Extended galerkin meshfree methods for fracture modeling in advanced funtional composite materials

Schematic of elastic stress waves von Mises propagating in an infinite plate with a semi-infinite edge crack at different steps using the present formulation with the linear ramp functio

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

Independent Examiner 1: Assoc Prof Dr Nguyễn Xuân Hùng

Independent Examiner 2: Assoc Prof Dr Nguyễn Mạnh Cường

Examiner 1: Assoc Prof Dr Nguyễn Đình Kiên

Examiner 2: Assoc Prof Dr Lê Văn Cảnh

Examiner 3: Assoc Prof Dr Lương Văn Hải

SCIENTIFIC SUPERVISORS:

1 Assoc Prof Dr Trương Tích Thiện

DECLARATION

The thesis content is based on my original research work in the Department of Engineering Mechanics, Faculty of Applied Science, Ho Chi Minh city University of Technology, VNU – HCM, Vietnam I declare surely that this document has been created by myself and that has not been submitted for any other degree or qualification except as specified

Author

Nguyễn Thanh Nhã

ABSTRACT

This thesis deals with the numerical computation of 2-D linear fracture problems using the two extended Galerkin meshless methods including radial point interpolation method (RPIM) and improved moving Kriging (MK) interpolation method Enrichment techniques including the use of step function for crack faces, standard branch functions and new linear ramp function for crack tip are first applied in RPIM and MK meshless frameworks The meshless moving Kriging method is improved by using three types of correlation function (i.e quartic polynomial, truncated quartic polynomial and Gaussian functions) to eliminate the effect of the user numerical experience parameter and applied

to crack problems The developed methods are applied for crack analyzing in several types of material including isotropic, orthotropic and functionally graded composite materials Various crack problems such as static, dynamic behavior of crack models and quasi-static crack propagation are numerically investigated and compared with solutions given by analytical, experiment or other numerical methods The agreements between the obtained results using extended meshless methods and those of other methods show the correction of the developed approaches

ACKNOWLEDGEMENTS

This doctoral dissertation is the outcome of many years working at the Department of Engineering Mechanics (DEM), Faculty of Applied Science, Ho Chi Minh City University of Technology I would like to sincerely thank my principal supervisor Assoc Prof Truong Tich Thien for his helpful advices and guidance during my study work I express my special gratitude to my scientific supervisor Assoc Prof Bui Quoc Tinh from the Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Japan, for his invaluable support, guidance and mentoring I deeply thank all my supervisors for offering me the opportunity to conduct this research work

I also would like to thank Assoc Prof Nguyen Luong Dung, Prof Ngo Kieu Nhi, Assoc Prof Vu Cong Hoa and other members for their useful advice and supports, and for creating such a friendly and comfortable working atmosphere

In addition, I am grateful for the support from my department colleagues, Nguyen Thai Hien, Tran Thai Duong, Nguyen Duy Khuong, Tran Kim Bang, Le Duong Hung Anh who has been always with me in difficult times Special thanks go to my close colleague and friend, Nguyen Ngoc Minh, for his useful discussions, ideas and programming experience

Last but not least, I would like to express my profound gratefulness to my family, especially my parents, my wife Nguyen Thi My Hien and my sons Nguyen Quang Khai, Nguyen Minh Quoc Without their continuous encouragement, support and love, I would not have been able to pursue my work and ambition

Ho Chi Minh City, July 2018

Nguyen Thanh Nha

CONTENTS

DECLARATION i

ABSTRACT ii

ACKNOWLEDGEMENTS iii

LIST OF FIGURES ix

LIST OF TABLES xvi

NOMENCLATURE xvii

CHAPTER 1 INTRODUCTION AND OBJECTIVE 1

1.1 Statement of crack problems 1

1.2 Advanced functional composite materials 3

1.3 Literature review 5

1.3.1 Extended Finite Element method (XFEM) 7

1.3.2 Extended Meshfree approach 8

1.4 Fundamental of Fracture Mechanics 9

1.4.1 Crack behavior in isotropic 9

1.4.2 Crack behavior in orthotropic materials 12

1.4.3 Crack behavior in functionally graded materials 14

1.5 Objective of the dissertation 15

1.6 Outline of the thesis 16

CHAPTER 2 EXTENDED MESHFREE GALERKIN METHODS FOR FRACTURE MECHANICS 18

2.2 Enrichment methods 21

2.2.1 Enrichment for discontinue crack faces 21

2.2.2 Standard enrichment for crack tip using branch functions 22

2.2.3 New enrichment for crack tip using ramp function 24

2.2.4 Apply to crack propagation problems 26

2.3 Meshfree Galerkin method for fracture problems and solution procedure28 2.3.1 Fundamental equations of elastic problems 28

