Luận văn thạc sĩ Cơ kỹ thuật: Phân tích đánh giá ổn định động của hệ thống điện khi có tác động của nguồn năng lượng điện gió

For this reason, theradial point interpolation method RPIM is used for modelling thin Kirchhoff-Love plate in this thesis.. 284.2 Non-dimensional frequency parameter ¯ω of a simply suppo

ĐẠI HỌC QUỐC GIA TP HCM

TRƯỜNG ĐẠI HỌC BÁCH KHOA

-

LỒ SÌU VẪY

FRACTURE ANALYSIS IN THIN PLATE USING KIRCHHOFF-LOVE PLATE THEORY AND AN

EXTENDED MESHFREE METHOD

Chuyên ngành: Cơ kỹ thuật

Mã số: 8520101

LUẬN VĂN THẠC SĨ

TP HỒ CHÍ MINH, tháng 7 năm 2021

CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI TRƯỜNG ĐẠI HỌC BÁCH KHOA –ĐHQG -HCM

Cán bộ hướng dẫn khoa học : PGS TS Trương Tích Thiện

5 TS Trương Quang Tri

Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý chuyên ngành sau khi luận văn đã được sửa chữa (nếu có)

ĐẠI HỌC QUỐC GIA TP.HCM CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM TRƯỜNG ĐẠI HỌC BÁCH KHOA Độc lập - Tự do - Hạnh phúc

NHIỆM VỤ LUẬN VĂN THẠC SĨ

Họ tên học viên: Lồ Sìu Vẫy MSHV: 1970507

Ngày, tháng, năm sinh: 16/11/1997 Nơi sinh: Đồng Nai

Chuyên ngành: Cơ kỹ thuật Mã số: 8520101

NHIỆM VỤ VÀ NỘI DUNG:

1 Nghiên cứu lý thuyết tấm Kirchhoff-Love

2 Nghiên cứu phương pháp không lưới RPIM

3 Phát triển công thức XRPIM dành cho tấm Kirchhoff-Love bị nứt

4 Phát triển một chương trình giải các bài toán tấm Kirchhoff-Love bằng phương pháp RPIM

5 Phát triển chương trình để phân tích ứng xử dao động tự do của tấm Kirchhoff-Love bị nứt

II NGÀY GIAO NHIỆM VỤ: 22/02/2021

III NGÀY HOÀN THÀNH NHIỆM VỤ: 05/12/2021

IV.CÁN BỘ HƯỚNG DẪN: PGS TS Trương Tích Thiện

I would like to express my endless thanks and gratefulness to my supervisor Assoc.Prof Dr Thien Tich Truong, Dr Nha Thanh Nguyen and Dr Minh Ngoc Nguyenfor their supports and advices during the process of completion of my thesis Without theirinstructions, the thesis would have been impossible to be done effectively

I really appreciate the lecturers of Department of Engineering Mechanics for theircomments and helps while I am doing this thesis, which leads me to the right direction Ialso acknowledge the support of time and facilities from Ho Chi Minh City University ofTechnology (HCMUT), VNU-HCM for this thesis

Last but not least, a special thanks to my parents for their love, care and have mostassistances and motivation me for the whole of my life

Tóm tắt luận văn

Đối với các bài toán tấm chịu uốn, sử dụng lý thuyết tấm để mô tả cấu trúc tấm mỏng sẽ ít tốnkém hơn so với sử dụng mô hình 3D Trong các lý thuyết tấm thì lý thuyết Kirchhoff-Loverất thích hợp để phân tích cấu trúc tấm mỏng Nếu bỏ qua các bậc tự do trong mặt phẳng củatấm thì mỗi nút chỉ có một bậc tự do - đó là độ võng Vì lý do đó, các thành phần của trườngchuyển vị chỉ tính theo độ võng Tuy nhiên, phương pháp phần tử hữu hạn cổ điển (FEM)cần các phép biến đổi toán học phức tạp để xây dựng một phần tử mới thỏa mãn các yêu cầucủa lý thuyết Kirchhoff-Love Vì vậy, phương pháp nội suy điểm hướng kính (RPIM) được

sử dụng để mô phỏng tấm mỏng Kirchhoff-Love trong luận văn này Bên cạnh đó, việc phântích các kết cấu bị nứt rất quan trọng vì nó liên quan đến tuổi thọ của kết cấu Do đó, luậnvăn này sử dụng phương pháp nội suy điểm hướng kính mở rộng (XRPIM) để khảo sát daođộng tự do của tấm Kirchhoff-Love bị nứt XRPIM được phát triển dựa trên RPIM nên yêucầu về đạo hàm cấp hai trong lý thuyết Kirchhoff-Love được xử lý một cách dễ dàng Kết quả

mô phỏng số từ nghiên cứu này được so sánh với các kết quả của các tác giả khác đã công bố

