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: To solve this problem, Navier assumed the transverse mechanical and temperature loads, q, in the form of a double trigonometric series as where λ = π/a, μ = π/b, and q 0 represents the intensity of the load at the plate center.Following the Navier solution procedure, we assume the following solution form for u 0 , v 0 , w b and w s that satisfies the following boundary conditions: U cos(λx) sin μy V sin(λx) cos μy W b sin(λx) sin μy W s sin(λx) sin μy where U, V , W b , and W s 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: where

Numerical Results and Discussions
The study has been focused on the static behavior of functionally graded plate based on the present new higherorder 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 aluminumalumina is considered.The following material properties are used in computing the numerical values (Bouazza et al. [14]).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 It is clear that the deflection increases as the side-tothickness 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 deflection w and the distribution across the plate thickness of in-plane longitudinal stress σ x and longitudinal tangential stress τ xy 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: 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.
In Table 4, 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 inplane longitudinal stress σ x while it decreases for in-plane normal stress σ y .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.
Tables 5 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 σ x and σ y 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.
Figures 2 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.
Figures 7 and 8 depict the through-the-thickness distributions of the shear stresses τ yz and τ xz , the in-plane longitudinal and normal stresses σ x and σ y , and the longitudinal tangential stress τ xy 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-thethickness distributions of the shear stresses τ yz and τ xz 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 Figures 5 and 6, the in-plane longitudinal and normal stresses, σ x and σ y , 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, τ xy (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 σ x , σ y and τ xy 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 Figure 4 for various ratios of module, E m /E c (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.

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 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.

= 1 wFigure 3 :Figure 4 :
Figure 3: Dimensionless center deflection (w) as a function of the side-to-thickness ratio (a/h) of an FGM square plate.

Figure 5 := 3 a/h = 10 k = 2 Figure 6 :
Figure 5: Variation of in-plane longitudinal stress (σ xx ) throughthe thickness of an FGM plate for different values of the side-tothickness ratio.

Table 3 :
Distribution of stresses across the thickness of isotropic homogenous plates (E; a/b = 1 and k = 0).

Table 4 :
Effects of volume fraction exponent and loading on the dimensionless stresses and displacements of a FGM square plate (a/h = 10).

Table 5 :
Comparison of normalized displacements and stresses of a FGM square plate (a/b = 1) and k = 0.

Table 6 :
Comparison of normalized displacements and stresses of a FGM rectangular plate (b = 3a) and k = 2.