A novel computational approach for functionally graded isotropic and sandwich plate structures based on a rotation-free meshfree method

Thin-Walled Structures 107 (2016) 473–488

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A novel computational approach for functionally graded isotropic and sandwich plate structures based on a rotation-free meshfree method Tan N. Nguyen a,b, Chien H. Thai b,c,n, H. Nguyen-Xuan d,e,n a

Department of Civil and Environmental Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Republic of Korea Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam c Falculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam d Duy Tan University, Da Nang, Vietnam e Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, South Korea b

art ic l e i nf o

a b s t r a c t

Article history: Received 29 November 2015 Accepted 13 June 2016

This paper investigates a rotation-free moving Kriging (MK) meshfree approach for isotropic and sandwich functionally graded material (FGM) plates based on a refined plate theory (RPT). The present formulation makes certain that the tangential stress-free boundary conditions at the top and bottom surfaces of the plate are satisfied with any nonlinear distribution functions. The basic idea of computational method behind this work is to exploit a quartic spline correlation function for establishment of MK basic shape functions. Our finding yields an improved meshfree formulation whose solution becomes certainly stable and no longer depends on a correlation parameter θ . In particular, it is interesting that boundary conditions related to the slopes can be eliminated by a simple rotation-free technique as addressed in isogeometric analysis. Our method uses only four variables for each node and hence the computational cost is completely well-controlled. Several numerical examples are provided to validate high reliability of the present method in comparison with other numerical methods. The present results also show that no shear locking is observed. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Moving Kriging Meshfree method Refined plate theory (RPT) FGM plate Isogeometric analysis

1. Introduction Functionally graded materials (FGMs) play an important role in thin-walled structures. FGM is a material with mixture of two distinct material phases: ceramic and metal, has greatly attractive. FGM is fully inherited the mechanical properties of each material component. This yields the special properties for FGM such as good strength, high insulation, high fatigue resistance and good corrosion resistance. Therefore, FGM has been widely used in aerospace engineering, nuclear reactors, medical, semiconductor technologies, etc. FGM plate structures consist of isotropic and sandwich FGM plates. There are two types of sandwich FGM plate structures: the plate with FGM core and two isotropic skins, or with isotropic core and two FGM skins. Many recent researches have been conducted to analyze behavior of FGM structures such as beams [1,2], plates [3–8] and shells [9,10]. n Corresponding authors at: Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam (Chien H. Thai). Duy Tan University, Da Nang, Vietnam (H. Nguyen-Xuan). E-mail addresses: [email protected] (T.N. Nguyen), [email protected] (C.H. Thai), [email protected] (H. Nguyen-Xuan).

http://dx.doi.org/10.1016/j.tws.2016.06.011 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

In this paper, we devote an effectively novel meshfree method for static, dynamic, buckling analyses of isotropic and sandwich FGM plates. We construct the displacement field based on the background of the classical plate theory (CPT). Therefore, shear locking can be avoided in a natural way. The CPT assumptions are only available for thin plates. The first-order shear deformation theory (FSDT) is suitable for moderately thick to thin plates [11,12]. Herein, the FSDT shear stresses need to be justified by shear correction factors (SCFs) to tune for the shear energy. As known, the choice of SCFs is a problem dependent [13]. Other difficulty using FSDT in association with numerical methods is of shear locking phenomenon in the thin plate limit. Some shear locking-free techniques can be, no means exhaustive, listed here as Mixed Interpolation of Tensorial Components (MITC) [14], reduced integration [15], Mindlin-type plate element with improved transverse shear (MIN) [16], discrete shear gap (DSG) elements [17–20] and so on. Thanks to higher-order shear deformation theories (HSDTs), more comprehensive shear stress/strain through the plate thickness are obtained and traction-free condition is satisfied at top and bottom surfaces as the displacement field takes into account higher-order terms. It should be emphasized that the results derived from HSDTs are more accurate and stable than those

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of FSDT [21]. Some HSDTs can be mentioned as the third-order shear deformation theory (TSDT) [3], the trigonometric shear deformation theory [22–25], the refined plate theory (RPT) [26–29], etc. The RPT was proposed by Senthilnathan et al. [30] with two unknown components of the transverse displacement w b and w s . Shimpi et al. [31–33] and Thai et al. [34,35] developed the RPT for isotropic and orthotropic plates. The RPT has a few variable and it has no shear locking phenomenon for thin plate case. However, the C1 continuous requirement obstructs the natural use of the C0 finite element method (FEM). Such a drawback of the FEM can be overcome by other numerical methods such as a meshfree method and an isogeometric analysis. Meshfree methods have been applied successfully so far [36]. However, most of these methods have same difficulty that shape functions do not satisfy kronecker-delta property. Hence, essential boundary conditions are not directly imposed as the popular way in the FEM. Some boundary correction techniques can be listed such as Lagrange multipliers [37], penalty methods [36,38] or coupling with FEM [39–42]. Alternatively, the moving Kriging basis functions satisfy the kronecker-delta property. The MK meshfree method was first proposed for solving one dimensional steady-state heat conduction problem [43] and then for one-dimensional bar, thick and thin beams and two-dimensional plane problems [44,45]. A wide range of various applications involved thin plates [46], shell structures [47], two-dimensional piezoelectric structures [48], free vibration analysis of Kirchhoff plates [49], bending, buckling and vibration analyses of laminated composite plates [50,51]. Unfortunately, the solution obtained from MK meshfree method depends heavily on the quality of the moving Kriging interpolation which is affected by a correlation parameter θ . The finding of an ‘optimal’ value θ is very challenging [43]. In this study, we exploit a new correlation function and numerically prove that by our finding the solution of MK meshfree method is stable and no longer depends on θ . More importantly, boundary conditions related to slopes can be imposed by a simple rotation-free technique borrowed from isogeometric analysis. Various numerical examples are given to show the reliability and effectiveness of the present method. Outline of this paper is organized as follows. Next section describes the refined plate theory for FGM plates. Section 3 presents a FGM plate formulation based on moving Kriging interpolation. Results and discussions are presented in Section 4. Some conclusions are remarked in Section 5.

2. Refined plate theory for FGM plates 2.1. Problem formulation

2.1.1. Isotropic FGM plate-type A Functionally graded material (FGM) is a special composite material type. It is made from two distinct material phases and is inherited fully mechanical properties of these materials. Most significant mechanical properties of FGM are the ability of insulation, high fatigue resistance and good corrosion resistance. Two well-known homogenous models of the rule of mixture and the Mori-Tanaka techniques are used. For the mixture model, the volume fraction of each component material can be expressed as follows [3] n ⎛1 z⎞ Vc (z ) = ⎜ + ⎟ , z ∈ ⎡⎣ −h/2, h/2⎤⎦; ⎝2 h⎠

Vm = 1 − Vc

z h/ 2 -h/ 2

y x

(1)

in which, Vc , Vm are the volume fraction of ceramic and metal, respectively. It can be seen that the volume fraction of each component material not only changes continuously through the thickness but also depends on the power index n. For the effective property of FGM plate according to the rule of mixture [3], one writes

Pe = PcVc (z ) + PmVm(z )

(2)

The effective properties Pe indicated in Eq. (2) can be Young's modulus (E), Poisson's ratio (ν) and density mass (ρ) while Pc , Pm are the properties of ceramic and metal, respectively. However, the rule of mixture does not consider the interactions between the particles [7,8]. Therefore, the Mori-Tanaka technique [52] was proposed. In this model, the effective bulk and shear modulus including the above interactions can be described as

Ke − Km Vc ; = K −K Kc − Km 1 + Vm K +c 4 / 3mμ m

where f1 =

m

μm (9Km + 8μm ) 6(Km + 2μm )

μe − μm μc − μm

=

Vc μ −μ 1 + Vm μc + fm m

1

Ee =

9Keμe 3Ke + μe

;

νe =

(3)

.

