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

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.

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 (Table 1).

Table 1 

Displacement models.

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.

2. 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).

Figure 1 

Geometry of rectangular plate composed of FGM.

2.1. Displacement Fields and Strains

The assumed displacement field is as follows: 𝑢𝑥,𝑦,𝑧=𝑢₀𝑥,𝑦−𝑧𝜕𝑤_𝑏(𝑧)𝜕𝑥−𝑓𝜕𝑤_𝑠,𝑣𝜕𝑥𝑥,𝑦,𝑧=𝑣₀𝑥,𝑦−𝑧𝜕𝑤_𝑏(𝑧)𝜕𝑦−𝑓𝜕𝑤_𝑠,𝑤𝜕𝑦𝑥,𝑦,𝑧=𝑤_𝑏𝑥,𝑦+𝑤_𝑠𝑥,𝑦, (1) where 𝑢₀ and 𝑣₀ 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 𝑓(𝑧)=𝑧1+3𝜋2secℎ²12−3𝜋2𝑧ℎtanhℎ. (2) 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: 𝜀_𝑥=𝜀⁰_𝑥+𝑧𝑘^𝑏_𝑥+𝑓(𝑧)𝑘^𝑠_𝑥,𝜀_𝑦=𝜀⁰_𝑦+𝑧𝑘^𝑏_𝑦+𝑓(𝑧)𝑘^𝑠_𝑦,𝛾_𝑥𝑦=𝛾⁰_𝑥𝑦+𝑧𝑘^𝑏_𝑥𝑦+𝑓(𝑧)𝑘^𝑠_𝑥𝑦,𝛾_𝑦𝑧=𝑔(𝑧)𝛾^𝑠_𝑦𝑧,𝛾_𝑥𝑧=𝑔(𝑧)𝛾^𝑠_𝑥𝑧,𝜀_𝑧=0, (3) where 𝜀⁰_𝑥=𝜕𝑢₀𝜕𝑥,𝑘^𝑏_𝑥𝜕=−²𝑤_𝑏𝜕𝑥²,𝑘^𝑠_𝑥𝜕=−²𝑤_𝑠𝜕𝑥²,𝜀⁰_𝑦=𝜕𝑣₀𝜕𝑦,𝑘^𝑏_𝑦𝜕=−²𝑤_𝑏𝜕𝑦²,𝑘^𝑠_𝑦𝜕=−²𝑤_𝑠𝜕𝑦²,𝛾⁰_𝑥𝑦=𝜕𝑢₀+𝜕𝑦𝜕𝑣₀𝜕𝑥,𝑘^𝑏_𝑥𝑦𝜕=−2²𝑤_𝑏𝜕𝑥𝜕𝑦,𝑘^𝑠_𝑥𝑦𝜕=−2²𝑤_𝑠,𝛾𝜕𝑥𝜕𝑦_𝑠𝑦𝑧=𝜕𝑤_𝑠𝜕𝑦,𝛾^𝑠_𝑥𝑧=𝜕𝑤_𝑠(𝑧)𝜕𝑥,𝑔=1−𝑓^′(𝑧),𝑓^′(𝑧)=𝑑𝑓(𝑧).𝑑𝑧 (4)

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 𝑃(𝑧)=𝑃_𝑡−𝑃_𝑏𝑧ℎ+12^𝑘+𝑃_𝑏, (5) 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 𝜈 is assumed to be a constant. The linear constitutive relations are 𝜎_𝑥𝜎_𝑦𝜏_𝑦𝑧𝜏_𝑥𝑧𝜏_𝑥𝑦=𝑄₁₁𝑄₁₂𝑄000₁₂𝑄₁₁00000𝑄₄₄00000𝑄₅₅00000𝑄₆₆𝜀_𝑥𝜀_𝑦𝛾_𝑦𝑧𝛾_𝑥𝑧𝛾_𝑥𝑦, (6) where 𝑄₁₁=𝐸(𝑧)1−𝜈²,𝑄₁₂=𝜈𝑄₁₁,𝑄₄₄=𝑄₅₅=𝑄₆₆=𝐸(𝑧)2().1+𝜈 (7)

