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Trung Thanh Tran, Quoc-Hoa Pham, Trung Nguyen-Thoi, "An Edge-Based Smoothed Finite Element for Free Vibration Analysis of Functionally Graded Porous (FGP) Plates on Elastic Foundation Taking into Mass (EFTIM)", Mathematical Problems in Engineering, vol. 2020, Article ID 8278743, 17 pages, 2020. https://doi.org/10.1155/2020/8278743
An Edge-Based Smoothed Finite Element for Free Vibration Analysis of Functionally Graded Porous (FGP) Plates on Elastic Foundation Taking into Mass (EFTIM)
In this paper, free vibration analysis of the functionally graded porous (FGP) plates on the elastic foundation taking into mass (EFTIM) is presented. The fundamental equations of the FGP plate are derived using Hamilton’s principle. The mixed interpolation of the tensorial components (MITC) approach and the edge-based smoothed finite element method (ES-FEM) is employed to avoid the shear locking as well as to improve the accuracy for the triangular element. The EFTIM is a foundation model based on the two-parameter Winkler–Pasternak model but added a mass parameter of foundation. Materials of the plate are FGP with a power-law distribution and maximum porosity distributions in the forms of cosine functions. Some numerical examples are examined to demonstrate the accuracy and reliability of the proposed method in comparison with those available in the literature.
The plate resting on the elastic foundation (EF) is one of the most common types of structures which have practical applications in civil and industrial constructions, especially in transportation and irrigation. In particular, the structures of beams and plates on the EF are subjected to moving loads of means of transport such as roadbeds affected by vehicles, railway tracks, and aircraft runways. In most publications, when investigating the mechanical behavior of structures on the EF, the researchers mainly use the one-parameter Winkler foundation model  or two-parameter Winkler–Pasternak foundation model [2, 3]. The analysis of plates resting on the Winkler–Pasternak foundation was previously addressed by several authors. For instance, Fazzolari  used an analytical method to consider free vibration and buckling of porous FG Sandwich beams resting on the EF with the Winkler–Pasternak foundation model. Leissa  presented results for the free vibration of rectangular plates. Xiang et al.  studied free vibration for Mindlin plates on the Winkler–Pasternak foundation using an analytical method. Omurtag et al.  used the finite element method (FEM) for the free vibration analysis of the Kirchhoff plates resting on the EF. Özçelikörs et al.  analyzed the exact solutions of bending, buckling, and vibration problems of a Levy-plate on the two-parameter foundation. Matsunaga  used a special higher-order plate theory (HSDT) to analyze vibration and buckling of thick plates on the EF. Ayvaz et al.  developed a modified Vlasov model to consider the earthquake response of thin plates on the EF. Shen et al.  based on the Rayleigh–Ritz method to study free and forced vibration of the Reissner–Mindlin plates resting on the EF. Liew et al. [11, 12] and Han and Liew  analyzed the free vibration of rectangular plates resting on the EF using a differential quadrature method. Zhou et al.  considered the vibration of rectangular plates on the EF using the Ritz method. Chucheepsakul and Chinnaboon  investigated plates by a two-parameter foundation model using a boundary element method. Civalek and Acar  investigated the bending of Mindlin plates on the EF by developing a singular convolution method. Ferreira et al.  presented bending and free vibration analyses of the FGP plates on the Winkler–Pasternak foundation by using radial basis functions. Shahsavari et al.  used a new quasi-3D hyperbolic theory for the free vibration analysis of the FGP plates resting on the EF. Zenkour and Radwan  proposed an exact analytical approach for free vibration analysis of laminated composite and Sandwich plates resting on the EF using a four-unknown plate theory. Duc et al.  presented the analysis of nonlinear thermal dynamic response of shear deformable functionally graded plates on the EF. Mahmoudi et al.  developed a refined quasi-three-dimensional shear deformation theory to analyze the functionally graded Sandwich plates resting on the two-parameter EF under thermomechanical loading. Duc et al.  used an analytical method to calculate static bending and free vibration of FG carbon nanotube-reinforced composite plate resting on Winkler–Pasternak foundations. Thang et al.  considered the effects of variable thickness on buckling and postbuckling behavior of imperfect sigmoid FGM plates on elastic medium subjected to compressive loading. Banh-Thien et al.  developed the isogeometric analysis for buckling analysis of nonuniform thickness nanoplates in an elastic medium.
