A polygonal finite element method for laminated composite plates

Accepted Manuscript

A polygonal finite element method for laminated composite plates Nam V. Nguyen , Hoang X. Nguyen , Duc-Huynh Phan , H. Nguyen-Xuan PII: DOI: Reference:

S0020-7403(17)31748-4 10.1016/j.ijmecsci.2017.09.032 MS 3943

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

28 June 2017 1 September 2017 19 September 2017

Please cite this article as: Nam V. Nguyen , Hoang X. Nguyen , Duc-Huynh Phan , H. Nguyen-Xuan , A polygonal finite element method for laminated composite plates, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.09.032

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ACCEPTED MANUSCRIPT Highlight 

A polygonal finite element method (PFEM) based on C0-type higher-order shear deformation theory (C0-HSDT) is proposed for static and free vibration analyses of laminated composite plates.



A piecewise-linear shape function which is constructed based on sub-triangles of polygonal element is considered. A simple numerical integration over polygonal elements is devised.

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Shear locking is addressed by a simple Timoshenko’s beam model.

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The numerical results show the efficiency and reliability of the present approach.

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ACCEPTED MANUSCRIPT

A polygonal finite element method for laminated composite plates Nam V. Nguyen1, Hoang X. Nguyen2, Duc-Huynh Phan3, H. Nguyen-Xuan4,5* 1

Faculty of Mechanical Technology, Industrial University of Ho Chi Minh City, Vietnam

2

Faculty of Engineering and Environment, Northumbria University, Newcastle upon Tyne NE1 8ST, United Kingdom

Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, Vietnam 4

5

Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

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3

Department of Architectural Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, Seoul 143-747, South Korea

Abstract

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In this study, a polygonal finite element method (PFEM) is extended and combined with the C0-type higher-order shear deformation theory (C0-HSDT) for the static and free vibration analyses of laminated composite plates. Only the piecewise-linear shape function which is constructed based on sub-triangles of polygonal element is considered. By using the analogous technique which relies on the sub-triangles to calculate numerical integration over

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polygonal elements, the procedure becomes remarkably efficient. The assumption of strain field along sides of polygons being interpolated based on Timoshenko’s beam leads to the

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fact that the shear locking phenomenon can be naturally avoided. In addition, the C0-HSDT theory, in which two additional variables are included in the displacement field, significantly

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improves the accuracy of the displacements and transverse shear stresses. Numerical examples are provided to illustrate the efficiency and reliability of the proposed approach.

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Keywords: Polygonal finite element method, laminated composite plates, shear locking,

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Wachspress, piecewise-linear shape function.

*

Corresponding author. Email address: [email protected] (H. Nguyen-Xuan).

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1 Introduction Thanks to outstanding engineering properties such as high strength, lightweight, strength-to-weight ratios, long fatigue life etc., laminated composite materials have been extensively applied in various fields of engineering including aerospace, automotive, civil, biomedical and other areas. As a result, numerous analysis models have been developed in order to study their mechanical behaviors under different loading conditions. In general, the

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laminated composite plate theories can be classified into the following categories: the threedimensional (3D) elasticity model [1-4] and the two-dimensional (2D) model such as equivalent single-layer (ESL) theories [5], layer-wise theories [6, 7], zigzag theories [8] and quasi-3D theories [9,10]. However, 3D solutions may not be feasible when solving practical problems due to its complex geometries, arbitrary boundary conditions and high

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computational cost. Consequently, various ESL plate theories have been devised and widely used in computational mechanics to predict the behaviors of plate structures [11-21]. In the ESL plate theories, the classical laminated plate theory (CLPT) which developed based on the Kirchhoff-Love assumptions is the simplest theory. However, as it neglects the

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effects of transverse shear, this theory provides acceptable solutions for thin plates only. In order to overcome this shortcoming, the first-order shear deformation theory (FSDT) based

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on Reissner–Mindlin theory [11, 12], which accounts for transverse shear effects, has been developed applicable for both thin and moderately thick laminated composite plates.

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Nevertheless, this theory requires an appropriate shear correction factor (SCF) to accurately predict the distribution of shear strain/stress along the plate thickness satisfying the tractionfree boundary conditions at the top and bottom surfaces of plate. Therefore, the accuracy of

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solutions based on FSDT theory will be strongly depended on the accuracy of the SCF. Unfortunately, the values of SCF are not trivial to calculate as it depends on types of loadings,

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geometric parameters, material coefficients and arbitrary boundary conditions of the problems. Therefore, a large number of significant higher-order shear deformation theories (HSDTs) have been proposed to surmount the limitations in CLPT and FSDT such as thirdorder shear deformation theory (TSDT) [13], refined plate theories (RPT) [14], trigonometric shear deformation theory (TrSDT) [15,16], exponential shear deformation theory (ESDT) [17,18], hyperbolic shear deformation theory (HSDT) [19-21]. However, these theories require the C1-continuity of the generalized displacement field which is not easy to derive the second-order derivative of deflection. This is really challenging in the framework of

ACCEPTED MANUSCRIPT traditional finite element analysis. In an effort to overcome this drawback, Shankara and Iyengar [22] proposed the C0-continuity of the generalized displacements (C0-HSDT) which two unknown terms are added to the displacement field. Therefore, only the first derivative of deflection is considered in this model. To the best of author’s knowledge, although various theories have been developed for the purpose of improving the quality of the numerical results, it seems consensual that almost

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existing techniques rely on typical triangular or quadrilateral meshes. So, in the last few decades, developing the generalizations of FEM based on arbitrary polygonal mesh has gained increasing attention of many researchers in computational solid mechanics. A polygonal element with an arbitrary number of nodes is able to provide greater flexibility, suitable in complex microstructures modeling, well-suited for material design and sometimes

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more accurate and robust results [23]. In recent years, polygonal finite elements have been widely implemented in mechanics problems such as nonlinear constitutive modeling of polycrystalline materials [24-26], nonlinear elastic materials [23,27], incompressible fluid flow [28,29], crack modeling [30,31], limit analysis [32], topology optimization [33-35], contact-impact problem [36], Reissner-Mindlin plate analysis [37] and so on. However, as far

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as authors are aware, analysis of laminated composite plate based on arbitrary polygonal meshes has not been found yet. Therefore, the goal of this study is to present a unified

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formulation which applies to arbitrary polygonal mesh including triangles and quadrilaterals associated with the C0-HSDT model for static and free vibration analyses of laminated

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composite plates.

Due to the complexity of the general convex polygonal elements in comparison with

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traditional finite elements, the construction of shape functions over arbitrary polygons is almost different from those of standard triangular or quadrilateral elements. In the literature, there are numerous approaches have been presented for the determination of polygonal shape

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functions. Among them, generalized barycentric coordinates have been widely used in computational solid mechanics in recent years. Wachspress [38] pioneered to develop rational polynomial interpolation functions over planar convex polygonal domain, which satisfy the Kronecker delta and reproducing properties. After that, Warren [39] further developed rational basis functions for arbitrary convex polytopes (3D) which Meyer et al. [40] then extended for irregular polygons. It is worthwhile noting that Wachspress’ coordinates are not well-defined for non-convex. Hence, Floater [41] introduced a method based on mean value coordinates with an ability to interpolate for both convex and non-convex polygonal domains.

ACCEPTED MANUSCRIPT Moreover, several other methods have been proposed, including metric coordinates by Malsch et al. [42, 43], maximum entropy coordinates by Sukumar [44], natural neighbor (Laplace) based on the natural neighbors Galerkin method by Sukumar and Tabarraei [45, 46], moving least squares coordinates [47], etc. Sukumar and Malsch [48] has presented an outline the construction of polygonal shapes functions. In addition to the aforementioned works, the sharp upper and lower bound piecewise-linear functions that satisfy the defining properties of barycentric coordinates have been reported by Floater et al. [49]. Accordingly,

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the piecewise-linear shape functions are defined based on these sub-triangles of polygonal element. In order to appreciate numerical integration over the polygonal elements, the same technique also based on that sub-triangles proposed by Sukumar [46]. As a result, there is consistent between the construction shape functions and the evaluation integration over

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polygonal element, producing a remarkable efficiency in numerical computation.