để kiểm chứng tính chính xác của phương pháp

For the plate bending problems, using a plate theory for modelling thin plate structure is lesscomputational cost than modelling it in 3D The Kirchhoff-Love plate theory is appropriatefor analysing thin plate structures If the membrane deformation is ignored in the Kirchhoff-Love plate, each node has only one degree of freedom – the deflection For that reason, thecomponents of the displacement field are calculated only in terms of deflection The clas-sical finite element method (FEM), however, needs complex mathematical transformations

to formulate a new element that satisfies the Kirchhoff-Love theory For this reason, theradial point interpolation method (RPIM) is used for modelling thin Kirchhoff-Love plate

in this thesis Besides, the analysis of cracked structures is important because it is related

to the lifetime of the structures Therefore, this thesis employs the extended radial point terpolation method (XRPIM) to investigate the free vibration of the cracked Kirchhoff-Loveplate The XRPIM is based on RPIM so the requirement for calculating the second-orderderivative in the Kirchhoff-Love theory is easily done The numerical results from this studyare compared with other published results to verify the accuracy of the method

I hereby declare that this master thesis represents my own work which has been doneafter registration for the degree of Master in Engineering Mechanics at Ho Chi Minh cityUniversity of Technology, VNU – HCM, Viet Nam and has not been previously included

in a thesis or dissertation submitted to this or any other institution for a degree, diploma orother qualifications

Author

List of Abbreviations and Nomenclatures x

1.1 State of the Art 1

1.2 Scope of study 2

1.3 Research objectives 3

1.4 Author’s contributions 3

1.5 Thesis outline 3

2 Methodology 4 2.1 The Kirchhoff-Love plate theory 4

2.1.1 Equilibrium equation 5

2.1.2 Constitutive equation 7

2.1.3 Governing equation 9

2.1.4 Finite element approximation 10

2.2 The Radial Point Interpolation Method 14

2.2.1 Brief introduction to the RPIM 14

2.2.2 RPIM shape functions construction 14

2.3 The extended RPIM for the cracked Kirchhoff-Love plate 19

3 Implementation 23 3.1 Compute stiffness matrix and mass matrix 23

3.2 Compute strain computing matrix 25

4 Numerical Results 27

4.1 Square plate under uniform pressure 27

4.2 Plate with a central crack 29

4.2.1 Simply supported square plate 29

4.2.2 Simply supported rectangular plate 31

4.2.3 Clamped circular plate 34

4.3 Plate containing a side crack 38

4.3.1 Simply supported square plate 38

4.3.2 Simply supported rectangular plate 38

4.3.3 Clamped annular plate 42

4.4 Square plate with an oblique crack 42

4.4.1 Central crack 44

4.4.2 Side crack 49

5 Conclusion and outlook 51 5.1 Conlusions 51

5.2 Future works 52

List of Figures

2.1 Internal and external forces on the plate 5

2.2 Plate before and after deformation 7

2.3 A rectangular element with three degrees of freedom w, ∂w ∂x1 , ∂w ∂x2 per node 11 2.4 Discrete nodes (gray dots) and support domains 15

2.5 Pascal triangle of monomials for 2D case 16

2.6 Split nodes and tip nodes around the crack curve 20

2.7 Illustration of tangential and normal direction for a crack 21

2.8 Global coordinate system and local coordinate systems 21

3.1 Algorithm for computing K and M matrices 24

3.2 Algorithm for computing B and N matrices 26

4.1 Geometry of the square plate and the coordinate system 28

4.2 Convergence of the result 29

4.3 Deflection of the simply supported square plate 30

4.4 Deflection at the center line y = 0 30

4.5 Geometries of a square plate and a rectangular plate containing a central crack 31 4.6 Mode shapes of five lowest modes of a square plate containing a central crack 33 4.7 Mode shapes of five lowest modes of a rectangular plate containing a central crack 35

