An edge-based smoothed finite element for buckling analysis of functionally graded material variable-thickness plates

Vietnam Journal of Mechanics, VAST, Vol.43, No. 3 (2021), pp. 221 – 235 DOI: https://doi.org/10.15625/0866-7136/15503 AN EDGE-BASED SMOOTHED FINITE ELEMENT FOR BUCKLING ANALYSIS OF FUNCTIONALLY GRADED MATERIAL VARIABLE-THICKNESS PLATES Tran Trung Thanh1, Tran Van Ke1, Pham Quoc Hoa2, Tran The Van2, Nguyen Thoi Trung3,∗ 1Le Quy Don Technical University, Hanoi, Vietnam 2Tran Dai Nghia University, Ho Chi Minh City, Vietnam 3Ton Duc Thang University, Ho Chi Minh City, Vietnam ∗E-

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-mail: nguyenthoitrung@tdtu.edu.vn Received: 22 September 2020 / Published online: 10 August 2021 Abstract. The paper aims to extend the ES-MITC3 element, which is an integration of the edge-based smoothed finite element method (ES-FEM) with the mixed interpolation of tensorial components technique for the three-node triangular element (MITC3 element), for the buckling analysis of the FGM variable-thickness plates subjected to mechanical loads. The proposed ES-MITC3 element is performed to eliminate the shear locking phe- nomenon and to enhance the accuracy of the existing MITC3 element. In the ES-MITC3 element, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent MITC3 triangular elements sharing the same edge. The numerical results demonstrated that the proposed method is reliable and more accurate than some other published solutions in the literature. The influences of some geometric parameters, material properties on the stability of FGM variable-thickness plates are examined in detail. Keywords: buckling analysis, critical load, variable thickness plate, edge-based finite ele- ment method, ES-MITC3. 1. INTRODUCTION The functionally graded materials (FGMs) can change the material properties grad- ually, continuously, and smoothly in different directions. Therefore, the delamination in laminated composites can be eliminated in these materials. They are made of two components, mainly metal with high toughness and ceramic with outstanding heat and corrosive resistance properties. Due to such excellent properties, they are applied in var- ious high-tech industries such as automotive, nuclear, civil engineering, and aerospace. There have been many studies on the mechanical behavior of FGM structures including the buckling problem. Some typical studies can be summarized as follows. Ramu et â 2021 Vietnam Academy of Science and Technology 222 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung al. [1] studied the stability of FGM under uniaxial and biaxial compression load using the finite element method (FEM) based on classical plate theory (CPT). Rohit et al. [2] used third-order shear deformation theories (TSDT) to analyze the buckling of the sim- ple supported FGM plates under uniaxial load. Wu et al. [3] studied the stability of FGM plates subjected to thermal and mechanical loads using FSDT. Javaheri et al. [4] based on the analytical method (AM) for the stability analysis of FGM plates subjected to in- plane compressive load. Zenkour [5] calculated the free vibration and buckling of FGM constant-thickness sandwich plates. Shariat et al. [6] studied the buckling of thick FGM plate by AM. Thai et al. [7] used an efficient and simple refined theory for buckling anal- ysis of FGM plates. Reddy [8] combined an analytical method and TSDT to analyze the buckling of the FGM plate. Thinh et al. [9] proposed an eight unknown higher-order shear deformation theory for vibration and buckling analysis of constant-thickness FGM plates. Variable-thickness structures are