An ES-MITC3 Finite Element Method Based on Higher-Order Shear Deformation Theory for Static and Free Vibration Analyses of FG Porous Plates Reinforced by GPLs

Faculty of Mechanical Engineering, Tran Dai Nghia University, Ho Chi Minh City, Vietnam Faculty of Mechanical Engineering, Ho Chi Minh City University of Technology, Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc &ang University, Ho Chi Minh City, Vietnam Faculty of Civil Engineering, Ton Duc &ang University, Ho Chi Minh City, Vietnam Faculty of Mechanical Engineering, Le Qui Don University, Hanoi, Vietnam


Introduction
In recent years, FG porous material has attracted a great interest from many researchers over the world due to their superior mechanical properties such as lightweight, wear resistance, and high strength.
ese properties make FG porous structures to be very suitable to apply for civil engineering, aerospace structures, nuclear plants, and other applications. In this regard, Rezaei et al. [1] presented the analytical approach based on Reddy's third-order shear deformation theory to analyse the free vibration of thick porous plates. Leclaire et al. [2] used the classical theory and the stress-strain relations of Biot to investigate the free vibration of the thin porous plates saturated by a fluid. Zhao et al. [3] introduced a semianalytical method based on a modified Fourier series to study the vibration of FG porous shells. Coskun et al. [4] developed the analytical solutions using the general third-order shear deformation theory (TSDT) for the static, free vibration, and buckling analyses of FG porous microplates. Nguyen et al. [5] studied the dynamic response of FG porous plates resting on elastic foundation under thermal and mechanical loads by the analytical method and the first-order shear deformation theory (FSDT). Phung et al. [6] combined between the thirdorder shear deformation theory and the isogeometric analysis to study nonlinear transient of the FG porous plates in hygro-thermo-mechanical environment. Barati used the analytical approach based on nonlocal strain gradient theory to investigate the free vibration of FG porous nanoshells [7] and the forced vibration of FG porous nanoplates [8]. Shahverdi and Barati [9] introduced a general nonlocal strain gradient (NSG) elasticity model to analyse the vibration of FG porous nanoplates resting on an elastic substrate. Forsat et al. [10] studied the transient response of FG porous nanoshell subjected to different impulsive loadings using the nonlocal strain gradient theory. Mirjavadi et al. [11] examined the dynamic response of FG porous cylindrical microshells under the moving loads using strain gradient size dependency. Fenjan et al. [12] used the differential quadrature method combined with the nonlocal strain gradient theory to analyse the mechanical-hygrothermal vibrations of the FG porous plates resting on the viscoelastic foundation. Ebrahimi et al. [13] provided an analytical approach for the free vibration of smart FG porous plates with different boundary conditions. Barati and Zenkour [14] explored the free vibration of FG porous nanoplates using the higher-order shear deformation theory, and they also used a refined four-variable plate theory to study the vibration behavior of FG porous plates with piezoelectric [15]. Also, Barati and Shahverdi [16] proposed the higher-order refined four-variable plate theory to examine the nonlinear vibration of the FG porous nanoplates.
Although there are many advantages as mentioned above, the FG porous materials are still limited in several engineering applications due to the existence of internal pores which cause the significant decrease of stiffness of structures. To deal with this problem (i.e., increase the stiffness of structure but still retain their lightweight properties), carbon nanotubes (CNTs) [17] or graphene platelets (GPLs) [18][19][20][21] have been used to reinforce the FG porous structures. However, in [22], Li and co-workers demonstrated that the reinforcement of GPLs is better than that of CNTs because GPLs have low manufacturing cost and help increase the load carrying capability by improving the bonding with the matrix. Many research studies have been conducted to investigate the effect of GPLs on behaviors of FG porous structures by using many various approaches. Seyed et al. [23] presented a semianalytical approach based on the FSDT to solve dynamic behavior of FG porous shells reinforced by GPLs subjected to radial dynamic load. Barati et al. [24] examined the free vibration response of FG porous shells using the analytical method and the FSDT. eir work showed that the porosity distribution and GPL distribution significantly affect the vibration frequencies of FG porous shells. Using the analytical approach via the FSDT and Chebyshev-Ritz method, Yang et al. [25] reported the buckling and free vibration behavior of FG porous plates reinforced by GPLs. Gao et al. [26] used the differential quadrature method to investigate the free vibration of FG porous plates reinforced with GPLs resting on the elastic foundation using different boundary conditions. Hamilton's principle and the classic plate theory with von Karman strains assumption are applied to derive the equations of emotion of plates in their paper. Recently, the isogeometric analysis (IGA) method [27][28][29][30] is also applied for analysis of FG porous structures by many researchers. Based on the FSDT and TSDT, Li et al. [31] introduced the static, free vibration, and buckling behaviors of FG porous plates reinforced by GPLs. Also, Nguyen et al. [32] developed the IGA approach based on Bézier extraction to examine the static bending and transient response of piezoelectric FG porous plates reinforced by GPLs. In [33], Wang and coworkers proposed an analytical solution and Navier technique using the higher-order shear deformation theory (HSDT) to study the FG porous doubly curved shell reinforced by GPLs. In addition, Nguyen et al. [34] proposed the polygonal finite element method to investigate the static bending, free vibration, and dynamic behavior of FG porous and piezoelectric FG porous plates with GPL reinforcement. In this study, they used the Halpin-Tsai model to establish the relationship between the porosity coefficient and Young's modulus.
In an effort to improve the convergence and accuracy for analysis of the plate and shell structures, the original MITC3 element [35] has been combined with the ES-FEM [36][37][38][39][40][41] to give the so-called ES-MITC3 element [42][43][44][45][46]. e numerical results in these works showed that the ES-MITC3 has the following superior properties [42]: (1) the ES-MITC3 uses only three-node triangular elements which are easy in mesh generation even for complicated geometry domains; (2) the ES-MITC3 can eliminate transverse shear locking even with the ratio of the thickness to the length of the structures reaching 10 −8 while many other elements are limited with 10 −5 ; (3) the accuracy of the ES-MITC3 is less affected by the badly shaped elements or mesh distortions; (4) the ES-MITC3 has better accuracy than the existing triangular elements such as MITC3 [35], DSG3 [47], and CS-DSG3 [48] and is a good competitor against the quadrilateral element MITC4 [49].
is article further extends the ES-MITC3 to investigate for static bending and vibration responses of FG porous plates reinforced by GPLs. e HSDT is utilized in approximation formulation of displacement field to avoid using the transverse shear correction factors and to enhance the accuracy of the shear stresses across the thickness of the FG porous plates. e different porosity distribution types and the GPL patterns including symmetry, asymmetry, and uniformity across the plate thickness are considered in the present work. e obtained results are compared with other published approaches in the literature to demonstrate the accuracy and reliability of the proposed method. Moreover, the influences of material parameters such as GPL weight fraction, the porosity coefficient, the geometrical sizes of GPLs, and the distribution patterns on the static and free vibration of FG porous plates are investigated in detail.