4.8 A circular plate containing a central crack 35

4.9 Discrete model of the circular plate 36

4.10 Mode shapes of five lowest modes of a circular plate containing a central crack 37 4.11 Geometry of square plate and rectangular plate with a side crack 38

4.12 Mode shapes of five lowest modes of a square plate with a side crack 40

4.13 Mode shapes of five lowest modes of a rectangular plate with a side crack 40

4.14 A annular plate containing two symmetric side cracks 42

4.15 Mode shapes of five lowest modes of a rectangular plate containing a centralcrack 434.16 Geometry of square plate with a central oblique crack and a side oblique crack 444.17 Mode shapes of five lowest modes of a square plate containing a centraloblique crack 444.18 Node distribution Left: Not aligned with the crack, Right: Aligned with thecrack 484.19 Mode shapes of five lowest mode of a square plate containing a side obliquecrack 50

List of Tables

2.1 Some radial basis functions 164.1 Convergence of the non-dimensional deflection of different methods 284.2 Non-dimensional frequency parameter ¯ω of a simply supported square platecontaining a central crack 324.3 Non-dimensional frequency parameter ¯ω of a simply supported rectangularplate containing a central crack 344.4 Non-dimensional frequency parameter ¯ω of the clamped circular plate with

a central crack 374.5 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a side crack 394.6 Non-dimensional frequency parameter ¯ω of a simply supported rectangularplate with a side crack 414.7 Non-dimensional frequency parameter ¯ω of the clamped annular plate withtwo side cracks 434.8 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a central oblique crack 454.9 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a central oblique crack 464.10 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a central oblique crack 474.11 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a central crack The c/a ratio is equal 0.6 and α = 45o 484.12 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a side oblique crack 494.13 Non-dimensional frequency parameter ¯ω of a simply supported square platewith a side oblique crack 50

List of Abbreviations and Nomenclatures

Abbreviation

2D Two dimensional

3D Three dimensional

CS-DSG3 Cell-based smoothed discrete shear gap method using triangular element

DDM Domain Decomposition Method

DOF Degree Of Freedom

FEM Finite Element Method

FSDT First-order Shear Deformation Theory

HSDT Higher-order Shear Deformation Theory

MQ Multiquadrics

PIM Point Interpolation Method

RBF Radial Basis Function

RPIM Radial Point Interpolation Method

SBTP4 Strain-based triangular plate with four nodes

TPS Thin Plate Spline

XCS-DSG3 Extended cell-based smoothed discrete shear gap method using triangular

ele-ment

XFEM Extended Finite Element Method

XRPIM Extended Radial Point Interpolation Method

ω Free vibation frequency

∂x1, ∂x2, ∂x3 Partial derivative in x1, x2, x3 directions

W The set containing all the nodes

Ws The set containing the nodes in the support domain which is cut by the crack

Wt The set containing the crack tip

θ Shape parameter

D Bending stiffness

E Young’s modulus

f (x) Sign distance funtion

G(x) Tip enrichment funtion

h Plate’s thickness

H(x) Heaviside funtion

Mij Moment resultant

mij Moment per unit length

n Number of nodes inside the support domainp(x) Polynomial basis funtion

Px3 Lateral force

Qi Shear force resultant

qi Shear force per unit length

R(x) Radial basis funtion

ui Displacement field

w Transverse deflection

x1, x2, x3 Cartesian coordinate

Chapter 1

Introduction

Plate and shell structures are very common in practice, its application can be mentioned as:ship’s hull, fluid container, aircraft fuselage and so on Therefore, investigating the mech-anical behavior of thin-walled structures is really important In addition to this, modellingplate and shell structure by using an appropriate plate theory can be more efficient than con-sidering it as a 3D model There are many types of plate theories can be employed to reduce

a 3D model to a 2D plate, such as the Kirchhoff-Love plate theory [1, 2, 3, 4, 5, 6, 7, 8],the Reissner-Mindlin plate theory (also known as the First-order Shear Deformation The-ory) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18], the Higher-order Shear Deformation Theory[19, 20, 21, 22, 23] In the scope of this thesis, only thin plate structures are examined.Hence, the Kirchhoff-Love plate theory which is widely used in the analysis of thin platestructure, is appropriate for this study

