On the general framework of high order shear deformation theories for laminated composite plate structures: A novel unified approach

Author’s Accepted Manuscript On the general framework of high order shear deformation theories for laminated composite plate structures: A novel unified approach Tuan N. Nguyen, Chien H. Thai, H. Nguyen-Xuan www.elsevier.com/locate/ijmecsci

PII: DOI: Reference:

S0020-7403(16)00018-7 http://dx.doi.org/10.1016/j.ijmecsci.2016.01.012 MS3205

To appear in: International Journal of Mechanical Sciences Received date: 17 August 2015 Revised date: 7 January 2016 Accepted date: 8 January 2016 Cite this article as: Tuan N. Nguyen, Chien H. Thai and H. Nguyen-Xuan, On the general framework of high order shear deformation theories for laminated composite plate structures: A novel unified approach, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.01.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the general framework of high order shear deformation theories for laminated composite plate structures: A novel unified approach Tuan N. Nguyena , Chien H. Thaib , H. Nguyen-Xuanc,d,∗ a

Department of Computational Engineering, Vietnamese-German University, Binh Duong New City, Vietnam b Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam c Center for Interdisciplinary Research in Technology (CIRTech), University of Technology (HUTECH), Ho Chi Minh City 700000, Vietnam d Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul, 143-747, South Korea

Abstract This paper brings to the readers a unified framework on higher order shear deformation theories (HSDTs), modelling and analysis of laminated composite plates. The major objective of this work is to 1) unify all higher order shear deformation theories in a unique formulation by a polynomial form; 2) propose the new higher shear deformation models systematically based on a unified formulation. In addition, the effect of thickness stretching is taken into account by considering a quasi-3D theory. The principle of virtual displacements is exploited to derive a weak form based on the generalized displacement fields of the higher order shear deformation theories. Numerical results are computed by using isogeometric analysis and verified to show the accuracy and reliability of the present approach. It is found that the unique formulation of a polynomial form can theoretically cover all existing HSDTs models and is thus sufficient to describe the nonlinear and parabolic variation of transverse shear stress. Moreover, the proposed higher order shear deformation theories predict the proper responses for laminated composite plates in comparison with the available ones in the literature. Keywords: laminated composite plates, higher order shear deformation theory, a unified formulation, quasi-3D theory, Isogeometric analysis

∗

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

Preprint submitted to International Journal of Mechanical Sciences

February 18, 2016

1. Introduction During the last haft century, laminated composite material is one of the most widely used materials in tremendous scientific and engineering applications such as vehicles, aerospace, aircraft, construction, submarine, vessels, etc. owing to their adaptability and flexibility in variation of stacking sequence of fibre and matrix. Laminated composite plates are made up of orthotropic laminae which are adhered together. In these structures, stress-included failures occur through three mechanisms which are dependent of the normal and transverse shear stresses [1]. Therefore, it is essential to measure the normal and transverse shear tresses through the plate’s thickness precisely. Pagano [2, 3] pioneered to analyse the simple static problem using the three-dimension (3D) elasticity method. Noor [4, 5] subsequently carried out the 3D solution for free vibration and stability of multi-layered composite plate. Nevertheless, it is straightforward that the 3D solution is not a feasible tool to deal with complex geometries and arbitrary boundary conditions. Consequently, several alternative approaches have been proposed to facilitate a 3D model into a 2D model: equivalent single layer (ESL) theories [6], layer-wise (LW) theories [7], zigzag (ZZ) theories [8], quasi-3D theories [9]. The ESL theories considered in this paper are acknowledged to obtain the adequate accuracy of the behaviour of thin or/and thick laminated composite plates [10]. The simplest ESL theories, Kirchhoff theory or classical plate theory (CPT) [11], was first developed based on Kirchhoff-Love hypothesis that plane section perpendicular to the mid-plane of plates before deformation remains plane, rigid, and perpendicular to the deformed mid-plane after deformation. Due to neglecting the effect of shear strain on the deformation, CPT is only applicable to the thin plates. For the moderate thick plate, CPT underestimates deflection and overestimates natural frequencies and buckling load [12]. The first order shear deformation theory (FSDT), or might namely as Reissner-Mindlin plate theory [13, 14] , was developed to surmount this limitation by accounting for transverse shear deformation effect. According to FSDT, plane section will still be plane after deformation, however, are not normal to mid-plane after deformation. This assumption causes the constant transverse shear stress through the plate’s thickness which violates the traction boundary conditions on the top and bottom surface of plates. As a result, FSDT requires a shear correction factor (SCF) to satisfy the traction boundary condition on the top and bottom surface of plates. The precision of FSDT in prediction the behaviour of laminated composite plate strongly relies on the accuracy of the SCF. Unfortunately, the exact value of SCF is rather cumbersome to define for general problem, especially for laminated composite structures. 2

Moreover, FSDT also encounter a shear locking phenomenon when the thickness to length ratio becomes very small. The disadvantage of CPT and FSDT urged researchers to propose the higher order shear deformation theories. The HSDTs possess transverse shear functions, the major objective of this paper, which are capable to describe the nonlinear, parabolic variation of transverse shear stresses through thickness of plate. The parabolic distributions of transverse shear stresses avoid the usage of the SCF. Ambartsumian [15] initially proposed the first HSDT for anisotropic plates and shallow shells. The third order polynomial was served as transverse shear function is this research. The third order polynomial was later exploited in [16, 17, 18, 19] for anisotropic materials and was adapted for composite materials in [20, 21]. Afterwards, there are a vast number of HSDTs using non polynomial function have been devised in the literature, i.g. trigonometric HSDTs [22, 23, 24, 25, 26, 27, 28, 29], exponential HSDTs [30, 31, 32], hyperbolic HSDTs [33, 34, 35, 36, 37], combined or mixed HSDTs [38, 39, 40, 41]. It was proved that HSDTs provide the much greater accurate deflections and normal stresses than FSDT ones [20]. Up to now, the drawback of HSDTs is that there still exists the error in comparison to the 3D solution, and the transverse shear stresses, computed by constitutive equation, appear discontinuity at interface of two adjacent layers due to the fact that the material properties are different for each layer. In fact, it can be managed by computing the transverse shear stresses based on equilibrium equation which represent the realistic distribution of transverse shear stresses of laminated composite plates [20, 42]. Moreover, in numerical method, HSDTs requires C 1 continuity which has been very significantly solved by isogeometric analysis [43, 44, 26]. It is known that all mentioned HSDTs neglect the thickness stretching effect (normal deformation εz = 0), which causes the independent transverse displacement through the plate thickness. As introduced by Koiter [45], the magnitude of thickness stretching plays the same role as the shear deformation effect. In order to consider thickness stretching effect, Carrera [46] introduced a Carrera Unified Formulation (CUF). A vast number of quasi-3D theoris [47, 48, 49, 50] have been utilized for analysing the thickness stretching effect in the literature based on the CUF. Being different with the CUF, Zenkour [51] has proposed the quasi-3D theory with four variables. Based on this theory, the authors have developed a quasi-3D model incorporating with nonlocal theory for size dependent analysis of nanoplates [52]. Although there are considerable research efforts to introduce a vast number of transverse shear functions, all available functions are seemingly dispersed. The two novel contributions of present study in HSDTs are classified as: firstly, a uni3