e Distribution of Porosity.
ere are three types of porosity distribution along the thickness of FG porous plates including symmetrical (P-S), asymmetrical (P-A), and uniform (P-U), as shown in Figure 1. Young's modulus E(z), shear modulus G(z), and mass density ρ(z) of FG porous plates with three porosity distribution types can be given as follows [34]: , asymmetry porosity distribution (A − P), λ, uniform porosity distribution (U − P), in which E 1 , G 1 , and ρ 1 denote the maximum values of Young's modulus, shear modulus, and mass density of the FG plates without internal pores, respectively. e coefficient of porosity e 0 can be determined by where E 1 ′ and E 2 ′ are the maximum and minimum values of Young's modulus for two nonuniform porosity distribution types (S-P and A-P). Furthermore, E ′ represents Young's modulus for U-P case as shown in Figure 2. e mechanical property of closed-cell cellular solids according to Gaussian random field can be defined as [50] From equation (4), the coefficient of mass density can be derived as follows: Using the closed-cell Gaussian random field scheme, Poisson's v(z) can be obtained as [51] v(z) � 0.22p ′ + v 1 0.342p '2 − 1.21p 1 + 1 , where v 1 denotes Poisson's ratio of the pure matrix material without pores and It is noted that the total mass M of the FG porous plates with different porosity distributions does not change and can be calculated by In equation (2), the coefficient of porosity λ for U-P case can be written as [26] 2.2. e GPL Distribution. In this paper, three dispersion patterns such as symmetry distribution (S-GPL), asymmetry distribution (A-GPL), and uniform distribution (U-GPL), as depicted in Figure 2, are considered. e volume fraction of GPLs varies smoothly through the place's thickness direction which can be given as [25] where S ij with j � 1,2,3 is the maximum value of GPL volume fraction and i � 1,2,3 corresponding to the S-P, A-P, and U-P cases, respectively.

