A Refined Simple First-Order Shear Deformation Theory for Static Bending and Free Vibration Analysis of Advanced Composite Plates

A refined simple first-order shear deformation theory is developed to investigate the static bending and free vibration of advanced composite plates such as functionally graded plates. By introducing the new distribution shape function, the transverse shear strain and shear stress have a parabolic distribution across the thickness of the plates, and they equal zero at the surfaces of the plates. Hence, the new refined theory needs no shear correction factor. The Navier solution is applied to investigate the static bending and free vibration of simply supported advanced composite plates. The proposed theory shows an improvement in calculating the deflections and frequencies of advanced composite plates. The formulation and transformation of the present theory are as simple as the simple first-order shear deformation. The comparisons of deflection, axial stresses, transverse shear stresses, and frequencies of the plates obtained by the proposed theory with published results of different theories are carried out to show the efficiency and accuracy of the new theory. In addition, some discussions on the influence of various parameters such as the power-law index, the slenderness ratio, and the aspect ratio are carried out, which are useful for the design and testing of advanced composite structures.


Introduction
Functionally graded materials (FGMs) are a class of advanced composite materials. The mechanical properties of FGMs change continuously over the thickness of structures. In general, FGM is made from a mixture of ceramic and metal. In recent years, they have gained significant attention in many engineering fields such as automotive, civil engineering, aerospace, and nuclear engineering. Hence, due to the exotic properties of FGMs, many researchers have been captivated to investigate the bending behaviors, free vibration, and dynamic and buckling behaviors of FGM beams, plates, and shells. According to the literature, the analysis of FGM plates can be investigated with some different theories such as the classical plate theory (CPT), the first-order shear deformation theory (FSDT), higher-order shear deformation theory (HSDT), the quasi-3D theory and Carrera unified formulation (CUF).
In the CPT, transverse shear deformation is neglected, so only thin plates can be regarded by this theory. Timoshenko et al. [1] used the CPT to analyze plates and shells. Liessa [2] applied the CPT for the free vibration of isotropic thin rectangular plates. Javahenri et al. [3] investigated the buckling developed a three-dimensional solution for rectangular plate bending. Batra and Vidoli [70] used a three-dimensional variational principle to derive an HSDT for the analysis of piezoelectric plates. Qian et al. [71,72] applied the HSDT and the normal deformable plate theory and meshless local Petrov-Galerkin (MLPG) method for the static bending, free vibration, and dynamic response of FGM plates. Gilhooley et al. [73] also used the HSDT and the normal deformable plate theory and MLPG with radial basis functions for the analysis of thick FGM plates. Talha and his co-authors [74] used the HSDT to study the bending behavior and free vibration of FGM plates-the effect of some geometric parameters and the power-law index were carried out. Nguyen et al. [75] applied the HSDT and IGA for the analysis of composite sandwich plates. Akavci [76,77] developed two new hyperbolic HSDTs for the analysis of laminated composite and FGM plates. Karama and his partners [78] employed the HSDT for analysis of laminated composite beams. In this study, the composite beam was modelled by the multi-layered model based on the HSDT. Matsunaga [79] analyzed the free vibration and stability of FGM plates. In his work, the FGM plates were modelled using a 2-D HSDT. Aydogdu [80] developed a new HSDT to analyze laminated composite plates. Mantari and his co-authors [81][82][83][84][85][86][87][88][89][90] developed various quasi-3D plate theories for the static bending, free vibration, and buckling of laminated composite plates, FGM plates, and sandwich FGM plates. Nguyen et al. [91] developed a new inverse trigonometric shear deformation theory for isotropic and FGM sandwich plates analysis. Thai et al. [92,93] applied IGA with the inverse trigonometric shear deformation theory and generalized shear deformation theory to investigate laminated composite and FGM sandwich plates. Zenkour [94] used the sinusoidal function to develop 3-D elasticity solutions to study bending behavior and free vibration of exponentially graded thick rectangular plates. Bui et al. [95] applied the TSDT and the FEM for the mechanical behaviors of heated FGM plates in a high-temperature environment. Do et al. [96] analyzed bi-directional FGM plates using the FEM and the TSDT. Mantari et al. [97,98] developed various quasi-3D theories which consisted of four unknowns to study FGM plates. Thai et al. [99] employed a sinusoidal function to develop a simple quasi-3D theory with only five unknowns to analyze FGM plates. Zenkour [100][101][102][103] developed many different quasi-3D theories which contained only four unknowns to study the bending behavior and vibration behavior of FGM plates and FGM sandwich plates. Neves and his co-authors [104] developed a new quasi-3D theory using a hyperbolic function to analyze FGM plates. Neves et al. [105] applied a quasi-3D HSDT and a meshless technique for the static bending, free vibration and buckling of sandwich FGM plates. In [106], Neves and his co-authors developed a quasi-3D SSDT to analyze FGM plates. Cerrera et al. [107] investigated the influences of thickness stretching in FGM plates and shells.
Furthermore, Carrera et al. [108,109] proposed the unified formulation (CUF) for multilayered composite structures. Brischetto et al. [110,111] studied the bending behavior of FGM plates and shells using CUF. Cinefra et al. [112] and Ferreira et al. [113] investigated the bending behavior and vibration behavior of laminated composite shells. In their works, the SSDT was developed using CUF. The bending behavior of FGM plates and shells was investigated by Cinefra and his co-authors [114]. In his work, the combination of the CUF and the mixed interpolation of tensorial components (MITC) was used to develop a nine-node shell element.
By decomposing the transverse displacements into two parts, the bending part and shear part, the simplified FSDT has less unknowns than the FSDT, the HSDT, the SSDT and the quasi-3D theory, so its computational expenses are reduced. Thus, the development of a simplified FSDT is still necessary. This paper developed a refined simple FSDT for the analysis of advanced composite plates, such as FGM plates. By introducing the distribution shape function to the shear strain, the proposed theory not only shows an improvement on expecting deflections but also accounts for a parabolic transverse shear strain distribution through the thickness of the plates. The Navier solution was applied to investigate the static bending and free vibration of simply supported plates. Several numerical examples are presented to illustrate the accuracy of the new refined plate theory.