fied formulation for transverse shear functions is presented which provides the systematic scheme to derive the transverse shear functions. Secondly, four new transverse shear functions are subsequently proposed for investigating laminated composite plates based on the first idea. In addition, the effect of thickness stretching is taken into account by the quasi-3D theory [53]. The two simple static analysis of symmetrically cross-ply laminated composite plates are performed to show the superiority in the accuracy and efficiency of the present study. The weak form are basically derived based on the principle of virtual displacements and solved by the-state-of-the-art numerical method, isogeometric analysis (IGA), which merges Computer Aided Design (CAD) geometric modelling with Finite Element Analysis (FEA) by means of non-uniform rational B-splines (NURBS) basis functions. The rest of paper is outlined as follows: a unified formulation of higher order shear deformation theories is presented and verified in next section for laminated composite plates. In Section 3, the efficiency of a unified formulation is examined by proposing four new transverse shear functions. The effect of normal deformation based on the quasi-3D theory is further discussed in Section 4. Finally, some remarkable conclusions are drawn. 2. On the general framework of high order shear deformation theories 2.1. A unified formulation Let consider Ω as a mid-plane of a laminated composite plate depicted in Fig. 1 T  in which u0 , v0 , w0 and β = βx βy stand for the displacement components in the x, y, z directions and the rotations in the x − z and y − z planes (or the y and the x axes), respectively. According to Zenkour [53], a generalized higher order shear deformation displacement field accounting for the thickness stretching effect can be expressed as ∂w0 + f (z) βx (x, y) ∂x ∂w0 v (x, y, z) = v0 (x, y) − z + f (z) βy (x, y) ∂y w (x, y, z) = w0 (x, y) + g (z) βz (x, y)

u (x, y, z) = u0 (x, y) − z

(1)

where an additional term βz accounts for the normal deformation and the function f (z) describes the distribution of thickness stretching effect through the g (z) = dz plate’s thickness.

4

Figure 1: Geometry of laminated composite plate.

Based on the displacement field Eq. (1), the normal strains and shear strains are given as ∂ 2 w0 ∂u0 ∂βx −z + f (z) ∂x ∂x2 ∂x ∂ 2 w0 ∂v0 ∂βy −z + f (z) εyy = ∂y ∂y 2 ∂y   ∂ 2 w0 ∂βx ∂βy ∂u0 ∂v0 + − 2z + f (z) + γxy = ∂y ∂x ∂x∂y ∂y ∂x εz = g  (z) βz ∂βz γxz = f  (z) βx + g (z) ∂x ∂βz  γyz = f (z) βy + g (z) ∂y εxx =

5

(2)

The constitutive equation of k th orthotropic layer in its local coordinate for the quasi-3D theory is given by ⎡ ⎤ ⎡ ⎤ (k) ⎤(k) ε(k) ⎡ σ1 0 C11 C12 0 C13 0 ⎢ 1 ⎥ ⎢ (k) ⎥ ⎥ ⎢ ε(k) ⎥ ⎢ σ2 ⎥ ⎢ C21 C22 0 C23 0 0 ⎢ (k) ⎥ ⎢ ⎥ ⎢ 2(k) ⎥ ⎢ ⎢ τ12 ⎥ ⎢ 0 ⎥ 0 0 ⎥ ⎢ γ12 ⎥ 0 C66 0 ⎥ ⎢ ⎥ ⎢ (3) = ⎢ σ (k) ⎥ ⎢ C31 C32 0 C33 0 ⎥ ⎢ ε(k) ⎥ 0 ⎢ 3 ⎥ ⎢ ⎥ ⎢ 3 ⎥ ⎢ (k) ⎥ ⎣ 0 (k) ⎥ 0 0 0 C55 0 ⎦ ⎢ ⎣ γ13 ⎦ ⎣ τ13 ⎦ (k) (k) 0 0 0 0 0 C44 τ23 γ23 where E1 (ν21 + ν31 ν23 ) E1 (ν31 + ν21 ν32 ) E1 (1 − ν23 ν32 ) C12 = C13 = Δ Δ Δ E2 (1 − ν13 ν31 ) E2 (ν32 + ν12 ν31 ) E3 (1 − ν12 ν21 ) C23 = C33 = = Δ Δ Δ = G23 C55 = G13 C66 = G12 Δ = 1 − ν12 ν21 − ν23 ν32 − ν31 ν13 − 2ν12 ν32 ν13

C11 = C22 C44

Using transformation equations, the constitutive equation of k th global coordinate system (x, y, z) is given by ⎡ ⎤(k) ⎡ ⎤(k) ⎡ C 11 C 12 C 16 C 13 0 0 εxx σxx ⎢ σyy ⎥ ⎢ C 12 C 22 C 26 C 23 0 ⎥ ⎢ 0 ⎥ ⎢ εyy ⎢ ⎢ ⎥ ⎢ τxy ⎥ ⎢ C 16 C 26 C 66 C 36 0 ⎢ 0 ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ γxy ⎢ σz ⎥ = ⎢ C 13 C 23 C 36 C 33 0 ⎥ 0 ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ εz ⎦ ⎣ γxz ⎣ τxz ⎦ ⎣ 0 0 0 0 C 55 C 45 τyz γyz 0 0 0 0 C 45 C 44

layer in the ⎤(k) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4)

where the transformed elastic coefficients C ij are given in [54] ⎡ ⎤ ⎡ 4 ⎤ C 11 2c2 s2 s4 4c2 s2 c ⎢ C 12 ⎥ ⎢ c2 s2 ⎥ c 4 + s4 c 2 s2 −4c2 s2 ⎢ ⎥ ⎢ 4 ⎥ 2 2 4 2 2 ⎢ C 22 ⎥ ⎢ s ⎥ 2c s c 4c s ⎢ ⎥=⎢ ⎥ 3 2 2 3 2 2 ⎢ C 16 ⎥ ⎢ −c s cs (c − s ) cs 2cs (c − s ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ C 26 ⎦ ⎣ −cs3 cs (s2 − c2 ) c3 s 2cs (s2 − c2 ) ⎦ 2 c2 s2 (c2 − s2 ) C 66 ⎤ ⎡ c2 s2 ⎤⎫−2c⎡2 s2 ⎧⎡ ⎤     C 13 ⎬ c 2 s2 ⎨ C 44 C C 44 13 ⎣ C 45 ⎦ ; ⎣ C 36 ⎦ = ⎣ cs −cs ⎦ ; ; C 33 = C33 C55 C23 ⎭ ⎩ 2 2 c s C 55 C 23 (5) 6

here c = cos θ, s = sin θ and θ is a stacking sequence. In case of HSDTs (g (z) = 0), the constitutive equation is simplified for plane stress condition as given in Reddy [10]. The weak form of static analysis of laminated composite plates is derived by using the principle of virtual displacements according to Reddy [10]. For sake of brevity, the weak form can be simply expressed as:        h T T δw0 + g (6) δβz qdΩ δεp Dp εp dΩ + δεs Ds εs dΩ = 2 Ω Ω Ω where q is the transverse loading per unit area and ⎤ ⎡ ∂u0 ⎡ ⎤ ⎥ ⎢ ∂x ε0 ⎥ ⎢ ∂v0 ⎥ ⎢ ⎢ ε1 ⎥ ⎥ ⎢ ⎢ ⎥ ∂y εp = ⎣ ; ε0 = ⎢ ⎥ ⎦ ε2 ⎢ ∂u0 ∂v0 ⎥ ⎥ ⎢ ε3 ⎣ ∂y + ∂x ⎦ 0 ⎤ ⎡ 2 ⎡ ∂ w0 ∂βx ⎢ ∂x2 ⎥ ⎢ ∂x ⎢ ∂ 2w ⎥ ⎢ ∂βy 0 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ∂y ε1 = − ⎢ ∂y 2 ⎥ ; ε2 = ⎢ ⎢ ∂ 2w ⎥ ⎢ ∂βx ∂βy 0 ⎥ ⎢ 2 ⎢ ⎣ ∂x∂y ⎦ ⎣ ∂y + ∂x 0 0 and

⎡

A ⎢ B Dp = ⎢ ⎣ E J

B D F L

E F H O

⎤ ⎡ ⎥ 0 ⎥ ⎥ ⎢ 0 ⎥ ⎥ ; ε3 = ⎢ ⎣ 0 ⎥ ⎥ βz ⎦

(7) ⎤ ⎥ ⎥ ⎦

⎤ J   s1 s12 D L ⎥ D ⎥;D = O ⎦ s Ds12 Ds2 P

(8)

here Aij , Bij , Dij , Eij , Fij , Hij, Jij , Lij , Oij , Pij = h  2  1, z, z 2 , f (z) , zf (z) , f (z)2 , g  (z) , zg  (z) , f (z) g  (z) , g  (z)2 C ij dz