S ij
To calculate S ij , equation (12) can be rewritten as follows: Mathematical Problems in Engineering in which Effective Young's modulus of the FG reinforced by GPLs without internal pores is determined by the Halpin-Tsai micromechanics model as [52] where Figure 1: e porosity distribution types.  Mathematical Problems in Engineering in which W GPL , L GPL , and t GPL are average width, length, and thickness of GPLs, respectively, and E GPL and E m are Young's moduli of GPLs and metal matrix, respectively. e mass density ρ 1 and Poisson's ratio v 1 of the GPL reinforced metal can be calculated as where ρ GPL , v GPL , and V GPL are the mass density, Poisson's ratio, and volume fraction of GPLs, respectively, and ρ M , v M , and V M are the mass density, Poisson's ratio, and volume fraction of the metal, respectively.

C 0 -Type Higher-Order Shear Deformation eory for FG Porous
Plates. e displacement of FG porous plates in present work based on the C 0 -HSDT [53] model can be expressed as in which u and v, w are middle-surface displacements and θ x and θ y are the independent rotations of the normal in xz and yz planes, respectively; ϕ x and ϕ y are warping functions which replace derivative of deflection according to Reddy's HSDT; and c � −4/3h 2 .
For a bending plate, the strain field can be expressed as follows: where e membrane strain is given as e bending strains are written as and the transverse shear strain is defined as with From Hooke's law, the linear stress-strain relations of FG porous plates can be expressed as where in which E 11 and E 22 are effective Young's moduli; G 12 , G 13 , and G 23 denote the effective shear moduli; and υ 12 and υ 21 represent Poisson's ratios.

Weak Form Equations.
To obtain the governing equations of FG porous plates reinforced by GPLs, Hamilton's principle is applied in the following form: where U, V, and K are the strain energy, the work done by external loads, and the kinetic energy of FG porous plates, respectively. e strain energy is expressed as in which ε T � [ε m κ (1) κ (2) ] and γ T � [γ (0) γ (1) ].
where A, B, D, E, F, H, A s , B s , D s can be given by (As, Bs, Ds) � e work done by external transverse loads is expressed by e kinetic energy is given by and m is the mass matrix which is defined by m � with (I 1 , I 2 , I 3 , I 4 , I 5 , (29), (33), and (34) into equation (28), weak formulation for static and free vibration of FG porous plates, respectively, is finally obtained as

Formulation of an ES-MITC3 Method for FG Porous Plates
where n ne is the number of nodes of the plate element and N(x) and d e j � [u e j , v e j , w e j , θ e xj , θ e yj , ϕ e xj , ϕ e yj ] T are the shape function and the nodal degrees of freedom (dof ) of u e associated with the jth node of the element, respectively. e linear membrane and the bending strains of a triangular element can be expressed in matrix forms as follows: where 6 Mathematical Problems in Engineering To eliminate the shear locking phenomenon as the thickness of the plate decreases, the MITC3 element based on the first-order shear deformation theory (FSDT) is proposed by Lee and Bathe [1]. In their study, the covariant transverse shear strains of the triangular element are independently interpolated from the covariant transverse shear strains which are evaluated at the middle of triangular element edges, named typing points. e transverse shear strain field associated to typing points with 7 dofs per node can be written as follows: in which where x 1 are pointed out in Figure 3 and A e is the area of the threenode triangular element. Substituting the discrete displacement field into equations (36) and (37), we obtained the discrete system equations for static and free vibration analysis of FG porous plates using MITC3 based on C 0 − HSDT formulation, respectively, as where K is the stiffness matrix of FG porous plates and F represents the load vector. with where where ω is the natural frequency and M is the mass matrix

Formulation of an ES-MITC3 Method for FG Porous
Plates. In the ES-FEM, a domain Ω is divided into n k smoothing domains Ω κ based on edges of elements, such as Mathematical Problems in Engineering smoothing domain Ω κ associated with the inner edge k is formed by connecting two end nodes of the edge to centroids of adjacent MITC3 triangular elements as shown in Figure 4. Now, applying the edge-based smoothed finite element [2], the smoothed strain ε κ , a smoothed shear strain c κ over the smoothing domain Ω κ can be created by computing the integration of the compatible strain ε and the shear strain c respectively, in equations (36) and (37) such as where ϕ k (x) is a given smoothing function that satisfies at least unity property Ω k ϕ k (x)dΩ � 1. In this study, we use the constant smoothing function in which A k is the area of the smoothing domain Ω κ and is given by where n ek is the number of the adjacent triangular elements in the smoothing domain Ω κ and A i is the area of the ith triangular element attached to the edge k. By substituting equations (39)-(41) and (51) into equations (69) and (70), the approximation of the smoothed strains on the smoothing domain V GPL can be expressed as follows: where n nk sh is the total number of nodes of the triangular MITC3 elements attached to edge k (i.e., n nk sh � 3 for boundary edges and n nk sh � 4 for inner edges as given in Figure 4); d k j is the nodal dofs associated with the smoothing domain Ω k ; B e global stiffness matrix of FG porous plates using the ES-MITC3 is assembled by where K k is the ES-MITC3 stiffness matrix of the smoothing domain Ω k and given by in which