The effective Young's modulus Ee and Poisson's ratio defined as

νe are

3Ke − 2μe 2(3Ke + μe )

(4)

We now consider a FGM (Al/ZrO2) plate. The effective Young's modulus is calculated from two homogenous models: the rule of mixture and the Mori-Tanaka technique with various power indices n as plotted in Fig. 2. It is observed that with inhomogeneous material (n4 0) the effective Young's modulus through the thickness from the rule of mixture is higher than that from the rest model and with homogeneous material (n¼ 0) the results from two models are identical. 2.1.2. Sandwich plate with FGM core and isotropic skins-type B Sandwich plate-type B includes three layers with the total

h

A plate is illustrated in Fig. 1. We consider three different types of FGM plate: (1) Isotropic FGM plate-type A, (2) Sandwich platetype B: FGM core, isotropic skins, (3) Sandwich plate-type C:

isotropic core, FGM skins.

ceramic metal a Fig. 1. Constitute of a typical FGM plate.

b

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475

⎛ z − z ⎞n ⎡ h⎤ Vc (z ) = ⎜ 4 ⎟ , z ∈ ⎢z3, ⎥, top skin; ⎣ 2⎦ ⎝ z4 − z3 ⎠ Vc (z ) = 1, z ∈ [z2, z3], core; ⎛ z − z ⎞n ⎡ h ⎤ 1 Vc (z ) = ⎜ ⎟ , z ∈ ⎢− , z2⎥, bottom skin; ⎣ 2 ⎦ ⎝ z2 − z1 ⎠ Vm = 1 − Vc

(6)

in which, h is total thickness of the sandwich plate. Layer thickness ratio is defined as: hb − hc − ht , where hb ,hc and ht are thickness of bottom layer, core and top layer, respectively. If this ratio is equal to 1-1-1, three layers have the same thickness.

2.2. On the refined plate theory To date, a lot of higher order shear deformation theories have been developed for FGM plates. In particular, a simple and effective theory proposed by Reddy [3] is often used and expressed as Fig. 2. The effective modulus of FGM (Al/ZrO2) plate is calculated by the rule of mixture (in solid line) and the Mori-Tanaka technique (in dash dot line).

ceramic, Ec

z4 0.1h z3

Ec FGM

0.8h

Em metal, Em

z2 0.1h z1

Fig. 3. The sandwich plate is constituted by FGM core and isotropic skins.

thickness h, as depicted in Fig. 3. The bottom layer is pure metal, the top layer is pure ceramic and FGM core is graded from metal to ceramic. Volume fraction of each component material can be defined as in Eq. (5). n

⎛1 z ⎞ Vc (z ) = ⎜ + c ⎟ ; Vm = 1 − Vc hc ⎠ ⎝2

(5)

⎛ ∂w (x, y) ⎞⎟ u(x, y , z ) = u0(x, y) + zβx(x, y) + g (z )⎜βx(x, y) + ; ⎝ ∂x ⎠ v(x, y , z ) = v0(x, y) + zβy(x, y) ⎛ ∂w (x, y) ⎞ ⎛ h h⎞ + g (z )⎜βy(x, y) + ⎟, ⎜− ≤ z ≤ ⎟; ∂y ⎠ ⎝ 2 2⎠ ⎝ w (x, y , z ) = w0(x, y)

(7)

where u0, v0, w0, βx and βy denote the displacement components in the x; y; z directions, the rotations in the-y and the-x axes, respectively. The number of variables in Eq. (7) can be reduced by the simple assumptions proposed by Senthilnathan et al. [30]:

w = w b + w s; where w ,bx =

βx = − w ,bx;

∂w b ∂x

and w ,by =

βy = − w ,by

(8)

∂w b . ∂y

Substituting Eq. (8) into Eq. (7), a refined form of HSDT is formed by

u(x, y , z ) = u0 − zw ,bx + g (z )w ,sx; v(x, y , z ) = v0 − zw ,by + g (z )w ,sy;

where hc = z3−z2 is the thickness of core and zc ∈ [z2, z3].

w (x, y , z ) = w b + w s or u¯ = u 0 + z u1 + g (z )u2

2.1.3. Sandwich plate with isotropic core and FGM skins-type C Fig. 4 shows composition of the sandwich plate-type C which has isotropic core. Two skin layers have metal-rich at surfaces z ¼z1, z ¼z4 and ceramic-rich at surfaces z¼ z2, z ¼z3. For FGM skins, the volume fraction of ceramic and metal can be expressed as [53,54]

metal ceramic

z4 ht z3

ceramic

hc

ceramic metal

z2 hb z1

Fig. 4. The sandwich plate is constituted by FGM skins and isotropic core.

(9)

where

⎧ u0 ⎫ ⎧ u⎫ ⎪ ⎪ ⎪ ⎪ v0 ¯u = ⎨ v ⎬ ; u 0 = ⎨ ⎬ ; u1 = ⎪ ⎪ ⎪ ⎪ ⎩w⎭ b s ⎩w + w ⎭

⎧−w b ⎫ ⎧w s ⎫ ,x ⎪ ⎪ ⎪ ,x ⎪ ⎪ ⎪ ⎨−w b ⎬ ; u2 = ⎨ w ,sy ⎬ ,y⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ⎭ ⎭ ⎩ 0 ⎪

(10)

The transverse shear stresses and strains through the plate thickness can be described properly by using distribution function g (z ) = f (z ) − z . The function f (z ) can be chosen so that the value of its tangential at z = ± h/2 are equal to zeros. In this work, three effective functions f (z ) are used and listed in Table 1 and are plotted in Fig. 5. The following strain components are expressed by

{

ε = εxx εyy

γxy

T

}

T

= ε 0 + z κb + g (z )κs ;

γ = { γxz γyz } = f ′(z )εs where

(11)

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Fig. 5. Distribution functions f (z) and their derivatives through the plate thickness.

⎧ ε 0 ⎫T ⎡ A B E ⎤⎧ ε 0 ⎫ ⎪ ⎪ ⎪ ⎪ δ⎨ κb ⎬ ⎢ B D F1⎥⎨ κb ⎬dΩ + ⎥⎪ ⎪ ⎢ Ω ⎪ ⎪ ⎩ κs ⎭ ⎣ E F1 H ⎦⎩ κs ⎭

Table 1. Three distribution functions and their derivatives. Model

f (z )

Reddy (TSDT) [3]

z−

Karama (ESDT) [55] Thai (ITSDT) [56]

ze

∫ f ′(z )

⎧ u 0 ⎫T ⎡ I1 I2 I4 ⎤ ⎧ u¨ 0 ⎫ ⎥⎪ ⎪ ⎪ ⎪ ⎢ + δ⎨ u1 ⎬ ⎢ I2 I3 I5 ⎥ ⎨ u¨ 1 ⎬dΩ = 0 Ω ⎪ ⎪ ⎪ ⎪ ⎩ u2 ⎭ ⎢⎣ I4 I5 I6 ⎥⎦ ⎩ u¨ 2 ⎭

1 − 4z 2/h2

4 3 2 z /h 3

−2(z / h)2

(1 −

2 h

harctan( z ) − z

∫

4 2 −2(z / h)2 z )e h2

2 h

∫Ω δ εTs D sεsdΩ

2 h

(1 − ( z )2)/(1 + ( z )2)

in which, h /2

( I1, I2, I3, I4, I5, I6) = ∫−h /2 ρ(z)( 1, z, z2, g (z), zg (z), g 2(z))dz ⎧ wb ⎫ ⎧ ws ⎫ ⎧ u0, x ⎫ , xx ⎪ , xx ⎪ ⎪ ⎪ ⎪ b ⎪ ⎪ ⎪ ⎪ ⎪ ϵ 0 = ⎨ v0, y ⎬ ; κb = − ⎨ w , yy ⎬ ; κs = ⎨ w ,syy ⎬ ; ⎪u + v ⎪ ⎪ ⎪ ⎪ ⎪ s ⎪ b ⎩ 0, y 0, x ⎭ ⎪ ⎩ 2w , xy ⎪ ⎭ ⎩ 2w , xy ⎪ ⎭

⎧ ε 0 ⎫T ⎡ A B E ⎤⎧ ε 0 ⎫ ⎪ ⎪ ⎪ ⎪ δ⎨ κb ⎬ ⎢ B D F1⎥⎨ κb ⎬dΩ + ⎥⎪ ⎪ ⎢ Ω ⎪ ⎪ ⎩ κs ⎭ ⎣ E F1 H ⎦⎩ κs ⎭

∫ (12)

A weak form for static problem subjected to a transverse uniform loading q0 is formulated as

∫

=

(17)

For buckling problem, a weak form associated with in-plane forces (axial force or bi-axial force) is formed by

s⎫ ⎧ ⎪ w, x ⎪ ⎬ ϵs = ⎨ s⎪ ⎪ ⎩ w, y ⎭

⎧ ε 0 ⎫T ⎡ A B E ⎤⎧ ε 0 ⎫ ⎪ ⎪ ⎪ ⎪ δ⎨ κb ⎬ ⎢ B D F1⎥⎨ κb ⎬dΩ + ⎥⎪ ⎪ ⎢ Ω ⎪ ⎪ ⎩ κs ⎭ ⎣ E F1 H ⎦⎩ κs ⎭

(16)

∫Ω δ εTs D sεsdΩ + ...