Another important issue is how to eliminate shear locking phenomenon when the plate becomes progressively thinner. In order to address this deficiency, many approaches have been introduced and assessed for triangular and quadrilateral elements including reduced integration [50], selective reduced integration [51], assumed natural strains (ANS) [52-54],

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the discrete Kirchhoff methods [55, 56], etc. In recent years, based on the Timoshenko’s beam formulas, various plate elements have been developed in order to analyze both thin and

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moderately thick plates. Accordingly, Ibrahimbegovic [57, 58] employed the Timoshenko’s beam formulas to develop two quadrilateral thin-thick plate elements PQ2 (quadratic displacement field) and PQ3 (cubic displacement field) based on the mixed interpolation

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method. With a similar approach, Wanji and Cheung derived a refined triangular Mindlin plate element [59] and refined quadrilateral Mindlin plate element [60] for linear analysis of

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thin and thick plates. In addition, Soh et al. have developed two thin to moderately thick plate elements including ARS-T9 [61] and ARS-Q12 [62], which applied the Timoshenko’s beam

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formulas along each edge of the plate element. This technique has recently been extended and applied to Reissner-Mindlin plate [63, 64] and laminated composite plate [65, 66] which rely on triangular and quadrilateral elements. Based on the ideas of Soh et al. [61] and Cen et al. [65], a unified formulation for both thin and moderately thick plate elements based on arbitrary polygonal meshes was coined [37]. Therefore, in this study, it is further developed to analyze the static and free vibration behaviors of laminated composite plates on arbitrary polygonal meshes.

ACCEPTED MANUSCRIPT The outline of this study is organized as follows. The next section presents a brief review of the C0-HSDT type and a weak form of governing equations for laminated composite plate for static and free vibration problems. Section 3 focuses on the formulation of the PFEM for laminated composite plate with barycentric coordinates. Section 4 details the technique which is able to overcome the shear locking phenomenon in PFEM based on the Timoshenko’s beam formulas. The numerical examples which cover static and free vibration analysis of given in Section 6.

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laminated composite plates are presented in Section 5. Finally, some concluding remarks are

2 C0-type higher-order shear deformation plate theory and weak form for laminated composite plates

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2.1 C0-type higher-order shear deformation plate theory

Considering a laminated composite plate consisting of nl orthotropic layers with uniform thickness h and the fiber orientation  of each layer, its coordinate system is shown in Fig.1. According to the C0-HSDT model [22], the displacement field at an arbitrary point in the plate can be defined as follows:

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u ( x, y, z )  u0  z  x  cz 3   x  x  ,

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v( x, y, z )  v0  z  y  cz 3   y   y  ,

 h   z   2

h , 2

(1)

w( x, y, z )  w,

where u0  u0 , v0  and w are the membrane displacements and the transverse displacement

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T

of a point in the mid-plane, respectively; β   x ,  y  are the rotations of the normal to the

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T

mid-plane around the y- and x-axes, respectively; and c  4 / 3h2 . It is worth commenting

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that, Eq. (1) is devised from the higher-order theory by Reddy [13], in which, derivative of deflection is replaced by warping function   x , y  . Thus, the generalized displacement T

vector with 5 degrees of freedom for C1-continuity element can be transformed to a vector with 7 degrees of freedom for C0-continuity element as: u  u0 , v0 , w,  x ,  y , x ,  y  . T

The in-plane vector of Green–Lagrange strain at any point in a plate can be expressed as

 ,  x

,  xy   ε0  zκ1  z 3κ2 , T

y

where the membrane strains ε0 and the bending strains κ1 , κ2 are, respectively, given by

(2)

ACCEPTED MANUSCRIPT   x    x         y κ1   , y      y   x  x   y  

 u0     x   v0  ε0   ,  y    u0 v0      y x 

   x x    x x           y y κ2  c   , y y      y   y   x  x  x y x   y  

(3)

and the shear strains can be given as γ   xz ,  yz   εs  z 2 κs ,

where   x  x  κ s  3c  ,  y  y 

(4)

(5)

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w   x    x   εs   ,   y  w  y    

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T

By performing the transformation rule between the local and the global coordinate system as in Fig.1b, the constitutive equation, which based on Hooke’s law, of a kth orthotropic layer in global coordinate system xOy, are given by

Q12

Q16

0

Q 22

Q 26

0

Q 26

Q 66

0

0

0

Q55

0

0

Q 54

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k

 x   Q11     y  Q12     xy   Q16     xz   0    0  yz  

0   0   0  Q54   Q 44 

k

k

 x     y     xy  ,    xz     yz 

(6)

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where Qij  i, j  1, 2, 4,5,6  are the transformed material constants of the kth orthotropic

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layer with respect to the global x- and y-axes [5]. 2.2 Weak form equations for laminated composite plates

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In this study, the weak forms of the static and free vibration problems are derived by

applying the Hamilton principal and conducting integration by parts. Firstly, the weak form of static analysis of the laminated composite plates under transverse loading can be briefly expressed as





 εTp D*ε p d     γ T Ds*γd     wqd ,

(7)

in which q is distributing transverse load applied on the plate and strain components ε p and γ are expressed by

ACCEPTED MANUSCRIPT ε p  ε0 , κ1 , κ2  ,

γ  εs , κs  ,

T

T

(8)

*

and the material constant matrices D* and Ds can be expressed by A D   B  E

E F  , H 

B

*

D F

 As D  s B

Bs  , Ds 

* s

(9)

in which

 A , B , D , E , F , H    1, z, z , z , z , z  Q dz,  A , B , D    1, z , z  Q dz, ij

ij

s ij

s ij

s ij

ij

ij

h /2

ij

2

3

4

2

ij

4

ij

 h /2

 i, j  1, 2, 6  ,

6

 h /2

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h /2

ij

 i, j  4,5 ,

(10)

The weak form of free vibration analysis of the laminated composite plates is of the compact form T

 εTp D*ε p d     γ T Ds*γd     u mud , 

where the mass matrix m is given as

0

I2

0 I1

0 0 I3

c / 3I 4

I2 0 0

0 0 c / 3I 5

I3

0 c / 9I7

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2

in which  I1 , I 2 , I 3 , I 4 , I 5 , I 7   

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0

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0  I1  I1    m    sym. 

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

h /2

 h /2

 c / 3I 4  0   0 , c / 3I 5   0  c 2 / 9 I 7 

(11)

0

(12)

 1, z, z 2 , z 3 , z 4 , z 6 dz .

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3 A polygonal finite element method for laminated composite plates

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3.1 Shape functions on arbitrary polygonal elements In PFEM, a given domain is discretized into polygonal elements with arbitrary number of

edges. Then, the interpolation functions are constructed over each polygonal element. In the literature, various approaches have been developed for the determination of the interpolation functions on arbitrary convex polygons [69-75]. Among them, Wachspress [38, 39, 40, 67], mean-value [41, 68] and Laplace [46] shape functions are widely applied to construct the interpolation functions. In addition, Floater et al. [49] used sharp upper and lower bound piecewise linear functions in order to show all barycentric coordinates which are continuous

ACCEPTED MANUSCRIPT in its interior, as shown in Fig. 2a and Fig. 2b. These shape functions are ith barycentric coordinates and also satisfy the properties of barycentric coordinates including non-negative, partition of unity, Kronecker-delta, and linear precision. With less computational effort than Wachspress shape function and requires only three integration points for each sub-triangle of polygonal element, only the piecewise linear shape function [37] will be used in this study. Accordingly, each polygonal element with n nodes is

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subdivided into n sub-triangles in a non-overlapping and no-gap manner. The sub-triangles are created by simply connecting the centroid of the polygonal element to two end points of the edges as shown in Fig. 2c. In this case, the number of sub-triangles are the same as the number of nodes of polygonal element. Firstly, the shape functions at the vertices of polygonal elements satisfy the Kronecker-delta property:

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1 xI  x J 0 x I  x J

IPL  x J    IJ  

(13)

Next, the shape functions at the centroid x take the average value such that

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IPL  x   1/ n

(14)

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Last but not least, the shape functions and their derivatives over sub-triangles, i.e. Gauss points, can then be easily constructed by using conventional FE shape functions of triangular

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elements as follows

3

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IPL ( x )  JT3 ( x )IPL ( xJ ) for x   T3

(15)

J 1

3

IPL ( x )   JT3 ( x )IPL ( xJ ) for x   T3

(16)

J 1

in which JT3 ( x ) and JT3 ( x ) denote three-node triangular shape functions and their derivatives on a sub-triangle T3 ; IPL ( xJ ) are the shape functions of node I at the node J of

 T3 . Fig. 3 illustrates the comparison of different shape functions of four regular polygonal elements.