Numerical Results
In this section, several numerical examples are verified to illustrate new contributions including (1) verifying the accuracy of the present method for the free vibration and static bending analyses of the FG porous plates reinforced by GPLs by comparing with results in [25] and (2) investigating the influences of the porosity distributions, the GPL dispersion patterns, GPL weight fraction, and the porosity coefficients on the free vibration and static response of the FG porous plates. In the research of Yang et al. [25], it was found that a multilayer plate model has the optimal total layer number n � 12 which is easy for manufacture and economic efficiency simultaneously. In addition, the error of results between n � 12 and n � 1000 is less than 2%. erefore, n � 12 is also used in the following examples in this work.

Free Vibration Analysis of FG Porous Plates Reinforced by
GPLs. In this section, for convenience in comparing to results in the literature, the nondimensional natural frequency of the plates is given by ω � ωa ����� � I 1 /A 11 , in which I 1 and A 11 are the values of the inertia moment in mass matrix and the extensional stiffness of FG plate without any pores and GPLs.

e FG Porous Plates with GPL Reinforcement.
Firstly, we consider the plate with the length a � b � 2 and the thickness h � 0.1 as shown in Figure 5. e material properties of FG porous plates reinforced with GPLs are expressed in Table 1. Tables 2 and 3 show nondimensional frequencies of fully clamped square FG porous plates reinforced by GPLs with varying porosity coefficients and various GPL weight fractions. e results of proposed method are compared with those of several various methods such as MITC3 [33], MITC4 [35], and analytical approach [25]. It can be seen that the ES-MITC3 is a good competitor to quadrilateral shell element MITC4 and gives better accuracy compared with the original triangular elements MITC3. Note that the results of the ES-MITC3 also match well with the analytical solution based on the Chebyshev-Ritz method [25]. It is seen clearly that the increase of porosity coefficient of FG porous plates reinforced with GPLs leads to the reduction of fundamental frequencies. It is because the increase of the porosity coefficient will decrease both the stiffness and mass density of the FG porous plates. In addition, Tables 4 and 5 provide the nondimensional frequencies of FG porous plates with the cases of S-P and U-P reinforced by GPL-A and the cases of A-P reinforced by GPL-S, GPL-A, and GPL-U, respectively. Table 6 shows the nondimensional natural frequencies of the FG porous plates reinforced by GPLs with the different thickness. It can be seen that the FG porous plates have the same thickness; however, the case of A-P reinforced by GPL-A gives the minimum natural frequencies. On the contrary, the case of S-P with GPL-S gives the maximum natural frequencies. Moreover, Figure 6 depicts the first six mode shapes of FG porous plate in the case of S-P reinforced by GPL-S (e 0 � 0.2, a/h � 20, and Λ GPL � 1.0 wt). Figure 7 demonstrates the influence of the porosity coefficient and the GPL weight fraction on the natural frequencies of the FG porous square plate. In this case, the FG porous plates are made from S-P reinforced by three different types of GPL patterns. It can be found that the natural frequencies increase gradually when the porosity coefficient reduces and the GPL weight fraction increases. is may be explained by the fact that the presence of the internal pores in the structures with the smaller size and inferior density in the metal matrix leads to the increase of the stiffness of the FG porous plates. In addition, it is clear that the stiffness of the plate structures is significantly improved when a small amount GPLs reinforcement is added in the metal matrix.

e FG Porous Circular Plates with GPL Reinforcement.
Next, we consider the FG porous circular plate with the fully clamped boundary. e radius of circular plate is R � 1 and the thickness h � 0.1 as shown in Figure 8. e material properties of FG porous reinforced by GPLs are given in Table 1. e domain of circular plate is divided into 318 triangular elements. Table 7 lists the natural frequencies of FG porous circular plate with variable parameters such as the porosity coefficient, the GPL distribution, and the porosity distribution. It can be observed that the FG porous circular plates with U-P and GPL-A have the minimum natural frequency in all cases of investigations. Moreover, Figure 9 illustrates the first six mode shapes of the FG porous plate which is constituted by S-P with the porosity coefficient e 0 � 0.2, R/h � 10, and GPLs with weight fraction Λ GPL � 1.0 wt.%.