⎧ w b + w s ⎫T ⎡ N 0 N 0 ⎤⎧ w b + w s ⎫ ⎪ ,x xy ⎥⎪ , x ,x ⎪ ⎢ x ,x ⎪ ⎨ ⎬dΩ = 0 ⎬ h δ⎨ 0 ⎥⎪ b s⎪ Ω ⎪ w b + w s ⎪ ⎢N 0 , y ⎭ ⎣ xy N y ⎦⎩ w , y + w , y ⎭ ⎩ ,y

∫

(18)

0 in which, Nx0 , N y0 and Nxy are the pre-buckling loads corresponding with x, y and x-y directions.

∫Ω δ εTs D sεsdΩ

∫Ω δ( w b + w s)q0dΩ

(13)

3. The FGM plate formulation based on moving Kriging interpolation 3.1. An introduction to moving Kriging (MK) interpolation

where h /2

Aij , Bij , Dij , Eij, F1ij , Hij =

2

∫−h /2 (1, z, z , g (z), zg (z), g Dijs =

2

(z ))Q ijdz;

h /2

∫−h /2 [f ′(z)]2 Gijdz

(14)

Let Ω be the domain in 2 with the boundary Γ and consider an arbitrary point x in a support domain Ω x ∈ Ω that is represented by n nodes as shown in Fig. 6. According to moving Kriging approximation [43], MK interpolation function uh(x) is defined as

in which the material matrices are

uh ( x) = ⎡⎣ pT ( x) A + r T ( x)B⎤⎦u

⎤ ⎡1 ν 0 e ⎥ ⎢ ⎡ 1 0⎤ Ee Q= 0 ⎥; G = ⎢ νe 1 ⎢⎣ ⎥⎦ 2 1 ( + ν ) 1 − νe2 ⎢ e 0 1 ⎥ ⎣ 0 0 (1 − νe )/2⎦

in which p(x) is the polynomial. In this paper, the second-order polynomial is used to construct the shape function while u, uh(x) are the displacement vector and approximation value vector at point x, respectively,

Ee

(15)

For free vibration problem, a weak form can be derived from the equations of motion,

p(x) = {p1(x) p2 (x) p3(x) ...

(19)

pm (x)}T

(20)

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

477

r(x) = { R(x1, x) R(x2, x) ⋯ R(x n, x)}T

(24)

where R in Eq. (21) and Eq. (22) is defined as a correlation matrix which has size n  n as

⎡ R(x1, x1) ... R(x1, x n) ⎤ ⎥ ... ... ... R=⎢ ⎢ ⎥ ⎣ R(x n, x1) ... R(x n, x n)⎦

(25)

Another form of interpolation function can be expressed as n

uh(x) =

∑ NI (x)uI

(26)

I=1

in which the shape function can be written as

Fig. 6. Support domain of 2D problem.

m

NI (x) = Table 2. Material properties.

E (GPa) ν ρ (kg/m3)

n

∑ pj (x)AjI

+

j=1

Al

Al2O3

ZrO2-1

ZrO2-2

SiC

70 0.3 2707

380 0.3 3800

200 0.3 5700

151 0.3 3000

427 0.17 –

∑ rk(x)BkI

(27)

k=1

The first and second-order derivatives of MK shape functions are computed by m

NI, α(x) =

n

∑ pj, α (x)AjI

+

j=1 m

NI, αα(x) =

∑ rk, α(x)BkI ; k=1 n

∑ pj, αα (x)AjI

+

j=1

∑ rk, αα(x)BkI

(28)

k=1

One of the most popular correlation functions used in MK interpolation for mechanics problems is the Gaussian function expressed by [43] 2

R(x I , x J) = e−θrIJ where rIJ = ||x I−x J||

(29)

where rIJ is the Euclidean distance between points xI and xJ, the correlation parameter θ 4 0 is used to fit the underlying model. In the MK meshfree method, two parameters that affect strongly on the quality of the MK shape functions are number of nodes in the support domain and the correlation parameter θ . It is worth noting finding an ‘optimal’ θ for meshfree method [43] is challenging and is a dependent problem. In fact, with a given value of the correlation parameter θ , we need to determine a mesh node such that the Kronecker delta property is guaranteed. Conversely, for a mesh node we need to find a suitable θ so that the Kronecker delta property is satisfied. To avoid such a shortcoming, we introduce a quartic spline correlation function [69] which is similar to the weight function used in moving least square interpolation [37]:

⎡ ⎛ θrIJ ⎞2 ⎛ θrIJ ⎞3 ⎛ θrIJ ⎞4 ⎤ R(x I , x J) = c0⎢ 1 − 6⎜ ⎟ + 8⎜ ⎟ − 3⎜ ⎟ ⎥; ⎢ ⎝ a0 ⎠ ⎝ a0 ⎠ ⎝ a 0 ⎠ ⎥⎦ ⎣ Fig. 7. Couple adjacent nodes on the boundary used to impose the slope conditions.

A =

−1

( P R P) T

−1

PT R−1

B = R−1 ( I − PA)

(30)

(21)

Note that c0 can be fixed at 1 because it does not affect significantly on the shape function while a0 is a scale factor to normalize the distance. Hence, the quartic spline correlation function becomes

(22)

⎛ θrIJ ⎞2 ⎛ θrIJ ⎞3 ⎛ θrIJ ⎞4 R(x I , x J) = 1 − 6⎜ ⎟ + 8⎜ ⎟ − 3⎜ ⎟ ; ⎝ a0 ⎠ ⎝ a0 ⎠ ⎝ a0 ⎠

and P is a matrix with size n  m obtained from the polynomial as

⎡ p (x1) ... p (x1) ⎤ 1 m ... ... ⎥ P = ⎢ ... ⎥ ⎢ x ... p p ( ) (x ) ⎣ 1 n m n ⎦

⎛ ⎞ θrIJ ≤ 1⎟ ⎜0 ≤ a0 ⎝ ⎠

(23)

The term rT (x) is a vector containing n correlation functions R(x i, x) between any node xi and x point and is denoted as

⎛ ⎞ θrIJ ≤ 1⎟ ⎜0 ≤ a0 ⎝ ⎠

(31)

For computation, a0 is chosen to be the maximum distance between the computational point and the farthest node in its support domain. Using this quartic spline correlation function, we numerically prove that the moving Kriging shape functions are insensitive to the change of the correlation parameter θ . It is worthwhile emphasizing that the shape functions of MK meshfree method satisfy the kronecker delta property as that of

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z

y

h

q

b

x a (a)

(b) Fig. 8. Square plate: (a) The plate geometry; (b) Mesh with 25 nodes per side.