extensively used in many types of high- performance surfaces like aircraft, civil engineering, and other engineering fields. Using these structures will help adjust the weight of structural, and hence help maximize the capacity of the material. For example, Thang et al. [10] investigated the effects of variable- thickness on buckling and post-buckling of imperfect sigmoid FGM plates on elastic foundation (EF) subjected to compressive loading. Eisenberger et al. [11] investigated the buckling of variable-thickness thin isotropic plates by using the extended Kantorovich method. Naei et al. [12] analyzed the buckling of the FGM variable-thickness circular- plate using FEM. Jalali et al. [13] investigated thermal buckling of the FGM nonuniform- thickness circular sandwich plates employed the pseudo-spectral method. Alipour and co-workers used semi-analytical to studied buckling of heterogeneous variable-thickness viscoelastic circular-plates lying on the EF [14], and variable-thickness bi-directional FGM circular-plates placed on nonuniform-EF [15]. Alinaghizadeh et al. [16] applied the gener- alized differential quadrature (GDQ) method for buckling analysis of variable-thickness radially FGM annular sector plates located on two parameters EF. Bouguenina et al. [17] conducted analyses of FGM variable-thickness plates under thermal loads using finite difference method. Benlahcen et al. [18] employed an analytical solution to examine buckling of simply supported FGM plates with parabolic-concave thickness variation. Minh and Duc [19] investigated the effect of cracks on the stability of the FGM variable- thickness plates using TSDT and phase-field theory. In addition, Zenkour [20] presented the hygrothermal mechanical bending of variable-thickness plates using the AM. Allam et al. [21] investigated thermoelastic stresses in FG variable-thickness rotating annular disks using infinitesimal theory. Thien et al. [22] developed the isogeometric analysis (IGA) to analyze the buckling of non-uniform thickness nanoplates resting on the EF. To improve the convergence and accuracy for classical triangular elements, the origin MITC3 element [23] is proposed to combine with the ES-FEM [24] to give the so-called ES-MITC3 element [25–30]. In the formulation of the ES-MITC3 element, the system stiff- ness matrix is employed using strains smoothed over the smoothing domains associated with the edges of the triangular elements. The numerical results of the present study demonstrated that the ES-MITC3 element has the following superior properties: (1) the ES-MITC3 element can avoid the transverse shear locking phenomenon even with the An edge-based smoothed finite element for buckling analysis of functionally graded material variable-thickness plates 223 ratio of the thickness to the length of the structures reach 10−8 (readers can see detail in Ref. [25]; (2) the ES-MITC3 element has higher accuracy than the existing triangular ele- ments such as MITC3 element [23], DSG3 element [31] and CS-DSG3 element [32]; and is a good competitor with the MITC4 element [33]. According to the best of authors’ knowledge, the stability of FGM variable-thickness plates using the ES-MITC3 element has not yet been studied. Therefore, this paper aims to extend the ES-MITC3 element for the buckling analysis of FGM variable-thickness plates. The formulation is based on the FSDT due to its simplicity and computational efficiency. The accuracy and reliability of the present approach are verified by comparing the present numerical results with those of other available methods. Finally, the influence of geometrical parameters, and material properties on the buckling of FGM plates are fully studied. 