Material Properties of Advanced Composite Plates
Advanced composite materials such as functionally graded materials can be produced by continuously varying the constituents of multi-phase materials in a predetermined profile. An FGM can be defined by the variation in the volume fractions. In this paper, FGM plates with the power-law function (P-FGM) and exponential function (E-FGM) were considered ( Figure 1). Advanced composite materials such as functionally graded materials can be produced by continuously varying the constituents of multi-phase materials in a predetermined profile. An FGM can be defined by the variation in the volume fractions. In this paper, FGM plates with the power-law function (P-FGM) and exponential function (E-FGM) were considered ( Figure 1). For the case of P-FGM plates, the materials properties of P-FGM depend on the volume fraction, which can be obtained as a power-law function as the following formula.  For the case of P-FGM plates, the materials properties of P-FGM depend on the volume fraction, which can be obtained as a power-law function as the following formula.

Metal
where p is the material parameter and h is the thickness of the plate. The material properties of a P-FGM can be determined as where P c , P m are the Young's modulus or density of the ceramic and metal, respectively. For the case of E-FGM plates, the material properties of E-FGM are defined as where P 0 is the Young's modulus or density of the bottom surface of the FGM plate and p is the material parameter.

Kinematics
Corresponding to the simple FSDT, the transverse displacement w is separated into two parts-the bending constituent w b and the shear constituent w s . The displacement fields of the plate can be expressed as The strains related to the displacement fields are Certainly, the simple FSDT theory was based on the statement of linear shear strain distribution across thickness, so a constant shear correction coefficient was needed to overcome the shear-locking phenomenon. Nevertheless, the shear stress was distributed parabolically across the thickness and disappeared on the top and bottom surfaces of the plate. In this paper, an assumption of shear distributed function is presented to improve the simple FSDT. Therefore, the shear strains vector becomes where f (z) is the assuming shear distributed function, which defines the distribution of the transverse shear strains across the thickness of the plate. The shear distributed function was chosen so it satisfies the following conditions: The shear strain is distributed parabolically over the thickness and equal to zero on the top and bottom surfaces of the plate; the integration through the thickness of the plate approximating with the constant shear correction factor of the FSDT (5/6). Inspired by the study of Zenkour [52], the shear distributed function can be chosen as The constitutive equations for the plate can be expressed as

Equations of Motion
The equations of motion can be quantified using the Hamilton's principle, that is where δU is the variation of strain energy, δV is the variation of work done by external forces, and δK is the variation of kinetic energy. The expression of δU is The expression of δV is The expression of the variation of kinetic energy δK is After integrating Equation (16) over the thickness direction, Equation (16) becomes where (I 0 , I 1 , Substituting Equations (11), (15) and (17) into Equation (9) and integrating by parts, the equations of motions are obtained as δv : where ∇ 2 = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 .