ACCEPTED MANUSCRIPT In addition, it should be noted that, the numerical integration is required to perform over the polygonal element for evaluating the integrals. Nonetheless, to our knowledge, the standard quadrature rule over arbitrary polygonal element with number of nodes n  4 is not available. There are several approaches for numerical integration are proposed such as Natarajan et al. [76] based on the Schwarz-Christoffel conformal mapping, Chin et al. [77] based on Lasserre’s method and Mousavi et al. [78] based on an optimization algorithm. Sukumar [46] proposed simple but perhaps efficient method, which based on these sub-

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triangles of polygonal elements, and is often used in order to evaluate the integrals over polygonal element. Fortunately, constructing a piecewise-linear shape function also based on these sub-triangles, which lead to consistent numerical integration with the shape functions and their derivatives.

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3.2 PFEM formulation for laminated composite plates

The bounded domain  is discretized into a set  , in which  including nonoverlapping polygonal elements. The set  has ne elements and nn nodes, such that e   h   e1 e . Let  be the nodal basis (shape functions) of polygonal element  . ne

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Herein, the finite element solution ue ( x ) of the displacement model for laminated composite

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plates can be expressed as nne

nne

I

I

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ue (x)  Ie I 7 d Ie  Ie d Ie , in  e ,

(17)

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where nne is the number of vertices of the polygonal element, I 7 is the unit matrix of 7th rank,

d Ie  uI , vI , wI ,  xI ,  yI , xI , yI  denotes the displacement vector of the nodal degrees of T

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freedom of ue (x) associated with the Ith vertex of the polygonal element; Ie is the shape function at the Ith vertex of polygonal element. Substituting Eq. (17) into Eqs. (3) and (5), the membrane, bending and shear strains in Eq.

(8) can be rewritten in compact forms as

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nne

nne

ε0e   BIm.e d Ie , I

nne

ε  B e s

s 0.e I

nne

κ1e   BIb1.e d Ie ,

κ 2e   BIb 2.e d Ie ,

I

nne

κ  B

e I

e s

d ,

I

s1.e I

I

(18)

e I

d ,

I

in which

B

I , x   0 I , y 

BIb1.e

 0 0 0 I , x   0 0 0 0  0 0 0 I , y 

0 0 0 0 0  0 0 0 0 0 , 0 0 0 0 0 

0

I , y I , x

B

I

0

0

I

0 0 , 0 0 

0 0 I , x 0 0

I , x

0

0 0 I , y 0 0 I

0

I

0 0

I

0

0

0   I , y  , I , x 

I , y 0 I , x I , y

0

(19)

0 , I 

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BIs1.e

0 c  0 3 0  0 c 0

I , y I , x

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b 2.e I

0 0  0 0 , 0 0 

0

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 0 0 I , x BIs 0.e    0 0 I , y

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m .e I

respectively.

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where I , x and I , y are the derivatives of the shape functions I with respect to x and y,

Now, substituting Eq. (18) into Eq. (7), the governing algebraic equations of the

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form

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laminated composite plate using in PFEM for static analysis can be obtained in the following

Kd = F ,

(20)

where K is the global stiffness matrix assembled from the element stiffness matrices K e , which can be computed as

Ke  

e

 B  D B d     S  D S d , e I

T

*

e I

in which the matrices BIe and S Ie are defined by

e

e I

T

* s

e I

(21)

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BIe   BIm.e

BIb1.e

SIe   BIs 0.e

T

BIb 2.e  ,

(22)

T

BIs1.e  ,

(23)

and F is the global load vector expressed by



(24)

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F   q d  f b ,

in which f b is the complementary term of F subjected to prescribed boundary loads. For the free vibration analysis problem, finite element formulation yields

 K   M  d  0,

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2

(25)

where  is the natural frequency; and M denotes the global mass matrix, which can be expressed as follows

M    T m d.

(26)

M



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It is known that the shear locking phenomenon will appear in the limit of thin plates. To avoid this shortcoming, a shear locking free polygonal laminated composite plate element approach based on an assumed strain field via the Timoshenko’s beam formulae is proposed,

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given in Section 4.

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4 A shear locking-free polygonal laminated composite plate element

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4.1 Locking-free Timoshenko’s laminated composite beam element Consider a Timoshenko’s thick laminated composite beam element as shown in Fig. 4.

The formulas of deflection w( ), rotation  ( ) and shear strain  ( ) of the thick laminated composite beam element which based on the Timoshenko’s beam theory are as follows [61, 62]

ACCEPTED MANUSCRIPT

l l  i   j   1      1  2   1   1  2  ,  2 2     i 1      j  3 1  2   1    , w    wi 1     w j 

(27)

    ,

in which 2  wi  wj   i   j , l

6 , 1  12



=

Dlb , Dls l 2

(28)

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

with Dlb , Dls and l being bending, shear stiffness constants and length of the beam element, respectively. Now, this Timoshenko’s laminated composite beam theory is extended for plate

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polygonal element. In general, the edge ˆjkˆ of a polygonal element with end nodes ˆjth and kˆth is considered, as shown in Fig. 5. An assumed rotation and shear strains along the edge ˆjkˆ of polygonal element are determined by Eqs. (27). Herein, the bending and shear stiffness

constants of edge ˆjkˆ can be rewritten by the matching quantities of the plate element as



1 nl  Q11 k  3 k 1

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Dbˆjkˆ 

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follows [65]

nl



k 1

CE

AC

where

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D ˆsjkˆ   Q11 k 

 Q    Q   k 11 k 55

 h

3 k

ˆjkˆ

 h



ˆjkˆ

k

 hk31  ,

(29)

 hk 1  ,

(30)



ˆjkˆ

 Q11 k ck4,ˆjkˆ  2 Q12 k   2Q66 k  ck2,ˆjkˆ sk2,ˆjkˆ  Q22 k  sk4,ˆjkˆ ,

ˆjkˆ

 Q55 k cˆjk2ˆ  Q44 k  sˆjk2ˆ ,





ck ,ˆjkˆ  cos  k   ˆjkˆ , cˆjkˆ  cos  ˆjkˆ ,





sk ,ˆjkˆ  sin  k   ˆjkˆ ,

sˆjkˆ  sin ˆjkˆ ,

(31)

ACCEPTED MANUSCRIPT in which  k denotes the angle between the x-axis and the fiber direction 1-axis of the kth orthotropic layer;  ˆ ˆ is the angle between the x-axis and the edge ˆjkˆ of a polygonal element, jk as shown in Fig. 6; Qij  i, j  1,2,4,5,6  are the material constants of kth orthotropic layer [5]. It should be noted that when the thickness h of the plate approaches zero,  in Eq. b (28) will tend to zeros ( lim   12 lim Dls  0 ). Therefore,  will also approach zero. As a

l

h 0

Dl

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h 0

result, the transverse shear strain  ( ) will be eliminated automatically. Consequently, shear locking issue of the interpolation can be suppressed based on the Timoshenko’s laminated composite beam theory.