2.3.2 Discrete equations for fracture problem 30

2.4 General J-integral (static and dynamic) 32

2.5 Numerical integration 36

2.6 Numerical implementation procedure 36

2.6.1 Implementation procedure for quasi-static crack growth problem 36

2.6.2 Implementation procedure for dynamic crack problem (stationary state) 38

2.7 Summary 39

CHAPTER 3 X-RPIM FOR QUASI-STATIC CRACK GROWTH SIMULATION OF 2-D SOLIDS 40

3.1 Introduction 40

3.2 Crack growth and the SIFs implementation in isotropic material 43

3.3 Accuracy study 46

3.3.1 Mode I: Single edge-crack plate under tensile loading 46

3.3.2 Mixed-mode: Single edge-crack plate under uniform shear loading 49

3.4 Numerical examples for crack growth problems 51

3.4.1 Crack growth from a fillet 53

3.4.2 Crack growth in a perforated panel with a circular hole 57

3.5 Conclusions 59

CHAPTER 4 TRANSIENT DYNAMIC CRACK ANALYSIS OF ISOTROPIC AND COMPOSITE MATERIALS 61

4.1 Introduction 61

4.2 Evaluation of dynamic stress intensity factors for isotropic solids 64

Remark 1: 66

4.3 Transient dynamic crack analysis of isotropic solids 66

Remark 2: 67

4.3.1 Accuracy study of the SIFs in cracked isotropic plates 67

4.3.2 A semi-infinite edge crack under dynamic loading 72

4.3.3 Mixed-mode analysis of a slanted edge-cracked rectangular plate 80

4.3.4 Mixed-mode analysis of a cracked pipe 84

4.3.5 A complex structure with an edge crack 89

4.4 Transient dynamic crack analysis of orthotropic composites 91

4.4.1 Orthotropic enrichment functions for crack 92

4.4.2 Evaluation of dynamic stress intensity factors for orthotropic composites 92 4.5 Numerical results and discussion 94