Analytical Solutions
In this study, a simply supported rectangular plate was considered. The length of the plate was a, the width of the plate was b, and the height of the plate was h. The plate was subjected to a distributed transverse load q. Employing the Navier solution, the solutions of the plate were assumed as U mn e iωt cos α m x sin β n y V mn e iωt sin α m x cos β n y W bmn e iωt sin α m x sin β n y W smn e iωt sin α m x sin β n y where i 2 = −1, α m = mπ/a, β n = nπ/b, (U mn , V mn , W bmn , W smn ) are quantities to be determined, m and n are mode numbers, and ω is the frequency of free vibration. The transverse distributed load q was also expanded in the following form For the case of a sinusoidal distributed load, we have For the case of uniformly distributed load, the coefficients Q mn are defined as follows By substituting Equations (23) and (24) into the equations of motion, Equations (19)- (22), analytical solutions can be obtained from the following equation.
where K and M are, respectively, the stiffness matrix and the mass matrix; f is the force vector; ∆ is the vector of unknown coefficients, and ω is the frequency of free vibration. The elements of the K, M, f, and ∆ are as follows For bending analysis, the closed-form solution could be obtained by setting the natural frequency ω equal to zero. For free vibration analysis, the closed-form solution was obtained by setting the transverse load q equal to zero.

Numerical Results and Discussion
In this section, some numerical illustrations are carried out and discussed to prove the efficiency and accuracy of the proposed theory in the static bending and free vibration responses of simply supported isotropic homogeneous and FGM plates. The non-dimensional entities were used as the following formulas

Static Bending Analysis
Example 1. Firstly, the results obtained using the present theory were compared with those of the classical plate theory [1] given by Timoshenko, the Navier-type three-dimensionally (3-D) exact solution given by Werner [69], and the generalized shear deformation theory by Zenkour [52] in Tables 1 and 2.
The geometric and material properties of plate were a = 1, b = 1, E = 1, q 0 = 1, ν = 0.3 with three cases of the thickness of plate h = 0.01, h = 0.03, and h = 0.1. The comparison exhibited the fact that that the present results were in good agreement with other published results. According to Table 2, the axial stress equaled zero at the mid-plane for the case of the isotropic plate. Therefore, the neutral surface was identical to mid-plane for the isotropic plate.  Tables 3 and 4. According to these tables, the solutions of the proposed plate theory were very close to the results of Zenkour [52]. To demonstrate the accuracy of the present theory for wide range of aspects and side-to-thickness ratios a/h, through the thickness distributions of the in-plane longitudinal and normal stresses σ x and σ y , the longitudinal tangential stress τ xy , the shear stresses τ xz and τ yz in the FGM plate under uniform load are explained in Figures 2-7, which show, respectively, the influence of the aspect ratio and the side-to-thickness ratio on the center deflection of the plates. The obtained results were compared with those reported by Zenkour [52]. The comparison shows that the results of the present theory and Zenkour are almost identical, except for the case of the transverse shear stresses τ xz and τ yz , where a small difference between the results can be seen. However, it should be noticed that the results of Zenkour were obtained using the generalized shear deformation theory, while the present results were obtained using the proposed refined simple FSDT. According to Figure 2, the axial stress did not equal to zero at the mid-plane of the FGM plates, so the neutral surface moved toward the ceramic surface of the FGM plates. From Figures 4 and 5, the shear stresses were asymmetric through the thickness of the FGM plates. In addition, Figures 6 and 7 show that the deflection of the plate decreased when the aspect ratio (a/b) and side-to-thickness ratio (a/h) increased. To demonstrate the accuracy of the present theory for wide range of aspects and side-to-thickness ratios / , a h through the thickness distributions of the in-plane longitudinal and normal stresses x σ and y σ , the longitudinal tangential stress xy τ , the shear stresses xz τ and yz τ in the FGM plate under uniform load are explained in Figures 2-7, which show, respectively, the influence of the aspect ratio and the side-to-thickness ratio on the center deflection of the plates. The obtained results were compared with those reported by Zenkour [52]. The comparison shows that the results of the present theory and Zenkour are almost identical, except for the case of the transverse shear stresses xz τ and yz τ , where a small difference between the results can be seen.
However, it should be noticed that the results of Zenkour were obtained using the generalized shear deformation theory, while the present results were obtained using the proposed refined simple FSDT. According to Figure 2, the axial stress did not equal to zero at the mid-plane of the FGM plates, so the neutral surface moved toward the ceramic surface of the FGM plates. From Figures 4  and 5, the shear stresses were asymmetric through the thickness of the FGM plates. In addition, Figures 6 and 7 show that the deflection of the plate decreased when the aspect ratio (a/b) and side-to-thickness ratio (a/h) increased.            [107,108], and Thai et al. [29], in which Neves and Carrera used different quasi-3D theories and Thai used a simple FSDT. In addition, it should be observed that the effect of thickness stretching is accounted in quasi-3D theories, while it is ignored in the simple FSDT of Thai and their proposed theory. According to Table 5, it can be noticed that results obtained of the present theory are in good agreement with published results for both thin and thick FGM plates.  (2). Three different values of the power-law index p = 1, p = 4 and p = 10 were used in this example. The results obtained using the present theory were compared with the solutions given by Neves et al. [104][105][106], Carrera et al. [107,108], and Thai et al. [29], in which Neves and Carrera used different quasi-3D theories and Thai used a simple FSDT. In addition, it should be observed that the effect of thickness stretching is accounted in quasi-3D theories, while it is ignored in the simple FSDT of Thai and their proposed theory. According to Table 5, it can be noticed that results obtained of the present theory are in good agreement with published results for both thin and thick FGM plates.