Table 3. ¯ of SSSS FGM (Al/Al2O3) square plate subjected to the sinusoidal load q, a/h ¼ 10 and the power index n¼ 4. The central normalized deflection w n

Model

CUF [60] MK-Gaussian 4 MK-Quartic

a

εz

≠0 0 0 0 0 0 0

TSDT [3] ESDT [55] ITSDT [56] TSDT [3] ESDT [55] ITSDT [56]

¯ w

θ=1

θ = 10

θ = 40

θ = 60

θ = 100

0.8821 #N/A #N/A #N/A 0.8801 (  0.23) 0.8806 (  0.17) 0.8806 (  0.17)

#N/A #N/A #N/A 0.8801 (  0.23) 0.8806 (  0.17) 0.8806 (  0.17)

 1.2628 (  43.16)a  22.3458 (  2433.25)  0.8379 (  5.01) 0.8800 (  0.24) 0.8806 (  0.17) 0.8806 (  0.17)

0.7832 (  11.21) 0.8073 (  8.48) 0.8052 (  8.72) 0.8800 (  0.24) 0.8806 (  0.17) 0.8806 (  0.17)

0.8750 0.8756 0.8756 0.8800 0.8806 0.8806

(  0.80) (  0.74) (  0.74) (  0.24) (  0.17) (  0.17)

θ = 1000

0.8757 (  0.72) 0.8763 (  0.66) 0.8763 (  0.66) 0.8800 (  0.24) 0.8806 (  0.17) 0.8806 (  0.17)

The error compared to the solution CUF [60] is given within parentheses (%).

Table 4. ¯ of SSSS isotropic FGM (Al/Al2O3) square plate The central normalized deflection w subjected to sinusoidal load q, a/h ¼ 10, n ¼ 4, and the reference solution CUF [60]: ¯ = 0.8821. w n

Model

MK-Quartic 4

εz

TSDT [3] ESDT [55] ITSDT [56]

0 0 0

¯ w 15  15

19  19

25  25

31  31

0.9024 0.9030 0.9030

0.8849 0.8855 0.8855

0.8801 0.8806 0.8806

0.8796 0.8802 0.8802

FEM, i.e,

ϕI (x J) = δIJ

(32)

Hence, the essential boundary condition is imposed directly and easily. Unlike FEM, the meshfree method needs a support domain to determine a set of nodes used to construct shape functions. In this work, a circle support domain is used with radius defined as

Fig. 9. Deflection convergence study of SSSS FGM (Al/Al2O3) square plate with a/ h ¼10, n¼ 4.

d m = αd c

accuracy and computational cost.

(33)

where dc is a characteristic length, being equal to the node spacing with regular distribution nodes, scale factor α , which has to be chosen such that the support domain has enough nodes to construct the shape function and α = [2 ÷ 4] for elastic problems [36]. Practically, we can be fixed at α = 2.4 for all examples for solution

3.2. A RPT formulation based on MK approximation The displacement field can be approximated by MK interpolation as

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479

Table 5. ¯ and axial stress σ¯x of SSSS FGM (Al/Al2O3) square plate subjected to sinusoidal load q. The central normalized deflection w n

Model

εz

GSDT [61] CUF [60] CUF [60] SSDT [62] HSDT [63] HSDT [63] MK

1

TSDT [3] ESDT [55] ITSDT [56]

GSDT [61] CUF [60] CUF [60] SSDT [62] HSDT [63] HSDT [63] MK

4

TSDT [3] ESDT [55] ITSDT [56]

GSDT [61] CUF [60] CUF [60] SSDT [62] HSDT [63] HSDT [63] MK

10

0 0 ≠0 ≠0 0 ≠0 0 0 0 0 0 ≠0 ≠0 0 ≠0 0 0 0 0 0 ≠0 ≠0 0 ≠0 0 0 0

TSDT[3] ESDT[55] ITSDT[56]

a/h ¼ 4

10

¯ w

σ¯x(z =

– 0.7289 0.7171 0.6997 0.7308 0.7020 0.7275 0.7262 0.7245 – 1.1673 1.1585 1.1178 1.1552 1.1108 1.1584 1.1613 1.1606 – 1.3925 1.3745 1.3490 1.3760 1.3334 1.3894 1.3891 1.3854

– 0.7856 0.6221 0.5925 0.5806 0.5911 0.5894 0.5882 0.5877 – 0.5986 0.4877 0.4404 0.4338 0.4330 0.4528 0.4481 0.4468 – 0.4345 0.1478 0.3227 0.3112 0.3097 0.3308 0.3262 0.3247

h ) 3

100

¯ w

σ¯x(z =

0.5889 0.5890 0.5875 0.5845 0.5913 0.5868 0.5880 0.5878 0.5876 0.8819 0.8828 0.8821 0.8750 0.8770 0.8700 0.8801 0.8806 0.8806 1.0089 1.0090 1.0072 0.8750 0.9952 0.9888 1.0072 1.0073 1.0069

1.4894 2.0068 1.5064 1.4945 1.4874 1.4917 1.5107 1.5100 1.5100 1.1783 1.5874 1.1971 1.1783 1.1592 1.1588 1.2001 1.1981 1.1976 0.8775 1.1807 0.8965 1.1783 0.8468 0.8462 0.8911 0.8893 0.8887

h ) 3

¯ w

σ¯x(z =

– 0.5625 0.5625 0.5624 0.5648 0.5647 0.5616 0.5616 0.5616 – 0.8286 0.8286 0.8286 0.8241 0.8240 0.8272 0.8272 0.8272 – 0.9361 0.9361 0.8286 0.9228 0.9227 0.9346 0.9346 0.9346

– 20.149 14.969 14.969 14.944 14.945 15.1767 15.1766 15.1766 – 16.047 11.923 11.932 11.737 11.737 12.1293 12.1291 12.1290 – 11.989 8.9077 11.932 8.6011 8.6010 9.0336 9.0335 9.0334

h ) 3

Fig. 10. The stresses through the thickness of SSSS FGM (Al/Al2O3) square plate with sinusoidal load, a/h ¼4, n¼1.

(34)

Bm I

⎡0 0 N ⎡N 0 0 0⎤ 0⎤ I , xx ⎥ ⎢ ⎥ ⎢ I, x b 1 NI, y 0 0⎥ ; B I = − ⎢ 0 0 NI, yy 0⎥, = ⎢0 ⎥ ⎢ ⎥ ⎢ ⎣ 0 0 2NI, xy 0⎦ ⎣ NI, y NI, x 0 0⎦

in which qI = u0I v0I wIb wIs is the vector including degrees of freedom of node I. Substituting Eq. (34) into Eq. (12), the strain components can be rewritten as

BbI 2

⎤ ⎡0 0 0 N I , xx ⎡0 0 0 N ⎤ ⎥ ⎢ I, x ⎥ = ⎢ 0 0 0 NI, yy ⎥; B sI = ⎢ ⎥ ⎢ ⎣⎢ 0 0 0 NI, y ⎥⎦ ⎣ 0 0 0 2NI, xy ⎦

n

uh(x, y) =

∑ NI (x, y)qI I=1 T

{

n

ε0 =

∑ BmI qI ; I=1

}

n

κb =

∑ BbI 1qI ; I=1

n

κs =

∑ BbI 2qI I=1

n

and εs =

∑ BsI qI I=1

(35)

Substituting Eq. (34) into Eq. (10), the displacement fields u 0 , u1 and u2 can be expressed as follows n

where

(36)

u0 =

∑ N0I qI ; I=1

where

n

u1 =

∑ N1I qI I=1

n

and u2 =

∑ N2I qI I=1

(37)

480

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

Fig. 11. The stresses through the thickness of SSSS FGM (Al/Al2O3) square plate with sinusoidal load, a/h ¼ 4 and various power indices n.