2. THEORETICAL FORMULATION 2.1. FGM material The FGM is made up of two components: ceramic and metal. The mechanical prop- erties of FGM are assumed to vary smoothly through the thickness of plates as follows [6] P (z) = (Pc − Pm) Vc (z) + Pm, (1)  z p  h (x, y) h (x, y)  V (z) = + 0.5 with z 2 − ; , (2) c h(x, y) 2 2 in which P (z) represents for Young’s modulus E(z), Poisson’s ratio u(z); subscripts m and c denotes the metal and ceramic constituents; Vc(z) is the volume fraction of ceramic which according to a power-law function with p is the power-law index. The value of p equals to 0 and +Ơ represents a fully ceramic and fully metal plate, respectively. Note that, the thickness of plate is different at various positions on the plate and depends on the law of thickness variation (h is the function of x- and y-variables). 2.2. Mindlin’s plate theory According to Mindlin’s plate theory, the displacement field of the plate is given by [1] 8 u (x, y, z) = u (x, y) + zq (x, y) 0 x v (x, y, z) = v0 (x, y) + zqy (x, y) (3) > : w (x, y, z) = w0 (x, y) in which u, v, w, qx, qy are five unknown displacements of the mid-surface of the plate. The strain field can be expressed as follows 8 9 8 9 8 9 8 9 > #x > > u,x > > u0,x > > qx,x > > > > > > > > > #y => v,y => v0,y => qy,y => # = #xy = u,y + v,x = u0,y + v0,x + z qx,y + qy,x . (4) > gxz > > w,x + u,z > > v0.x + qx > > 0 > > > > > > > > > : gyz ; : w,y + v,z ; : w0,y + qy ; : 0 ; 224 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung Eq. (3) may be written by  #   # + zk  = 1 = m . (5) #2 g From Hooke’s law, the linear stress-strain relations can be determined by a formulation 8 9 2 3 8 9 s Q Q 0 0 0 # > x > 11 12 > x > > sy > 6 Q Q 0 0 0 7 > #y > < = 6 21 22 7 < = sxy = 6 0 0 Q66 0 0 7 #xy , (6) 6 7 > txz > 4 0 0 0 Q55 0 5 > gxz > > > > > : tyz ; 0 0 0 0 Q44 : gyz ; in which E(z) u(z)E(z) E(z) Q = Q = , Q = Q = , Q = Q = Q = . (7) 11 22 1 − u(z)2 12 21 1 − u(z)2 44 55 66 2(1 + u(z)) The force and moment resultants are obtained as follows [30]  T Nx Ny Nxy = A#m + Bk, (8a)  T Mx My Mxy = B#m + Ck, (8b)  T s Qxz Qyz = A g, (8c) with 2 3 Z h(x,y)/2 Q11 Q12 0 2 (A, B, C) = 4 Q21 Q22 0 5 1, z, z dz, (9) −h(x,y)/2 0 0 Q66 Z h(x,y)/2  Q 0  As = 55 dz. (10) −h(x,y)/2 0 Q44 It should be noted that compared to uniform thickness plates, all the matrices in Eqs. (9) and (10) depend on the law of thickness variation and thus the limits of integrations also depend on the position of points on plates. 2.3. Finite element formulation for buckling analysis of FGM variable-thickness plates The bounded domain W of the FGM plate is discretized into ne three-node triangular ne n elements with n nodes such that y ≈ ∑ ye and yi \ yj = ?, i 6= j. Then the gen- e=1 T e h e e e e e i eralized displacements at any point u = uj , vj , wj , qxj, qyj of the element ye can be approximated as [23] 2 3 NI (x) 0 0 0 0 nne 6 0 NI (x) 0 0 0 7 nne e 6 7 e e u (x) = 6 0 0 NI (x) 0 0 7 d = N (x) d , (11) ∑ 6 7 j ∑ j j=1 4 0 0 0 NI (x) 0 5 j=1 0 0 0 0 NI (x) An edge-based smoothed finite element for buckling analysis of functionally graded material variable-thickness plates 225 ne e where n is the number of nodes of ye; N (x) is the shape function matrix; and dj = T h e e e e e i th uj , vj , wj , qxj, qyj are the nodal degrees of freedom (DOF) associated with the j node of ye. The membrane bending strains of MITC3 element can be expressed in the matrix form as follows [23] e  e e e  e e e #m = Bm1 Bm2 Bm3 d = Bmd , (12a) e  e e e  e e e k = Bb1 Bb2 Bb3 d = Bbd . (12b) The smoothing domains yk is constructed based on the edges of the triangular el- nk k k k