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4.2 A novel PFEM formulation based on C0-HSDT type

Now, consider a laminated composite plate polygonal element  e with n nodes,  n  3 as shown in Fig. 5, the generalized nodal displacement vector of the polygonal element is , dn  , T

(32)

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d e  d1 , d 2 ,

in which di  ui , vi , wi ,  xi ,  yi , xi ,  yi  with  i  1, 2,..., n  . Using the similar technique [61,

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T

62], Nguyen-Xuan [37] constructed the interpolation procedure for shear and bending strain fields along the polygonal element edges are obtained based on Timoshenko’s beam theory.

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Accordingly, the assumed bending and shear strains of the polygonal element can be written

ε ep  Bed e   B m.e , B b1.e , B b2.e  d  , e

(33)

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in the matrix form as follows

in which Bb1.e  B1b1.e  Bˆ b1.e with Bˆ b1.e   H b   I b  G  . And

 e  S ed e   S s 0.e , B s1.e  d  , e

(34)

where B s 0.e   H s   I s  G  . The matrices Bm.e , Bb1.e , B b2.e , B s1.e are presented in Eq. (19). In b b s addition, the matrices G, H , I , H and I s are expressed by, respectively

ACCEPTED MANUSCRIPT

G n,7 n   2iˆ,7 ˆj 4 , ciˆ,7 ˆj 3 , biˆ,7 ˆj    2iˆ,7 kˆ 4 , ciˆ,7 kˆ 3 , biˆ,7 kˆ , iˆ , ˆj

(35)

     ,   kˆ  ˆj      ˆj kˆ   biˆ  biˆ  ciˆ  kˆ     ˆj   ciˆ  y  x  y  x      

(36)

 I b    iiˆˆ 1  2 iˆ  , nn

(37)

kˆ  3ciˆ   ˆj   ˆ   ˆ liˆ2  x k x j  kˆ  3biˆ   ˆj   ˆ  ˆ 2  liˆ  y k y j 

b ˆjkˆ   bmˆ  ˆj     ciˆbmˆ  cmˆ biˆ c ˆj biˆ  ciˆb ˆj     , cmˆ  ˆj c ˆjkˆ  ˆ iˆ , ˆj , kˆ , m     ciˆbmˆ  cmˆ biˆ c ˆj biˆ  ciˆb ˆj   

(38)

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 H s  2n

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       iˆ , ˆj , kˆ   3  liˆ2 

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 H b  3n

iˆ , kˆ

 I s      , nn  iiˆˆ iˆ 

ED

(39)

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in which liˆ  x ˆj  xkˆ denotes the length of the iˆth edge and  iˆ is presented in Eq. (28) and

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(29); iˆe is the shape function at the iˆth node of polygonal element  e and

b  y

ˆj

 ykˆ , ciˆ  xkˆ  x ˆj

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iˆ



iˆ  1, 2, , n  2, n  1, n  with  ˆj  2,3, , n  1, n,1 and mˆ  n,1, ˆ k  3, 4, , n,1, 2

n  2, n  1.

(40)

5 Numerical results In this section, the accuracy and stability of the proposed approach are performed through several numerical examples. There are different two types of boundary conditions are considered as simply supported (S) and clamped (C). For purpose of comparison, a list of elements which will be considered is as follows

ACCEPTED MANUSCRIPT 

MITC4: Four-node mixed interpolation of tensorial component element [54].



PRMn-T3: The present unified formulation for three-node triangular mesh. This element is identical to the ARS-T9 element [61].



PRMn-Q4: The present formulation for four-node quadrilateral meshes. Notice that the present element is different from the ARS-Q12 element reported in [62].



PRMn-PL: The present formulation based on piecewise-linear shape functions for n-

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node polygonal meshes. In all the following examples, the material properties of all layers are assumed to be the same thickness, mass density, and made of the same linearly elastic composite material but the fiber orientations may be different among the layers. The material properties are given as follows:

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Material type I: E1 / E2  25, G12  G13  0.5E2 , G23  0.2E2 , 12  0.25,   1.

Material type II: E1 / E2  40, G12  G13  0.6E2 , G23  0.5E2 , 12  0.25,   1. Unless mentioned otherwise, the material type I and II are employed to study the static and free vibration analysis of plate, respectively.

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5.1 Static analysis

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5.1.1 Square isotropic plate under uniform load In order to evaluate the convergence and reliability of the approach proposed, the model

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of an isotropic square plate with the length a and the thickness h will be investigated. The plates are subjected to uniform load q  1 with fully simply supported (SSSS) and fully

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clamped (CCCC) boundary conditions, respectively, which is shown in Fig. 7. The elastic material parameters used in the analysis are: Young’s modulus E  1,092,000 and Poisson’s

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ratio  0.3. Owing to the symmetric properties, only the lower-left corner of square plate is modeled. The square plate domain is discretized into three mesh types of three-node triangular, four-node quadrilateral and n-node polygonal elements. For three-node triangular and four-node quadrilateral elements, uniform meshes N  N with N = 8, 13, 21 and 37 are used. For purpose of comparison, the total number of nodes in all domain at each level of mesh for three element types is nearly similar. Fig. 8 shows four meshes level of plate using n-node polygonal element. Firstly, in order to illustrate the accuracy of the proposed formulation, the convergence of the central deflection and the moment of the central point are investigated. Fig. 9 and Fig. 10

ACCEPTED MANUSCRIPT show the convergence of the normalized deflection, central moments of CCCC and SSSS boundary conditions, respectively, with ratio a / h  1000 . It can be seen that all the present results converge well to the analytical solution [1], both for deflection and moment. As seen, in both CCCC and SSSS cases, PRMn-PL element shows more accurate results than those of PRMn-T3, PRMn-Q4 elements. In addition, the normalized deflection and the normal stresses obtained directly from constitutive equation at the center node of a SSSS square plate defined as w

100 E2 h3  a a  w , , 0 ; qa 4 2 2 

 xx 

h2 a a h  xx  , ,  . 2 qa 2 2 2

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for various a / h ratios are listed in Table 1. The normalized deflection and normal stress are

The obtained results are compared with finite element method (FEM-Q9) [79] based on FSDT reported by Reddy, Ferreira [80] using wavelets functions and the exact solutions [81]

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by Reddy. As expected, the present results are in good agreements with an exact solution for all various a / h ratios.

In order to demonstrate shear locking free of polygonal plate element when the thickness 10 becomes very thin, the thick square plate (a / h  10) to very thin one (a / h  10 ) with

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simply supported conditions is studied. It is well known that, solutions of Reissner–Mindlin theory will close to solutions of Kirchhoff theory when the length-to-thickness ratio ( a / h )

ED

increase gradually. Fig. 11 performed the deflection and moment at central respect to the

a / h ratios based on mesh level 4th. As expected, the proposed elements exhibit stable, high

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reliability and closed to the Kirchhoff solution when the thickness becomes very thin. As a result, all present elements can be locking-free when the plate thickness becomes progressively very thin. Hence, the present elements can be effectively applied for thin or

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moderately thick plate. In addition, it can be seen that PMRn-PL shows the best central

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moment in comparison with other elements. 5.1.2 Four-layer cross-ply square laminated plate under sinusoidal load Next, a fully simply supported square laminated plate is composed of four layers with the

stacking sequence of 00 / 900 / 900 / 00  will be investigated. The plate is subjected to a distributed sinusoidal load defined as q  q0 sin  x / a  sin  y / a  , as depicted in Fig. 12a. The following normalized deflection and stresses are used

ACCEPTED MANUSCRIPT 100 E2 h3  a a  h2 h2 a a h a a h w , , 0 ;    , , ;    yy  , ,  ; xx xx  yy    4 2 2 q0 a q0 a q0 a 2 2  2 2 2 2 2 4 h h h h   a  a    xy  0, 0,  ;  xz   xz  0, , 0  ;  yz   yz  , 0, 0  . q0 a 2 q0 a  2  q0 a  2 

w

 xy

The square laminated plate is discretized into three mesh types including three-node triangular, four-node quadrilateral and n-node polygonal elements using three meshes level: 2nd, 3rd and 4th, as given in Fig. 8. Plates whose length-to-thickness ratio is a / h  4 are

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chosen in this investigation. The convergence of the normalized deflection and stresses at central point of four-layer square laminated composite plate is presented in Table 2. As can be observed, the present results converge very well to an exact solution reported by Reddy [5]. Table 2 also reveals that the nearly same accuracy of displacement and stresses solutions are obtain at mesh level 3rd and 4th. The maximum error (%) of the present solutions compared

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with the exact solution [5] between two mesh levels is less than 0.44%. Thereby, for a practical point of view, the mesh level 3rd is used to model the laminated plates for the next sections.