−h 2

Dijs1 , Dijs12 , Dijs2

h

=

2  −h 2

 f  (z)2 , f  (z) g (z) , g(z)2 C ij dz (9) 7

Obviously, the weak form for HSDTs can be easily obtained from Eq. (6) by omitting ε3 in Eq. (7) and the function g (z) in Eq. (9). Additionally, the second derivative of displacement variable exists in the weak form. Hence, the quasi-3D and HSDTs require C 1 -continuity of the approximation of displacement variables. In this paper, the C 1 -continuity requirements will be naturally acquired by exploiting IGA with its non-uniform rational B-Spline (NURBS) functions. For sake of brevity, interested readers are encouraged to refer to several technical papers [44, 26, 55] for complete application of NURBS functions in analysis of composite plates. As setting g (z) = 0, the popular HSDT models are well known. Indeed, there are an enormous number of HSDTs which neglect the effect of thickness stretching have been presented in the literature. Moreover, it can be seen in Eq. (1) that the function g (z) is explicitly equal to the derivative of function f (z). As a consequence, for sake of simplicity, we consider a generalized approach for the transverse shear function f (z) of the HSDT models. The effect of thickness stretching based on the quasi-3D theory would be extensively investigated in Section 4. From Eq. (2), it is straightforward that f (z) represents the nonlinear distribution of the transverse shear strains and stresses through the thickness of laminated composite plates. The transverse shear functions f (z) are determined to fulfill the tangential stress-free boundary conditions at the top and bottom surfaces of the plates. Initially, polynomial functions were used by Ambartsumain [15] and afterwards Kaczkowski [16], Panc [17], Reissner [18] and Levinson [19], Murthy [21], Reddy [20] also proposed the different polynomial functions, namely KPR model and LMR model in Table 1, respectively. In the aspect of mathematics, the polynomial form is the simplest and most convenient function, thus be used to facilitate mathematical difficulty of HSDTs in closed-form solution. Consequently, the LMR model [19, 21, 20] was widely used in the analysis of laminated composite plate due to its simplicity. Moreover, in order to find the optimal function yielding the best accurate solutions, it is easy to optimize this kind of function by choosing polynomial coefficients. Recently, Nguyen-Xuan [44] proposed a fifth order shear deformation theory for analysis of composite sandwich plates. Following polynomial functions, the trigonometric functions were subsequently proposed in the work of Levy [22], Stein [23], Touratier [24] (LST model). Then numerous trigonometric functions and their inverse functions were presented in literature [25, 26, 27, 28, 29]. Based on Fourier development series, Kamara [30] implied that exponential function is very much richer than trigonometric function. Therefore, Kamara [30], Aydogdu [31] and Mantari3 [32] proposed three exponential function to consider the shear deformation effects. In the process of 8

investigating the shear deformation effects, many complex functions were further introduced in the literature such as hyperbolic functions [36], mixed functions [39]. A variety of transverse shear functions are listed in Table 1. Table 1: Various transverse shear functions of analysis shear deformation effect

Model

f (z)

Polynomial functions z3 h2 z− 8 6 Kaczkowski [16], Panc [17] 5z 5z 3 − 2 f (z) = and Reissner [18] (KPR) 4 3h Levinson [19], Murthy [21] 4z 3 f (z) = z − 2 and Reddy [20] (LMR) 3h 7z 2z 3 2z 5 − 2 + 4 Nguyen-Xuan [44] f (z) = 8 h h Trigonometric functions  πz  Levy [22], Stein [23] h f (z) = sin π h and Touratier [24] (LST)  πz  Arya [25] f (z) = sin h   2z Thai1 [26] f (z) = −z + htan−1 h  Ambartsumain [15]

Mantari1 [27] Mantari2 [27] Grover1 [28] Grover2 [28] Nguyen [29] Hyperbolic functions Soldatos [33]

f (z) =

 1 mh ;m= f (z) = tan (mz) − mzsec 5h  2  mh π ;m= f (z) = tan (mz) − mzsec2 2 2h   4r rh −z ; r = 0.46 f (z) = cot−1 2 h (4r  rzz  r +  1) r r f (z) = z sec − z sec 1 + tan ; r = 0.1 h  2 2 2 16rz 3 −1 rz f (z) = htan − 2 2 ;r=1 h 3h (r + 4) 2

  1 f (z) = h sinh − z cosh h 2 z 

9

Meiche [34] Akavci1 [35] Akavci2 [35] Mahi [36] Grover3 [37] Exponential functions

 πz  h sinh −z  πh f (z) = π −z −1 cosh 2    π 3π 1 3π 2 f (z) = h tanh − zsech 2 2 2  2h   π  π   πz π f (z) = zsech 1 − tanh − zsech 2 4 2 4 h  z3 h 2z 4 f (z) = tanh − 2 h 3cosh2 (1) h2  rz  2r ;r=3 −z √ f (z) = sinh−1 h h r2 + 4 

z 2 h f (z) = z × e −2  z 2 f (z) = z × 3 log 3 h  2 z −2 h + 0.028z f (z) = z × 2.85 −2

Karama [30] Aydogdu [31] Mantari3 [32] Combination functions Mantari4 [38]

Mantari5 [39]

Mantari6 [39]

1  πz  cos h + πz f (z) = sin × e2 h 2h   z z  m cosh h ×e f (z) = sinh h ⎛ ⎞ 1      m cosh⎝ ⎠ 1 1 z 2 ; m = −6 + msinh2 ×e − cosh h 2 2  z z  m cosh h f (z) = sinh ×e h ⎛ ⎞ 1      m cosh⎝ ⎠ 1 1 z 2 ; m = −7 + msinh2 ×e − cosh h 2 2  πz 

10

 m cos

f (z) = ⎡ z × e

nz  h

 ⎤ n n   m cos m cos n 1 2 − mn sin 2 ⎦ ; m = 1, n = 2.9 ×e −z × ⎣e 2 2   πz  −1 f (z) = tan sin   hπz  f (z) = sinh−1 sin h

Mantari7 [40]

Thai2 [41] Thai3 [41]

0.5 0.4 0.3 0.2

z/h

0.1 0 -0.1

LMR Model Thai1 Model Mahi Model Kamara Model

-0.2 -0.3 -0.4 -0.5 0

0.2

0.4

0.6

0.8

1

f'(z) Figure 2: The derivative of typical transverse shear functions.

It can be seen from Table 1 that the numerous transverse shear functions are proposed to give the mechanical response of the laminated composite plates as much close as possible to the exact 3D elasticity solution [2, 3]. Nevertheless, from the first simple polynomial function Ambartsumain [15], these functions become more and more complicated (e.g., [40, 39, 35]). Consequently, it turns out unnecessary burdensome to address the shear deformation effects in analysis 11

of laminated composite plates. From the beginning our objective was so robust, to propose a unified, simple formulation which can represent all existing transverse shear functions. For this purpose, Fig. 2 illustrates the first derivative of typical transverse shear functions. In general, despite the fact that the models presented Fig. 2 are completely different from the mathematical perspective, their derivative which describe the shear deformation effect are symmetric and even through the thickness of plates. On account of this observation, it is supposed that all existing transverse shear functions can be approximated by a unified and novel polynomial forms as follows f (z) ≈

∞ 

d2n−1 z 2n−1 = d1 z + d3 z 3 + d5 z 5 + d7 z 7 . . .

(10)

n=1

which yields the derivative f  (z) is even and symmetric function through thickness direction. 

f (z) ≈

∞ 

(2n − 1) d2n−1 z 2n−2 = d1 + 3d3 z 2 + 5d5 z 4 + 7d7 z 6 . . .