Besides, fracture analysis is also an important task because it is related to the lifetime ofthe structures However, the number of researches on cracked plate is still limited, especially,thin plate using Kirchhoff-Love theory Therefore, more investigations on the modelling ofcracked thin plates are necessary In fracture analysis, a powerful method to model crackdiscontinuity without remeshing is the eXtened Finite Element Method (XFEM) proposed by[24] and has been widely used [25, 26, 27, 28, 29] In XFEM formulation, the discontinuity(jump) in displacement fields and the singularity in the stress fields are described by usingenrichment functions XFEM has been developed to examined the behavior of the thickcracked plates using the Reissner-Mindlin theory [30, 31, 32] and thin cracked plates usingthe Kirchhoff-Love theory [33, 34, 35] However, in the finite element analysis of Kirchhoff-Love plate, it is necessary to construct higher-order shape function for the requirement of

the second-order derivatives in the Kirchhoff-Love formulation [36] Therefore, it bringscomputational difficulties which lead to the need for an alternative numerical method, takethe XIGA [37, 38] for example.

In this study, the alternative numerical method employed is the Meshfree method - theRadial Point Interpolation Method (RPIM) [39, 40, 41] The shape function in the RPIM isformulated from the radial basis and the polynomial basis so that the second-order derivative

of the shape function is easily obtained Hence, RPIM is easier to model the Kirchhoff-Loveplate than FEM Similar to the formulation of XFEM, Nguyen et al introduced the extendedRPIM by combining RPIM and enrichment functions [42, 43, 44] to investigate the fracturemechanics They used this method for 2D fracture analysis The XRPIM has also beenemployed to examined the cracked Reissner-Mindlin plate [45] Nevertheless, the number

of report on fracture analysis in Kirchhoff-Love plate by using XRPIM is still limited

This thesis investigates the fracture behavior of the cracked Kirchhoff-Love plate ing the XRPIM In the scope of this study, only the free vibration behavior of cracked plate

us-is investigated Thus-is thesus-is analyzes the free vibration behavior of thin plates with thickness crack The method used for computing is the XRPIM and the Kirchhoff-Love platetheory is used for modelling thin plate behavior The validity of the proposed method is ex-amined through numerous numerical examples, demonstrating the accuracy of the approach

In this study, the author concentrates on the following aspects:

• The Kirchhoff-Love plate theory: only thin plate structures are consider in this thesisand the Kirchhoff-Love theory is employed for modelling thin plate

• The eXtended Radial Point Interpolation Method: the Radial Point Interpolation Method

is used as the main numerical method in this thesis, the XRPIM is based on RPIM andused for fracture analysis

• The free vibration behavior of the cracked structure: the free vibration frequencies areconsidered, while the evaluation of the stress intensity factors is not mentioned in thisstudy

Other issues not mentioned above are beyond the scope of this study and will not be discussed

in this thesis

1.3 Research objectives

The target of this research is to examine the free vibration behavior of the cracked Love plate using the XRPIM To achieve this goal, the following tasks must be fulfilled:

Kirchhoff-• Develop a RPIM-based program to analyze behavior of Kirchhoff-Love plate

• Construct XRPIM formulation for the Kirchhoff-Love plate

• Develop a program to analyze the behavior of the cracked plate

The achievement of this thesis will be the premise for further study in related topics

The scientific aspects that this thesis contributes to:

• Formulating the Kirchhoff-Love theory in RPIM

• The analysis of cracked Kirchhoff-Love plate using XRPIM

• A MATLAB program for the analysis

The structure of this thesis is following this outline Chapter 2 introduces the related ories employed in this thesis First is the Kirchhoff-Love theory for thin plate structure.Then, a brief summary of the Radial Point Interpolation Method is provided In the last ofthis chapter, the eXtended Radial Point Interpolation Method is presented which providesmathematical formulation of XRPIM for thin-cracked plates in Kirchhoff-Love theory Theimplementation of the XRPIM is shown in Chapter 3 Chapter 4 brings various numericalexamples, demonstrating the accuracy of the approach Finally, some concluding remarksand outlooks are given in Chapter 5

the-Chapter 2

Methodology

This section is written based on Publication B

A three-dimensional structure is considered to be a plate structure when it has onedimension much smaller than the other two In other words, a plate is a flat structure that has

a small thickness compared to the in-plane dimensions In practice, analyzing this structure

by using a 3D model is not absolutely necessary and the problem can be reduced to a 2Danalysis However, when considering it as a 2D analysis, an appropriate plate theory isrequired so that the 3D behavior of the structure is reconstructed For thin plate structure,the Kirchhoff-Love plate theory is widely used in the analysis