Table 6. The central normalized deflection of FGM (Al/ZrO2-1) square plate with uniform load, a/h ¼5, Mori-Tanaka model and various boundary conditions. BCs

n

HOSNDPT [64]

ceramic 0.5 1 2 4 8 metal

SFSF

SFSF

Model

0.5019 0.7543 0.8708 0.9744 – – 1.4345

MK TSDT [3]

ESDT [55]

ITSDT [56]

0.5098 0.7621 0.8793 0.9846 1.0717 1.1602 1.4563

0.5089 0.7613 0.8791 0.9845 1.0714 1.1587 1.4553

0.5088 0.7609 0.8781 0.9840 1.0703 1.1576 1.4544

SSSS CCCC

ceramic 0.5 1 2 4 8 metal

SSSS

0.1671 0.2505 0.2905 0.3280 – – 0.4775

0.1712 0.2549 0.2949 0.3326 0.3647 0.3949 0.4892

0.1710 0.2546 0.2946 0.3325 0.3646 0.3944 0.4886

0.1707 0.2542 0.2942 0.3321 0.3641 0.3937 0.4878

Fig. 12. The central normalized deflection of FGM (Al/ZrO2-1) square plate with various boundary conditions, a/h ¼ 5 and the power indices: n ¼0, 0.5, 1, 2.

the weak forms of the static, free vibration and buckling analyses are rewritten as ceramic 0.5 1 2 4 8 Metal

CCCC

N 0I

0.0731 0.1073 0.1253 0.1444 – – 0.2088

⎡N 0 0 0⎤ ⎥ ⎢ I = ⎢ 0 NI 0 0 ⎥; N1I = ⎢ 0 0 N N⎥ ⎣ I I⎦ N2I

0.0713 0.1045 0.1219 0.1404 0.1571 0.1705 0.2036

0.0708 0.1038 0.1212 0.1397 0.1562 0.1692 0.2022

0.0703 0.1032 0.1205 0.1389 0.1553 0.1680 0.2010

Kq = F

(38)

by

⎤ ⎡0 0 N I , x NI , x ⎥q = ⎢ 0 0 NI, y NI, y ⎥⎦ I I=1 ⎣ n

∑⎢

(41)

( K − λcrKg)q = 0

(42)

in which, K and F are the global stiffness matrix and the global load vector, respectively,

The derivations of the transverse displacements are also given

⎧ wb + ws⎫ ⎪ ,x ,x ⎪ ⎨ ⎬= b s ⎪ ⎩ w, y + w, y ⎪ ⎭

( K − ω M)q = 0 2

⎡ 0 0 −N 0 ⎤ I, x ⎥ ⎢ ⎢ 0 0 −NI, y 0⎥ and ⎥ ⎢ ⎣ 0 0 0 0⎦

⎡0 0 0 N ⎤ I, x ⎥ ⎢ = ⎢ 0 0 0 NI, y ⎥ ⎥ ⎢ ⎣0 0 0 0 ⎦

(40)

⎧ m⎫ ⎧ Bm ⎫T I ⎤ BI ⎪ ⎡ ⎪ ⎪ ⎪ ⎪ b1 ⎪ ⎪ ⎢ A B E ⎥⎪ ⎨ B I ⎬ B D F1 ⎨ BbI 1 ⎬ + (B sI )T D sB sI dΩ K= ⎥⎪ Ω⎪ ⎪ ⎪ ⎢⎣ b2 E F1 H ⎦⎪ Bb2 ⎪ ⎪ ⎩ I ⎭ ⎩ BI ⎪ ⎭

∫

(43)

n

∑ B IgqI I=1

F= (39)

Substituting Eqs. (35), (37) and (39) into Eqs. (13), (16) and (18),

∫Ω q0RIdΩ

where

(44)

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

a)

b)

481

c)

Fig. 13. Deflection profiles of FGM (Al/ZrO2-1) square plate with n¼4, a/h¼ 5 and various boundary conditions: (a) SSSS; (b) SFSF; (c) CCCC.

RI =

{0

T

0 NI NI }

(45)

4.1. Effect of the correlation parameter θ on the solution

and M is the global mass matrix given by

⎧ 0⎫ ⎧ N 0 ⎫T ⎡ I I I ⎤ NI ⎪ I ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 1 2 4⎥ ⎪ ⎨ N1I ⎬ ⎢ I2 I3 I5 ⎥ ⎨ N1I ⎬dΩ M= Ω⎪ ⎪ ⎢ ⎥ ⎪ 2⎪ 2 ⎪ ⎩ NI ⎪ ⎭ ⎩ NI ⎪ ⎭ ⎣ I4 I5 I6 ⎦ ⎪

∫

(46)

Kg is the geometric stiffness matrix,

Kg = h

⎡ N0 N0 ⎤ x xy ⎥ g B dΩ 0 0⎥ I ⎣ Nxy N y ⎦

∫Ω (B Ig) ⎢⎢ T

{

}

(47)

(48)

For computations, a background mesh needs to be used with 4  4 Gaussian quadrature points per cell.

4. Results and discussions Static, free vibration and buckling analyses are conducted for isotropic and sandwich FGM plates. Three distribution functions listed in Table 1 are used. Material properties are given in Table 2, except several cases directly noted. Boundary conditions (BCs) are enforced as Simply supported (S):

v0 = w b = w s = 0 b

s

u0 = w = w = 0

We consider an isotropic FGM (Al/Al2O3) square plate-type A with the length a, the thickness h and the fully simply supported boundary condition (SSSS). The plate is subjected to the sinusoidal πy πx load q = q0 sin( a )sin( a ) as shown in Fig. 8(a). The effective properties are calculated by the rule of mixture model. The central normalized displacement and axial stress are defined as

¯ = w

and ω and λcr ∈ + are known as the natural frequency and the critical buckling load, respectively. b2 From Eq. (36), it is observed that two matrices Bb1 I and BI contain the second-order derivatives of the shape function. This means that the approximate displacements require the C1-continuity. Hence, a second-order polynomial basis given in Eq. (48) is enough to establish the shape functions which fulfill the C1-continuity naturally.

pT ( x) = 1 x y x2 xy y2 , (m = 6)

boundary nodes and its adjacent nodes are assigned value zeros.

10h3Ec a4q0

⎛a b ⎞ h ⎛ a b h⎞ w ⎜ , , 0⎟ and σ¯x = σx⎜ , , ⎟ ⎝2 2 ⎠ aq0 ⎝ 2 2 3 ⎠

(51)

The plate is represented by a set of 25  25 nodes. The central normalized displacement is investigated with a wide range of θ ¼ [1C1000]. Table 3 shows the results using Gaussian and quartic spline correlation functions. It can be seen that the obtained results based on the Gaussian model are unstable, and depend strongly on the parameter θ and are even undefined in many cases θ = [1 ÷ 10]. Conversely, our results based on the quartic spline model are almost unchanged although a wide range of θ . Therefore, the correlation parameter can be fixed at 1 for all the examples. 4.2. Convergence study We now investigate the influence of the mesh nodes on the ¯ of the plate as given in Section 4.1. central normalized deflection w The same data is re-used. The obtained results are shown in Table 4 and Fig. 9. It is observed that the present solution converges to that of the Carrera's unified formulation (CUF) [60]. The error of the present solution and the reference solution is small. Table 4 and Fig. 9 show that good results are obtained with the mesh 25  25 nodes (error 0.17%), and therefore a mesh of 25  25 nodes can be used for all next examples.

on x = 0, a on y = 0, b

(49)

Clamped (C):

u0 = v0 = w b = w s = w ,bn = w ,sn = 0

(50)

The BCs for u0 , v0, w b, w s can be treated similar to the finite element method. Besides, the slopes w ,bn, w ,sn are not defined as approximation variables. Hence, they are not imposed directly as displacement variables. As an alternative technique, a simple rotation-free technique addressed in isogeometric analysis is applied [57–59]. To fulfill this simple technique, we need to design a distribution of nodes as the same as mesh construction of isogeometric analysis as shown in Fig. 7. Then, the displacements at the

4.3. Static analysis 4.3.1. Isotropic FGM plates-type A In this section, we study the central normalized deflection ¯ (z ¼0) and stress σ¯x (z¼ h/3) of the problem presented in Section w 4.1. Static analysis is conducted with various power indices n and thickness ratios. The results obtained are compared with those of generalized shear deformation theory (GSDT) [61], quasi-3D sinusoidal shear deformation theory (SSDT) [62], CUF [60], quasi-3D higher-order shear deformation theory by a meshfree technique (HSDT) [63]. Table 5 shows that the present results are close to the reference ones. As seen, the accuracy of solution is always ensured for various power indices n. The stresses through the plate thickness are shown in Figs. 10 and 11.