(11)

n=1

The fact is that there are the variety forms of transverse shear functions are proposed in the literature, however, their derivatives are in the parabolic shape as shown in Fig. 2 which can be covered by the polynomial function given in Eq. (11). Furthermore, since it is possible to infinitely select the coefficients d1 , d3 , d5 . . ., the transverse shear function given in Eq. (10) might contain many unknown functions. For example, let us determine the coefficients d1 , d3 , d5 in Eq. (10) in order define an equivalent model of the first trigonometric model (LST model [22, 23, 24]). In order to determine three coefficients, three independent equations which are obtained by equalizing Eq. (10) and an original function at three specific value are required. Therefore, the polynomial coefficients d1 , d3 , d5 can be defined by

12

solving the system equations as follows   ⎧ π (0) h 2 4 ⎪ ⎪ z = 0 : d1 + 3d3 (0) + 5d5 (0) = sin ⎪ ⎪ π h ⎡  ⎤ ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪  2  4 π ⎪ ⎪ ⎢ h h h h 4 ⎥ ⎪ ⎪ ⎥ + 5d5 = sin ⎢ ⎨ z = : d1 + 3d3 ⎣ h ⎦ 4 4 4 π ⎪ ⎪ ⎡  ⎤ ⎪ ⎪ h ⎪ ⎪  2  4 π ⎪ ⎪ ⎢ ⎪ h h h h 2 ⎥ ⎪ ⎥ ⎪ z = : d1 + 3d3 + 5d5 = sin ⎢ ⎪ ⎣ h ⎦ ⎪ 2 2 2 π ⎪ ⎩

(12)

1.6384 0.732 which gives d1 = 1, d3 = − , d5 = . 2 h h4 The list of three coefficients d1 , d3 , d5 of the fifth order approximation for all existing transverse shear functions are given in Table 2. It is clear that the transverse shear function proposed by Kamara [30] and Aydogdu [31] are exactly identical from viewpoint of mathematical equivalence. Fig. 3 shows the transverse shear functions of two typical models and their cubic, quintic approximation. It is evident that two models well converge with quintic approximation and LST model [22, 23, 24] converges more rapidly than Grover3 counterpart [37] due to its simplicity. 0.5

0.5 0.4 0.3 0.2

LST Model LST Cubic App. LST Quintic App. Grover3 Model Grover3 Cubic App. Grover3 Quintic App.

0.4 0.3 0.2 0.1

z/h

z/h

0.1 0

0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -0.5 -0.4 -0.3 -0.2 -0.1

0

0.1

0.2

0.3

0.4

0.5

f(z)

-0.5 0

LST Model LST Cubic App. LST Quantic App. Grover3 Model Grover3 Cubic App. Grover3 Quantic App.

0.2

0.4

0.6

0.8

1

f'(z)

Figure 3: The transverse shear functions and their approximated functions.

13

1.2

1.4

Table 2: The approximated quintic polynomial functions f (z) = d1 z + d3 z 3 + d5 z 5 .

Model

d3

d5

Polynomial functions Ambartsumain [15] 0.125h2 Kaczkowski [16], Panc [17] 1.25 and Reissner [18] (KPR)

−0.1667 1.6667 − h2

0

Levinson [19], Murthy [21] and Reddy [20] (LMR)

1

−

Nguyen-Xuan [44]

0.875

Trigonometric functions Levy [22], Stein [23] and Touratier [24] (LST)

1

Arya [25] Thai1 [26] Mantari1 [27] Mantari2 [27] Grover1 [28]

d1

1.3333 h2 2 − 2 h

Grover2 [28]

−3.7565 × 10−3

Nguyen [29]

1

−0.1276

Meiche [34]

−1

Akavci1 [35]

1.0063

Akavci2 [35]

1.0227

14

2 h4 0.732 h4 2.2998 h5 2.56 h4 4.2971 × 10−5 h5 2.1025 h5

0.005 h2 1.3961 − h2

2.087 × 10−5 h4 0.15059 h4

0.1667 h2 1.0851 h2 1.5629 − h2 0.2115 − h2

8.4205 × 10−3 h4 0.5958 h4 0.5306 h4 2.7653 − h4

Hyperbolic functions Soldatos [33]

0

1.6384 h2 5.147 − 3 h 2.4 − 2 h 2.6667 × 10−3 h3 1.2184 h3 −

3.1416 h 1 2.0134 × 10−3 − h 1.5708 − h

0

1 1.3359 h

1.8208 h2 3.6729 − h3

1.1698 h4 4.5402 h5

Karama [30]

1

−

Aydogdu [31]

1

Mantari3 [56]

1.028

1.96 h2 1.96 − 2 h 2.0491 − h2

1.504 h4 1.504 h4 1.6282 h4

19.29 h3 6.8003 × 10−3 − h3 2.9078 × 10−3 − h3 10.097 − h2 10.413 − h3 8.0458 − h3

24.694 h5 6.5385 × 10−3 h5 3.1373 × 10−3 h5 13.947 h4 14.937 h5 9.2569 h5

Mahi [36] Grover3 [37]

−

Exponential functions

Combination functions Mantari4 [38] Mantari5 [39] Mantari6 [39] Mantari7 [40] Thai2 [41] Thai3 [41]

6.7504 h 3.0569 × 10−3 h 1.2004 × 10−3 h 3.214 3.1416 h 3.1416 h

−

According to Table 2, it is obvious that all available functions listed in Table 1 can be explicitly approximated in form of a unified polynomial form. It is predicted that the present approach is theoretically capable of covering all remaining shear functions in literature as well as new unknown shear functions. The accuracy and efficiency of the proposed unified formulation for transverse shear functions is demonstrated by two following numerical examples. 2.2. Numerical validations In order to show the accuracy of the present unified framework, let us consider symmetric cross-ply three layers [00 /900 /00 ] and four layers [00 /900 /900 /00 ] square composite plates. The length to thickness ratio is taken as a/h = 4 unless specified mention. The composite plates   πy  are subjected to sinusoidally distributed load  πx q (x, y) = q0 sin sin with full simply supported boundary conditions a a 15

as shown in Fig. 4. The material properties, the normalized deflection and stresses are defined according to the exact 3D elasticity solution of Pagano [2, 3] • Material properties: E1 = 25E2 , E3 = E2 , G12 = G13 = 0.5E2 , G23 = 0.2E2 , ν12 = ν23 = ν13 = 0.25. • For three layers [00 /900 /00 ]:   a a h 100E2 h3  a a  h2 , , 0 , σ xx = , ,− w= w0 σxx q 0 a4  2 2  q 0 a2 2 2 2  a a h h h2 h2 , , − , σ yy = σ τ = τxy 0, 0, − yy xy 2 2 q0 a 2 2 6 q0 a 2   a  a h h τxz 0, , 0 , τ yz = τyz , 0, 0 τ xz = q0 a 2 q0 a 2 • For four layer [00 /900 /900 /00 ]:   a a h 100E2 h3  a a  h2 , , 0 , σ xx = , , w= w0 σxx q 0 a4  2 2  q0 a2  2 2 2 a a h h h2 h2 , , , 0, 0, σ yy = σ τ = τ yy xy xy q 0 a2 2 2 4 q 0 a2 2   a  a h h τxz 0, , 0 , τ yz = τyz , 0, 0 τ xz = q0 a 2 q0 a 2 According to Reddy [20] and Wu et al. [42], it is stated that the shear stress obtained by using the constitutive equation are on the low accuracy than using the equilibrium equation compared to 3D solution [2, 3]. In addition, the shear stress computed by the equilibrium equation gives the unique value transverse shear stress which means no stress discontinuity. Hence, in this study the shear stresses are computed by integrating the equilibrium equation as follows  τxz = −  (σxx,x + τxy,y )dz (13) τyz = − (τxy,x + σyy,y )dz The convergence of the normalized deflection and stresses of three layers [0 /900 /00 ] square composite plate is first investigated in Table 3. In this paper, the composite plate is modelled with 17 × 17 elements as tested in [44]. Table 3 presents the normalized results with respect to the order (2n − 1) of the approximation in Eq. (10). The trigonometric model (Mantari1 [27]), exponential model 0

16

Figure 4: Geometry of square laminated plate under sinusoidally distributed load.