In recent years, the FGP materials have attracted great interest from many researchers over the world due to their lightness and high strength. As a result, they are widely applied for civil engineering, aerospace structures, nuclear plants, and other applications. Kim et al.  investigated bending, vibration, and buckling of the FGP microplates using a modified couples stress based on the analytical method. Coskun et al.  presented analytical solutions to analyze the bending, vibration, and buckling of the FG microplates based on the third-order shear deformation theory (TSDT). Chen et al.  investigated the static bending and buckling of the FGP beams by using the Timoshenko beam theory. Rezaei and Saidi [28, 29] studied the vibration of rectangular and porous-cellular plates based on an analytical method. The vibration of the FGP shallow shells using an improve Fourier method was examined by Zhao et al. . Moreover, the dynamics of the FGP doubly-curved panels and shells were also investigated in . Li et al.  analyzed the nonlinear vibration and dynamic buckling of the Sandwich FGP plates with graphene platelet reinforcement (GPL) on the EF. For nonlinear problems, Sahmani et al.  used the nonlocal method to analyze nonlinear large-amplitude vibrations of the FGP micro-/nanoplates with GPL reinforce. Wu et al.  studied the dynamic of the FGP structures by using FEM. Thang et al.  investigated the elastic buckling and free vibration of porous cellular plates based on the first-order shear deformation theory (FSDT). Although the FGP materials have many different types, in this paper, the authors only use the distribution of porosity as presented in [25, 26].
In the other front of the development of numerical methods for computational mechanics, Liu et al.  have recently proposed an edge-based smoothed FEM (ES-FEM) using triangular elements which show some following excellent properties for the 2D solid mechanics analyses such as (1) the numerical results are often found superconvergent and very accurate; (2) the method is stable and works well for dynamic analysis; and (3) the implementation of the method is straightforward and no penalty parameter is used. The ES-FEM has been developed for n-sided polygonal elements , viscoelastoplastic analyses , 2D piezoelectric , primal-dual shakedown analyses , fluid structure interaction [41, 42], and various applications [43–45]. Recently, in an effort to improve the accuracy of the plate and shell structural analyses, the classical MITC3 element  incorporated with the ES-FEM , has been proposed to give the so-called ES-MITC3 element [47–51]. In the formulation of the ES-MITC3, the system stiffness matrix is employed using strains smoothed over the smoothing domains associated with the edges of the MITC3 elements. The numerical results demonstrated that the ES-MITC3 has the following great properties : (1) the ES-MITC3 can eliminate transverse shear locking even with the ratio of the thickness to the length of the structures reach and (2) the ES-MITC3 has better accuracy than the existing triangular elements such as MITC3 , DSG3 , and CS-DSG3  and is a good competitor with the quadrilateral element MITC4 element .
The objective of this research now is to further extend the ES-MITC3 method for free vibration analyses of the FGP plates resting on the EFTIM. The governing equations are derived from the FSDT and the Reissner–Mindlin plate theory due to simplicity and computational efficiency. Besides, the EFTIM is modelled based on a two-parameter Winkler–Pasternak foundation model and added in a mass parameter of foundation. The plate is made from the FGP materials with a power-law distribution () and maximum porosity distributions () in the forms of cosine functions. The accuracy and reliability of the present formulation are verified by comparing with those of other available numerical results. Moreover, the effects of some geometric parameters and material properties on the free vibration of the FGP plates are also examined in detail.
2. Functionally Graded Porous Material Plates on Elastic Foundation
Let us consider an FGP plate resting on EFTIM, as shown in Figure 1. The FGP materials with a variation of two constituents and three different distributions of porosity through-thickness are presented as follows [25, 26]:where is the maximum porosity value. A typical material property of the FGP materials can be considered as in the following power-law relations:where and are the typical material properties at the top and the bottom surfaces, respectively; and is the power-law index. The normalized distributions of porosity through the thickness are shown in Figure 2(a). As shown in Figure 2(a), the porosity distribution of Case 1 is symmetric with respect to the midplane of plates. Case 2 and Case 3 are bottom and top surface-enhanced distributions, respectively. Besides, Figures 2(b)–2(d) show the distributions of a normalized typical property associated with three different cases of porosity distributions with parameters: , k = 1, 5, 10, and Pt/Pb = 10.
The EFTIM is built based on the Winkler–Pasternak foundation by adding a mass parameter of foundation:where is the displacement of FGP plate; and are, respectively, Winkler foundation stiffness and shear layer stiffness of the Pasternak foundation. In order to mention the effectiveness of the foundation mass involved in the oscillation as well as the continuous interaction of the spring with the plate, the parameter β with unit kg−1 is added. It characterizes the effective level of the foundation mass involved in vibration, which is determined based on an experimental basis and the ratio of the density of the foundation to the density of plate material which is defined as . Thus, the density of mass involved vibration with the foundation is determined . From equation (3) we see that, for the static problem, the EFTIM model and Winkler–Pasternak foundation model are the same. But, for the dynamic problems, these two models have differences, and when omitting the influence parameters of the foundation mass, the EFTIM model is equivalent to the Winkler–Pasternak foundation model. In addition, this foundation model also covers the Winkler foundation model when the influence of shear parameters and foundation mass parameters ignored. It was found that the EFTIM model closely resembles the true feature of the foundation, including the Pasternak and Winkler foundation models.