Besides the Kirchhoff-Love theory, there are many other plate theories, for instance,the First-order Shear Deformation Theory (FSDT), the Higher-order Shear DeformationTheory (HSDT)and so on Each plate theory has its own assumptions for the purpose of sim-plifying the computational work while preserving the accuracy of the solutions Particularly,the Kirchhoff-Love plate is based on the assumption that the transverse shear deformation

of the plate is neglected The presumption above can be explicitly written in two separatedassumptions [1]:

1 Kinematic assumption: The cross-section remains straight and perpendicular to themiddle surface of the plate

2 Static assumption: The transverse normal stresses are negligible

In this section, the formulation of the Kirchhoff-Love plate is introduced through theequilibrium equations, the constitutive equations and the governing equations The con-struction of finite element analysis for the Kirchhoff-Love plate element is performed in thelatter of this section as well

2.1.1 Equilibrium equation

Giving consideration to a piece of cut out of the plate as shown in Figure 2.1 This tial element has the thickness h in x3 direction and the lengths of dx1 and dx2 in x1 and x2directions On the top surface, the plate is subjected to lateral loads Px 3 only Also shown

differen-in the figure, Qi and Mij (i, j = 1, 2) are the shear force resultant and the moment ant, respectively All the internal and external forces acting on the plate must satisfies thefollowing equilibrium equations

result-X

Mx1 = 0, XMx2 = 0 and XPx3 = 0 (2.1)The meaning of the first two equations is that the summation of the moments around

x1and x2 axis must be zeros While the last equation means that the sum of all lateral forces

in the x3direction must equal zero

Figure 2.1: Internal and external forces on the plate

Taking all the moments into the first equation of Eq (2.1), this gives the followingexpression

force is changed into transverse shear force per unit length The equation demonstrating thesum of the moments of all forces around the x2 axis becomes

∂q1

∂x1 +

∂q2

∂x2 = −px3 (2.8)Observing that the twisting moment m12 = m21, then substituting Eqs (2.5) and (2.6)into (2.8) The newly obtained equation shows the relation between the moment and theexternal force

∂2m1

∂x2 1+ 2 ∂

2m12

∂x1∂x2 +

∂2m2

∂x2 2

= −px 3 (2.9)

2.1.2 Constitutive equation

Constitutive equations are the equations that exhibit the relation between stress and strain

in the material And remember that the strain field is derived from the displacement fieldthrough the strain-displacement relation, so the formulation of the displacement field is firstperformed in this section

Considering an arbitrary point located in the middle surface of the plate, this point hasthe vertical movement only Hence, the transverse displacement component is formulated interm of the transverse deflection w(x1, x2) [3]

u3(x1, x2, x3) = w(x1, x2) (2.10)where u3(x1, x2, x3) is the displacement in x3direction of an arbitrary point x(x1, x2, x3) inthe plate

Figure 2.2: Plate before and after deformation

Figure 4.3 shows the kinematics of deformation of the plate The in-plane displacementfield is in the form of a linear function of coordinate x3[3] and can be written as

From the strain-displacement relation and the displacement field in Eqs 2.10 and 2.11,the strain field is obtained as the equation below

∂2w

∂x2 2

κ1 = −∂

2w

∂x2 1, κ2 = −∂

2w

∂x2 2and χ = − ∂

2w

∂x1∂x2 (2.15)For an isotropic, homogeneous and linear elastic material, the stress-strain relationshipfollows Hooke’s law The constitutive equation is shown below

where E denotes the Young’s modulus and ν is the Poisson’s ratio of the material

Substituting Eq (2.12) to Eq (2.16), the equation shows the relationship of stress anddisplacement

∂2w

∂x2 2

From the above equations, it is seen that the units of these moments are moment perunit length, N (or N m/m) in the SI system.