4.5.1 Accuracy study of the SIFs of orthotropic composite 94

4.5.2 An edge crack in an orthotropic composite plate under dynamic loading 95

4.5.3 A center crack in an orthotropic composite plate under dynamic loading 98

4.6 Crack growth in orthotropic model 101

4.6.1 Criterion for crack growth direction in orthotropic model 101

4.6.2 Predicting for propagation angle in an edge crack orthotropic plate 101

4.7 Conclusions 103

CHAPTER 5 EXTENDED MESHLESS RADIAL POINT INTERPOLATION METHOD FOR FRACTURE ANALYSIS OF FGMs 105

5.1 Introduction 105

5.2 The interaction integral formulation for non-homogenous materials 108

5.2.1 Non-equilibrium formulation for FGM model 109

5.2.2 Extract SIFs for FGM model 111

5.3 Accuracy study of SIFs in FGM crack models 113

5.3.1 Single edge crack plate under mode I 113

5.3.2 Mixed-mode edge crack problem 116

5.3.3 Slant edge crack problem 119

5.4 Dynamic SIFs calculation for FGM crack models 121

5.4.1 FGM plate with center crack under dynamic tensile loading (case 1: x1-x2 FGM) 121

5.4.2 FGM plate with center crack under dynamic tensile loading (case 2: x2 FGM) 124

5.5 Conclusions 129

CHAPTER 6 IMPROVED EXTENDED MESHLESS MOVING KRIGING FOR FRACTURE MODELING OF SOLIDS AND FGMs 130

6.1 Introduction to the moving Kriging method 130

6.1.1 The moving Kriging shape function 131

Remark 3: 133

6.1.2 The improved moving Kriging shape functions 133

6.2 Improved X-MK for crack analysis of isotropic material 136

6.2.1 Accuracy study on static SIFs in solid 137

6.2.2 Dynamic crack analysis of isotropic material 141

6.3 Improved X-MK for dynamic crack analysis of FGM material 146

6.3.1 Rectangular x1-x2-FGM plate with center crack under dynamic tensile loading 146

6.3.2 Inclined center crack FGM plate under dynamic tensile loading 149

6.3.3 Dynamic crack in complex FGM model 152

6.4 Discussion 155

CHAPTER 7 CONCLUSIONS AND OUTLOOKS 157

7.1 Conclusions 157

7.2 Outlooks 158

LIST OF PUBLICATIONS 160

REFERENCES 162

LIST OF FIGURES

Figure 1.1 Cracks observed in Song Tranh hydropower dam 1

Figure 1.2 Cracks observed on road surface of Thang Long Bridge 2

Figure 1.3 Cracks in an airplane windshield 2

Figure 1.4 The three basic modes of fracture 2

Figure 1.5 Cracks observed on an aircraft body that made from composite material 4

Figure 1.6 Crack growth in a FGM specimen [4] 4

Figure 1.7 Definition of the coordinate axis ahead of a crack tip 11

Figure 1.8 2D orthotropic body with crack 13

Figure 1.9 2-D FGM body with crack 15

Figure 2.1 Schematic representation of the distance r and angle  at a crack-tip 22

Figure 2.2 Definition of the sets of nodes Wb and WS, respectively, in our meshfree method 23

Figure 2.3 Schematic of a crack tip coordinates in terms of linear ramp function Crack is represented by a black curve (X 1 , X 2 ) is the global coordinate system whereas (x 1 , x 2) is the local coordinate system at the crack-tip In practice, for simplicity the value of l c is taken as the radius of the support domain Green nodes are the split nodes enriched by the Heaviside function only, the red nodes enriched by the linear ramp function associated with Heaviside function 25

Figure 2.4 Visualization of enrichments for crack tip location using Heaviside function associated with linear ramp function The Heaviside function (left), the linear ramp function (middle) and the Heaviside function along with linear ramp function (right) 26

Figure 2.5 Geometric description for set S a 27

Figure 2.6 The projection of a point xI belonging to a S onto the advance vector tn 28 Figure 2.7 Notation representation of a cracked model 29

Figure 2.8 Definition of integral paths around the crack tip, normal unit vectors and coordinate systems 34

Figure 2.9 Sub-triangle cell description for crack edge (a) and crack tip (b) 36

where  is the crack angle with respect to the X1 axis (see Figure 2.1.) 44

Figure 3.1 (a) Schematic configuration of an edge-crack plate subjected to the uniformed tensile loading (b) Discretization of the problem and definition of enriched nodes at the crack-faces (star) and at the crack-tip (empty circle) 47

Figure 3.2 The deformed shape and normal stress yy distribution of the cracked plate with 20 40 nodes, a=3.5, enlarged by a factor of 50 48

Figure 3.3 Schematic configuration of mixed-mode single edge-crack in a rectangular plate under a uniform shear loading 50Figure 3.4 Distributed nodes and deformed shape of the cracked plate subjected to a uniform shear loading (a) 10 20 and (b) 20 40 enlarged by 10 times 50Figure 3.5 A center crack plate subjected to the uniformed tensile loading 51Figure 3.6 Distributed nodes and deformed shape of the square plate with center inside crack subjected to a uniform tensile loading 52Figure 3.7 Schematic configuration of a fillet with crack Type-1 (a) and type-2 (b) boundary conditions 54Figure 3.8 Distributed nodes and evolution of the crack path from a fillet: the type-1 boundary condition 55Figure 3.9 Distributed nodes and evolution of the crack path from a fillet: the type-2 boundary condition 56Figure 3.10 Zoom of the crack paths for both types of the boundary conditions at the vicinity of the fillet 57Figure 3.11 Schematic configuration of a perforated plate with a circle hole subjected

to a uniform tensile loading 58Figure 3.12 Distributed nodes and propagation of the crack path of a perforated panel with a circular hole subjected to a uniform tensile loading 58Figure 3.13 Zoom of the crack paths of a perforated panel with a circular hole subjected

to a uniform tensile loading 59

Figure 4.1 Schematic of different dynamic loadings used in the analysis 66Figure 4.2 Geometry of two static crack problems: (a) single mode with an edge-cracked plate subjected to tensile loading and (b) mixed-mode with an edge-cracked plate subjected to shear loading 68Figure 4.3 Convergence rate of the relative error vs number of nodes in each direction