3. The First-Order Shear Deformation Theory and Weak Form of the FGP Plates
3.1. First-Order Shear Deformation Theory for FGP Plates
The displacement of the FGP plates in the present work based on the FSDT model can be expressed aswhere , , , , and are five unknown displacements of the midsurface of the plate. For a bending plate, the strain field can be expressed as follows:where the membrane strain is given as
The bending and transverse shear strains are written as
From Hooke’s law, the linear stress-strain relations of the FGP plates can be expressed aswherewhere presents for effective Young’s modulus and represents Poisson’s ratio.
3.2. Weak Form Equations
To obtain the motion equations of the FGP plates for the free vibration analysis, Hamilton’s principle is applied with the following form:where and are the strain and kinetic energies, respectively. The strain energy is expressed aswhere is the strain energyand is the strain energywhere andand , , , and can be given by
The kinetic energy in equation (11) is given bywhere is the kinetic energywhere is the displacement field and is the mass matrix defined bywhere
In equation (11), the kinetic energy of the mass of foundation is defined as
4. Formulation of an ES-MITC3 Method for FGP Plates
4.1. Formulation of the Finite Element Using the MITC3 Element
The middle surface of plate is discretized into finite three-node triangular elements with nodes such that and . Then, the generalized displacements at any point of element can be approximated aswhere is the number of nodes of ; is the shape function matrix; and is the nodal degrees of freedom (d.o.f) associated with the jth node of .
The membrane bending strains of MITC3 element can be expressed in matrix forms as follows:where
To eliminate the shear locking phenomenon as the thickness of the plate becomes small, the formulation of the transverse shear strains of the MITC3 element based on FSDT  in this study can be written as follows:wherewithwhere
Here, , , , and are pointed out in and is the area of the three-node triangular element as shown in Figure 3.
Substituting the discrete displacement field into equation (22), we obtained the discrete system equations for free vibration analysis of FGP plate resting on the EF, respectively, aswhere and are the stiffness and mass matrices, respectively.
The stiffness matrix in equation (38) can be written aswherewhere
Next, the mass matrix in equation (38) can be defined aswhere
4.2. Formulation of an ES-MITC3 Method for FGP Plates
The smoothing domains is constructed based on edges of the triangular elements such that and . An edge-based smoothing domain for the inner edge is formed by connecting two end-nodes of the edge to the centroids of adjacent triangular MITC3 elements, as shown in Figure 4.
Applying the edge-based smooth technique , the smoothed membrane, bending, and shear strain , over the smoothing domain can be created bywhere , , and the compatible membrane, bending, and the shear strains, respectively; is a given smoothing function that satisfies at least unity property .
In this study, we use the constant smoothing functionwhere is the area of the smoothing domain and is given bywhere is the number of the adjacent triangular elements in the smoothing domain ; and is the area of the triangular element attached to the edge
By substituting equations (47)–(49) into equations (24), (25), and (32), the approximation of the smoothed strains on the smoothing domain can be expressed as follows:where is the total number of nodes of the triangular MITC3 elements attached to edge k ( for boundary edges and for inner edges as given in Figure 4; is the nodal d.o.f associated with the smoothing domain ; and , , and are the smoothed membrane, bending, and shear strain gradient matrices, respectively, at the jth node of the elements attached to edge k computed by
The stiffness matrix of the FGP plate using the ES-MITC3 is assembled bywhere is the ES-MITC3 stiffness matrix of the smoothing domain and given bywhere
5. Accuracy of the Proposed Method
In this section, the various numerical examples are solved to verify the reliability and accuracy of the proposed method. For convenience, the stiffness factors and nondimensional frequencies of the plates are defined as the following equations:
To demonstrate the performance of numerical results, the relative frequency error is defined bywhere and are nondimensional frequencies of present method and nondimensional frequencies in [17, 18], respectively.
The results of the convergence of the first two nondimensional frequencies of the plate in the case of fully simple support (SSSS) plate and a fully clamped (CCCC) plate with h/a = 0.1, K1 = 100, K2 = 10, respectively, are shown in Figure 5. From these results, it can be seen that almost all frequencies corresponding to different cases of boundary conditions (BC) converge with 18 × 18 element mesh. For 18 × 18 mesh, we compare the first three nondimensional frequencies of a plate resting on the Winkler–Pasternak foundation with the published results as shown in Table 1. It can be seen that the present results agree well with the results of the authors using analytical methods [5, 14, 17] and are more accurate than those using the original MITC3 element and FEM . In addition, from Table 2, it is obvious that the relative error of the present results compared to  is less than 2%. In , they used a new quasi-3D hyperbolic theory to investigate the free vibration of the FGP plate resting on the EF. These results are the basis to analyze the free vibration of FGP plates on the EFTIM.