(Kamara [30]), hyperbolic model (Grover3 [37]) and mixed model (Mantari6 [39]) are used to show the convergence of the present unified approach. The percentage errors compared between the equivalent polynomial with original model are given in parenthesis. It is observed that trigonometric model (Mantari1 [27]) converges rapidly with only fifth order polynomial whereas the others require the 7th order of approximated polynomial functions for an excellent convergence. Henceforth the following numerical results are gained by approximating 7th order polynomial functions. Table 3: The convergence of the deflection and stresses of three layer [00 /900 /00 ] square composite plate with respect to the approximation order (2n − 1).

Model Order Mantari1 [27] w σ xx σ yy τ xy τ xz τ yz w Kamara [30] σ xx σ yy τ xy τ xz τ yz

3 5 7 Original 1.9218 (0.0208) 1.9214 (0.0000) 1.9214 (0.0000) 1.9214 0.7310 (0.0821) 0.7304 (0.0000) 0.7304 (0.0000) 0.7304 0.5017 (0.0000) 0.5017 (0.0000) 0.5017 (0.0000) 0.5017 0.0495 (0.0000) 0.0495 (0.0000) 0.0495 (0.0000) 0.0495 0.2813 (0.0710) 0.2815 (0.0000) 0.2815 (0.0000) 0.2815 0.2023 (0.0000) 0.2023 (0.0000) 0.2023 (0.0000) 0.2023 1.9218 (1.1013) 1.9416 (0.0823) 1.9433 (0.0051) 1.9432 0.7310 (5.3599) 0.7767 (0.5567) 0.7722 (0.0259) 0.7724 0.5017 (0.0199) 0.5012 (0.1196) 0.5018 (0.0000) 0.5018 0.0495 (3.6965) 0.0516 (0.3891) 0.0514 (0.0000) 0.0514 0.2813 (4.4560) 0.2686 (0.2599) 0.2693 (0.0000) 0.2693 0.2023 (0.1481) 0.2025 (0.0494) 0.2027 (0.0494) 0.2026 17

Grover3 [37]

w σ xx σ yy τ xy τ xz τ yz Mantari6 [39] w σ xx σ yy τ xy τ xz τ yz

1.9218 (1.6932) 1.9201 (1.7801) 1.9625 (0.3888) 1.9549 0.7310 (9.1248) 0.8369 (4.0403) 0.7916 (1.5912) 0.8044 0.5017 (0.2598) 0.491 (1.8785) 0.5022 (0.3597) 0.5004 0.0495 (6.6038) 0.0535 (0.9434) 0.0528 (0.3774) 0.053 0.2813 (9.1156) 0.2540 (1.4740) 0.2583 (0.1939) 0.2578 0.2023 (0.1981) 0.1990 (1.4364) 0.2027 (0.3962) 0.2019 1.9218 (1.2334) 1.9370 (0.4523) 1.9464 (0.0308) 1.9458 0.7310 (8.7391) 0.8120 (1.3733) 0.8001 (0.1124) 0.8010 0.5017 (0.4807) 0.4969 (0.4807) 0.4995 (0.0401) 0.4993 0.0495 (5.8935) 0.0528 (0.3802) 0.0526 (0.0000) 0.0526 0.2813 (7.7778) 0.2597 (0.4981) 0.2610 (0.0000) 0.2610 0.2023 (0.1981) 0.2011 (0.3962) 0.2019 (0.0000) 0.2019

For comparison, Table 4 expresses the numerical results of the rest of the transverse shear functions for three layers [00 /900 /00 ] square composite plate. The unified models which are equivalent to original models are called U-Model. The normalized results of four layers [00 /900 /900 /00 ] square composite plate are obtained in Table 5. Two significant observations are drawn from Table 4 and Table 5 as follows (i) The present unified approach is highly effective and applicable, irrespective of the complexity of transverse shear functions. There always exists a polynomial function corresponding to an original transverse shear function so that the results are almost coincide. (ii) In the current HSDT framework, the deflection and stresses obtained based on the generalized displacement field Eq. (1), are not affected by the linear combinations of transverse shear functions f1 (z) = αf (z) where α are constants. For instant, the KPR Model [16, 17, 18] and LMR [19, 21, 20] 10 are the linear combination of Ambartsumain [15] with coefficient α = 2 h 8 and α = 2 , respectively. As an consequence, the static results of three h models are exactly identical as shown in Table 4 and Table 5. Similarly, the model of LST [22, 23, 24] and Arya [25] are equivalent with coefficient π α= . h

18

Table 4: The normalized deflection and stresses of three layers [00 /900 /00 ] square composite plate using original and unified model.

Model 3D Pagano [2]) Ambartsumain [15] KPR [16, 17, 18] LMR [19, 21, 20] Nguyen-Xuan [44] LST [22, 23, 24] U-LST Arya [25] U-Arya Thai1 [26] U-Thai1 Mantari2 [27] U-Mantari2 Grover1 [28] U-Grover1 Nguyen.V.H [29] U-Nguyen.V.H Soldatos [33] U-Soldatos Meiche [34] U-Meiche Akavci1 [35] U-Akavci1 Akavci2 [35] U-Akavci2 Mahi [36] U-Mahi Aydogdu [31] U-Aydogdu Mantari3 [56] U-Mantari3 Mantari4 [38] U-Mantari4 Mantari5 [39]

w 2.0060 1.9218 1.9218 1.9218 1.9405 1.9346 1.9346 1.9346 1.9346 1.9515 1.9541 1.8859 1.8860 1.9526 1.9566 1.9248 1.9248 1.9204 1.9204 1.9082 1.9082 1.9315 1.9315 1.8598 1.8600 1.9410 1.9413 1.9432 1.9433 1.9438 1.9439 1.9424 1.9457 1.9461

σ xx 0.7550 0.7310 0.7310 0.7310 0.8015 0.7520 0.7520 0.7520 0.7520 0.7914 0.7867 0.6878 0.6885 0.7981 0.7914 0.7350 0.7350 0.7291 0.7291 0.7136 0.7136 0.7457 0.7457 0.6871 0.6859 0.7599 0.759 0.7724 0.7722 0.7745 0.7743 0.8208 0.8165 0.7892 19

σ yy 0.5560 0.5017 0.5017 0.5017 0.4986 0.5021 0.5021 0.5021 0.5021 0.5012 0.5018 0.4986 0.4986 0.5007 0.5016 0.5019 0.5019 0.5016 0.5016 0.5007 0.5007 0.5021 0.5021 0.4956 0.4956 0.5024 0.5025 0.5018 0.5018 0.5017 0.5017 0.4964 0.4972 0.5006

τ xy 0.0505 0.0495 0.0495 0.0495 0.0525 0.0505 0.0505 0.0505 0.0505 0.0524 0.0523 0.0474 0.0474 0.0527 0.0526 0.0497 0.0497 0.0495 0.0495 0.0487 0.0487 0.0502 0.0502 0.0468 0.0468 0.0510 0.0509 0.0514 0.0514 0.0515 0.0515 0.0534 0.0533 0.0521

τ xz 0.2560 0.2813 0.2813 0.2813 0.2623 0.2753 0.2753 0.2753 0.2753 0.2626 0.2628 0.2934 0.2933 0.2604 0.2606 0.2801 0.2801 0.2819 0.2819 0.2863 0.2863 0.2770 0.2770 0.2977 0.2977 0.2724 0.2725 0.2693 0.2693 0.2687 0.2687 0.2546 0.2547 0.2644

τ yz 0.2170 0.2023 0.2023 0.2023 0.2017 0.2027 0.2027 0.2027 0.2027 0.2024 0.2027 0.2008 0.2008 0.2022 0.2026 0.2024 0.2024 0.2023 0.2023 0.2018 0.2018 0.2026 0.2026 0.1997 0.1997 0.2028 0.2028 0.2026 0.2026 0.2026 0.2026 0.2007 0.2010 0.2023

U-Mantari5

1.9463

0.7888

0.5007

0.0521

0.2644

0.2023

Table 5: The normalized deflection and stresses of four layers [00 /900 /900 /00 ] square composite plate using original and equivalent model.