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^ℎ/2_−ℎ/2Ω𝜎_𝑥𝛿𝜀_𝑥+𝜎_𝑦𝛿𝜀_𝑦+𝜏_𝑥𝑦𝛿𝛾_𝑥𝑦+𝜏_𝑦𝑧𝛿𝛾_𝑦𝑧+𝜏_𝑥𝑧𝛿𝛾_𝑥𝑧𝑑Ω𝑑𝑧−_Ω𝑞𝛿𝑤𝑑Ω=0, (8) 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 _Ω𝑁_𝑥𝛿𝜀⁰_𝑥+𝑁_𝑦𝛿𝜀⁰_𝑦+𝑁_𝑥𝑦𝛿𝜀⁰_𝑥𝑦+𝑀^𝑏_𝑥𝛿𝑘^𝑏_𝑥+𝑀^𝑏_𝑦𝛿𝑘^𝑏_𝑦+𝑀^𝑏_𝑥𝑦𝛿𝑘^𝑏_𝑥𝑦+𝑀^𝑠_𝑥𝛿𝑘^𝑠_𝑥+𝑀^𝑠_𝑦𝛿𝑘^𝑠_𝑦+𝑀^𝑠_𝑥𝑦𝛿𝑘^s_𝑥𝑦+𝑆^𝑠_𝑦𝑧𝛿𝛾^𝑠_𝑦𝑧+𝑆^𝑠_𝑥𝑧𝛿𝛾^𝑠_𝑥𝑧𝑑Ω−_Ω𝑞𝛿𝑤𝑑Ω=0, (9) where𝑁_𝑥,𝑁_𝑦,𝑁_𝑥𝑦𝑀^𝑏_𝑥,𝑀^𝑏_𝑦,𝑀^𝑏_𝑥𝑦𝑀^𝑠_𝑥,𝑀^𝑠_𝑦,𝑀^𝑠_𝑥𝑦=^ℎ/2_−ℎ/2𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦1𝑧𝑓(𝑧)𝑆𝑑𝑧,_𝑠𝑥𝑧,𝑆^𝑠_𝑦𝑧=^ℎ/2_−ℎ/2𝜏_𝑥𝑧,𝜏_𝑦𝑧𝑔(𝑧)𝑑𝑧, (10a)(10b)The governing equations of equilibrium can be derived from (9) by integrating the displacement gradients by parts and setting the coefficients 𝛿𝑢₀, 𝛿𝑣₀, 𝛿𝑤_𝑏, and 𝛿𝑤_𝑠 zero separately. Thus one can obtain the equilibrium equations associated with the present shear deformation theory as follows: 𝛿𝑢∶𝜕𝑁_𝑥+𝜕𝑥𝜕𝑁_𝑥𝑦𝜕𝑦=0,𝛿𝑣∶𝜕𝑁_𝑥𝑦+𝜕𝑥𝜕𝑁_𝑦𝜕𝑦=0,𝛿𝑤_𝑏∶𝜕²𝑀^𝑏_𝑥𝜕𝑥²𝜕+2²𝑀^𝑏_𝑥𝑦+𝜕𝜕𝑥𝜕𝑦²𝑀^𝑏_𝑦𝜕𝑦²+𝑞=0,𝛿𝑤_𝑠∶𝜕²𝑀^𝑠_𝑥𝜕𝑥²𝜕+2²𝑀^𝑠_𝑥𝑦+𝜕𝜕𝑥𝜕𝑦²𝑀^𝑠_𝑦𝜕𝑦²+𝜕𝑆^𝑠_𝑥𝑧+𝜕𝑥𝜕𝑆^𝑠_𝑦𝑧𝜕𝑦+𝑞=0. (11) 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 𝑁𝑀^𝑏𝑀^𝑠=𝐴𝐵𝐵^𝑠𝐴𝐷𝐷^𝑠𝐵^𝑠𝐷^𝑠𝐻^𝑠𝜀𝑘^𝑏𝑘^𝑠,𝑆=𝐴^𝑠𝛾, (12) where 𝑁=𝑁_𝑥,𝑁_𝑦,𝑁_𝑥𝑦^𝑡,𝑀^𝑏=𝑀^𝑏_𝑥,𝑀^𝑏_𝑦,𝑀^𝑏_𝑥𝑦^𝑡,𝑀^𝑠=𝑀^𝑠_𝑥,𝑀^𝑠_𝑦,𝑀^𝑠_𝑥𝑦^𝑡,𝜀=𝜀⁰_𝑥,𝜀⁰_𝑦,𝛾⁰_𝑥𝑦^𝑡,𝑘^𝑏=𝑘^𝑏_𝑥,𝑘^𝑏_𝑦,𝑘^𝑏_𝑥𝑦^𝑡,𝑘^𝑠=𝑘^𝑠_𝑥,𝑘^𝑠_𝑦,𝑘^𝑠_𝑥𝑦^𝑡,𝐴A=¹¹𝐴₁₂0𝐴₁₂𝐴₂₂000𝐴₆₆𝐵,𝐵=¹¹𝐵₁₂0𝐵₁₂𝐵₂₂000𝐵₆₆,𝐷𝐷=¹¹𝐷₁₂0𝐷₁₂𝐷₂₂000𝐷₆₆,𝐵^𝑠=𝐵^𝑠₁₁𝐵^𝑠₁₂0𝐵^𝑠₁₂𝐵^𝑠₂₂000𝐵^𝑠₆₆,𝐷^𝑠=𝐷^𝑠₁₁𝐷^𝑠₁₂0𝐷^𝑠₁₂𝐷^𝑠₂₂000𝐷^𝑠₆₆,𝐻^𝑠=𝐻^𝑠₁₁𝐻^𝑠₁₂0𝐻^𝑠₁₂𝐻^𝑠₂₂000𝐻^𝑠₆₆,𝑆=𝑆^𝑠_𝑥𝑧,𝑆^𝑠_𝑦𝑧^𝑡,𝛾=𝛾_𝑥𝑧,𝛾_𝑦𝑧^𝑡,𝐴^𝑠=𝐴^𝑠₄₄00𝐴^𝑠₅₅, (13a)(13b)(13c)(13d)(13e)where 