of an edge-cracked plate under tensile loading between the standard and ramp approaches 70Figure 4.4 Comparison of convergence of computational time (in second) versus several nodal distributions in each direction of an edge-cracked plate under tensile loading between standard and ramp approaches 70Figure 4.5 Edge crack problem of tensile loading Condition numbers of the system matrix produced by approaches based on either the standard or ramp functions versus

the number of nodes per side The crack length is a=3.5 Dash lines are averaged curves

of the two methods 71

Figure 4.6 Geometry and boundary conditions of an infinite plate with a semi-infinite mode I crack 72Figure 4.7 Regular meshless distributions of an infinite plate with a semi-infinite edge crack 74Figure 4.8 Convergence of the normalized mode-I DSIF as a function of time with respect to nodal densities using the standard branch function 74Figure 4.9 Convergence in percentage errors for the normalized mode-I DSIF as a function of time with respect to nodal densities using the standard branch function Large error can be found for the coarsest model as usual 75Figure 4.10 Schematic of elastic stress waves (von Mises) propagating in an infinite plate with a semi-infinite edge crack at different steps using the present formulation with the linear ramp function It graphically shows that the elastic stress waves start, touch and open the crack, then reflect, repeatedly 76Figure 4.11 Comparison of the normalized mode-I DSIF as a function of time obtained

by the standard branch function, the linear ramp function and the exact solution, showing

a very good agreement among three approaches The coarser model of the ramp function shows slightly less accuracy at the late state of the response 77Figure 4.12 Errors of the normalized mode-I DSIF as a function of time obtained by the standard branch function, the linear ramp function and the exact solution It is obvious that the poorest result is found for the coarse model using the ramp function 77Figure 4.13 Comparison of the normalized mode-I DSIF versus the normalized time among different methods including the exact solution, the singular finite element, the standard XFEM, the meshfree coupled XFEM and the present method based on the linear ramp function 78Figure 4.14 Effects of load intensities on the mode-I DSIF of a semi-infinite edge crack

in an infinite plate under step and blast loadings In the blast case, the dash-dot lines represent the time t and 1 t corresponding to the peak load and duration end of the blast 2

loading 80Figure 4.15 Geometry and boundary condition of a rectangular plate containing a slanted crack with three edges supported (right) and one bottom edge clamped (left) 81Figure 4.16 Comparison of the normalized DSIFs (mode-I & mode-II) as a function of time between the present method (standard & ramp) and moving least square meshless

by Wen and Aliabadi [63], showing a good agreement 82Figure 4.17 Convergence of the normalized DSIFs (mode-I & mode-II) as a function of time with respect to nodal densities using the linear ramp function 82Figure 4.18 Convergence of the normalized DSIFs (mode-I & II) as a function of time against different nodal densities using the present method Comparison is also included with the reference result given by Wen and Aliabadi [63] 84Figure 4.19 Geometry, boundary condition and loading of an edge crack in a quarter of the pipe 85

Figure 4.20 Meshfree discretization for a quarter of the pipe using 1406 scattered nodes, showing the crack (red curve), the enriched nodes (cut and tip nodes) 85Figure 4.21 Comparison of the normalized DSIFs as a function of time of an edge crack

in a quarter of the pipe using the present method These results are obtained using a

crack length of a=3m Both enrichment models offer the curves of mode-I and mode-II

with a close agreement between each other 86Figure 4.22 Effects of the crack lengths on the normalized DSIFs of an edge crack of a pipe using the present method 87Figure 4.23 Schematic of elastic stress waves propagating in a 1/4 pipe at different steps using the present formulation with the standard (left) and with the linear ramp function (right) It graphically shows that the elastic stress waves start, touch and open the crack, then reflect, repeatedly 88Figure 4.24 Geometry, boundary and loading conditions of a complex structure with an edge crack 90Figure 4.25 Meshfree discretization for a complex structure using 1944 irregular scattered nodes 90Figure 4.26 Comparison of the normalized DSIFs as a function of time between the standard and linear ramp function enrichments under the step loading type 91Figure 4.27 Comparison of the normalized DSIFs as a function of time between the standard and linear ramp function enrichments under the sine loading type 91Figure 4.28 Geometry and loading of an orthotropic composite plate with a center crack subjected to a static tensile loading This example is used for the accuracy analysis 94Figure 4.29 Geometry and load condition of a single edge crack in a rectangular orthotropic plate 96Figure 4.30 Comparison of the normalized mode-I DSIF as a function of time among the present X-RPIM, the XFEM [78], the time-domain BEM [79], and the FEM (ANSYS) [79] A close agreement is obtained 97Figure 4.31 Convergence of the normalized mode-I DSIF with respect to nodal densities using the developed method Similar behaviors are found 97Figure 4.32 Effects of different crack lengths on the normalized mode-I DSIF for an edge crack in an orthotropic composite plate using the developed method This result indicates a significant oscillation of the normalized mode-I DSIF with crack lengths 98Figure 4.33 Geometry and loading condition of a single center crack in a rectangular orthotropic plate 99Figure 4.34 Normalized mode-I DSIF for a single center crack in a rectangular orthotropic plate with several orientations of the axes of orthotropy obtained by the developed method These responses can be directly compared with those shown in Fig