ements such that Any =edge[-kbased=1y smoothedand yi \ finiteyj =element? for fori 6buckling= j. An analysis edge-based smoothing domain yk for the inner of functionally edge k is grade formedd material by connecting variable- twothickness end-nodes plates of the edge to the centroids of adjacent triangular MITC3 elements as shown in Fig.1. ả Fig.Fig. 1 1. .The The smoothing smoothing domain domain •yk isis formedformed byby triangular triangular elements elements. Applying the edge-based smooth technique [24], the smoothed membrane, bending and shear ả Applyingả ả the edge-based smooth techniqueả [24], the smoothed membrane, bending strain Mòđ, [â , \â over thek smoothingk k domain • can be created by:k and shear strain# ˜m,k ˜ ,g ˜ over the smoothing domain y can be created by Z k k ỷ #˜m = #mỷF (x) dy, (13a) Mò, = y My,k ™ (x)dψ, (13a) ´† Z k˜ k = kFk (x) dy, (13b) yk ỷ Z ỷ [â g=˜ k y= [ ™gF(xk )(dxψ) d, y, (13c) (13b) ´† yk where #m, k and g the compatible membrane, bending and the shear strains, respectively; Z k k F (x) is a given smoothing functionỷ that satisfiesỷ( ) at least the unity property F (x) dy (13c) \â = y \ ™ x dψ, yk † = 1. ´ ỷ where M,, [ and \ the compatible membrane, bending and the shear strains, respectively; ™ (x) is a ỷ( ) given smoothing function that satisfies at least the unity property∫´† ™ x dψ = 1. In this study, we use the constant smoothing function [24]: 1 x ∈ ψỷ ™ỷ(x) = E≠ỷ (14) 0 x ∉ ψỷ in which ≠ỷ is the area of the smoothing domain ψỷ and is given by 1 ẫỏ† ≠ỷ = y dψ = ∞ ≠õ (15) ´† 3 õàX where Åầỷ is the number of the adjacent triangular elements in the smoothing domain ψỷ; and ≠õ is the area of the ỗth triangular element attached to the edge Đ. The stiffness matrix of the FGM plate using the ES-MITC3 is assembled by [24]: † ẫà∂ ỷ ≥Ơ 8 = ∞ ≥Ơ ầ (16) ỷàX ỷ ỷ where Ơ≥ầ is the ES-MITC3 stiffness matrix of the smoothing domain ψ and given by 226 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung In this study, we use the constant smoothing function [24] 8 1 < , x 2 yk Fk (x) = Ak (14) : 0, x 2/ yk in which Ak is the area of the smoothing domain yk and is given by ek Z 1 n Ak = dy = Ai, (15) k ∑ y 3 i=1 where nek is the number of the adjacent triangular elements in the smoothing domain yk; and Ai is the area of the ith triangular element attached to the edge k. The stiffness matrix of the FGM plate using the ES-MITC3 element is assembled by [24] k nsh ˜ ˜ k Kp = ∑ Ke , (16) k=1 k k where Kee is the ES-MITC3 element stiffness matrix of the smoothing domain y and given by Z       ˜ k ˜ kT AB ˜ k ˜ kT s ˜ k ˜ kT AB ˜ k k ˜ kT s ˜ k k Ke = B B + Bs A Bs dy = B B A + Bs A Bs A , (17) yk BC BC in which h i ˜ kT ˜ k ˜ k B = Bmj Bbj , (18) and the strain-displacement matrices are presented in detail in [30]. The geometric stiffness matrix of the FGM plate using the ES-MITC3 element is de- termined by [28] nk sh Z   ˜ ˜ ek ˜ e ˜ T ¯ ˜ Kg = Kg with Kg = Y NYi dy, (19) ∑ k i k=1 y where  N N  N = x xy , (20) Nxy Ny with  Z h(x,y)  Nx, Ny, Nxy = sx, sy, sxy dz, (21) −h(x,y) and Y˜i is presented in [28]. It is noted that the integrations in Eq. (21) also depend on the law of thickness variation, therefore the limits of integrations will depend on the position of points on plates. Apply the principle of minimum total potential energy, the stability problem in- volves the solution of the following eigen problem in which Pcr is the critical load K˜ p + PcrK˜ g = 0. (22) An edge-based smoothed finite element for buckling analysis of functionally graded material variable-thickness plates 227 3. CONVERGENCE AND ACCURACY OF THE PROPOSED METHOD In order to evaluate the convergence and accuracy