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T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

Distributions of the axial and shear stresses across the plate thickness based on three plate models are shown in Fig. 10. It is seen that three distributions of axial stress have the same path while those of shear stresses have a small difference. Fig. 11 shows also the good distributions of axial and shear stresses with various power indices n based on TSDT [3]. Next, static analysis is conducted for isotropic FGM (Al/ZrO2-1) square plates subjected to uniformly distributed load with various boundary conditions. Mori-Tanaka model is adopted. The central normalized deflection is defined as ¯c = (100wcEmh3) /(12(1 − ν 2)q0a4 ). The boundary conditions are inw vestigated as simply-simply-simply-simply (SSSS), simply-freesimply-free (SFSF), clamped-clamped-clamped-clamped (CCCC).

Table 7. ¯ and stress σ¯xz of SSSS sandwich square plate-type B The normalized deflection w subjected to sinusoidal load. n

Model

CUF [60] CUF [60] SSDT [62] 1 SSDT [62] HSDT [63] HSDT [63] TSDT [3] ESDT [55] MK ITSDT [56] CUF [60] CUF [60] SSDT [62] SSDT [62] 4 HSDT [63] HSDT [63] TSDT [3] ESDT [55] MK ITSDT [56] CUF [60] CUF [60] SSDT [62] SSDT [62] 10 HSDT [63] HSDT [63] TSDT [3] ESDT [55] MK ITSDT [56]

εz

0 ≠0 0 ≠0 0 ≠0 0 0 0 0 ≠0 0 ≠0 0 ≠0 0 0 0 0 ≠0 0 ≠0 0 ≠0 0 0 0

a/h ¼ 4

σ¯xz (z =

0.7735 0.7628 0.7744 0.7416 0.7746 0.7417 0.7716 0.7701 0.7684 1.0977 1.0930 1.0847 1.0391 1.0826 1.0371 1.0805 1.0866 1.0879 1.2240 1.2172 1.2212 1.1780 1.2183 1.1752 1.2143 1.2243 1.2257

0.2596 0.2604 0.2703 0.2742 0.2706 0.2745 0.2554 0.2661 0.2641 0.2400 0.2400 0.2699 0.2723 0.2671 0.2696 0.2544 0.2662 0.2668 0.1935 0.1932 0.1998 0.2016 0.1996 0.1995 0.1856 0.1953 0.1973

h ) 6

4.3.2. Sandwich plates with FGM core-type B A sandwich square plate-type B includes three layers with a total thickness h. The bottom layer is aluminum (the thickness is hb ¼ 0.1h, Em ¼ 70 GPa), the top layer is alumina (the thickness ht ¼0.1h, Ec ¼380 MPa) and the FGM core which has the volume fraction defined in Eq. (5), is graded from aluminum to alumina. The plate is subjected to sinusoidal load and has fully-simply supported at four boundaries. The normalized deflection and stress are defined as

¯ = w

10

¯ w

Table 6 and Fig. 12 show present results in comparison with those of another method using more variables such as the higherorder shear and normal deformable plate theory with 18 degrees of freedom per node (HOSNDPT) [64]. A good agreement is found for all the BCs and the power indices n. It is evident that increasing the plate stiffness, the central deflection decreases. Fig. 13 shows the deflection profiles with various boundary conditions.

¯ w

σ¯xz (z =

0.6337 0.6324 0.6356 0.6305 0.6357 0.6305 0.6326 0.6324 0.6321 0.8308 0.8307 0.8276 0.8202 0.8272 0.8199 0.8265 0.8276 0.8279 0.8743 0.8740 0.8718 0.8650 0.8712 0.8645 0.8714 0.8732 0.8735

0.2593 0.2594 0.2718 0.2788 0.2720 0.2789 0.2484 0.2518 0.2509 0.2398 0.2398 0.2726 0.2778 0.2695 0.2747 0.2397 0.2543 0.2566 0.1944 0.1944 0.2021 0.2059 0.2018 0.2034 0.1756 0.1873 0.1894

h ) 6

10h3Ec a4q0

⎛ b h⎞ ⎛a b ⎞ h w ⎜ , , 0⎟ and σ¯xz = σxz ⎜ 0, , ⎟ ⎝2 2 ⎠ aq0 ⎝ 2 6 ⎠

(52)

¯ and stress σ¯xz corTable 7 shows the normalized deflection w responding to various thickness ratios and power indices n. Several results obtained are compared with those of quasi-3D sinusoidal shear deformation theory (SSDT) [62], CUF [60] and a meshfree-based HSDT [63]. A good agreement is found. Fig. 14 draws the shear stress of SSSS sandwich square plate under the sinusoidal load with a/h ¼4, 10 and various power indices n. 4.4. Free vibration analysis 4.4.1. Isotropic FGM plates Free vibration analysis is carried out for a SSSS isotropic FGM (Al/ZrO2-1) square plate. The effective properties are calculated by the Mori-Tanaka model. The normalized frequencies ω¯ of the first mode shape and the first-ten mode shapes are shown in Tables 8 and 9, respectively. The obtained results are compared with analytical solutions [65], HOSNDPT [7] and quasi-3D solutions using SSDT and HSDT [62,63]. As expected, obtained results match well the exact values for all cases of the power index n and thickness ratio. This proves that the present method retains high effective for free vibration analysis. The first-six mode shapes of a SSSS isotropic FGM square plate are plotted in Fig. 15.

Fig. 14. The shear stress based on TSDT [3] of SSSS sandwich square plate-type B subjected to sinusoidal load, a/h ¼ 4, 10 and various power indices n.

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

483

Table 8. The first normalized frequency ω¯ = ωh ρm /Em of SSSS FGM (Al/ZrO2-1) square plate, thickness ratio a/h ¼5. Model

εz

Exact [65] HOSNDPT [7] SSDT [62] SSDT [62] HSDT [63] HSDT [63] TSDT [3] ESDT [55] MK ITSDT [56] a

≠0 0 ≠0 0 ≠0 0 0 0

n 0

0.5

1

– – – – 0.2459 0.2469 0.2463 0.2465 0.2467

– – – – 0.2219 0.2228 0.2225 0.2226 0.2228

0.2192 0.2152 0.2184 0.2193 0.2184 0.2193 0.2187 0.2188 0.2190

(  1.82)a (  0.36) (0.04) (  0.36) (0.05) (  0.23) (  0.18) (  0.09)

2

3

5

10

0.2197 0.2153 (  2.00) 0.2189 (  0.36) 0.2198 (0.04) 0.2191 (  0.27) 0.2200 (0.14) 0.2192 (  0.23) 0.2193 (  0.18) 0.2194 (  0.13)

0.2211 0.2172 (  1.76) 0.2202 (  0.41) 0.2212 (0.04) 0.2206 (  0.23) 0.2215 (0.18) 0.2206 (  0.23) 0.2206 (  0.23) 0.2208 (  0.14)

0.2225 0.2194 (  1.39) 0.2215 (  0.45) 0.2225 (0.00) 0.2220 (  0.22) 0.2230 (0.22) 0.2218 (  0.31) 0.2219 (  0.27) 0.2221 (  0.18)

– – – – 0.2219 0.2229 0.2213 0.2215 0.2217

The error (%) compared to the exact solution is given within parentheses.