Model 3D Pagano [3]) Ambartsumain [15] KPR [16, 17, 18] LMR [19, 21, 20] Nguyen-Xuan [44] LST [22, 23, 24] U-LST Arya [25] U-Arya Thai1 [26] U-Thai1 Mantari1 U-Mantari1 [27] Mantari2 [27] U-Mantari2 Nguyen.V.H [29] U-Nguyen.V.H Soldatos [33] U-Soldatos Meiche [34] U-Meiche Akavci1 [35] U-Akavci1 Akavci2 U-Akavci2 [35] Mahi [36] U-Mahi Grover3 [37] U-Grover3 Karama [30] U-Kamara

w 1.9540 1.8936 1.8936 1.8936 1.9218 1.9088 1.9088 1.9088 1.9088 1.9258 1.9274 1.8931 1.8931 1.8520 1.8522 1.8970 1.8970 1.8920 1.8920 1.8779 1.8779 1.9049 1.9049 1.8332 1.8331 1.9147 1.9148 1.9257 1.9303 1.9193 1.9193

σ xx 0.7200 0.6617 0.6617 0.6617 0.7232 0.6796 0.6796 0.6796 0.6796 0.7121 0.7072 0.6612 0.6612 0.6249 0.6256 0.6651 0.6651 0.6601 0.6601 0.6469 0.647 0.6742 0.6742 0.6272 0.626 0.6859 0.6850 0.7220 0.7086 0.6970 0.6967 20

σ yy 0.6630 0.6305 0.6305 0.6305 0.6341 0.6332 0.6332 0.6332 0.6332 0.6366 0.6373 0.6304 0.6304 0.6229 0.6229 0.6311 0.6311 0.6302 0.6302 0.6276 0.6276 0.6325 0.6326 0.6184 0.6185 0.6345 0.6346 0.6373 0.6393 0.6350 0.6350

τ xy 0.0467 0.0439 0.0439 0.0439 0.0469 0.0449 0.0449 0.0449 0.0449 0.0466 0.0465 0.0439 0.0439 0.0418 0.0418 0.0441 0.0441 0.0438 0.0438 0.0430 0.0430 0.0446 0.0446 0.0415 0.0414 0.0452 0.0452 0.0471 0.0468 0.0458 0.0458

τ xz 0.2190 0.2269 0.2269 0.2269 0.2174 0.2237 0.2237 0.2237 0.2237 0.2183 0.2186 0.2270 0.2270 0.2339 0.2338 0.2262 0.2262 0.2272 0.2272 0.2297 0.2297 0.2246 0.2246 0.2349 0.2349 0.2225 0.2225 0.2170 0.2176 0.2208 0.2208

τ yz 0.2910 0.2804 0.2804 0.2804 0.2768 0.2799 0.2799 0.2799 0.2799 0.2768 0.2770 0.2804 0.2804 0.2796 0.2796 0.2803 0.2803 0.2804 0.2804 0.2803 0.2803 0.2801 0.2801 0.2806 0.2806 0.2794 0.2794 0.2745 0.2750 0.2789 0.2789

Aydogdu [31] U-Aydogdu Mantari3 [56] U-Mantari3 Mantari4 [38] U-Mantari4 Mantari5 [39] U-Mantari5 Mantari6 [39] U-Mantari6

1.9193 1.9193 1.9201 1.9201 1.9196 1.9221 1.9237 1.9238 1.9242 1.9246

0.697 0.6967 0.6988 0.6985 0.7377 0.7331 0.7114 0.7109 0.7215 0.7205

0.6350 0.6350 0.6351 0.6351 0.6344 0.6353 0.6355 0.6356 0.6353 0.6355

0.0458 0.0458 0.0459 0.0458 0.0476 0.0475 0.0465 0.0464 0.0469 0.0469

0.2208 0.2208 0.2205 0.2205 0.2153 0.2154 0.2187 0.2187 0.2173 0.2173

0.2789 0.2789 0.2788 0.2788 0.2730 0.2733 0.2776 0.2776 0.2763 0.2764

It is worth noting that the proposed unified polynomial function has endowed with a unified linkage for HSDTs models available in literature. It brings to readership a new idea for further researches on laminated composite plate analysis. As aforementioned discussion, the superiority of the proposed unified formulation is the flexibility in choosing the polynomial coefficients. Furthermore, it is essential to observe from Table 4 and Table 5 that although extensive research efforts have been made to propose various transverse shear functions, there is a relatively gap between all existing model and the benchmark results (exact 3D elasticity solution of Pagano [2, 3]), especially transverse shear stresses. Based on the idea of polynomial functions, in next Section, we illustrate a typical way to determine the transverse shear function of polynomial form. In fact, this choice is not a general case, and therefore the readership could utilize an optimization tool such as genetic algorithms, particle swarm optimization, etc, to find new optimal functions in future researches.

3. A novel approach to transverse shear function 3.1. On polynomial transverse shear function According to two observations of the preceding numerical examples, we propose the novel polynomial transverse shear function presented as the following form f (z) = z + d3 z 3 + d5 z 5 + d7 z 7 . . . (14) The traction boundary condition at top and bottom surfaces of plate is apparently the requisite requirement. Therefore, there are not much more two unknown 21

coefficients needed to be determined. For instant, let us consider the quintic polynomial as follows f (z) = z + d3 z 3 + d5 z 5 (15) Due to the zero-traction boundary condition at top and bottom of plate, Eq. (15) yields   3h2 16 d3 (16) d5 = − 4 1 + 5h 4 Consequently, there is only the value of d3 of the quintic polynomial in Eq. (15) that needs to be optimized to achieve the most accurate solution in comparison with the exact 3D solution of Pagano [2, 3]. Here we find out the optimal d3 to minimize the average error of symmetric cross-ply three layers [00 /900 /00 ] plates subjected a sinusoidal load. The exact 3D solutions are given in Table 4. The variation of average error with respect to d3 is illustrated in Fig. 5. The average error of two polynomial models derived from Reddy [20] (PHSDT [20]) and NguyenXuan et al. [44] (PHSDT [44]) in comparison with the present method is given in Fig. 5. It can be observed that the minimum average error is obtained as d3 is 17 equal to − . 10h2 In case of using 7th polynomial form, two coefficients d3 and d5 need to be optimized and the last one is computed by   3d3 64 5d5 (17) d7 = − 6 1 + 2 + 7h 4h 16h4 The variation of average error with respect to two coefficients d3 and d5 of symmetric cross-ply three layers [00 /900 /00 ] square plates is depicted in Fig. 6. 87 169 and , As a consequence, two coefficients d3 and d5 are decided as − 2 20h 10h4 respectively. Generally, in order to obtain an optimal polynomial function for multiobjective optimization of general layers and stacking sequences, the optimization tool should be strongly recommended. Eventually, with aforementioned ideas, we introduce four new transverse shear functions as follows • present 1: f (z) = z −

17 3 22 5 z + z 2 10h 25h4

• present 2: f (z) = z −

11 3 52 5 z + z 5h2 25h4 22

25

Average error (%)

20

present PHSDT [] PHSDT [4]

15

10

5

0 -5

-4

-3

-2

-1 d3/h2

0

1

2

3

d3 Figure 5: The variation of average error with respect to 2 of three layers [00 /900 /00 ] square h composites plates.