𝐴_𝑖𝑗, 𝐵_𝑖𝑗, and so forth are the plate stiffness, defined by𝐴₁₁𝐵₁₁𝐷₁₁𝐵^𝑠₁₁𝐷^𝑠₁₁𝐻^𝑠₁₁𝐴₁₂𝐵₁₂𝐷₁₂𝐵^𝑠₁₂𝐷^𝑠₁₂𝐻^𝑠₁₂𝐴₆₆𝐵₆₆𝐷₆₆𝐵^𝑠₆₆𝐷^𝑠₆₆𝐻^𝑠₆₆=^ℎ/2_−ℎ/2Q₁₁1,𝑧,𝑧²,𝑓(𝑧),𝑧𝑓(𝑧),𝑓²(𝑧)1𝜈1−𝜈2𝐴𝑑𝑧,₂₂,𝐵₂₂,𝐷₂₂,𝐵^𝑠₂₂,𝐷^𝑠₂₂,𝐻^𝑠₂₂=𝐴₁₁,𝐵₁₁,𝐷₁₁,𝐵^𝑠₁₁,𝐷^𝑠₁₁,𝐻^𝑠₁₁,𝐴^𝑠₄₄=𝐴^𝑠₅₅=^ℎ𝑛_ℎ𝑛−1𝑄₄₄𝑔(𝑧)²𝑑𝑧. (14a)(14b)(14c)Substituting from (12) into , we obtain the following equation:𝐴₁₁𝑑₁₁𝑢₀+𝐴₆₆𝑑₂₂𝑢₀+𝐴₁₂+𝐴₆₆𝑑₁₂𝑣₀−𝐵₁₁𝑑₁₁₁𝑤_𝑏−𝐵₁₂+2𝐵₆₆𝑑₁₂₂𝑤_𝑏−𝐵^𝑠₁₂+2𝐵^𝑠₆₆d₁₂₂𝑤_𝑠−B^𝑠₁₁𝑑₁₁₁𝑤_𝑠𝐴=0,₂₂𝑑₂₂𝑣₀+𝐴₆₆𝑑₁₁𝑣₀+𝐴₁₂+𝐴₆₆𝑑₁₂𝑢₀−𝐵₂₂𝑑₂₂₂𝑤_𝑏−𝐵₁₂+2𝐵₆₆𝑑₁₁₂𝑤_𝑏−𝐵^𝑠₁₂+2𝐵^𝑠₆₆𝑑₁₁₂𝑤_𝑠−𝐵^s₂₂𝑑₂₂₂𝑤_𝑠𝐵=0,₁₁𝑑₁₁₁𝑢₀+𝐵₁₂+2𝐵₆₆𝑑₁₂₂𝑢₀+𝐵₁₂+2𝐵₆₆𝑑₁₁₂𝑣₀+𝐵₂₂𝑑₂₂₂𝑣₀−𝐷₁₁𝑑₁₁₁₁𝑤_𝑏−2𝐷₁₂+2𝐷₆₆𝑑₁₁₂₂𝑤_𝑏−𝐷₂₂𝑑₂₂₂₂𝑤_𝑏−𝐷^𝑠₁₁𝑑₁₁₁₁𝑤_𝑠−2𝐷^𝑠₁₂+2𝐷^𝑠₆₆𝑑₁₁₂₂𝑤_𝑠−𝐷^𝑠₂₂𝑑₂₂₂₂𝑤_𝑠𝐵=𝑞,_𝑠11𝑑₁₁₁𝑢₀+𝐵^𝑠₁₂+2𝐵^𝑠₆₆𝑑₁₂₂𝑢₀+𝐵^𝑠₁₂+2𝐵^𝑠₆₆𝑑₁₁₂𝑣₀+𝐵^𝑠₂₂𝑑₂₂₂𝑣₀−𝐷^𝑠₁₁𝑑₁₁₁₁𝑤_𝑏−2𝐷^𝑠₁₂+2𝐷^𝑠₆₆𝑑₁₁₂₂𝑤_𝑏−𝐷^𝑠₂₂𝑑₂₂₂₂𝑤_𝑏−𝐻^𝑠₁₁𝑑₁₁₁₁𝑤_𝑠−2𝐻^𝑠₁₂+2𝐻^𝑠₆₆𝑑₁₁₂₂𝑤_𝑠−𝐻^𝑠₂₂𝑑₂₂₂₂𝑤_𝑠+𝐴^𝑠₅₅𝑑₁₁𝑤_𝑠+𝐴^𝑠₄₄𝑑₂₂𝑤_𝑠=𝑞, (15a)(15b)(15c)(15d)where 𝑑_𝑖𝑗, 𝑑_𝑖𝑗𝑙, and 𝑑_{𝑖𝑗𝑙𝑚} are the following differential operators: 𝑑_𝑖𝑗=𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗,𝑑_𝑖𝑗𝑙=𝜕³𝜕𝑥_𝑖𝜕𝑥_𝑗𝜕𝑥_𝑙,𝑑_{𝑖𝑗𝑙𝑚}=𝜕⁴𝜕𝑥_𝑖𝜕𝑥_𝑗𝜕𝑥_𝑙𝜕𝑥_𝑚,𝑑_𝑖=𝜕𝜕𝑥_𝑖,𝑖,𝑗,𝑙,𝑚=1,2. (16)