Table 9. The first-ten normalized frequencies ω¯ of SSSS FGM (Al/ZrO2-1) square plate, n¼ 1. a/h

Model

Modes 1

5

10

20

Exact [65] HOSNDPT [7] TSDT [3] MK ESDT [55] ITSDT [56] Exact [65] HOSNDPT [7] SSDT [62] HSDT [63] TSDT [3] MK ESDT [55] ITSDT [56] Exact [65] HOSNDPT [7] SSDT [62] HSDT [63] TSDT [3] MK ESDT [55] ITSDT [56]

0.2192 0.2152 0.2187 0.2188 0.2190 0.0596 0.0584 0.0596 0.0596 0.0596 0.0596 0.0596 0.0153 0.0149 0.0153 0.0153 0.0153 0.0153 0.0153

(2,3)

4

5

6

7

8

9

– 0.4114 0.4108 0.4108 0.4108 – 0.1410 0.1426 0.1426 0.1422 0.1423 0.1423 – 0.0377 0.0377 0.0377 0.0377 0.0377 0.0377

– 0.4761 0.4793 0.4799 0.4806 – 0.2058 0.2058 0.2059 0.2054 0.2054 0.2054 – 0.0593 0.0596 0.0596 0.0598 0.0598 0.0598

– 0.4761 0.4793 0.4799 0.4806 – 0.2058 0.2058 0.2059 0.2054 0.2054 0.2054 – 0.0747 0.0739 0.0739 0.0733 0.0733 0.0733

– 0.5820 0.5828 0.5828 0.5828 – 0.2164 0.2193 0.2193 0.2194 0.2195 0.2197 – 0.0747 0.0739 0.0739 0.0733 0.0733 0.0733

– 0.6914 0.6972 0.6985 0.7001 – 0.2646 0.2676 0.2676 0.2646 0.2647 0.2650 – 0.0769 0.0950 0.0950 0.0951 0.0952 0.0952

– 0.8192 0.8171 0.8190 0.8202 – 0.2677 0.2676 0.2676 0.2646 0.2648 0.2650 – 0.0912 0.0950 0.0950 0.0951 0.0952 0.0952

– 0.8217 0.8171 0.8190 0.8203 – 0.2913 0.2910 0.2912 0.2914 0.2914 0.2914 – 0.0913 0.1029 0.1030 0.1027 0.1027 0.1027

Fig. 15. The first-six mode shapes of SSSS FGM (Al/ZrO2-1) square plate with n¼ 2, a/h ¼ 5.

10 – 0.8242 0.8202 0.8202 0.8212 – 0.3264 0.3363 0.3364 0.3354 0.3356 0.3360 – 0.1029 0.1029 0.1030 0.1027 0.1027 0.1027

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Table 10. The first-five normalized frequencies ω¯ of SSSS sandwich square plate, hb-hc-ht ¼ 21-2, a/h¼ 10. n

Model

Frequency

Table 12. The normalized critical load p¯cr of FGM (Al/ZrO2-2) circular plate with various of n, h/R. n

ω¯ 1

ω¯ 2

ω¯ 3

ω¯ 4

ω¯ 5

1.3018

3.1588

3.1588

4.9166

6.0405

1.3016 1.3019 1.3022

3.1484 3.1500 3.1513

3.1484 3.1500 3.1513

4.9141 4.9178 4.9209

5.3609 5.3609 5.3609

3D elasticity [66]

0.9404

2.2862

2.2862

3.5647

4.3844

TSDT [3] ESDT [55] ITSDT [56]

0.9441 0.9447 0.9451

2.2997 2.3030 2.3052

2.2997 2.3030 2.3052

3.6119 3.6198 3.6249

4.3984 4.4063 4.4063

Method

0 3D elasticity [66]

1

MK

10

MK

TSDT [3] ESDT [55] ITSDT [56]

Table 11. The first normalized frequency ω¯ of SSSS sandwich square plate with various layer thickness ratios, a/h¼ 10. n

Method

TSDT [3] a SSDT [62] a 3D elasticity [66] 0.5 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] a SSDT [62] a 3D elasticity [66] 1 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] a SSDT [62] a 3D elasticity [66] 5 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] a SSDT [62] a 3D elasticity [66] 10 TSDT [3] MK ESDT [55] ITSDT [56] a

1-0-1

2-1-2

2-1-1

1-1-1

2-2-1

1-2-1

1.4442 1.4443 1.4461 1.4460 1.4463 1.4466 1.2432 1.2433 1.2447 1.2447 1.2450 1.2453 0.9460 0.9463 0.9448 0.9471 0.9478 0.9482 0.9284 0.9288 0.9273 0.9295 0.9303 0.9309

1.4841 1.4842 1.4861 1.4859 1.4861 1.4864 1.3001 1.3002 1.3018 1.3017 1.3019 1.3022 0.9818 0.9820 0.9810 0.9830 0.9835 0.9838 0.9430 0.9433 0.9418 0.9441 0.9447 0.9451

1.5125 1.5126 1.5084 1.5082 1.5084 1.5086 1.3489 1.3489 1.3351 1.3350 1.3352 1.3354 1.0743 1.0744 1.0294 1.0318 1.0322 1.0324 1.0386 1.0455 0.9893 0.9933 0.9937 0.9939

1.5192 1.5193 1.5213 1.5210 1.5212 1.5214 1.3533 1.3534 1.3552 1.3549 1.3551 1.3553 1.0447 1.0448 1.0453 1.0459 1.0462 1.0464 0.9955 0.9952 0.9952 0.9967 0.9971 0.9973

1.5520 1.5520 1.5493 1.5490 1.5491 1.5494 1.4079 1.4079 1.3976 1.3973 1.3974 1.3976 1.1473 1.1474 1.1098 1.1103 1.1105 1.1187 1.1053 1.0415 1.0610 1.0624 1.0626 1.0628

1.5745 1.5745 1.5766 1.5764 1.5764 1.5766 1.4393 1.4393 1.4413 1.4410 1.4410 1.4411 1.1740 1.1740 1.1757 1.1754 1.1754 1.1755 1.1231 1.1346 1.1247 1.1245 1.1247 1.1247

2

10

0.2

0.25

0.3

14.0890 14.0890 14.0333 14.0368 14.0418 19.4110 19.4130 19.3360 19.3403 19.3471 23.0740 23.0750 22.9848 22.9912 22.9991 27.1330 27.1310 27.0241 27.0293 27.0374

12.5740 12.5750 12.5299 12.5423 12.5590 17.3110 17.3100 17.2488 17.2644 17.2870 20.8030 20.8050 20.7312 20.7540 20.7811 24.4230 24.4220 24.3350 24.3541 24.3817

11.6380 11.6390 11.6002 11.6182 11.6414 16.0130 16.0120 15.9602 15.9830 16.0144 19.3770 19.3780 19.3144 19.3472 19.3851 22.7250 22.7250 22.6487 22.6767 22.7155

10.6700 10.6700 10.6380 10.6616 10.6909 14.6720 14.6720 14.6280 14.6583 14.6981 17.8820 17.8810 17.8287 17.8718 17.9201 20.9480 20.9490 20.8839 20.9214 20.9710

4.5. Buckling analysis 4.5.1. Isotropic FGM plates We calculate the normalized critical load 3

p¯cr = pcr R2/Dm ,

νm2 )

Dm = Emh /12(1 − of FGM (Al/ZrO2-2) circular plate with the radius R and the thickness h. The plate is fully clamped at the boundary subjecting to uniform radial pressure p0 as shown in Fig. 16(a). The mesh is shown in Fig. 16(b). The rule of mixture is adopted as defined by Eq. (53). n ⎛1 z⎞ Pe = PcVc (z ) + PmVm(z ) where Vm(z ) = ⎜ + ⎟ , Vc = 1 − Vm ⎝2 h⎠

(53)

The obtained results are shown in Table 12. It is seen that the present solutions have a good agreement with those based on TSDT [67] and UTSDT [68]. Although various thickness ratios h/R and power indices n are assessed, the solution accuracy is always ensured. Fig. 17 also plots the first-four buckling mode shapes of FGM clamped circular plate.

Results are solved in [66].