of the proposed method, the au- thors consider the following two examples: Example 1. Firstly, we consider a fully clamped (CCCC) FGM constant-thickness plate with material properties given by the metal (Al) Em = 70 GPa, um = 0.3 and ce- ∗ ramic (Al2O3) Ec = 380 GPa, uc = 0.3. The non-dimensional critical load Pcr of FGM plates with different mesh-size are listed in Table1. It can be seen that, in all cases, the An edgeAn- basededge- basedsmoothed smoothed finite elementfinite element for buckling for buckling analysis analysis results by the ES-MITC3 element converge faster and are more accurate than those by of functionally of functionally grade dgrade materiald material variable variable-thickness-thickness plates plates the MITC3 element. Specifically, at the 18ì18 mesh-size, the ES-MITC3 element gives ∗ ∗ ; ;ừ ừ TableTable 1. The 1. convergence The convergence of mesh of -meshsize of-size non of-dimensional non-dimensional critical critical load %load*ứ = %%*ứ*ứ=Œ %/*ứ?*ŒℎI/ ?of* ℎCCCCI of CCCC ∗ 2 3 Table 1. The convergence of mesh-sizesquaresquare ofFGM non-dimensional FGM plate splate. s. critical load Pcr = Pcrb /Ech0 of CCCC square FGM plates a/h a/h p pMesh Mesh size sizeES -MITC3ES-MITC3 Error Error(%) (%) MITC3MITC3 Error Error(%) (%) Wu [3]Wu [3] 100a/h 100 p1 Mesh1 12x12 size12x12 ES-MITC34.62124.6212 0.12 Error 0.12 (%) 4.6265 MITC34.6265 Error0.23 0.23 (%) Wu [3] 14x1414x14 4.61854.6185 0.06 0.06 4.62104.6210 0.11 0.11 12ì12 4.6212 0.12 4.6265 0.23 1416x16ì1416x16 4.6167 4.61854.6167 0.02 0.06 0.02 4.6194 4.62104.6194 0.08 0.11 0.08 4.61584.6158 100 1 1618x18ì1618x18 4.6160 4.61674.6160 0.01 0.02 0.01 4.6180 4.61944.6180 0.05 0.08 0.05 4.6158 1820x20ì1820x20 4.6160 4.61604.6160 0.01 0.01 0.01 4.6176 4.61804.6176 0.04 0.05 0.04 20ì20 4.6160 0.01 4.6176 0.04 40 40 5 5 12x1212x12 3.00413.0041 0.15 0.15 3.00553.0055 0.20 0.20 1214x14ì1214x14 3.0020 3.00413.0020 0.08 0.15 0.08 3.0030 3.00553.0030 0.11 0.20 0.11 14ì14 3.0020 0.08 3.0030 0.11 16x1616x16 3.00063.0006 0.03 0.03 3.00163.0016 0.07 0.07 2.99962.9996 40 5 16ì16 3.0006 0.03 3.0016 0.07 2.9996 1818x18ì1818x18 2.9998 2.99982.9998 0.01 0.01 0.01 3.0005 3.00053.0005 0.03 0.03 0.03 2020x20ì2020x20 2.9998 2.99982.9998 0.01 0.01 0.01 3.0001 3.00013.0001 0.02 0.02 0.02 (a) The FGM plate with a/h = 100 and p = 1 (b) The FGM plate with a/h = 40 and p = 5 a) Thea) FGM The FGM plate platewith a/hwith=100 a/h =100and p and=1. p=1. b) Theb) FGM The FGM plate platewith a/hwith=40 a/h and=40 p and=5. p=5. Fig. 2. The convergence of mesh-size to non-dimensional∗ critical∗ ; load;ừ ừ Fig. 2.Fig. The 2. convergenceThe convergence of mesh of mesh-size -tosize non to-dimensional non-dimensional critical critical load %load= %%*ứ =Œ %/*ứ?Œℎ/ ?of* ℎsquareI of square ∗ 2 3 *ứ *ứ * I Pcr = Pcrb /FGMEch0FGM plateof square .plate . FGM plate ∗ ∗ ; ;ừ ừ TableTable 2. Comparison 2. Comparison of nondimensional of nondimensional critical critical load %load*ứ = %%*ứ*ứ=Œ %/*ứ?*ŒℎI/ ?of* ℎrectangularI of rectangular FGM FGM plate splate. s. (h=a/40;(h= aa/40;=1 is a =1fixed) is fixed). . b/a b/a p p Wu [3]Wu [3] MITC3MITC3 Error Error(%) (%) PresentPresent Error Error(%) (%) 1.5 1.5 0 0 11.851611.8516 11.891311.8913 0.33 0.33 11.863311.8633 0.10 0.10 2 2 0 0 17.529917.5299 17.568617.5686 0.22 0.22 17.548117.5481 0.10 0.10 3 3 0 0 35.123935.1239 35.216835.2168 0.26 0.26 35.153035.1530 0.08 0.08 1.5 1.5 2 2 4.64004.6400 4.63344.6334 0.14 0.14 4.64304.6430 0.06 0.06 2 2 2 2 6.858106.85810 6.84466.8446 0.20 0.20 6.84986.8498 0.12 0.12 3 3 2 2 13.769713.7697 13.720313.7203 0.36 0.36 13.753813.7538 0.12 0.12 228 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung the converging