Static Analysis of FG Porous Plates Reinforced by GPLs.
In order to validate the accuracy and stability of the proposed ES-MITC3 based on TSDT for the static analysis problem of the FG porous plate reinforced by GPLs, a fully simply supported FG porous plate subjected to uniform load q � 10 3 N/m 2 is investigated. e FG porous plate has the thickness-to-width ratio (h/a) of 0.1, with the porosity coefficient e 0 � 0.5, S-P, and GPL-S as shown in Figure 5. Two GPL weight fractions Λ GPL (1.0 wt.%) and (0.5 wt.%) are studied for five metal matrices given in Table 8. Table 9 presents the percentage of central deflection of the FG porous plate reinforced by GPLs with various metal matrices δw � (|w 1 − w 0 |/w 0 ) * 100(%), in which w 1 and w 0 are the   Next, we consider the deflection of simply supported FG porous square plate reinforced by GPLs under sinusoidally distributed load q � q 0 sin(πx/a)sin(πy/b) with q 0 � 0.1 MPa. e plate's length (a � b � 1 m), thickness      (h � 0.05 m), and material properties as well as the dimensions of GPLs are given in Table 1. e number of layers of FG porous plates n � 12 is applied again in the static analysis. Table 10 shows nondimensional central deflection (w � w/h) of the FG porous plates reinforced by GPLs with GPL weight fraction Λ GPL � 1.0 wt.%. It can be observed that the FG porous plates with A-P and GPL-A for all cases using the same porosity coefficient give the largest displacement while the S-P and GPL-S give the lowest displacement. e influences of porosity coefficient and GPL weight fracture on central deflection of the FG porous plates reinforced by three various types of GPLs distribution and the porosity distribution S-P are shown in Figure 10. It can be found that the central displacement of FG porous plate increases gradually when the porosity coefficient increases. As expected, the internal pores existing in the metal matrix of the FG plates with larger size and higher density lead to the reduction of stiffness of plate structure. Besides, we can see that the GPL weight fracture in the metal matrix also has a significant effect on the central deflection of the plates. Accordingly, with the same porosity distribution (S-P) and porosity coefficient, when the GPL weight fracture increases,        the central displacement will decrease. It can be concluded that the increase of porosity coefficient of the FG porous plates will decrease the density and increase the GPL weight fracture and hence makes the plate become stiffer. Figure 11 demonstrates the plots of the axial stress σ xx (a/2, a/2, z) through the thickness of FG porous plates reinforced by GPLs with three types of porosity distribution and various GPL dispersion patterns under sinusoidally distributed load from top surface. It can be seen that the maximum compressive stress at the top surface of FG porous plate is in the case of porosity distribution S-P and GPL-S reinforcement. On the contrary, the maximum tensile stress is in the case of the porosity distribution S-A and GPL-A reinforcement. Note that the FG porous structures reinforced by GPLs are weaker in compression than in tension in most cases of porosity distributions and GPL dispersion patterns.

Conclusions
In this paper, the static and free vibration analyses of FG porous plates reinforced by GPLs are studied by using the ES-MITC3 based on the HSDT. In this formulation, the strain smoothing technique over the smoothing domains is associated with edges of MITC3 triangular elements to improve the accuracy of the original MITC3 element for structure analysis. From the numerical results, we can draw the following conclusions: (i) e ES-MITC3 element is simply obtained from linear approximations using 3-node triangular elements, and thus high computational cost is avoided.
(ii) e ES-MITC3 uses three-node triangular elements that are much easily generated automatically even for complicated geometry domains. (iii) e results obtained by ES-MITC3 element are more accurate than those obtained by using the original MITC3 element. (iv) e proposed method uses the HSDT model which provides more accurate solutions with no demand of the shear correct factors. (v) e porosity distribution, the porosity coefficient, and the GPL dispersion patterns significantly influence the static and free vibration response of the FG porous plates with GPL reinforcement. (vi) Numerical results in the present study are useful for calculation, design, and testing of geometrical parameters and material properties in engineering and technologies. (vii) e present approach can be developed to investigate the FG porous plate reinforced by GPLs subjected to other loads as well as analysis of FG porous shells with GPL reinforcement.

Data Availability
e data used to support this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  16 Mathematical Problems in Engineering