5 by using the time-domain BEM [79] 99Figure 4.35 Normalized mode-II DSIF for a single center crack in a rectangular orthotropic plate with several orientations of the axes of orthotropy obtained by the

developed method These responses of mode-II DSIF can be directly compared with those shown in Fig 6 by using the time-domain BEM [79] 100Figure 4.36 Convergence of the normalized mode-I & mode-II DSIFs as a function of time with respect to nodal densities using the developed method for an orthotropic angle

60

  100Figure 4.37 Geometry and loading of an orthotropic composite plate with an edge crack subjected to a static tensile loading 102Figure 4.38 Crack paths with various values of angle  102

Figure 5.1 Edge crack plate with material gradation in the x1 direction 113

Figure 5.2 Comparison of the normalized mode-I SIF versus the ratio a/W among

different methods including element free Galerkin method [110, 113], analytical [114], FEM [103] and X-RPIM (tension load) 114

Figure 5.3 Comparison of the normalized mode-I SIF versus the ratio a/W among

different methods including element free Galerkin method [110], analytical [114], FEM [103] and X-RPIM (bending load) 115Figure 5.4 Normalized SIF KI with various sizes of support domain  (tension load

case) 116Figure 5.5 Edge crack plate with material gradation in the x direction, under tension 2

load 117Figure 5.6 Nodal distribution for edge crack plate with material gradation in the x 2

direction 117Figure 5.7 Comparison of the normalized mode-I SIF versus the dimensionless gradient parameter L 118Figure 5.8 Comparison of the normalized mode-II SIF versus the dimensionless gradient parameter L 118Figure 5.9 Convergence of the normalized mode-I SIF with respect to the dimensionless gradient parameter L 119Figure 5.10 Slanted edge crack plate under mixed-mode 120Figure 5.11 Nodal distribution for FGM plate with some different  values 120Figure 5.12 Normalized SIFs results for the slanted edge crack plate under mixed-mode 121

Figure 5.13 Center crack FGM plate with material distribution in both x1 and x2

directions 123

Figure 5.14 Nodal distribution for FGM plate with material distribution in both x1 and

x2 directions 123Figure 5.15 Comparison of the normalized mode-I & mode-II DSIF as a function of time between the proposed X-RPIM and FEM [105] 124

Figure 5.16 Convergence of the normalized mode-I & mode-II DSIFs as a function of time with respect to nodal densities 124Figure 5.17 Center crack FGM plate with material distribution in x2- directions 126Figure 5.18 Nodal distribution for FGM plate (right half model) with material

distribution in x2 directions with 2 0,0.05 and 0.1, respectively 126Figure 5.19 Normalized mode-I DSIF with several values of the material parameter 2 These results can be directly compared with those shown in Fig 19 (a) by using FEM [105] 127Figure 5.20 Normalized mode-II DSIF with several values of the material parameter 2 These results can be directly compared with those shown in Fig 19 (b) by using FEM [105] 127Figure 5.21 Nodal displacement at time points: t  1.01,2.52, 5.03 and 6.04 s 128