Extracting the normal stress σ1and σ2 from Eq (2.17)

σ1 = − Ex3

1 − ν2

 ∂2w

∂x2 1+ ν∂

2w

∂x2 2

2w

∂x2 1



(2.21)Substituting Eqs (2.20) and (2.21) into Eq (2.18) and then integrating these equations,the bending moment resultants are obtained as the following equations

m1 = − Eh

312(1 − ν2)

 ∂2w

∂x2 1+ ν∂

2w

∂x2 2



= −D ∂2w

∂x2 1+ ν∂

2w

∂x2 2



= D(κ1 + νκ2) (2.22)and

m2 = −D ∂2w

∂x2 2+ ν∂

2w

∂x2 1



= D(κ2+ νκ1) (2.23)where

D = Eh

312(1 − ν2) (2.24)

is the bending stiffness (or flexural rigidity) of the plate

Similar to the bending moments, the twisting moment resultant is acquired by rating Eq (2.19)

integ-m12= m21 = −(1 − ν)D ∂

2w

∂x1∂x2

= D(1 − ν)χ (2.25)The substitution of Eqs (2.22), (2.23) and (2.25) into Eq (2.9) gives the governingdifferential equationof the plate [2]

This equation can be shortened by using the 2D Laplacian operator

∇2(·) = ∂

2(·)

∂x2 1+∂

2(·)

∂x2 2

(2.27)Now Eq (2.26) can be written in a shorter form

D∇2∇2

w(x1, x2) = px 3 (2.28)This fourth-order governing equation is also called the biharmonic equation [4]

2.1.4 Finite element approximation

It can be observed that the governing equation is a fourth-order differential equation, so it iscomplicated to find a analytical solution For that reason, analyzing plate bending problems

in the discrete form is necessary This section presents the formulation of the Kirchhoff-Loveplate element in the finite element method

The deflection of the plate is approximated by the following equation

w =

nXj=1

Njdej = Nde (2.29)

where Nj is the shape functions and dej are the terms related to w and its derivatives

The bilinear shape functions are not appropriate in this scenario, because the ment of the second-order derivatives in the strain-displacement relation (Eq (2.12)) is notfulfilled For that reason, the Hermite function is employed to construct the new shape func-tion [5]

require-The construction of the not conforming Hermite approximation function is shown low and the illustration of a rectangular not conforming element is displayed in Figure 2.3.The figure also shows that each node has three degrees of freedom, the deflection and itsderivatives (can be viewed as the rotation angle), w, ∂w

be-∂x1,

∂w

∂x2 The deflection in Eq (2.29)

is now written as the summation of polynomial terms

w = a1+ a2x1+ a3x2+ a4x1x2+ a5x21+ a6x22+ a7x21x2+ a8x1x22+ a9x31+ a10x32+ a11x31x2+ a12x1x32

Figure 2.3: A rectangular element with three degrees of freedom w, ∂w

∂x1,

∂w

∂x2 per nodeBack to Eq (2.29), the displacement vector de containing the DOFs at four cornernodes can be expressed as

de =w1

w2 w3w4w1,1w,12 w3,1w,14 w1,2w2,2w,23 w4,2T (2.32)Each term in Eq (2.32) is obtained by substituting the coordinate of the four cornernodes into Eqs (2.30) and (2.31), and then rewrite dein the matrix form

where A is a coefficient matrix containing already known components, a is the vector taining coefficients aj for j = 1, 2, , 12 Matrix a can be acquired by matrix inversion as

con-a = A−1de (2.34)Taking the solution and substituting it into the approximation function in Eq (2.30),the shape function for Kirchhoff-Love plate element is obtained as [6]

1,ηdet J−1det Jξ− x1,ξx1,ηη+ x1,ξx1,ηdet J−1det Jη

η,11 = det J−2(−x2,ηx2,ξξ+ x2,ηx2,ξdet J−1det Jξ+ x2,ξx2,ξη − x2

2,ξdet J−1det Jη

η,22 = det J−2(−x1,ηx1,ξξ+ x1,ηx1,ξdet J−1det Jξ+ x1,ξx1,ξη − x2

1,ξdet J−1det Jη

ξ,12 = det J−2(−x2,ηx1,ξη+ x2,ηx1,ηdet J−1det Jξ+ x2,ξx1,ηη− x2,ξx1,ηdet J−1det Jη

η,12 = det J−2(−x2,ξx1,ξη − x2,ηx1,ξdet J−1det Jξ+ x2,ηx1,ξξ+ x2,ξx1,ξdet J−1det Jη

(2.39)where the determinant of Jξ and Jη are defined as

det Jξ = x1,ξx2,ξη− x2,ξx1,ξη+ x2,ηx1,ξξ− x1,ηx2,ξξdet Jη = −x1,ηx2,ξη + x2,ηx1,ξη− x2,ξx1,ηη+ x1,ξx2,ηη

Kirchhoff-in this study for modellKirchhoff-ing the Kirchhoff-Love plate In the next section, the formulation ofRPIM and why using it is easier to model the Kirchhoff-Love plate are presented