Several length-to-thickness ratios a / h of 4, 10, 20 and 100 are chosen to investigate the effects of the plate’s geometry on the responses of plates. The numerical results which are

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generated from the proposed approached are presented in Table 3 comparing with those of other methods such as the finite strip method (FSM) based on FSDT by Akhras et al. [82],

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finite element method (FEM) based on HSDT by Reddy [13], the meshfree method based on the first order layerwise deformation theory (LW) proposed by Ferreira et al. [83,84], a

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generalized layerwise higher-order shear deformation theory and isogeometric analysis by Thai et al. [85] and the elasticity solution reported by Pagano [1]. It can be observed that the

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obtained results agree well with other reference values for all ratios a / h . Generally, in comparison with the elasticity solution [1], the proposed approach performs slightly better than FSM-HSDT [82] and FEM-HSDT [13].

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Fig. 13 illustrates the distribution of stresses, which are computed using the constitutive

equations, through the thickness of plates with a / h  10. It can be observed that although the shear stresses which are free at the plate’s surfaces distribute parabolically through the thickness of each laminae, it experiences discontinuity at the interface of two adjacent layers which is reported by Reddy [5].

ACCEPTED MANUSCRIPT 5.2 Free vibration analysis 5.2.1 Square laminated plates In this subsection, the free vibration analysis of a four-layer 00 / 900 / 900 / 00  square laminated composite plate with SSSS boundary condition is discussed. The plate has the length a and thickness h. The normalized frequency is defined by   a 2 / h   / E2 . In

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order to evaluate the convergence of the natural frequency of the proposed approach, the full laminated composite plate is modeled using three uniform meshes N x N with N = 8, 16 and 24, corresponding mesh level 1st, 2nd and 3rd using three-node triangular and four-node quadrilateral elements, respectively. The n-node polygonal meshes are also constructed with nearly the same numbers of node corresponding to each level of uniform mesh. The length-

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to-thickness ratio a / h  5 and elastic modulus ratio E1 / E2  40 are employed in this computation. The convergence of the first fundamental frequencies is listed in Table 4. These results are compared with approximate 3-D solution by Noor [3] and the analytical solutions based on TSDT theory derived by Reddy [5]. It should be noted that the differences of normalized frequencies between mesh level 2nd and 3rd are not significant. For computation,

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mesh level 2nd can be therefore used for the following tests.

In the next step, the influence of the length-to-thickness ratios on the frequency is also

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investigated. The obtained results are presented in Table 5 and are compared with Cho et al. using higher-order individual-layer theory [87], Wu et al. based on local higher-order theory

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[88], Matsunaga based on global higher-order theory [89] and Zhen and Wanji using globallocal higher-order theory [90]. As expected, the present results show good agreement with

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other published literatures.

The first six normalized frequencies of a CCCC three-layer 00 / 900 / 00  square

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laminated composite plate with E1 / E2  40 and various length-to-thickness ratio a / h are



also given in Table 6. The normalized frequencies are obtained as   a 2 /  2

  h / D  0

3 with D0  E2 h /12 1 12 21  . The results of the proposed approach are compared with the p-

Ritz method by Liew [91], the refined three-node triangular finite element using global-local higher-order theory (GLHOT) reported by Zhen and Wanji [90] and the multiquadric radial basis function pseudospectral (MRBF-PS) based on FSDT [92] and wavelet function [80] by Ferreira et al.. It can be seen that the results given in Table 6 are in compliance with other

ACCEPTED MANUSCRIPT solutions. The first six mode shapes of a CCCC three-layer square laminated composite plate with a / h  10 are depicted in Fig. 14. 5.2.2 Circular laminated plates In this part, a circular four-layer  0 /  0 /  0 /  0  laminated composite plates with fully clamped boundary are analyzed. The geometry and polygonal mesh of the circular plate

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are depicted in Fig. 15. The radius-to-thickness ratio R / h  5 with various fiber orientation angles  0  00 ,150 ,300 , 450 are considered. The natural frequency is normalized as

   4 R2 / h 

  / E2  . The normalized frequencies calculated from the present approach

corresponding with various fiber orientation angles are presented in Table 7. These results are compared with the isogeometric analysis based on TSDT (IGA-TSDT) [85], the moving least

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squares differential quadrature method based on FSDT (MLSDQ) [86], the four-node quadrilateral element using stabilized nodal integration based on FSDT (MISQ20) [93], and the isogeometric analysis based on the layer-wise FSDT (LW-FSDT) model [94]. As can be seen, the present results are in an excellent agreement with available solutions. In addition, the frequencies increase according to increment of fiber orientation angles. Fig. 16 illustrates

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the first six mode shapes of a full clamped circular laminated plate 450 / 450 / 450 / 450  .

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5.2.3 Square plate with a complicated cutout In order to demonstrate the capability of the present method deal with complex

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geometries, a thin isotropic square plate with a complicated cutout is further studied in this section. Fig. 17a show the geometry of the plate and its dimensions. The material parameters

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of plate are considered as follows: Young’s modulus E  200x109 Pa , Poisson’s ratio  0.3, 3 mass density   8000 kg/m . The length-to-thickness ratio is a / h  200. The square plate

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with a complicated cutout is discretized into polygonal elements with 589 nodes in all domain, as depicted in Fig. 17b. Two different types of boundary conditions as SSSS and CCCC are



investigated. The normalized natural frequency is given by   a 2  h / D





1/2

where



D  Eh3 / 12 1  2  is the flexural rigidity of the plate. The first ten normalized frequencies of a thin isotropic square plate with two different types of boundary conditions are listed in Table 8 and Table 9. The present results are compared with those of several published results such as the IGA using the classical plate theory (IGA-CPT) with eight patches [95], MKI

ACCEPTED MANUSCRIPT method [96], EFG method [97], node-based smoothing RPIM (NS-RPIM) method [98] and IGA based on level sets (IGA-LS) [99]. It can be seen that the present results agree very well with other reference solutions for both boundary conditions. Of course, the natural frequencies, which a CCCC boundary condition is considered, are generally higher than those with SSSS boundaries. Next, a SSSS three-layer laminated composite square plate with a complicated cutout is studied. The geometry and polygonal mesh of the plate are the same as those in Fig. 17. The

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material properties for the laminated composite plate are considered as follows: ratio of elastic constants E1 / E2  2.45 , G12 / E2  0.48 , Poisson’s ratio 12  0.23 , mass density

  8000 kg/m3 and plate thickness h  0.06 m. The normalized natural frequency is given by 

  ha 2

4

/ D0.1  in which D0.1  E1h3 / 12 1 12 21   . The results show in Table 10 are

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compared with other published works corresponding with various fiber orientations. Once again, the present results are in excellent agreement with the reference solutions for all considered orientations. Fig. 16 shows the first ten mode shapes of three layers complicated shaped composite plate with fiber orientations 300 / 300 / 300  .

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In the last example, a CCCC antisymmetric angle-ply 300 / 450  laminated composite relatively thick square plate with a complicated cutout is investigated. The plate has the same

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geometry and polygonal mesh of the previous section, whereas plate thickness h  0.5 m . The lamination scheme consists of a two layer 300 / 450  of Graphite-Epoxy: E1  137.9GPa;

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E2  8.96GPa; G12  G13  7.1GPa; G23  6.21GPa;12  13  0.3;   1450kg/m3 . The first ten

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frequencies of plate, presented in Table 11, are compared with a 3-D FEM model based on 20-node elements in Abaqus and strong formulation isogeometric analysis (SFIGA) [100] reported by Fantuzzi et al. As expected, the present results have in excellent agreement with

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other reference solutions for all the ten frequencies.