4.4.2. Sandwich plates with FGM skins and isotropic core-type C We study the normalized frequency ω¯ of a SSSS sandwich square plate-type C. The plate is made of a pure ceramic core, FGM skins with metal - rich at top and bottom surfaces. The effective properties are calculated by the rule of mixture. Material properties are given as: Ec = 380E0, ρc = 3800ρ0 , Em = 70E0 and ρm = 2707ρ0 ,

ρ0 = 1kg/m3 and E0 = 1GPa. The normalized frequency is defined as

ω¯ =

0.5

TSDT [67] UTSDT [68] TSDT [3] ESDT [55] MK ITSDT [56] TSDT [67] UTSDT[68] TSDT [3] ESDT [55] MK ITSDT [56] TSDT [67] UTSDT [68] TSDT [3] ESDT [55] MK ITSDT [56] TSDT [67] UTSDT [68] TSDT [3] ESDT [55] MK ITSDT [56]

h/R 0.1

ωa2 h

ρ0 E0

The normalized frequencies are calculated with the ratio of layer (bottom, core, top) thickness hb-hc-ht ¼ 2-1-2 and compared with the 3D elasticity solution used Chebyshev polynomials [66], TSDT [3], SSDT [62]. It is observed from Table 10 that a very good agreement is found. Regarding various power indices n and layer thickness ratios, Table 11 also shows an excellent agreement with the reference solutions.

4.5.2. Sandwich plates with FGM skins and isotropic core-type C In the last problem, we continue to calculate the normalized critical buckling load P¯cr of SSSS sandwich square plate-type C (cf. Fig. 18) with the thickness ratio a/h¼ 10. The rule of mixture is adopted. Material properties are assumed as Ec ¼ 380E0, Em ¼ 70E0, E0 ¼1 GPa. The uni-axial and bi-axial normalized critical loads are defined as

P¯cr =

Pcra2 100h3E0

Tables 13 and 14 show the critical buckling load with two cases: uni-axial and bi-axial loads. Again, the present solutions are in very good agreement with all the reference solutions. It is observed that when the power index increases, the critical buckling load decreases.

5. Conclusions This paper has for the first time investigated an improved MK

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

485

y

p0

R

x

(a)

(b)

Fig. 16. The clamped circular plate: (a) Geometry; (b) Mesh used to discrete the plate.

Fig. 17. The first-four buckling mode shapes of FGM clamped circular plate with h/R ¼ 0.1, n¼ 2.

Ny y

a

x

Nx

a

y

a

a

(a)

(b)

x

Nx

Fig. 18. The SSSS sandwich square plates are subjected to uni-axial and bi-axial compressions. (a). Uni-axial compression; (b). Bi-axial compression.

meshfree formulation for FGM isotropic and sandwich plates. The basic idea is to replace the Gaussian correlation function by a quartic spline correlation function in the moving Kriging shape functions. As a simple yet efficient way, we confirmed by numerical

examples that the present approach is stable and no longer depends on the correlation parameter θ . Other interesting point is to bring a simple rotation-free technique from isogeometric analysis back the meshfree method for imposing boundary conditions of the slopes.

486

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

Table 13. The uni-axial critical buckling load of SSSS sandwich square plate-type C with a/h ¼ 10. n

0

1

5

10

Theory

P¯cr

TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼0 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼0 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼0 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼0 TSDT [3] MK ESDT [55] ITSDT [56]

1-0-1

2-1-2

2-1-1

1-1-1

2-2-1

1-2-1

13.0050 13.0061 12.9529 13.0051 13.0164 13.0209 13.0271 5.1671 5.1685 5.0614 5.0785 5.1721 5.1751 5.1776 2.6582 2.6601 2.6365 2.6468 2.6608 2.6647 2.6674 2.4873 2.4893 2.4722 2.4822 2.4896 2.4940 2.4972

13.0050 13.0061 12.9529 13.0051 13.0164 13.0209 13.0271 5.8401 5.8412 5.7114 5.7302 5.8456 5.8483 5.8506 3.0426 3.0441 3.0079 3.0187 3.0456 3.0487 3.0507 2.7463 2.7484 2.7205 2.7308 2.7490 2.7525 2.7548

13.0050 13.0061 12.9529 13.0051 13.0164 13.0209 13.0271 6.1939 6.1946 6.0547 6.0736 6.1999 6.2017 6.2036 3.4035 3.4045 3.3626 3.3720 3.4070 3.4091 3.4106 3.0919 3.1344 3.0607 3.0694 3.0951 3.0974 3.0988

13.0050 13.0061 12.9529 13.0051 13.0164 13.0209 13.0271 6.4647 6.4654 6.3150 6.3356 6.4709 6.4727 6.4744 3.5796 3.5806 3.5301 3.5415 3.5831 3.5854 3.5867 3.1947 3.1946 3.1576 3.1684 3.2021 3.2005 3.2021

13.0050 13.0061 12.9529 13.0051 13.0164 13.0209 13.0271 6.9494 6.9498 6.7841 6.8055 6.9561 6.9574 6.9591 4.1121 4.1129 4.0507 4.0616 4.1163 4.1181 4.1194 3.7075 3.1457 3.6617 3.6715 3.7114 3.7134 3.7148

13.0050 13.0061 12.9529 13.0051 13.0164 13.0209 13.0271 7.5066 7.5063 7.3200 7.3437 7.5136 7.5136 7.5146 4.7347 4.7349 4.6470 4.6606 4.7393 4.7398 4.7402 4.2799 4.3818 4.2055 4.2179 4.2841 4.2850 4.2856

Table 14. The bi-axial critical buckling load of SSSS sandwich square plate- type C with a/h ¼10. n

0

1

5

10

Theory

TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼ 0 TSDT [3] MK ESDT [55] ITSDT[56] TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼ 0 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼ 0 TSDT [3] MK ESDT [55] ITSDT [56] TSDT [3] SSDT [54] HSDT [63] εz≠0 HSDT [63] εz ¼ 0 TSDT [3] MK ESDT [55] ITSDT [56]

P¯cr 1-0-1

2-1-2

2-1-1

1-1-1

2-2-1

1-2-1

6.5025 6.5030 6.4764 6.5025 6.5092 6.5113 6.5144 2.5836 2.5842 2.5307 2.5392 2.5863 2.5878 2.5890 1.3291 1.3300 1.3183 1.3234 1.3305 1.3325 1.3338 1.2436 1.2448 1.2361 1.2411 1.2449 1.2471 1.2487

6.5025 6.5030 6.4764 6.5025 6.5092 6.5113 6.5144 2.9200 2.9206 2.8557 2.8651 2.9231 2.9244 2.9256 1.5213 1.5220 1.5040 1.5093 1.5229 1.5244 1.5254 1.3732 1.3742 1.3602 1.3654 1.3746 1.3764 1.3775

6.5025 6.5030 6.4764 6.5025 6.5092 6.5113 6.5144 3.0970 3.0973 3.0273 3.0368 3.1003 3.1012 3.1021 1.7018 1.7022 1.6813 1.6860 1.7036 1.7047 1.7054 1.5460 1.5672 1.5303 1.5347 1.5477 1.5488 1.5495

6.5025 6.5030 6.4764 6.5025 6.5092 6.5113 6.5144 3.2324 3.2327 3.1575 3.1678 3.2358 3.2366 3.2375 1.7898 1.7903 1.7650 1.7707 1.7917 1.7928 1.7935 1.5974 1.5973 1.5788 1.5842 1.5991 1.6003 1.6011

6.5025 6.5030 6.4764 6.5025 6.5092 6.5113 6.5144 3.4747 3.4749 3.3920 3.4027 3.4784 3.4790 3.4799 2.0561 2.0564 2.0254 2.0308 2.0583 2.0592 2.0598 1.8538 1.5729 1.8308 1.8358 1.8558 1.8568 1.8575

6.5025 6.5030 6.4764 6.5025 6.5092 6.5113 6.5144 3.7533 3.7531 3.6600 3.6718 3.7572 3.7572 3.7577 2.3673 2.3674 2.3235 2.3303 2.3698 2.3701 2.3703 2.1400 2.1909 2.1028 2.1090 2.1422 2.1427 2.1429

We only used four variables per node for the plate formulation and thus no high computational cost is required. The numerical results are provided to demonstrate the reliability and efficiency of the

present method for static, free vibration, buckling analyses of isotropic and sandwich FGM plates. Finally, the present formulation is immune from shear locking in the thin plate limit.

T.N. Nguyen et al. / Thin-Walled Structures 107 (2016) 473–488

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