ISOGEOMETRIC APPROACH FOR STATIC ANALYSIS OF LAMINATED COMPOSITE PLATES PHƯƠNG PHÁP ĐẲNG HÌNH học CHO PHÂN TÍCH TĨNH tấm COMPOSITE NHIỀU lớp

ISOGEOMETRIC APPROACH FOR STATIC ANALYSIS OF LAMINATED COMPOSITE PLATES PHƯƠNG PHÁP ĐẲNG HÌNH HỌC CHO PHÂN TÍCH TĨNH TẤM COMPOSITE NHI ỀU LỚP Nguyen Thi Bich Lieu1a, Nguyen Xuan Hung2

ISOGEOMETRIC APPROACH FOR STATIC ANALYSIS OF LAMINATED

COMPOSITE PLATES

PHƯƠNG PHÁP ĐẲNG HÌNH HỌC CHO PHÂN TÍCH TĨNH TẤM COMPOSITE

NHI ỀU LỚP

Nguyen Thi Bich Lieu1a, Nguyen Xuan Hung2b

1HCMC University of Technology and Education, Vietnam

2Vietnamese-German University, Binh Duong Province, Vietnam

a

lieuntb@hcmute.edu.vn; b

hung.nx@vgu.edu.vn

ABSTRACT

A generalized higher-order shear deformation theory for static analysis of laminated composite plates using isogeometric analysis (IGA) is presented The present theory not only

is derived from the classical plate theory (CPT) but also includes the first-order shear deformation theory (FSDT) The displacement field depends on arbitrary distributed function The shear locking phenomenon can be ignored and hence the shear stress free surface conditions are naturally satisfied Although it has same number of degrees of freedom as the FSDT, it does not require shear correction factors Galerkin weak form of static analysis model for laminated composite plates is used to obtain the discreted system of equations It can be solved by isogeometric approach based on the non-uniform rational B-splines (NURBS) basic functions Two numerical examples of the laminated composite plates including symmetric and non-symmetric with various boundary conditions are presented to illustrate the effectiveness of the proposed method compared to other methods reported in the literature

Keywords: Laminated composite plates, isogeometric analysis (IGA), higher-order

shear deformation theory, NURBS, static analysis

TÓM T ẮT

Một lý thuyết biến dạng cắt tổng quát bậc cao cho phân tích tĩnh tấm composite nhiều

lớp sử dụng phương pháp phân tích đẳng hình học (IGA) được đưa ra Lý thuyết hiện tại không chỉ được suy ra từ lý thuyết cổ điển (CPT) mà còn bao gồm lý thuyết biến dạng cắt bậc

nhất (FSDT) Nó được định nghĩa sao cho trường chuyển vị phụ thuộc vào hàm phân bố bất

kỳ Không có hiện tượng khóa cắt và vì thế điều kiện ứng suất cắt bằng không tự nhiên được

thỏa mãn Mặc dù lý thuyết đề xuất có cùng bậc tự do với FSDT nhưng nó không yêu cầu hệ

số hiệu chỉnh cắt Dạng yếu Galerkin của mô hình phân tích tĩnh cho tấm composite nhiều lớp được sử dụng để đạt được hệ thống các phương trình rời rạc Nó được giải quyết bằng phương pháp đẳng hình học dựa trên các hàm cơ sở NURBS Hai ví dụ số của tấm composite nhiều

lớp bao gồm tấm đối xứng và tấm bất đối xứng với các điều kiện biên khác nhau được đưa ra

để minh họa hiệu quả của phương pháp đề xuất so với các phương pháp khác

Từ khóa: tấm composite nhiều lớp, phân tích đẳng hình học (IGA), lý thuyết biến dạng

cắt bậc cao, hàm NURBS, phân tích tĩnh

1 INTRODUCTION

Due to its dominant role in many engineering structures and modern industries, laminated composite and sandwich plates are widely used in adiverse array of structures in areas such as aviation, shipbuilding, civil engineering and so on [1] Therefore, the development of efficient and reliable mathematical models, deformation theories, and analysis

methods to predict the short and long-term behavior of the multilayer composite structures is extremely important