6 Conclusions In this study, a novel PFEM formulation, based on the C0-type HSDT model, has been presented to investigate the static and free vibration analysis of laminated composite plates. The proposed formulations are valid for arbitrary polygonal meshes, in which the triangular and quadrilateral elements are considered as special cases. PFEM utilizes the piecewise-linear shape function which allow us to calculate easily numerical integration of polygonal element.

ACCEPTED MANUSCRIPT By using C0- HSDT theory, the numerical results are more accuracy and describe exactly the distribution of shear stress without employing shear correction factors. Numerous numerical examples with different geometries, stiffness ratio, number of layers and boundary conditions are investigated. Numerical results demonstrated that the present approach is valid for both thick and thin laminated composite plates with accuracy and high reliability compared with

Acknowledgment

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other methods available.

The first author would like to thank Mr. Son Nguyen-Hoang and Mr. Khai Nguyen-Chau for their assistance while doing this study. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number

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107.02-2016.19.

ACCEPTED MANUSCRIPT References 1. Pagano N. Exact solutions for rectangular bidirectional composites and sandwich plates. J Composite Materials. 1970; 4:20-34. 2. Pagano N, Hatfield HJ. Elastic behavior of multilayered bidirectional composites. AIAA journal. 1972;10(7):931-3. 3. Noor AK. Free vibrations of multilayered composite plates. AIAA journal.

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1973;11(7):1038-9.

4. Noor AK. Stability of multilayered composite plates. Fibre Science and Technology. 1975;8(2):81-9.

5. Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis: CRC press; 2004.

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6. Carrera E. Evaluation of layerwise mixed theories for laminated plates analysis. AIAA journal. 1998;36(5):830-9.

7. Ferreira A. Analysis of composite plates using a layerwise theory and multiquadrics discretization. Mechanics of Advanced Materials and Structures. 2005;12(2):99-112 8. Sahoo R, Singh B. A new trigonometric zigzag theory for static analysis of laminated

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composite and sandwich plates. Aerospace science and technology. 2014; 35:15-28. 9. Mantari JL, Soares CG. Four-unknown quasi-3D shear deformation theory for advanced

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composite plates. Composite Structures. 2014;109: 231–239. 10. Thai CH, Zenkour A, Wahab MA, Nguyen-Xuan H. A simple four-unknown shear and

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normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis. Composite Structures. 2016; 139:77-95.

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61. Soh A, Long Z, Cen S. A new nine DOF triangular element for analysis of thick and thin plates. Computational Mechanics. 1999;24(5):408-17.

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62. Soh A-K, Cen S, Long Y-Q, Long Z-F. A new twelve DOF quadrilateral element for analysis of thick and thin plates. European Journal of Mechanics-A/Solids. 2001;20(2):299-326.

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ACCEPTED MANUSCRIPT 82. Akhras G, Cheung MS, Li W. Finite strip analysis of anisotropic laminated composite plates using higher-order shear deformation theory. Computers & structures. 1994;52(3):471-7. 83. Ferreira A, Fasshauer G, Batra R, Rodrigues J. Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter. Composite Structures. 2008;86(4):328-43. 84. Ferreira A. Analysis of composite plates using a layerwise theory and multiquadrics

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discretization. Mechanics of Advanced Materials and Structures. 2005;12(2):99-112. 85. Thai CH, Ferreira A, Wahab MA, Nguyen-Xuan H. A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis. Acta Mechanica. 2016;227(5):1225-50.

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on FSDT using the moving least squares differential quadrature method. Computer Methods in Applied Mechanics and Engineering. 2003;192(19):2203-22. 87. Cho K, Bert C, Striz A. Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory. Journal of Sound and Vibration. 1991;145(3):42942.

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88. Wu C-P, Chen W-Y. Vibration and stability of laminated plates based on a local high order plate theory. Journal of Sound and Vibration. 1994;177(4):503-20.

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89. Matsunaga H. Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Composite Structures. 2000;48(4):231-44.

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90. Zhen W, Wanji C. Free vibration of laminated composite and sandwich plates using global–local higher-order theory. Journal of Sound and Vibration. 2006;298(1):333-49.

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91. Liew K. Solving the vibration of thick symmetric laminates by Reissner/Mindlin plate theory and thep-Ritz method. Journal of Sound and Vibration. 1996;198(3):343-60.

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92. Ferreira A, Fasshauer G. Computation of natural frequencies of shear deformable beams and plates by an RBF-pseudospectral method. Computer Methods in Applied Mechanics and Engineering. 2006;196(1):134-46.

93. Nguyen-Van H, Mai-Duy N, Tran-Cong T. Free vibration analysis of laminated plate/shell structures based on FSDT with a stabilized nodal-integrated quadrilateral element. Journal of Sound and Vibration. 2008;313(1):205-23. 94. Thai CH, Ferreira A, Carrera E, Nguyen-Xuan H. Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory. Composite Structures. 2013; 104:196-214.

ACCEPTED MANUSCRIPT 95. Shojaee S, Izadpanah E, Valizadeh N, Kiendl J. Free vibration analysis of thin plates by using a NURBS-based isogeometric approach. Finite Elements in Analysis and Design. 2012; 61:23-34. 96. Bui TQ, Nguyen MN. A moving Kriging interpolation-based meshfree method for free vibration analysis of Kirchhoff plates. Computers & structures. 2011;89(3):380-94. 97. Liu G, Chen X. A mesh-free method for static and free vibration analyses of thin plates of complicated shape. Journal of Sound and Vibration. 2001;241(5):839-55.

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98. Cui X, Liu G, Li G, Zhang G. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells. International Journal for Numerical Methods in Engineering. 2011;85(8):958-86.

99. Yin S, Yu T, Bui TQ, Xia S, Hirose S. A cutout isogeometric analysis for thin laminated composite plates using level sets. Composite Structures. 2015; 127:152-64.

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100. Fantuzzi N, Tornabene F, Strong formulation isogeometric analysis (SFIGA) for

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laminated composite arbitrarily shaped plates, Composites Part B. 2016; 96:173-203.

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(a) Geometry of a laminated composite plate

(b) Global xOy and local 1O2 coordinate

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of a laminated composite plate

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Fig.1. Geometry and coordinate systems of a laminated composite plate.

(b) Lower bound shape function

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(a) Upper bound shape function

(c) Definition for piecewise-linear shape function Fig. 2. Linear shape functions for polygonal elements.

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a)

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(b)

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(c)

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(d)

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Fig. 3. The shape functions of four regular polygonal elements using (a) Wachspress shape functions, (b) mean-value shape functions,

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(c) Laplace shape functions and (d) piecewise-linear shape functions.

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Fig. 4. Timoshenko laminated composite beam element.

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Fig. 5. The normal and tangential direction of each edge of polygonal element.

Fig. 6. The orientation of the edge ˆjkˆ of a polygonal element.

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Fig. 7. Geometries and boundary conditions: SSSS and CCCC of an isotropic square plate

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under uniform load.

2nd mesh

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1st mesh

3rd mesh

4th mesh

Fig. 8. Polygonal meshes of an isotropic square plate under uniform load.

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(b) Central moment

(a) Central deflection

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Fig. 9. The rate convergence of a CCCC isotropic square plate with a ratio a / h  1000.

(a) Central deflection

(b) Central moment

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Fig. 10. The rate convergence of a SSSS isotropic square plate with a ratio a / h  1000.

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(a) Central deflection

(b) Central moment Fig. 11. The shear locking test for a SSSS isotropic square plate with varying a/h ratio

 a / h  10 10  . 10

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(b) Polygonal mesh

(a) Geometry and sinusoidal load

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Fig. 12. A square laminated composite plate subjected to sinusoidal load.

Fig. 13. The distribution of stresses through the thickness of a square laminated composite plate subjected to sinusoidal load with a / h  10 based on HSDT and FSDT.

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Fig. 14. The first six mode shapes of a three-layer 00 / 900 / 00  square laminated composite plate with CCCC boundary condition  a / h  10  .