This study concentrates on investigating the free vibration behavior of the crackedplate, so the equation that describes the free vibration behavior of the structure is performed

in the following The weak form for the dynamic undamped equation is given as [46]Z

S

δTp : σpdS +

ZV

δwT · ρ ¨wdV −

ZV

Ke =ZA

where N contains shape function and its first-order partial derivatives, at Ith node NI isdefined as

NI =nNI NI,1 NI,2oT (2.47)and m is the inertia matrix

The solution of Eq (2.43) can be obtained by solving the eigenvalue equation [47], theequation is now rewritten in the new form

K − ω2M
¯w = 0 (2.49)where ω is the free vibration frequency (eigenvalue) and ¯w indicates the mode shape of thefree vibration (eigenvector)

This section is written based on Publication A and Publication B

2.2.1 Brief introduction to the RPIM

The Point Interpolation Method which is proposed by Liu and Gu, is a branch of the MeshfreeMethod There are two types of PIM that related to the method of this thesis: PIM usingpolynomial basis function [41] and PIM using radial basis function [48]

The polynomial PIM uses the polynomial basis for constructing the shape functions.Following the work of Liu [49], the polynomial PIM is high accuracy and simple to employ

It has the ability of constructing any order shape function by increasing the number of nodesfor interpolation However, the singular polynomial moment matrix issue may sometimesoccurs To deal with this drawback of the polynomial PIM, the radial PIM is introduced.The radial PIM is developed from the polynomial PIM to solve the singularity issue.The radial basis is employed for constructing the shape function The radial PIM, however, isfacing with a major drawback - not consistent It means that the radial PIM cannot reproducethe linear field exactly

To solve this problem, the polynomial basis is brought back for establishing the shapefunction so that the consistency of the radial PIM shape function is satisfied The singularityissue when using polynomial basis function is also solved This new type of shape functionswhich can handle the drawbacks of the polynomial basis and the radial basis, is known as theRadial PIM with Polynomial Reproduction This type of shape function is used throughoutthis study For convenience, in this thesis, the ”RPIM” represents the ”Radial PIM withPolynomial Reproduction”

2.2.2 RPIM shape functions construction

The construction of RPIM with polynomial reproduction shape functions is introduced in thissection Considering a support domain contains n nodes (see Figure 2.4), the interpolation

function uh(x) consists of radial Ri(x) and polynomial pj(x) basis functions in the form[49]

uh(x) =

nXi=1

Ri(x) ai+

mXj=1

pj(x) bj

= RT(x) a + pT (x) b

(2.50)

where n is the number of nodes inside the support domain and also determines the number of

Figure 2.4: Discrete nodes (gray dots) and support domains

radial basis functions (RBF), m is the number of polynomial basis function The relationship

m < n must be satisfied so that the shape function is not in the singularity issue Thecoefficients aiand bi are for the radial and polynomial basis functions, respectively

Vector a is defined as

a = {a1, a2, , an}T (2.51)and the vector b is defined as

b = {b1, b2, , bm}T (2.52)The radial basis vector R is determined as

RT(x) = [R1(x), R2(x), , Rn(x)] (2.53)

There are various type of radial basis functions as listed in Table 2.1 [49] The Quarticfunction is employed as the radial basis function in this study, the meaning of the parameters

in the Quartic function in Table 2.1 is given in the following: θ is the shape parameter, ls

is the length scale parameter and defined by the normalized distance between x(x1, x2) and

Table 2.1: Some radial basis functions

2

2i1/2

(2.54)Following the research of [50], the advantage of Quartic function is the independent ofshape parameter θ In other words, the computational results are stable and not affected bychanging the shape parameter In this study, for convenience, θ = 1 is used in all numericalexamples

The polynomial basis vector in a 2D domain is defined as

pT (x) = [p1(x), p2(x), , pm(x)] =1, x1, x2, x1x2, x21, x22, , xn1, xn2 (2.55)all the components in pT(x) are chosen symmetrically from the Pascal triangle in Figure 2.5

Figure 2.5: Pascal triangle of monomials for 2D case

To determine the value of the coefficients ai and bj, the interpolation equation Eq.(2.50) is passed through all n nodes inside the support domain [49] The interpolation at the

kthpoint is given

uk = uh xk1, xk2
=

nXi=1

Ri xk1, xk2
ai+

mXj=1

pj xk1, xk2
bj (2.56)

in the matrix form

Us = RQa + Pmb (2.57)where Uscontains all the nodal variables of all n nodes inside the support domain