A large number of plate theories have been developed to significantly contribute to advances in computational mechanics of the plate problem For instance, the classical plate theory (CPT) [2] is a highly effective means of analyzing of thin plates with no accounting transverse shear strains Emergence of the first-order shear deformation theory (FSDT) [3] has generally been viewed as an improvement over the CPT for both moderately thick and thin plates which takes shear effect into account The generalized displacement field in FSDT is quite simple However, a shear correction factor (SCF) is required to rectify the unrealistic shear strain energy component To overcome the limitations of the FSDT, the higher-order shear deformation theories (HSDTs) have been then developed These theories disinterest SCFs and give more accurate and stable solutions (e.g., inter-laminar stresses and displacements) than the FSDT ones These higher-order shear deformation theories include, third-order shear deformation theory (TSDT) [4], fifth-order shear deformation theory (FiSDT) [5], trigonometric shear deformation theory (TrSDT) [6], exponential shear deformation theory (ESDT) [7]and others The HSDT model requires C1-continuity of generalized displacement field leading to the second-order derivative of the stiffness formulation and thus causes difficulties in the standard finite element formulations In recent years, C1-continous elements based on mesh free method [8] were proposed to solve the plate and shell problems In this paper, we demonstrate that the C1-continuous elements can be easily achieved by adopting isogeometric analysis without any additional variables

As we knew, finite element method (FEM) is an efficient computational tool for various classes of engineering problems and is the reliable choice for solving partial differential equations in the complex domains However, it also existssomedis advantages depended on

meshing process Therefore, Hughes et al.developed a highly effective numerical technique

so called isogeometric analysis (IGA)[9] that is capable of integrating finite element analysis (FEA) into conventional NURBS-based CAD design tools Following a decade of development, isogeometric analysis has surpassed the standard finite elements in terms of effectiveness and reliability for problems from simple to complex In IGA, the non-uniform rational B-splines (NURBS) basis functions are used not only for geometric description but also for approximation of the displacement field, subsequently allowing us to describe the curved geometry precisely by using only a few elements and yielding solutions with a higher accuracy Some remarkable references for plates can be listed as Kirchhoff plates [10], Mindlin-Reissner plates [11], plates based on HSDT [12] and plates based on the layer wise theory [13]

In this paper, a higher-order displacement fields in which displacement field is defined

in general form of distributed functions varying across the plate thickness The finite element formulation based on the HSDT requires elements with at least C1-inter-element continuity It

is difficult to achieve such elements for free-form geometries when using the standard Lagrangian polynomials as basis functions However, in IGA can be easily obtained because NURBS basis functions are Cp-1 continuous Two numerical examples are provided to illustrate the effectiveness and reliability of the present method in comparison with other results from the literature

2 A GENERALIZED HIGHER-ORDER SHEAR DEFORMATION PLATE THEORY

Figure 1 Plate geometry and coordinate system

Let Ω be the domain in R2 occupied by the mid-plane of the plate and u0, v0, w and β

=(βx;βy)T denote 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 Figure 1 shows the geometry of plate

and coordinate system A higher-order shear deformation theory derived from the classical plate theory is defined as follows [14]:

0

0

, ,

, ,

x

y

w x y t

x

w x y t

y

w x y z t w x y t

β β

∂

∂

∂

∂

=

;

z

−

 ≤ ≤ 

(1)

where f (z) is shape function determining the distribution of the transverse shear strains

and stresses through the thickness of plates This distributed function is chosen so that tangential stress-free boundary conditions at the top and bottom surfaces of the plates are satisfied

In the present formulation, if distributed function f(z) is chosen to be equal zero, the

higher-order shear deformation theory will take the form of classical plate theory (CPT) (see

in Eq.(2))

0

0

( , , ) ( , )

( , , ) ( , )

( , , ) ( , )

w

u x y z u x y z

x w

v x y z v x y z

y

w x y z w x y

∂

∂

∂

∂

=

;

z

−

≤ ≤

(2)

By defining f(z) = z and substituting φ = −x w,x + βx into Eq (1), the first order shear deformation theory (FSDT) is obtained as

0 0

( , , ) ( , )

( , , ) ( , )

( , , ) ( , )

x

y

u x y z u x y z

v x y z v x y z

w x y z w x y

φ φ

=

;

z

−

≤ ≤

In this paper, we use Eq.(1) with some functions f(z) in Table 1

Table 1:Various forms of shape function and its derivatives

2

4 3

z z h

h

−  

 

 

Arya (TrSDT) [6]

sin z

h

π

z c

π π 

2 z

h ze

 

 

 

 

1 4

z

e h

 

 

 

−

−

 

Nguyen (FiSDT) [5]

2 4

8

2 4

8

tan ( )

h

The in-plane strain vectorεpis thus expressed by the following equation

p= ε ε γxx yy xy = +z f z

and the transverse shear strain vector γ has the following form

' 1 ( )

xz yz f z

where f z' ( )is derivative of f z( ) function and

0,

0 0,

0, 0,

x

y

x y

u v

,

, 2

xx

yy

xy

w w w

 − 

= − 

− 

,

, ,

x x

y y

y x x y

β β

y

β β

 