(a) Geometry and boundary condition

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(b) Polygonal mesh

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Fig. 15. A circular laminated composite plate under uniform load and clamped boundary

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condition.

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Fig. 16. The first six mode shapes of a four-layer 450 / 450 / 450 / 450  clamped circular laminated composite plate with R / h  5 .

(a) Geometry parameters of a square plate with a complicated cutout

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(b) Polygonal mesh

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Fig. 17. A geometry and polygonal mesh of a square plate with a complicated cutout.

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Fig. 18. The first ten mode shapes of a three-layer 300 / 300 / 300  CCCC laminated composite plate with a complicated cutout.

ACCEPTED MANUSCRIPT Table 1 The normalized deflection and the normal stress of a SSSS isotropic square plate under uniform load. Method

w

 xx

10

FEM-FSDT [79]

4.7700

0.2899

Wavelets [80]

4.7912

0.2763

Exact [81]

4.7910

0.2762

PRMn-T3

4.7659

0.2737

PRMn-Q4

4.8226

PRMn-PL

4.8119

FEM-FSDT [79]

4.5700

Wavelets [80]

4.6254

Exact [81]

4.6250

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20

PRMn-T3

0.2792 0.2683 0.2763 0.2762 0.2734

4.6581

0.2748

4.6438

0.2773

4.4820

0.2664

4.5716

0.2762

4.5790

0.2762

PRMn-T3

4.5560

0.2742

PRMn-Q4

4.6105

0.2736

4.5984

0.2761

PRMn-PL

Wavelets [80]

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Exact [81]

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FEM-FSDT [79]

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PRMn-PL

AC

0.2763

4.6055

PRMn-Q4

100

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a/h

ACCEPTED MANUSCRIPT Table 2 Convergence of the normalized deflection and stresses of a square laminated composite plate 00 / 900 / 900 / 00  under sinusoidal load with a / h  4 .

3 1.9008

4 1.9020

PRMn-Q4

1.9052

1.9034

1.9027

PRMn-PL

1.9123

1.9034

1.9031

Method

w

 xx

1.8937

PRMn-T3

0.6995

PRMn-Q4

0.6999

PRMn-PL

0.7190

Exact-HSDT [5]  yy

PRMn-T3 PRMn-Q4 PRMn-PL

 xz

PRMn-T3

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PRMn-Q4 PRMn-PL

0.7076

0.7035

0.7049

0.7106

0.7078 0.6651

0.6210

0.6274

0.6298

0.6267

0.6293

0.6303

0.6358

0.6351

0.6323

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Exact-HSDT [5]

0.7067

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Exact-HSDT [5]

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PRMn-T3

Mesh index 2 1.8973

Normalized

0.2118

0.2130

0.2135

0.2133

0.2136

0.2137

0.2057

0.2058

0.2062

Exact-HSDT [5]  xy

0.0465

0.0463

PRMn-Q4

0.04592

0.0460

0.0461

PRMn-PL

0.04618

0.0461

0.0462

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Exact-TSDT [5]

AC

0.2064 0.04659

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PRMn-T3

0.6322

0.0440

ACCEPTED MANUSCRIPT Table 3 The normalized deflection w and stresses  of a SSSS square laminated composite plate 00 / 900 / 900 / 00  under sinusoidal load.

a/h

Method

w

 xx

 yy

 xy

 xz

4

FEM-HSDT [13]

1.8937

0.6651

0.6322

0.0440

0.2064

0.2389

FSM-HSDT [82]

1.8939

0.6806

0.6463

0.045

0.2109

0.2444

RBF-PS [83]

1.9091

0.6429

0.6265

0.0443

0.2173

--

Layerwise [84]

1.9075

0.6432

0.6228

0.0441

0.2166

--

IGA-TSDT [85]

1.9060

0.7334

0.6984

0.0434

0.2298

--

Elasticity [1]

1.954

0.72

0.666

0.0467

0.27

--

PRMn-PL

1.9034

0.7106

0.6351

0.0462

0.2058

0.2432

FEM-HSDT [13]

0.7147

0.5456

0.3888

0.0268

0.2640

0.1531

FSM-HSDT [82]

0.7149

0.5589

0.3974

0.0273

0.2697

0.1568

RBF-PS [83]

0.7203

0.5487

0.3966

0.0273

0.2993

--

Layerwise [84]

0.7309

0.5496

0.3956

0.0273

0.2888

--

IGA-TSDT [85]

0.7359

0.5598

0.4074

0.0274

0.3138

--

Elasticity [1]

0.743

0.559

0.403

0.0276

0.301

--

PRMn-PL

0.7218

0.5657

0.3934

0.0274

0.2717

0.1519

20

FEM-HSDT [13]

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10

 yz

0.5060

0.5393

0.3043

0.0228

0.2825

0.1234

0.5061

0.5523

0.3110

0.0233

0.2883

0.1272

0.5113

0.5407

0.3256

0.0230

0.3073

--

Layerwise [84]

0.5121

0.5417

0.3056

0.0230

0.3248

--

IGA-TSDT [85]

0.5170

0.5430

0.3090

0.0230

0.3280

--

Elasticity [1]

0.517

0.543

0.309

0.0230

0.328

--

PRMn-PL

0.5096

0.5471

0.3070

0.0230

0.2900

0.1152

100 FEM-HSDT [13]

0.4343

0.5387

0.2708

0.0213

0.2897

0.1117

FSM-HSDT [82]

0.4343

0.5507

0.2769

0.0217

0.2948

0.1180

RBF-PS [83]

0.4348

0.5391

0.2711

0.0214

0.3359

--

Layerwise [84]

0.4374

0.5420

0.2697

0.0216

0.3232

--

IGA-TSDT [85]

0.4346

0.5381

0.2707

0.0214

0.3519

--

Elasticity [1]

0.4347

0.539

0.271

0.0214

0.339

--

PRMn-PL

0.4363

0.5385

0.2719

0.0214

0.3320

0.1000

FSM-HSDT [82]

AC

CE

PT

RBF-PS [83]

ACCEPTED MANUSCRIPT Table 4 Convergence of first normalized frequency of a SSSS square laminated composite plate 00 / 900 / 900 / 00  with a / h  5 and E1 / E2  40 .

Method

Mesh level 2

3

PRMn-T3

11.248

10.857

10.779

PRMn-Q4

10.801

10.736

10.724

PRMn-PL

10.905

10.759

Noor-3D [3]

CR IP T

1

10.734 10.752

Exact-HSDT [5]

10.787

AN US

Table 5

The first normalized frequency of a SSSS square laminated composite plate 00 / 900 / 900 / 00  with varying a/h ratio  E1 / E2  40 .

Method

a/h 10

20

25

50

100

Cho [87]

10.673

15.066

17.535

18.054

18.670

18.835

Wu [88]

10.682

15.069

17.636

18.055

18.670

18.835

Matsunaga [89]

10.6876

15.0721

17.6369

18.0557

18.6702

18.8352

Zhen [90]

10.7294

15.1658

17.8035

18.2404

18.9022

19.1566

PRMn-T3

10.857

15.3290

17.9248

18.3269

18.8763

19.0187

10.736

15.0866

17.6492

18.0691

18.6853

18.8496

10.759

15.1066

17.6546

18.0709

18.6753

18.8177

AC

ED

CE

PRMn-PL

PT

PRMn-Q4

M

5

ACCEPTED MANUSCRIPT Table 6 The first six normalized frequencies of a CCCC square laminated composite plate 00 / 900 / 00  with varying a/h ratio.