Example 4.
Continuously, an exponential FGM plate with thickness ratio a/h = 2 and a/h = 4 were investigated. The Poisson's ratios were constant and equal to 0.3. Young's modulus was evaluated using the exponential distribution. The results of the present theory were compared with those of the 3D elasticity solution [94], quasi-3D theories [88,94], the HSDT [85], and the simple HSDT [56]. From Table 6, the present results are in excellent agreement with literature results for medium thick plates. For the very thick FGM plates (a/h = 2), the deflections obtained of the proposed theory were slightly larger than those of 3D results and quasi-3D results, because the thickness stretching effect was neglected in the present theory.

Free Vibration Analysis
Example 5. The next verification was performed for the free vibration of an isotropic homogeneous rectangular plate with a simply supported boundary condition. The length-to-height ratios of the plates were a/h = 1000 and 5. The first six non-dimensional frequencies ω * of the present theory were compared with the available published results of Manna [27], Leissa [2], Liew et al. [12] and Raju [11], in which, Manna [27] used a family of higher-order triangular element, Leissa [2] used an analytical solution, Liew et al. [12] used the pb-2 Rayleigh-Ritz method, and Raju [11] used a nine-node Lagrangian quadrilateral isoparametric plate element. The comparision was shown in Table 7. According to Table 7, it can be concluded that the present solutions are in good agreement with published solutions. Example 6. The next example was carried out for an isotropic Al/Al 2 O 3 square plate. The Young's modulus and density of aluminum were E m = 70 GPa and ρ m = 2702 kg/m 3 , respectively, and those of alumina were E c = 380 GPa and ρ c = 3800 kg/m 3 , respectively. The Poisson's ratio of the plate was assumed to be constant through the thickness, and it equaled 0.3. In this example, Young's modulus and density were obtained using Equation (2). The length-to-thickness ratio a/h varied from 2 to 10, and the power-law index varied from 0 to 10. The first two non-dimensional frequenciesω for different values of length-to-thickness ratio a/h and the power-law index p using the present theory and those of other theories are given in Table 8. From this table, it can be found that the present theory has an excellent accuracy to determine the frequency for FGM plates. It was also observed that the non-dimensional frequencies of FGM plates decreased as the value of the power-law index increased. Example 7. The first four non-dimensional frequencies ω of an FGM rectangular plate with length-to-thickness ratio varied from 5 to 20 and the power-law index varied from 0 to 10 are compared in Table 9. The plate was made from aluminum (as metal) and alumina (as ceramic). The material properties of aluminum were E m = 70 GPa and ρ m = 2702 kg/m 3 , and those of alumina were E c = 380 GPa and ρ c = 380 kg/m 3 . The Poisson's ratio of the plate was assumed to be constant through the thickness, and it equaled to 0.3. Equation (2) was used to evaluate the Young's modulus and density of the plate. The first four non-dimensional frequencies ω obtained by using the present theory were compared with those given by Hosseini-Hashemi et al. [24] based on the FSDT, Reddy [36] based on the TSDT, and Thai et al. [58] based on the SSDT. In addition, the variations of the non-dimensional fundamental frequency of FGM square plate with respect to the power-law index p and length-to-thickness ratio a/h are compared in Figures 8 and 9, respectively. According to Table 9 and Figures 8 and 9, the non-dimensional frequencies achieved by the proposed theory are in excellent agreement with those obtained by the FSDT, TSDT and SSDT. From Table 9 and Figure 8, the first frequencies of the FGM plate decreased when the power-law index increased. When the length-to-thickness ratio increased, the first frequencies of the FGM plate increased, as shown in Figure 9.    