=  

 

By neglecting σz for each orthotropic layer, the constitutive equation of k th layer in the local coordinate system derived from Hooke’s law for a plane stress is given by

66

55 54

45 44

Q

(7)

in which reduced stiffness components, Q ij k, are expressed by

ν

where E1,E2 are the Young’s modulus in the 1 and 2 directions, respectively;

12, 13, 23

G G G are the shear modulus in the 1-2, 3-1, 2-3 planes, respectively and νij are the Poisson’s ratios

The stress - strin relationship in the global reference system (x,y,z) is computed by

11 12 16

12 22 26

61 62 66

55 54

45 44

(9)

where Q k ijis the transformed material constant matrix

A weak form of the static model for the plates under transverse loading q0 can be written as

0

whereq0is the transverse loading per unit area and

A B E

D = B D F

E F H

(11)

in which

/ 2 ( ij, ij, ij, ij, ij, ij) h (1, , , ( ), ( ), ( )) ijd

h

A B D E F H z z f z zf z f z Q z

−

/ 2 2 ' / 2 ( ij) h ( ) d

h

−  

=∫   i j, = 4,5

(12)

GENERALIZED HIGER-ORDER SHEAR DEFORMATION THEORY

By using the NURBS basis functions defined in [9,14], both the description of the

geometry (or the physical point) and the displacement field u of the plate are approximated as

follows

m n h

A R

ξ η =∑× ξ η

m n h

A R

ξ η =∑× ξ η

where n×m is the number of basis functions, and T ( )

x y

=

x is the physical coordinate vector

InEq.(13),R A( )ξ η, is a rational basic function, PA is the control point and

0 0

T

A=  u A v A w A βxA βyA 

control point A

By substituting Eq (13)with Eq.(4), the in-plane and shear strains can be rewritten as

1 2 1

m n

A

×

=

in which

, ,

, , , ,

0 0 0 0 , 0 0 0 0 ,

, 0 0 0 0

0 0 0 0

0 0 0

A x A

A

A y A x

R R

R R

−

−

(15)

By substituting Eq (14)with Eq.(10),the formulation of static analysis is obtained in the following form

where the global stiffness matrix K is given by

T

Ω

(17)

and the load vector F is calculated as

0 d

q

Ω

in which

0 0 R A 0 0

=

4 RESULTS AND DISCUSSIONS

This section, due to the limitation of number of pages of the conference, we consider two examples through a series of benchmark problems for laminated composite plates including one for symmetric plate and one for anti-symmetric plate

4.1 Four-layer [0 0 /90 0 /90 0 /0 0 ] square laminated plate under a sinusoidally distributed load

A four-layer fully simply supported square laminated plate subjected to a sinusoidal pressure defined asq(x, y) q0sin( x)sin( y)

= is considered, as shown in Figure 2 All layers

of the laminated plate are assumed to be of the same thickness and made of the same linearly

elastic composite materials The length to width ratio is a/b = 1 and the length to thickness ratios are a/h = 4, 10, 20 and 100, respectively Material is used

1 25 2 , 12 13 0.5 2 , 23 0.2 2 , 12 0.25.

E = E G =G = E G = E ν =

The normalized displacement and stresses are defined as

2 2

(100 ) ( , , 0) / ; ( , , ); ( , , )

(0, 0, ); (0, , 0); ( , 0, 0)

Figure 2 Geometry of a laminated plate under a sinusoidally distributed load

Table 2: Normalized displacement and stresses of a simply supported [0 0 /90 0 /90 0 /0 0 ]