results with the maximum error of 0.01% compared to those by Wu et al. [3] using the analytical method (AM). In contrast, the MITC3 element at the18x18 mesh-size has not yet converged as shown in Fig.2. Furthermore, the obtained results by the ES-MITC3 element are compared to those of other published results as shown in Table2. It should be noted that the error is determined by the following formula: jPresent − [3]j Error (%) = 100 ì and types of boundary conditions are defined as fol- j[3]j lows: 1) Simply supported edge boundary condition (S): u0 = w = jx = 0 at y = 0, y = b or v0 = w = jy = 0 at x = 0, x = a; and 2) Clamped edge boundary condition (C): at y = 0, y = b or v0 = w = jx = jy = 0 at x = 0, x = a. ∗ 2 3 Table 2. Comparison of non-dimensional critical load Pcr = Pcrb /Ech0 of rectangular FGM plates. (h = a/40; a = 1 is fixed) b/a p Wu [3] MITC3 Error (%) Present Error (%) 1.5 0 11.8516 11.8913 0.33 11.8633 0.10 2 0 17.5299 17.5686 0.22 17.5481 0.10 3 0 35.1239 35.2168 0.26 35.1530 0.08 1.5 2 4.6400 4.6334 0.14 4.6430 0.06 2 2 6.8581 6.8446 0.20 6.8498 0.12 3 2 13.7697 13.7203 0.36 13.7538 0.12 Example 2. Secondly, a simply supported (SSSS) isotropic plate with linearly variable y thickness h = h (1 + a ) is considered. The non-dimensional critical load is calculated 0 b ∗ 2 2 3 by Pcr = Pcrb /(p D) with D = Eh0/12. The obtained results of the present work are ∗ Table 3. Comparison of non-dimensional critical load Pcr of SSSS isotropic plates with variable thickness a a/b Method 0.125 0.25 0.5 0.75 1 IGA-FSDT [22] 7.4621 8.7531 11.5687 14.6953 18.1368 0.5 Kantorovich method [11] 7.4645 8.7633 11.6112 14.7942 18.3175 ES-MITC3 7.4625 8.7601 11.5989 16.6987 18.2981 IGA-FSDT [22] 5.4194 6.3869 8.5627 11.0657 13.9017 0.7 Kantorovich method [11] 5.4199 6.3891 8.5741 11.0979 13.9730 ES-MITC3 5.4198 6.3885 8.5738 11.0889 13.9865 IGA-FSDT [22] 4.8428 5.7224 7.7327 10.0858 12.7877 0.9 Kantorovich method [11] 4.8413 5.7165 7.7111 10.0460 12.7381 ES-MITC3 4.8418 5.7203 7.7198 10.0683 12.7524 Tran Trung Thanh, Tran Van Ke, Pham Quoc Hoa, Tran The Van, Nguyen Thoi Trung Example 2. Secondly, a simply supported (SSSS) isotropic plate with linearly variable thickness 4 ℎ = ℎ (1 + α ) is considered. The non-dimensional buckling load is calculated by %∗ = I ự *ứ ; ; ừ %*ứŒ /(“ ”) with ” = ?ℎI/12. The obtained results of the present work are compared to those by Thien et al. [22] using the IGA based on FSDT and Eisenberger et al. [11] employed Kantorovich method. These results are listed in Table 3. It is observed that the obtained results by the proposed method are in a good agreement with those published in the literature. From the above two examples, it can be concluded that the proposed method is reliable for further analyses. ∗ Table 3. Comparison of non-dimensional critical load %*ứ of SSSS isotropic plates with variable thickness. α a/b Method 0.125 0.25 0.5 0.75 1 0.5 IGA-FSDT [22] 7.4621 8.7531 11.5687 14.6953 18.1368 Kantorovich 7.4645 8.7633 11.6112 14.7942 18.3175 method [11] ES-MITC3 7.4625 8.7601 11.5989 16.6987 18.2981 0.7 IGA-FSDT [22] 5.4194 6.3869 8.5627 11.0657 13.9017 Kantorovich 5.4199 6.3891 8.5741 11.0979 13.9730 method [11] ESAn-MITC3 edge-based smoothed finite5.4198 element for buckling6.3885 analysis of functionally8.5738 graded material variable-thickness11.0889 plates 22913.9865 0.9 IGA-FSDT [22] 4.8428 5.7224 7.7327 10.0858 12.7877 compared to those by Thien et al. [22] using the IGA based on FSDT and Eisenberger et Kantorovich al. [11] employed Kantorovich4.8413 method.5.7165 These results are7.7111 listed in Table10.04603. It is observed 12.7381 thatmethod the obtained [11] results by the proposed method are in a good agreement with those publishedES-MITC3 in the literature.4.8418 From the above5.7203 two examples,7.7198 it can be concluded10.0683 that the12.7524 proposed method is reliable for further analyses. 