Figure 6.1 The MK shape function with quartic polynomial function (Formula 1) 134Figure 6.2 The MK shape function with internal length parameter (Formula 2) 135Figure 6.3 The MK shape function with truncated quartic polynomial function (Formula 3) 136Figure 6.4 Model of an edge-crack plate subjected to the uniformed tensile loading137Figure 6.5 Schematic configuration of a center crack plate subjected to the uniformed tensile loading 140Figure 6.6 Comparison of the mode-I SIF versus the crack angle x1 among different

improved X-MK methods and analytical solution 141Figure 6.7 Comparison of the mode-II SIF versus the crack angle x1 among different

improved X-MK methods and analytical solution 141Figure 6.8 Geometry and boundary conditions of a three-point bending beam 142Figure 6.9 Comparison of normalized mode-I DSIF as a function of time between the proposed improved X-MK methods and X-FEM [18] 143Figure 6.10 Comparison of the effect of size of J domain using in improved X-MK with formula 1 144Figure 6.11 Geometry and boundary conditions of a rectangular plate with inside slant crack under tensile loading 145Figure 6.12 Comparison of normalized mode-I DSIF as a function of time among the proposed improved X-MK methods with BEM [100] and FEM [125] 145Figure 6.13 Comparison of normalized mode-IIs DSIF as a function of time among the proposed improved X-MK methods with BEM [100] and FEM [125] 146Figure 6.14 Comparison of the normalized DSIFs as a function of time between the proposed improved X-MK methods and FEM [105] 147

Figure 6.15 Comparison of the normalized DSIFs between the proposed improved

X-MK method (formula 1) and FEM [105] 148Figure 6.16 Comparison of the normalized DSIFs between the proposed improved X-

MK method (formula 2) and FEM [105] 148Figure 6.17 Comparison of the normalized DSIFs between the proposed improved X-

MK method (formula 3) and FEM [105] 149Figure 6.18 Inclined center crack x1-FGM plate 150Figure 6.19 Nodal distribution for FGM plate with inclined center crack 151Figure 6.20 Comparison of the normalized DSIFs for the left crack tip between the proposed improved X-MK methods (formula 1, 2, 3) and SBFEM [126] 151Figure 6.21 Comparison of the normalized DSIFs for the right crack tip between the proposed improved X-MK methods (formula 1, 2, 3) and SBFEM [126] 152Figure 6.22 (a) Geometry and boundary conditions of a cracked FGM model (b) Nodal distribution for the meshfree model 153

Figure 6.23 Young’s modulus distribution along the x2 direction 154

Figure 6.24 Normalized DSIFs as function of time given by X-MK (formula 2) for complex crack FGM model with different material gradient 154Figure 6.25 Normalized DSIFs as function of time given by X-MK (formula 2) for complex crack FGM model with different crack lengths 155

LIST OF TABLES

Table 3.1 Comparison of the 11 factor between the analytical and the proposed

X-RPIM solutions 48

Table 3.2 Influence of the support domain size on the SIFs 49

Table 3.3 Comparison of the SIFs for shear edge-cracked plate between the FEM and the X-RPIM 51

Table 4.1 Comparison of the SIF KI of mode I for an edge-cracked plate under tensile loading 69

Table 4.2 Comparison of the mixed-mode SIFs for an edge-cracked plate under shear loading 69

Table 4.3 Comparison of the normalize SIF for a square orthotropic plate with a center crack among different approaches The “Cells” means the number of background cells used for numerical integration of the meshfree methods 95

Table 4 4 Initial crack propagation angles (degree) 103

Table 6.1 Comparison of the mode-I SIF using X-MK with formula 1 138

Table 6.2 Comparison of the SIF of mode I using X-MK with formula 2 138

Table 6.3 Comparison of the SIF of mode I using X-MK with formula 3 139

Table 6.4 Variation of the support domain size with respect to the SIF 139

E Young’s modulus in x2 direction (orthotropic material)

G Shear modulus (isotropic material)

12

G Shear modulus x1 x2 plane (orthotropic material)

H () Heaviside step function

K Normalized stress intensity factor

 Poisson’s ratio in x1 x2 plane (orthotropic material)