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:𝑣₀=𝑤_𝑏=𝑤_𝑠=𝜕𝑤_𝑠𝜕𝑦=𝑁_𝑥=𝑀^𝑏_𝑥=𝑀^𝑠_𝑥𝑎=0at𝑥=−2,𝑎2,𝑢₀=𝑤_𝑏=𝑤_𝑠=𝜕𝑤_𝑠𝜕𝑥=𝑁_𝑦=𝑀^𝑏_𝑦=𝑀^𝑠_𝑦𝑏=0at𝑦=−2,𝑏2, (17a)(17b)To solve this problem, Navier assumed the transverse mechanical and temperature loads, 𝑞, in the form of a double trigonometric series as 𝑞=𝑞₀sin(𝜆𝑥)sin𝜇𝑦, (18) 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 for 𝑢₀, 𝑣₀, 𝑤_𝑏 and 𝑤_𝑠 that satisfies the following boundary conditions: 𝑢₀𝑣₀𝑤_𝑏𝑤_𝑠=()()𝑊𝑈cos𝜆𝑥sin𝜇𝑦𝑉sin𝜆𝑥cos𝜇𝑦_𝑏()𝑊sin𝜆𝑥sin𝜇𝑦_𝑠sin(𝜆𝑥)sin𝜇𝑦, (19) 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: [𝐶]{Δ}={𝑃}, (20) where {Δ}={𝑈,𝑉,𝑊_𝑏,𝑊_𝑠}^𝑡 and [𝐶] is the symmetric matrix given by [𝐶]=𝑎₁₁𝑎₁₂𝑎₁₃𝑎₁₄𝑎₁₂𝑎₂₂𝑎₂₃𝑎₂₄𝑎₁₃𝑎₂₃𝑎₃₃𝑎₃₄𝑎₁₄𝑎₂₄𝑎₃₄𝑎₄₄, (21) in which 𝑎₁₁=𝐴₁₁𝜆²+𝐴₆₆𝜇²,𝑎₁₂=𝜆𝜇𝐴₁₂+𝐴₆₆,𝑎₁₃=−𝜆𝐵₁₁𝜆²+𝐵₁₂+2𝐵₆₆𝜇²,𝑎₁₄=−𝜆𝐵^s₁₁𝜆²+𝐵^𝑠₁₂+2𝐵^s₆₆𝜇²,𝑎₂₂=𝐴₆₆𝜆²+𝐴₂₂𝜇²,𝑎₂₃=−𝜇𝐵₁₂+2𝐵₆₆𝜆²+𝐵₂₂𝜇²,𝑎₂₄=−𝜇𝐵^𝑠₁₂+2𝐵^𝑠₆₆𝜆²+𝐵^𝑠₂₂𝜇²,𝑎₃₃=𝐷₁₁𝜆⁴+2𝐷₁₂+2𝐷₆₆𝜆²𝜇²+𝐷₂₂𝜇⁴,𝑎₃₄=𝐷^𝑠₁₁𝜆⁴+2𝐷^𝑠₁₂+2𝐷^𝑠₆₆𝜆²𝜇²+𝐷^𝑠₂₂𝜇⁴,𝑎₄₄=𝐻^𝑠₁₁𝜆⁴+2𝐻^𝑠₁₁+2𝐻^𝑠₆₆𝜆²𝜇²+𝐻^𝑠₂₂𝜇⁴+𝐴^𝑠₅₅𝜆²+𝐴^𝑠₄₄𝜇², (22)