The following constraints must be satisfied to ensure the polynomial term is uniqueapproximation

nXi=1

pj(xi) ai = PTma = 0 (2.58)Equation (2.57) and (2.58) are combined into

"

RQ Pm

PTm 0

# (ab

)

=

(

Us0

b = SbUs (2.63)

Sb =PT

mR−1Q Pm−1PTmR−1Q (2.64)and

Sa = R−1Q − R−1Q PmSb (2.65)Substituting the values of a and b into Eq (2.50), the interpolation function (2.50) isrewritten as

uh(x) =

nXk=1

φk(x) uk= Φ (x) Us =RT (x) Sa+ pT (x) Sb Us (2.66)

where Φ (x) is the matrix of shape function containing n shape function

Φ (x) = [φ1(x), φ2(x), , φi(x), , φn(x)] (2.67)and the shape function at kthnode φk(x) is defined by

φk(x) =

nXi=1

Ri(x) Sika +

mXj=1

pj(x) Sjkb (2.68)

where Sa

ikdenotes the (i, k) element of matrix Saand Sb

jk is the (j, k) element of matrix SbThe RPIM shape function possesses the Kronecker delta function property

φi(xj) = δij (2.69)

So the essential boundary conditions can be directly imposed on nodes Meanwhile, in thewell-known Element Free Galerkin (EFG) method, the Lagrange multiplier is employed toenforce the boundary conditions because the shape function does not satisfy the Kroneckerdelta function property [49]

The first-order derivatives of the RPIM shape function can be easily obtained

∂φk

∂x1 =

nXi=1

∂pj

∂x1S

b jk

∂φk

∂x2 =

nXi=1

∂pj

∂x2S

b jk

∂x2 1

=

nXi=1

∂2Ri

∂x2 1

Sika +

mXj=1

∂2pj

∂x2 1

Sjkb

∂2φk

∂x2 2

=

nXi=1

∂2Ri

∂x2 2

Sika +

mXj=1

∂2pj

∂x2 2

∂2Ri

∂x1∂x2

Sika +

mXj=1

∂2φI

∂x2 2

plate

This section is written based on Publication A

The extended RPIM (or XRPIM) is based on the extended finite element method(XFEM) concept The crack path is not explicitly defined in the geometry of the object,but in the mathematical functions The discontinuity (jump) in the displacement field alongthe crack path and the singularity in the stress fields at the crack tip are described by usingenrichment functions

As shown in Section 2.1.4, the deflection w of any points in the plate can be ated by Eq (2.29) In the ”extended” concept, the enrichment functions are inserted in theapproximation function Therefore, the deflection of an arbitrary point in the plate is nowexpressed as

#

(2.73)

where H and G are the enrichment functions, H (or H (x)) is the Heaviside function of point

x and HI (or H xI
) is the Heaviside function of the Ith node, Gl (or Gl(x)) is the lth tipenrichment function of point x and GlI (or Gl(x)) is the lthtip enrichment function of the Ithnode

Figure 2.6: Split nodes and tip nodes around the crack curve

There are three node sets in Eq (2.73), W is the set containing all the nodes in thecomputational domain, Ws is the set containing the nodes in the support domain which iscut by the crack and Wtis the set containing the crack tip The node in Wsis called the ”splitnode”, see the blue dot in Figure 2.6 And the node inside Wtis known as the ”tip node”, it

is illustrated by the red square in Figure 2.6

The Heaviside function is defined as

H (f (x)) =

(+1 if f (x) > 0

−1 if f (x) < 0 (2.74)where f (x) denotes the sign distance function Considered a point x near the crack curve,the closest point from x to the crack curve is x∗ (see Figure 2.7), the sign distance function

f (x) is given as

f (x) = (x − x∗) · sn (2.75)where snis the vector normal to the curve and stis the tangential vector (see Figure 2.7).The tip enrichment function Gl is defined as below [51]

Figure 2.7: Illustration of tangential and normal direction for a crack

Figure 2.8: Global coordinate system and local coordinate systems

The enrichment functions are applied into the Kirchhoff-Love plate in order to modelthe cracked plate From the approximation equation (2.73), the strain field is easily obtained.After that, the strain computing matrices are obvious defined, the split enrichment B-operator