20

2

3

4

5

6

Wavelets [80]

4.4466

6.6422

7.6996

9.1851

9.7393

11.3988

FEM-GLHOT [90]

4.450

6.524

8.178

9.473

9.492

11.769

p-Ritz [91]

4.447

6.642

7.7

9.185

9.738

11.399

RBF-PS [92]

4.5141

6.508

8.0361

9.3468

9.3929

11.5749

PRMn-PL

4.5115

6.4917

8.0663

9.3704

9.4217

11.665

Wavelets [80]

7.4106

10.3944

13.9128

15.4403

15.8061

19.5797

FEM-GLHOT [90]

7.484

10.207

14.34

14.863

16.07

19.508

p-Ritz [91]

7.411

10.393

13.913

15.429

15.806

19.572

RBF-PS [92]

7.4727

10.2544

14.244

14.9363

15.9807

19.4129

PRMn-PL

7.4626

10.226

14.29

14.932

16.031

19.535

Wavelets [80]

10.9528

14.036

20.4533

23.1974

24.9827

29.2795

FEM-GLHOT [90]

11.003

14.064

20.321

23.498

25.35

29.118

p-Ritz [91]

10.953

14.028

20.388

23.196

24.978

29.237

10.968

13.9636

20.0983

23.3572

25.0859

28.6749

10.918

13.909

20.034

23.307

25.072

28.674

RBF-PS [92]

Wavelets [80]

14.4455

17.5426

25.1868

37.8851

39.5489

39.6519

FEM-GLHOT [90]

14.601

17.812

25.236

37.168

38.528

40.668

p-Ritz [91]

14.666

17.614

24.511

35.532

39.157

40.768

RBF-PS [92]

14.4305

17.3776

24.2662

35.5596

37.7629

39.3756

PRMn-PL

14.252

17.237

24.061

34.762

36.829

38.596

AC

CE

100

PT

PRMn-PL

CR IP T

1

AN US

10

Modes

M

5

Method

ED

a/h

ACCEPTED MANUSCRIPT Table 7 The first six normalized frequencies of a clamped circular laminated composite plate  0 /  0 /  0 /  0  .

30

2

3

4

5

6

IGA-TSDT [85]

23.2939

30.8547

42.2709

46.1549

53.8977

56.4591

MLSDQ [86]

22.110

29.6510

41.1010

42.6350

50.3090

54.5530

MISQ20 [93]

22.1230

29.7680

41.7260

IGA-LW [94]

22.9131

30.3926

41.7512

PRMn-PL

22.197

29.727

41.339

IGA-TSDT [85]

23.6474

32.2171

44.2845

MLSDQ [86]

22.774

31.455

43.35

MISQ20 [93]

22.698

31.568

IGA-LW [94]

23.2593

PRMn-PL

42.8050

50.7560

56.9500

44.6381

52.4653

55.8996

43.081

43.563

51.604

46.1658

55.2677

58.9615

43.469

52.872

57.386

43.635

44.318

53.468

60.012

31.678

43.7111

44.8855

53.8963

58.2989

22.6346

31.1864

43.6688

44.0442

53.5376

59.1995

IGA-TSDT [85]

24.7528

36.3588

46.0955

51.2387

57.9977

68.1487

MLSDQ [86]

24.071

36.153

43.968

51.074

56.315

66.224

MISQ20 [93]

24.046

36.399

44.189

52.028

57.478

67.099

IGA-LW [94]

24.2403

35.482

44.902

50.008

56.274

66.284

23.782

35.333

44.407

50.586

56.62

68.335

PT

PRMn-PL IGA-TSDT [85]

25.3764

38.7885

45.8500

55.8943

58.6486

69.8247

MLSDQ [86]

24.752

39.181

43.607

56.759

56.967

65.571

MISQ20 [93]

24.766

39.441

43.817

57.907

57.945

66.297

IGA-LW [94]

24.7957

37.7292

44.675

54.0674

56.8998

67.741

PRMn-PL

24.398

37.744

44.244

55.041

57.296

67.977

AC

CE

45

CR IP T

1

AN US

15

Modes

M

0

Method

ED

0

ACCEPTED MANUSCRIPT Table 8 The first ten normalized frequencies of a SSSS isotropic square plate with a complicated cutout. MKI [96]

1

5.193

5.390

2

6.579

3

NS-RPIM [98]

IGA-LS[99]

PRMn-PL

4.919

4.912

4.8696

7.502

6.398

6.396

6.2437

6.597

8.347

6.775

6.770

6.6307

4

7.819

10.636

8.613

8.561

8.4208

5

8.812

11.048

9.016

8.992

8.749

6

9.420

12.894

10.738

10.670

10.475

7

10.742

13.710

10.930

10.888

10.72

8

10.776

14.062

11.601

11.590

11.329

9

11.919

16.649

12.903

12.806

12.586

10

13.200

17.364

13.283

13.180

12.906

CR IP T

IGA [95]

AN US

Mode

M

Table 9

The first ten normalized frequencies of a CCCC isotropic square plate with a complicated

ED

cutout. IGA [95]

EFG [97]

IGA-LS [99]

PRMn-PL

1

7.621

7.548

7.410

7.428

7.2441

2

9.810

10.764

9.726

9.829

9.5667

9.948

11.113

9.764

9.858

9.6019

11.135

11.328

10.896

10.960

10.616

11.216

12.862

11.114

11.178

10.796

12.482

13.300

12.353

12.367

11.925

7

12.872

14.168

12.781

12.835

12.429

8

13.650

15.369

13.368

13.433

12.922

9

14.676

16.205

14.485

14.440

13.920

10

14.738

17.137

14.766

14.743

14.163

PT

Mode

4 5

AC

6

CE

3

NS-RPIM [98]

ACCEPTED MANUSCRIPT Table 10 The first six normalized frequencies of a SSSS laminated composite square plate with complicated cutout for various orientations. Method

Mode

15 / 150 /150 

300 / 300 / 300 

AC

5

6

18.169 30.303

36.581

57.429

64.145

85.656

EFG [97]

18.226 31.127

36.237

56.874

62.390

83.565

IGA-LS [99]

18.192 30.936

36.082

56.420

62.024

82.966

PRMn-PL

17.886 29.645

34.367

54.663

58.836

80.606

MKI [96]

18.323 31.472

37.617

63.077

66.538

86.486

EFG [97]

19.177 32.445

37.238

58.716

63.994

86.500

IGA-LS [99]

19.100 32.149

36.458

57.573

63.361

84.776

PRMn-PL

18.780 30.772

34.633

55.689

59.978

82.162

MKI [96]

20.310 33.987

39.898

58.111

69.699

92.099

EFG [97]

20.926 34.915

39.101

62.222

67.054

92.715

IGA-LS [99]

20.606 33.997

37.610

59.797

65.688

88.809

PRMn-PL

20.25

32.477

35.728

57.747

61.98

85.889

MKI [96]

20.987 34.897

39.269

63.375

69.017

96.588

21.736 36.079

39.975

63.897

68.525

96.767

IGA-LS [99]

21.313 34.801

38.289

60.897

66.885

91.601

PRMn-PL

20.941 33.231

36.37

58.781

62.995

88.966

MKI [96]

18.027 32.506

37.268

57.698

70.768

92.998

EFG [97]

18.278 32.264

36.134

57.151

65.853

90.678

IGA-LS [99]

18.201 31.082

36.096

56.473

62.523

83.660

PRMn-PL

17.897 29.761

34.4

54.739

59.254

81.256

EFG [97]

PT

CE

00 / 900 / 00 

4

MKI [96]

ED

450 / 450 / 450 

3

AN US

00 / 00 / 00 

2

M

1

CR IP T

Angle ply

ACCEPTED MANUSCRIPT Table 11 The first ten frequencies of a CCCC laminated composite square plate 300 / 450  with a FEM 3D [100]

SFIGA [100]

PRMn-PL

1

64.8417

64.5668

63.4943

2

93.3058

92.6564

90.5482

3

102.856

102.207

101.098

4

117.567

116.698

5

126.622

125.770

6

134.543

133.591

7

150.225

149.149

8

155.213

154.032

9

182.663

10

186.809

AN US

Mode

CR IP T

complicated cutout.

125.017

132.312 148.820 153.510

181.219

182.115

185.304

186.593

M ED PT CE AC

114.971

ACCEPTED MANUSCRIPT

Polygonal mesh

AC

CE

PT

ED

M

AN US

A square plate with a complicated cutout

CR IP T

Graphical Abstract

Several mode shapes of a three-layer 300 / 300 / 300  CCCC laminated composite plate with a complicated cutout.