Φ Vector of shape functions

BEM Boundary element method

DOFs Degrees of freedom

DSIF Dynamic Stress Intensity Factors

EFG Element-Free Galerkin method

FEM Finite Element Method

FGM Functionally graded material

LEFM Linear elastic fracture mechanics

MLPG Meshless local Petrov-Galerkin method

MLS Moving least squared

PIM Point interpolation method

RBFs Radial basis functions

RKPM Reproducing Kernel Particle method

RPIM Radial Point Interpolation Method

SBFEM Scaled boundary finite element method SIFs Stress Intensity Factors

TPS Thin plate spline

XFEM Extended Finite Element Method

X-MK Extended moving Kriging

X-RPIM Extended Radial Point Interpolation Method

CHAPTER 1 INTRODUCTION AND OBJECTIVE

1.1 Statement of crack problems

Many catastrophic accidents have been reported to be related to the initiation and propagation of cracks such as the collapse of buildings during earthquake, failure of working mechanical components and damage of transport facilities These accidents caused a great loss both in terms of economic properties and human lives In cases where cracks were detected early, long before collapse occurs, lots of efforts and money still have to be spent for maintenance Some examples of cracks appearing in engineering structures in Vietnam involve the Song Tranh hydropower dam and the Thang Long bridge (see Figure 1.1 and Figure 1.2) Most recently, in June 2016, a domestic flight from Ha Noi to Can Tho had to make an emergency landing in Tan Son Nhat Airport because the airplane windshield was cracked (see Figure 1.3)

Figure 1.1 Cracks observed in Song Tranh hydropower dam

Nowadays, fracture analysis has become more and more important in various fields, in order to predict and prevent failure events of structures Crack opening in structures subjected to arbitrary loading conditions could be described as a combination of the three following modes (illustrated in Figure 1.4)

- Mode I: Opening mode (in-plane tension)

- Mode II: Sliding mode (in-plane shearing)

- Mode III: Tearing mode (out-of-plane shearing)

a) Opening mode b) Sliding mode c) Tearing mode

Figure 1.4 The three basic modes of fracture

Figure 1.2 Cracks observed on road surface of Thang Long Bridge

Figure 1.3 Cracks in an airplane windshield

Mode I

Mode II

Mode III

Early studies have introduced fundamental concepts of fracture mechanics such as stress intensity factor and energy release rate Once a crack is opened, new free surfaces are created The energy release rate determines the energy dissipated per unit of newly created fracture surface area, which should be equal to the work done to create the fracture surfaces That is the idea of the Griffith criteria for crack opening in theory of linear elastic fracture mechanics (LEFM) for brittle materials High stress value is usually observed in region near the crack tip It has been proved theoretically that for the case of a small circular hole inside an infinite plate subject to remote tensile stress, one could expect the stress near the hole could be expected to be three times the remote tensile stress Determination of stress intensity factor provides a finite evaluation of the stress state near the crack tip LEFM theory infers that stress field is singular at the crack tip, i.e stress tends to infinity, which is in fact not realistic Later authors like Irwin and Dugdale supplement the LEFM theory by adding a local plastic zone surrounding the crack tip Hence, the stress at the crack tip is not infinity but has a finite value Further details on theory could be referred to the book by Gdoutos [1]

In cases where the crack propagates rapidly, such that the material inertia is significant and cannot be ignored, one should consider dynamic fracture The situation is much more complicated than that of static cracks The feature that distinguishes dynamic crack behavior from the static or quasi-static one is the stress waves Stress waves can arise

by two ways: they are emitted at the crack tip or from the external load applied on domain boundaries The waves can be reflected or scattered as they encounter any free surfaces When stress waves reflected from domain boundaries are back to crack tip, the stress state at crack tip is altered and the crack speed could be changed If the spatial sizes are small, a number of reflective waves arrive at the crack tip successively, increasing the complexity

1.2 Advanced functional composite materials

Orthotropic composite materials and their structures are used widely in various fields in engineering One of the most preeminent property of composite is the high strength to weight ratio in comparison with conventional engineering materials In many cases, orthotropic composites are fabricated in thin plate forms which are so susceptible to

fault A typical fault in composite structure is cracking due to imperfection in fabrication process or hard working conditions such as overload, fatigue, corrosion and so on Functionally graded materials (FGMs) are advanced composites, which have been manufactured based on the concept of continuous variation of the material properties in one or more specified directions [2] In recent years, the FGMs have been applied in the manufacture of structure parts that subjected to non-uniform working requirements [3] For instance, in a thermal protection system, FGMs evolve the advantage of typical ceramics such as corrosion and heat resistance and of typical metal such as mechanical strength and stiffness FGMs can be applied to make thermal barrier coating for space applications, transport system, energy conversion system, thermal-electric and piezoelectric devices, dental and medical implants, and many others