23

55 50

Average error (%)

45 40 35 30 25 20 20 19 18 17 4 16 d5/h 15

15 10 5 0 0

-1

-2

-3

-4

-5

d3/h2

-6

-7

-8

14

d3 d5 Figure 6: The variation of average error with respect to 2 and 4 of three layers [00 /900 /00 ] h h square composites plates.

24

• present 3: f (z) = z −

87 3 169 5 138 7 z + z − 6z 2 20h 10h4 5h

• present 4: f (z) = z −

11 3 2 8 7 z + 4 z5 + z 2 5h h 35h6

0.5 0.4 0.3 0.2

z/h

0.1 0 -0.1

LMR Nguyen-Xuan present 1 present 2 present 3 present 4

-0.2 -0.3 -0.4 -0.5 0

0.2

0.4

0.6

0.8

1

f'(z) Figure 7: The derivative of the novel polynomial transverse shear functions and two existing polynomial functions.

It is apparent that such proposed functions above are completely flexible by mean of polynomial coefficients. Thus, the readership could choose a better polynomial function for analysis of multilayer composite plates. Despite the fact that these functions are not defined by an optimization tool, the following results obtained are greatly accurate and reliable. In fact, using optimization tool is out of our research’s scope in this paper. 3.2. Numerical validations In order to show the accuracy of the present formulation, let us consider symmetric cross-ply three layers [00 /900 /00 ] and four layers [00 /900 /900 /00 ] square composite plates as tested in the previous section. Fig. 7 depicts the derivative of 25

the proposed functions in comparison to existing ones. It is conscious of the fact that all proposed models in Fig. 7 are normalized to be the same unit peak. It is interesting that the present model 3 behaves quite specially, expand at the top, bottom surface and contract at the centre which leads to the prediction of improving the accuracy of the present model 3. Let us consider the three layers [00 /900 /00 ] square composite plates. The loading type and boundary conditions are sinusoidal load and full simply supported, respectively. Table 6 presents the comparison of the deflection and five stresses term. Various researches on investigating laminated composite plates using zigzag theory (ZZ [57, 58, 8]), layer-wise theory (LW [59]), global-local theory using four-node quadrilateral and three-node triangular element (GL [42]) and exponential high order shear deformation theory (EHSDT [32]), hyperbolic high order shear deformation theory (HHSDT [39, 37]), trigonometric high order shear deformation theory (THSDT [27, 26]), polynomial high order shear deformation theory (PHSDT [60, 20]) are compared to the present results. The exact 3D elasticity solution of Pagano [2] are taken as the benchmark results. The normalized forms are used as given in the previous example. The percentage errors are computed for deflection and stresses in parenthesis and the average error is also computed in the last column. Table 6: Comparison of the normalized deflection and stresses of three layer [00 /900 /00 ] square composite plate subjected to sinusoidal load

Model Exact [2] ZZ [57] ZZ [58] ZZ [8] LW [59] GL-PT [42] GL-PQ [42] EHSDT [32]

w 2.0060 1.9646 (2.0638) 1.9597 (2.3081) 1.9254 (4.0179) 2.0140 (0.3988) 2.0724 (3.3101) 2.0557 (2.4776) 1.9427 (3.1555)

σ xx 0.7550 0.7843 (3.8808) 0.7819 (3.5629) 0.7770 (2.9139) 0.8310 (10.0662) 0.7669 (1.5762) 0.8435 (11.7219) 0.7910 (4.7682)

σ yy 0.5560 0.5220 (6.1151) 0.5195 (6.5647) 0.5145 (7.4640) 0.5490 (1.2590) 0.5764 (3.6691) 0.5610 (0.8993) 0.5010 (9.8921) 26

τ xy τ xz τ yz Avr. err. 0.0505 0.2560 0.2170 0.0515 0.2387 0.1901 (1.9802) (6.7578) (12.3963) 5.5323 0.0510 0.2335 0.1885 (0.9901) (8.7891) (13.1336) 5.8914 0.0505 0.2243 0.1797 (0.0000) (12.3828) (17.1889) 7.3279 0.0526 0.2330 0.2140 (4.1584) (8.9844) (1.3825) 4.3749 0.0523 0.2576 0.2152 (3.5644) (0.6250) (0.8295) 2.2624 0.0522 0.2569 0.2205 (3.3663) (0.3516) (1.6129) 3.4049 0.0520 0.2230 0.1930 (2.9703) (12.8906) (11.0599) 7.4561

HHSDT [39]

1.9470 (2.9412) HHSDT [37] 1.9550 (2.5424) THSDT [27] 1.9222 (4.1775) THSDT [26] 1.9515 (2.7168) PHSDT [44] 1.9405 (3.2652) PHSDT [20] 1.9218 (4.1974) present 1 1.9364 (3.4696) present 2 1.9417 (3.2054) present 3 1.9816 (1.2164) present 4 1.9410 (3.2403)

0.7920 (4.9007) 0.8079 (7.0066) 0.7330 (2.9139) 0.7955 (5.3642) 0.8056 (6.7020) 0.7340 (2.7815) 0.7573 (0.3046) 0.7950 (5.2980) 0.7538 (0.1589) 0.7966 (5.5099)

0.5010 (9.8921) 0.5015 (9.8022) 0.5020 (9.7122) 0.5020 (9.7122) 0.4994 (10.1799) 0.5020 (9.7122) 0.4995 (10.1619) 0.5064 (8.9209) 0.4992 (10.2158)

0.0524 (3.7624) 0.0532 (5.3465) 0.0500 (0.9901) 0.0526 (4.1584) 0.0526 (4.1584) 0.0507 (0.3960) 0.0522 (3.3663) 0.0523 (3.5644) 0.0523 (3.5644)

0.2280 (10.9375) 0.2438 (4.7656) 0.2020 (21.0938) 0.1974 (22.8906) 0.1945 (24.0234) 0.1830 (28.5156) 0.2739 (6.9922) 0.2639 (3.0859) 0.2583 (0.8984) 0.2636 (2.9688)

0.1950 (10.1382) 0.2019 (6.9585) 0.1830 (15.6682) 0.2331 (7.4194) 0.2295 (5.7604) 0.2027 (6.5899) 0.2020 (6.9124) 0.2036 (6.1751) 0.2019 (6.9585)

Comparing the average error, it indicates that all present results are in excellent agreement with the exact solution [2] for both fifth order and seventh order polynomial. As would be expected, the present model 3 attain the best solution among four present models and there is only GL-PQ model [42] can obtain the better results than the present model 3. Among all high order shear deformation models, all four present models gains the lower average errors and also obtain the more precise shear stress. The distribution of stresses through the plate’s thickness are plotted in Fig. 8 for the present model 3 and two existing polynomial transverse shear functions by [20, 44]. Obviously, the present model 3 not only obtains the most precise results at specific point but also provides the almost exact stress’s distribution through the plate’s thickness in comparison with 3D elasticity [2] as illustrated in Fig. 8. Since the shear stress τ xz and τ yz are computed by the equilibrium equation Eq. (13), there is no shear stress discontinuity phenomenon at interface between two adjacent layers. It should be noticed that the plotting of 3D solutions [2] are extracted from the paper of Wu et al. [42]. Next the static behaviours of a four layers [00 /900 /900 /00 ] square simply 27

7.0953 6.0703 9.0926 8.7103 9.0149 11.8315 4.5774 5.3383 3.4890 5.4096

0.5

0.5 3D Pagano LMR Nguyen-Xuan present 3

0.4

0.3

0.2

0.2

0.1

0.1 z/h

z/h

0.3

0

0

-0.1

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

3D Pagano LMR Nguyen-Xuan present 3

0.4

-0.5 -0.6

0.8

-0.4

-0.2

0

ZZ

3D Pagano LMR Nguyen-Xuan present 3

0.6

0.2

0.25

0.3

0.4 0.3 0.2

0.1

0.1 z/h

0.2

0

-0.1

-0.2

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -0.05 -0.04 -0.03 -0.02 -0.01