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): 𝐸_𝑀=70 GPa; 𝜈=0.3.(ii)Ceramic (alumina, Al₂O₃): 𝐸_𝐶=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 𝑤=10ℎ³𝐸𝑎⁴𝑞₀𝑤𝑎2,𝑏2,𝑢_𝑥=100ℎ³𝐸𝑎⁴𝑞₀𝑢_𝑥𝑎2,𝑏2ℎ,−4,𝑢_𝑦=100ℎ³𝐸𝑎⁴𝑞₀𝑢_𝑦𝑎2,𝑏2ℎ,−6,𝜎_𝑥=ℎ𝑎𝑞₀𝜎_𝑥𝑎2,𝑏2,ℎ2,𝜎_𝑦=ℎ𝑎𝑞₀𝜎_𝑦𝑎2,𝑏2,ℎ3,𝜏_𝑥𝑦=ℎ𝑎𝑞₀𝜏_𝑥𝑦ℎ0,0,−3,𝜏_𝑦𝑧=ℎ𝑎𝑞₀𝜏_𝑦𝑧𝑎2ℎ,0,6,𝜏_𝑥𝑧=ℎ𝑎𝑞₀𝜏_𝑥𝑧𝑏0,2,0. (23)

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 deflection 𝑤 and the distribution across the plate thickness of in-plane longitudinal stress 𝜎_𝑥and 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: 𝑎=1,𝑏=1,𝐸_𝑚=𝐸_𝑐=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).

Table 3 

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

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 in-plane 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.

Table 4 

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

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.

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.

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.

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.

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-the-thickness 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.

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.