square laminated plate under a sinusoidally distributed load

4 Exact 3D [15] 1.9540 0.7200 0.6660 0.0467 0.2700 0.2910

IGA-Reddy[4] 1.8936 0.6607 0.6300 0.0440 0.2064 0.2389

IGA-Arya[6] 1.9088 0.6796 0.6332 0.0450 0.2162 0.2462

IGA-Thai[14] 1.9258 0.7164 0.6381 0.0467 0.2396 0.2624

IGA-Soldatos[12] 1.8920 0.6644 0.6316 0.0439 0.2055 0.2382

10 Exact 3D [15] 0.7430 0.5590 0.4030 0.0276 0.3010 0.1960

IGA-Reddy[4] 0.7147 0.5440 0.3881 0.0267 0.2640 0.1530

IGA-Arya[6] 0.7198 0.5486 0.3905 0.0270 0.2787 0.1588

IGA-Thai[14] 0.7272 0.5552 0.3937 0.0273 0.3133 0.1704

IGA-Soldatos[12] 0.7142 0.5449 0.3881 0.0267 0.2627 0.1526

20 Exact 3D [15] 0.5170 0.5430 0.3090 0.0230 0.3280 0.1560

IGA-Reddy[4] 0.5060 0.5383 0.3038 0.0228 0.2825 0.1234

IGA-Arya[6] 0.5070 0.5395 0.3090 0.0228 0.2989 0.1272

IGA-Thai[14] 0.5098 0.5412 0.3058 0.0229 0.3372 0.1366

IGA-Soldatos[12] 0.5059 0.5385 0.3038 0.0228 0.2810 0.1231

100 Exact 3D [15] 0.4347 0.5390 0.2710 0.0214 0.3390 0.1410

IGA-Reddy[4] 0.4342 0.5379 0.2704 0.0213 0.2897 0.1116

IGA-Arya[6] 0.4344 0.538 0.2705 0.0213 0.3069 0.1148

IGA-Thai[14] 0.4345 0.538 0.2705 0.0213 0.3467 0.1229

IGA-Soldatos[12] 0.4343 0.5379 0.2704 0.0213 0.2882 0.1114

Figure 3 Comparison of the normalized stress distributions through the thickness of a

four-layer [0 0 /90 0 /90 0 /0 0] laminated composite square plate (a/h = 4)

Table 2 displays the obtained results using various distributed functions with IGA for

the normalized displacement and stresses The obtained results including some f(z) functions

of Reddy[4] (IGA-Reddy), Arya [6] (IGA-Arya), Thai et al.[14] (IGA-Thai), Soldatos [12] (IGA- Soldatos) and the exact 3D elasticity approach of Pagano [15] According to this table,

IGA conforms well to the exact 3D elasticity solution for all ratios a/h, especially for thick plates For a thick plate with a/h = 4 and 10, the IGA-Thai [14] is accurate than other

solutions using IGA It even moves beyond TSDT by Reddy [4] Figure 3 plots the

distribution of stresses through the thickness of a four-layer square plate with a/h = 4

Obviously, obtained results have the similarity with each other

4.2 Two-layer [0 0 /90 0 ] square laminated plate under a sinusoidally distributed load

Let us consider a two-layer (00/900) squareplate subjected to sin load and material as

same above example witha/b = 1; a/h = 10 Three boundary conditions are presented SCSC,

SSSS và SFSF, where S = simply supported, C= clampedand F = free

The normalized displacement is defined as:

/ 2, / 2, 0

100 w a a

qa

Table 3: Normalized displacement w and stresses of a [00 /90 0 ] square laminated plate

under a sinusoidally distributed load with various boundary conditions

IGA-Reddy[4] 0.6146 -0.4917 0.3778 0.0183 0.1530

IGA-Nguyen[5] 0.5971 -0.5009 0.3733 0.0075 0.1472

IGA-Reddy[4] 1.2161 -0.7446 0.7446 0.0533 0.2837

IGA-Nguyen[5] 1.2044 -0.7492 0.7492 0.0534 0.2766

IGA-Reddy[4] 1.9925 -0.2606 1.2262 0.0098 0.3993

IGA-Nguyen[5] 1.9736 -0.2632 1.2315 0.0032 0.3872

Figure 4: The normalized stress distributions through the thickness of a two-layer [0 0 /90 0] laminated composite square plate (a/h = 10) with IGA-Reddy

Table 3 displays the obtained results from IGA-Reddy, IGA-Nguyen and IGA-Thai These results are compared with analytical solutions of Vel et al [16] It can be seen that the normalized displacement and stresses are close to those of the analytical solutions for all various boundary conditions, especially IGA-Reddy solution Figure 4 illustrates the

normalized stress distributions through the thickness with a/h = 10 using IGA-Reddy for three

differential boundary conditions

5 CONCLUSIONS

This work presents an isogeometric finite element method for static of laminated composite plates The results obtained using the generalized higher-order shear deformation theory showed high reliability and matched well for all test cases from thin to thick plates when compared with analytical solutions and exact three-dimensional elasticity As described

in this paper, the distributed function f(z) along the thickness of plate can be easily modified

to obtain the most optimum solution for the targeted engineering problem The choice of the

distributed function f(z) is still an open question The authors want to emphasize the

higher-order continuity of NURBS basic functions and the flexibility and generality of generalized displacements when combining IGA with arbitrary distribution functions in this paper

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