0

3D Pagano LMR Nguyen-Xuan present 3

0

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0.01 0.02 0.03 0.04 0.05

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  Figure 8: Stress distribution of three layers 00 /900 /00 square composite plate under sinusoidal load.

supported composite plate under sinusoidal load is investigated. Table 9 presents some referred results using zigzag theory (ZZ [57, 58, 8]), layer-wise theory ([61, 62, 63, 64, 65]),e.g. Although present model 3 is nominated as the best choice for three layers composite plates as discussed in previous example, for four layers the present model 4 attains the most accurate results, the average error is merely 1.8766%, even better than some LW and ZZ results as expressed in Table 7. The rest of all present models also agree very well to 3D solution and the present model 2 yields the better solution than other HSDT models even though it is a simple fifth polynomial function. Fig. 9 illustrates the distribution of stress through thickness of plates based on two existing polynomial function by [20, 44] and the present model 3 and 4. Table 7: Comparison of the normalized deflection and stresses of four layer [00 /900 /900 /00 ] square composite plate subjected to sinusoidal load

Model Exact [3] ZZ [57]

w 1.9540 1.9283 (1.3153) ZZ [58] 1.9273 (1.3664) ZZ [8] 1.9016 (2.6817) LW [61] 1.9060 (2.4565) LW [62] 1.9075 (2.3797) LW [63] 1.9091 (2.2979) LW [64] 1.8931 (3.1167) LW [65]) 1.9366 (0.8895) HHSDT [39] 1.9247 (1.4995) HHSDT [28] 1.9257 (1.4483) THSDT [27] 1.8940 (3.0706)

σ xx 0.7200 0.7115 (1.1806) 0.7089 (1.5417) 0.7149 (0.7083) 0.6419 (10.8472) 0.6432 (10.6667) 0.6429 (10.7083) 0.6408 (11.0000) 0.7201 (0.0153) 0.7140 (0.8333) 0.7255 (0.7639) 0.6640 (7.7778)

σ yy τ xy τ xz τ yz 0.6630 0.0467 0.2190 0.2910 0.6510 0.0474 0.2362 0.2758 (1.8100) (1.4989) (7.8539) (5.2234) 0.6454 0.0490 0.2332 0.2746 (2.6546) (4.9251) (6.4840) (5.6357) 0.6391 0.0467 0.2366 0.2913 (3.6048) (0.0000) (8.0365) (0.1031) 0.6257 0.0443 0.2169 (5.6259) (5.1392) (0.9589) 0.6228 0.0441 0.2166 (6.0633) (5.5675) (1.0959) 0.6265 0.0443 0.2173 (5.5053) (5.1392) (0.7763) 0.8506 0.0436 0.2160 (28.2956) (6.6381) (1.3699) 0.6605 0.0467 0.2194 0.2915 (0.3710) (0.0000) (0.1689) (0.1821) 0.6370 0.0467 0.2350 0.2580 (3.9216) (0.0000) (7.3059) (11.3402) 0.6390 0.0472 0.2500 0.2698 (3.6199) (1.0707) (14.1553) (7.2852) 0.6310 0.0440 0.2060 0.2390 (4.8265) (5.7816) (5.9361) (17.8694) 29

Avr. err.

3.1470 3.7679 2.5224 5.0056 5.1546 4.8854 10.0841 0.2711 4.1501 4.7239 7.5437

THSDT [28] THSDT [26] PHSDT [44] PHSDT [20] present 1 present 2 present 3 present 4

1.9262 (1.4227) 1.9258 (1.4432) 1.9218 (1.6479) 1.8939 (3.0757) 1.9114 (2.1801) 1.9220 (1.6377) 1.9247 (1.4995) 1.9218 (1.6479)

0.7210 (0.1389) 0.7164 (0.5000) 0.7274 (1.0278) 0.6806 (5.4722) 0.6842 (4.9722) 0.7173 (0.3750) 0.6673 (7.3194) 0.7189 (0.1528)

0.6386 (3.6802) 0.6381 (3.7557) 0.6355 (4.1478) 0.6463 (2.5189) 0.6336 (4.4344) 0.6345 (4.2986) 0.6435 (2.9412) 0.6343 (4.3288)

0.0471 (0.8565) 0.0467 (0.0000) 0.0470 (0.6424) 0.0450 (3.6403) 0.0451 (3.4261) 0.0466 (0.2141) 0.0456 (2.3555) 0.0467 (0.0000)

0.2442 (11.5068) 0.2396 (9.4064) 0.2371 (8.2648) 0.2109 (3.6986) 0.2229 (1.7808) 0.2181 (0.4110) 0.2215 (1.1416) 0.2180 (0.4566)

0.2654 (8.7973) 0.2624 (9.8282) 0.2589 (11.0309) 0.2390 (17.8694) 0.2798 (3.8488) 0.2775 (4.6392) 0.2721 (6.4948) 0.2774 (4.6735)

4. An enhancement with thickness stretching effect This section aims to additionally investigate the effect of thickness stretching based on the quasi-3D theory with an arbitrary function f (z) from the previous f (z) sections and the function g (z) = . Without the loss of generality, we only ildz lustrate the performance of the present approach using the function f (z) of HSDT model 3 derived from the previous section f (z) = z −

87 3 169 5 138 7 z + z − 6z 2 20h 10h4 5h

(18)

In order to assess the effect of thickness stretching, let us consider the example of symmetric cross-ply three layers [00 /900 /00 ] in the previous Section. The effect of thickness stretching is presented in Table 8 by comparing the normalized deflection and stresses. As shown in Table 8, considering the thickness stretching effect provides the excellent agreement of transverse shear stresses τxz and τyz . Also, the average error given by the quasi-3D theory is smaller than one obtained by the HSDT model 3.

30

4.4004 4.1556 4.4603 6.0459 3.4404 1.9293 3.6253 1.8766

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Figure 9: Stress distribution of four layers [00 /900 /900 /00 ] square composite plate under sinusoidal load.

0.25

Table 8: The effect of thickness stretching on deflection and stresses of three layer [00 /900 /00 ] square composite plate subjected to sinusoidal load

Model Exact [2] present HSDT 3

w 2.0060 1.9816 (1.2164) present quasi-3D 1.9505 (2.7661)

σ xx 0.7550 0.7538 (0.1589) 0.7554 (0.0562)

σ yy 0.5560 0.5064 (8.9209) 0.4939 (11.1733)

τ xy 0.0505 0.0523 (3.5644) 0.0515 (1.9733)

τ xz τ yz Avr. err. 0.2560 0.2170 0.2583 0.2036 (0.8984) (6.1751) 3.4890 0.2567 0.2138 (0.2568) (1.4649) 2.9484

5. Conclusions We have presented a novel unified framework of HSDTs for laminated composite plate structures. All existing transverse shear functions of HSDT models are mathematically represented by a unique polynomial formulation regardless of the complexity of these functions. In other words, a polynomial form is completely sufficient to consider the nonlinear and parabolic distribution of transverse shear stresses. The weak form is constructed by using the principle of virtual displacements and the numerical results are obtained by using the state-of-the-art of isogeometric analysis. The numerical results show the accuracy and reliability of the unified formulation. It is significant that the proposed unified polynomial is potentially capable to provide the readership a general perspective on the close connection of all transverse shear function existing in the literature. Based on a unified formulation, the unknown coefficients of a polynomial function can be optimized to obtain the more accurate results. Therefore, in order to bring to the readership a new viewpoint on constructing a new higher order shear deformation theory, we depicted a typical way to devise several new transverse shear functions. The quasi-3D theory is developed based on the new transverse shear function to involve the normal deformation in the displacement field. The proposed transverse shear functions and the proposed quasi-3D theory anticipate the better agreement to 3D elasticity solution in comparison with other HSDT models, even yield the more accurate results compared to some zigzag theories or layer-wise theories. Finally, for further researches, the paper opens more promising ideas for finding optimal transverse shear functions under the polynomial form. Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2014.24. 32

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