Table 10. It can be seen that a significant agreement between the results of the present theory and different approaches for the first non-dimensional frequencies is found for all length-to-thickness ratios and the power-law index.  Example 8. This example aimed to verify the obtained results of thin and thick plates. A fully simply supported Al/Al 2 O 3 square thick plate with different length-to-thickness ratios a/h was analyzed. The material properties of aluminum were E m = 70 GPa and ρ m = 2707 kg/m 3 , and those of alumina were E c = 380 GPa and ρ c = 3800 kg/m 3 . The Poisson's ratio of the plate was assumed to be constant through the thickness, and it equaled 0.3. Equation (2) was used to evaluate the Young's modulus and density of the plate. The first non-dimensional frequenciesω obtained by the present theory and different methods for some values of the power-law index and length-to-thickness ratios are tabulated in Table 10. It can be seen that a significant agreement between the results of the present theory and different approaches for the first non-dimensional frequencies is found for all length-to-thickness ratios and the power-law index. Example 9. In this last example, the results of free vibration of a square plate made of Al/Al 2 O 3 using the proposed theory were compared with those of Brischetto [115] using the exact elasticity solution. The material properties of Al and Al 2 O 3 were: E m = 73 GPa, ν m = 0.3, ρ m = 2800 kg/m 3 , E c = 380 GPa, ν c = 0.3, and ρ c = 3800 kg/m 3 . The three cases of dimensions and length-to-thickness ratios were a = b = 100, a/h = 100; a = b = 20, a/h = 20, and a = b = 5, a/h = 5. The mass density and Young's modulus were obtained by the power-law function. The comparison of the first three non-dimensional frequencies ω obtained by the proposed theory and those of Brischetto using the exact elasticity solution are given in Table 11. According to Table 11, the results of the proposed theory are in good agreement with those of Brischetto using the exact elasticity solution.

Conclusions
In this paper, a refined simple first-order shear deformation plate theory was developed for the static bending and free vibration of advanced composite plates such as functionally graded plates. By introducing the distributed shape function to the shear strain, the refined theory accounted for a variable transverse shear strain distribution through the thickness of the plate, and it satisfied the traction free boundary conditions at the top and bottom surfaces of the plate. Moreover, the presented theory retained the simplicity of the FSDT. Analytical solutions were obtained for simply supported FGM plates using the Navier technique. Some numerical examples were carried out to verify the convenience and accuracy of the proposed theory. According to these examples, some remarkable information can be given: • The proposed theory is efficient and accurate for the static bending and free vibration analysis of FGM plates.

•
For FGM plates, the neutral surface is not identical to the mid-plane surface. It moves toward the ceramic surface, and it is different from the isotropic plates.

•
The power-law index, aspect ratio, and side-to-thickness ratio have a great effect on the bending behavior and free vibration of FGM plates.
This theory can be applied to the analysis of other structures such as beams and shells made of advanced composite plates. In addition, the proposed theory can be improved by optimizing the distributed shape function to achieve results that are close to the 3D solution, which is a good idea for further work.