4. BUCKLING4. BUCKLING ANALYSIS ANALYSIS OF OF FGM FGM VARIABLE VARIABLE-THICKNESS-THICKNESS PLATES PLATES a In this section, we consider the FGM variable-thickness plate’ ( = 100, a is fixed) In this section, we consider the FGM variable-thickness plate ( = h100, a is fixed) as shown in 1ữ 0 as shown in Fig.3. The plate thickness varies along the x-direction following the law 2 Fig. 3. The plate thickness variesx along the x-direction following the law ℎ = ℎ(G) = ℎ (1 + ). The h = h (x) = h (1 + ). The material parameters of the FGM plate are given by:I metal ’ 0 a material parameters(Al) Em = of70 the GPa, FGMum =plate0.3 are and given ceramic by: (Al metal2O3) (Al)Ec = ?380, = GPa,70 ◊%œuc =, @0.3., = The0.3 non-and ceramic 2 l ∗ Pcrb ∗ ÀÿŸự (Al2O3) ?*dimensional= 380 ◊%œ critical, @* = load0.3. isThe introduced non-dimensional by Pcr = critical. load is introduced by %*ứ = ⁄ . 3 Jÿ1 Ech0 ữ y p p a hx() ceramic x h0 x metal b z Fig. 3. Fig.The 3 .FGM The FGM variable variable-thickness-thickness plate plate under under in in-plane-plane force force along along the thex-direction x-direction. 4.1. Effect of power-law index p In order to study the effect of the power-law index p on buckling of FGM plates, we consider a square FGM plate with different boundary conditions (BCs), and the power- law index p is changed from 0 to 100. The non-dimensional critical load of the FGM plates is listed in Table4 and displayed in Fig.4. It can be seen that the critical force of the plate depends not only on the BC but also on the power-law index p. The rich ceramic FGM plates have a higher hardness than the rich metal FGM plates, so the critical force is higher. The critical force decreases when the power-law index p increases, and the rate decreases faster when the index p increases from 0 to 1, and slower when p > 1. An edgeAn edge-based-based smoothed smoothed finite finite element element for buckling for buckling analysis analysis of functionally of functionally grade graded materiald material variable variable-thickness-thickness plates plates 4.1. Effect4.1. Effect of power of power-law- lawindex index p p In orderIn order to study to study the effectthe effect of the of powerthe power-law- lawindex index p on p buckling on buckling of FGM of FGM plates, plates, we considerwe consider a a squaresquare FGM FGM plate plate with with different different boundary boundary conditions conditions (BCs) (BCs), and, andthe powerthe power-law- lawindex index p is pchange is changed d fromfrom 0 to 0100. to 100. The Thenon -ndimensionalon-dimensional critical critical load load of the of FGMthe FGM plate plates is slisted is listed in T inable Table 4 and 4 anddisplayed displayed in Fig.in Fig.4. It 4can. It becan seen be seen that thatthe criticalthe critical force force of the of platethe plate depends depends not onlynot only on the on BCthe BCbut alsobut also on theon the powerpower-law- lawindex index p. The p. Therich richceramic ceramic FGM FGM plates plates have have a higher a higher hardness hardness than than the richthe richmetal metal FGM FGM plates,plates, so the so criticalthe critical force force is higher. is higher. The The critical critical force force decreases decreases when when the the power power-law- law index index p p increaseincreases, ands, andthe ratethe ratedecreases decreases faster faster when when the indexthe index p increases p increases from from 0 to 01 ,to and 1, andslower slower when when C > C > 1. 