Next, we consider an SSSS FGP plate (Al/Al2O3) with its material properties as follows: metal (Al) , and ceramic (Al2O3) , . Poisson’s ratio is fixed at . The FGP plate with even porosities is expressed as in :where presents the porosity volume fraction. The stiffness factor and nondimensional frequencies of the plates are shown in equation (58) with and . The first nondimensional frequencies of present work compared with  are shown in Table 2.
6. Numerical Results and Discussions
For free vibration problems, a fully simple support (SSSS) FGP plate is considered, wherein , , , , , and . The first six nondimensional frequencies of the FGP plate with porosity distribution of Case 1 and stiffener of foundation are shown in Table 3; and the first six mode shapes are presented in Figure 6. The stiffness factors and nondimensional frequencies of FGP plate are shown in equation (58) with .
6.1. Influence of the Parameters of the EFTIM to the Free Vibration for the FGP Plate
Firstly, in order to investigate the effect of the featured-index of the mass of foundation β to free vibration of the FGP plate, the featured-index of mass is changed from 0 to 1. In Figure 7 and Table 4, it is seen that all cases of porosity distribution featured-index of the mass of foundation β significantly influence to free vibration of the FGP plate. As β increases, the mass of the plate increases, and the frequencies of the plate decrease. For all cases of porosity distribution of the FGP plate, the porosity distribution of Case 3 leads to the maximum values of frequencies of the plate, while the porosity distribution of Case 2 leads to the minimum values. It can be observed that the FGP plate with porosity distribution Case 3 is stiffer than plates with other porosity distributions.
Next, the influence of nondimensional parameters of foundation stiffness K1 and K2 is investigated. We change from 100 to 1000 and from 10 to 100 with respect to and . The first nondimensional frequency of the FGP plate with three cases of porosity distribution is presented in Table 5 and shown in Figure 8. As shown in these figure and table, when K1 and K2 increase, the nondimensional frequency of plate also increases. We also examine the effect of (β, K1) and (β, K2) parameters to nondimensional frequency. The numerical result is presented in Table 6 and Figure 9. Consequently, Winkler foundation stiffness K1 and shear layer stiffness of Pasternak foundation K2 make stiffness of plate become greater and the mass of the EF involved in the plate’s vibration makes reduce frequencies.
6.2. Influence of the Parameters-FGP to Free Vibration of the Plate on EFTIM
Let us consider the effect of materials property to free vibration of the FGP plate. The power-law index k is changed from 0 to 10, and maximum porosity distributions Ω has the value from 0 to 1. We examine the SSSS FGP plate and fully clamped (CCCC) plate resting on EFTIM. The parameters of EFTIM are given by , , , and . The first nondimensional frequencies of plate with three cases of porosity distribution is shown in Table 7 and Figure 10.
As shown in these figures and tables, when k and change, the values of nondimensional frequency change with no rule. It is understandable because with each case of change in porosity distributions k and , the stiffness and the weight of the plate changes. From Figure 10, in the case of the CCCC plate, nondimensional frequency depending on k and value varies more complex than the case of the SSSS plate. If k and values are same, the frequency of the CCCC plate is larger than that of the SSSS plate. The results are quite reasonable because the SSSS boundary condition inherently offers more flexible boundary conditions than the CCCC boundary condition.
In this paper, new numerical results of free vibration of the FGP plate resting on EFTIM are studied. We used the ES-MITC3 to establish the fundamental equation of the FGP plate. The computed results obtained by this approach are in excellent agreement with others published. Our work has the following advantages.
The novel ES-MITC3 which computes the free vibration of the plate on EF takes into account the mass of foundation.
The numerical results obtained by ES-MITC3 show good agreement with the reference solutions and are more accurate than those obtained by the original MITC3.
The elastic foundation of Pasternak with three-parameters is developed by adding the featured-index of mass β to accurately describe the actual elastic foundation.
The mass of the elastic foundation involved in the vibration of the plate reduces the frequency of vibration, while two parameters K1 and K2 effect the stiffness of the plate.
The material parameters k, and the case of porosity distribution effect of vibration of the plate. Numerical results are useful for calculation, design, and testing of material parameters in engineering and technologies.
This study suggests some further works on the dynamic response and heat transfer problems of the FGP plate resting on EF using different plate theories.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 107.02-2019.330.
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