1. TableTable 4. The 4. Thecritical critical load load of the of squarethe square FGM FGM variable variable-thickness-thickness plate plate. . NonNon-dimensional-dimensional critical critical load load %∗ %∗ p p *ứ *ứ SSSSSSSS SCSCSCSC CSCSCSCS CCCCCCCC 0 0 5.35085.3508 10.215810.2158 9.24229.2422 13.796713.7967 0.5 0.5 3.47693.4769 6.62996.6299 6.01426.0142 8.95598.9559 1 1 2.67902.6790 5.10485.1048 4.63754.6375 6.89656.8965 2 2 2.09412.0941 3.98723.9872 3.62673.6267 5.38665.3866 2305 5 Tran1.7712 Trung1.7712 Thanh, Tran Van Ke,3.3705 Pham3.3705 Quoc Hoa, Tran The Van,3.0662 Nguyen3.0662 Thoi Trung 4.55234.5523 10 10 1.61191.6119 3.06833.0683 2.78812.7881 4.14324.1432 Table 4. The critical load of the square FGM variable-thickness plate 20 20 1.42311.4231 2.7112.711 2.45982.4598 3.66053.6605 50 50 1.21091.2109 2.30912.3091 2.09212.0921 ∗ 3.11803.1180 Non-dimensional critical load Pcr 100 100p 1.11151.1115 2.86392.8639 SSSS2.1208 SCSC2.1208 CSCS1.92021.9202 CCCC 4.2. Effect4.2. Effect of 0length of length to width to 5.3508width ratio ratio b/a b/a 10.2158 9.2422 13.7967 0.5 3.4769 6.6299 6.0142 8.9559 NextNext, a rectangular,1 a rectangular FGM 2.6790FGM plate plate with with the powerthe 5.1048 power-law- lawindex index p=2 4.6375 pand=2 anddifferent different BC BCis 6.8965 considered. is considered. The The lengthlength to width to width2 ratio ratio b/a isb/a taken 2.0941 is taken from from 0.5 to0.5 5 towhile 3.9872 5 while the widththe width of 3.6267the of platethe plate a is aassumed is assumed 5.3866 to be to constant. be constant. The Thenon -nondimensional-dimensional5 critical critical 1.7712 load loads ares providedare provided 3.3705 in T inable Table 5 and 5 andpresented 3.0662 presented in Fig. in Fig.5. It 4.5523 5 can. It canbe seen be seen that that the lengththe length to10 width to width ratio ratio b/a 1.6119 b/astrongly strongly alters alters the 3.0683 criticalthe critical load load of 2.7881 the of theFGM FGM variable variable 4.1432-thickness-thickness plate. plate. Specifically,Specifically, 20 as ratio as ratio b/a b/aincrease 1.4231 increases, thes, criticalthe critical 2.7110 load load dec reases decreases 2.4598 rapidly. rapidly. Moreover, Moreover, 3.6605 Tables Tables 4, 54 ,and 5 and FiguresFigures 4, 5 4show50, 5 show that thatthe fully 1.2109the fully clamped clamped FGM FGM 2.3091 plate plate has hasthe greatestthe 2.0921 greatest critical critical force, force, 3.1180while while the criticalthe critical forceforce is smallest is 100smallest in the in casethe 1.1115case of the of fullythe fully simple simple 2.1208 support. support. This This is 1.9202easy is easy to understand to understand 2.8639because because the fullythe fully clampedclamped plate plate leads leads to an to increase an increase in the in stiffnessthe stiffness of the of FGMthe FGM plate plate. . Fig. 4. TheFig. effectFig.4. The 4. of Theeffectp on effect theof p critical ofon p on buckling Fig.Fig. 5. The Fig.5. The effect 5. Theeffect of effectb /ofa b/aratio of ratiob/a on ratio theon the criticalon criticalthe critical the criticalthe critical bucklingload buckling of load the FGMload of the plateof FGMthe FGM plate. plate. buckingbuckingbucking load load of load theof the FGM of FGMthe plate FGM plate. plate. 4.2. Effect of length to width ratio b/a Next, a rectangular FGM plate with the power-law index p = 2 and different BCs are considered. The length to width ratio b/a is taken from 0.5 to 5 while the width of the plate a is assumed to be constant. The non-dimensional critical loads are provided in

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