Figure 1.5 Cracks observed on an aircraft body made from composite material

Figure 1.6 Crack growth in a FGM specimen [4]

Along with the applications of these new advanced materials in engineering, their structural behavior such as vibration, buckling and fracture are under question For such reason, crack behavior of orthotropic composite materials and functionally graded materials has become an interesting study subject (see Figure 1.7 and Figure 1.8)

1.3 Literature review

Studying fracture mechanics, including quantitative analysis of the initiation and propagation of cracks, is essential in analyzing the durability of engineering structures However, the fracture problems are usually complicated Analytical solution is only available for basic problems with simple geometry and simple boundary conditions, which are mostly served for studying purpose For engineering problems, numerical solutions seem to be more suitable Thanks to the development of technology such as high-performance computing, various computational methods and algorithms have been developed and intensively investigated for crack problems

The currently most popular numerical method is the Finite Element Method (FEM) Based on the variational principle and Galerkin method to numerically approximate solution for partial differential equations, FEM has been introduced since the 50s of twentieth century which still be favored in both academic and industrial communities The main idea of FEM is discretizing the problem domain into non-overlapping sub-domains, namely elements [5] An element usually has a simple geometric shape such

as a line (1D element), a triangle or a quadrilateral (2D element), a tetrahedron or a hexahedron (3D element) Equilibrium is enforced locally at nodes, which are the vertices of elements The to-be-solved unknowns, termed by degrees of freedom (DOFs) are associated with nodes The field variables, which could be displacement, pressure, temperature, etc., depending on the specific problems, are approximated as a linear combination of the corresponding nodal values, i.e the DOFs Usually, the numerical error between approximated results with the “true” one reduces when more elements (more nodes) are used to discretize the domain FEM attracts lots of attention due to many desirable properties, such as simple in implementation, rather fast computation and reasonable accuracy

In crack problems, cracks could be directly modeled as discontinuities in geometry When crack propagation is considered, the geometric discontinuities have to be updated, following by an update on domain discretization, i.e re-meshing [6] In practice, re-meshing is a challenging and burdensome task due to the arbitrary crack path and require lots of computational efforts Therefore, approaches that can model crack propagation without re-meshing have been introduced and gained favors [7] Instead of directly create new surfaces to model cracks, the effect of cracks as a jump in displacement and discontinuous stress fields are mathematically induced in the formulation of numerical solution, namely enrichments, with the help of the principle of partition of unity Depending on how the enrichments are taken into account, the approaches can be classified into two categories: intrinsic and extrinsic In the class of intrinsic enriched methods, the enrichment terms are added by replacing the standard shape functions by some special shape functions, at least at some specific regions, to capture the jumps caused by a discontinuity, which could be a crack, a void or an inclusion Using the intrinsic enrichments, the number of degrees of freedom does not change, with the price that the approximation space is changed In contrast, the extrinsic enrichment approach does not change the standard shape functions Instead, the jumps are induced by adding additional degrees of freedom locally in region surrounding the discontinuities

Besides modeling crack as sharp discontinuities, there are approaches that attempt to develop “smeared crack”, in which a damage zone is considered Some representatives

of smeared crack models can be listed as the gradient-enhanced damage model [8] and the phase field model [9, 10] In smeared crack models, a damage parameter is used to simulate the state of the material The damage parameter is usually ranged between 0 and 1, in which value 0 stands for intact material and value 1 stands for totally broken material Hence, instead of a sharp crack, a damage zone is obtained The size of damage zone is controlled by a length-parameter whose physical meaning is still being investigated The “smeared crack” converges to the “sharp crack” when this length-parameter is close to zero One advantage of damage model is that the problem can be formulated as continuum problem and damage parameter can be solved as a field variable From computational point of view, the damage model is rather simple and easy

an empirical function for description of damage evolution The recently introduced phase field model for fracture is formulated from energy conservation, hence it alleviates the limitation of empirical functions A disadvantage of phase field model is that it requires very fine element mesh and therefore it is time-consuming and computationally costly

1.3.1 Extended Finite Element method (XFEM)

Extended Finite Element method (XFEM) [11] is a well-known computational method

in modeling cracks, which “extends” the standard Finite Element method by extrinsic enrichments to represent the jumps at discontinuities Specific functions should be used

as enrichments for each type of discontinuities, i.e cracks, voids or inclusions For simulation of cracks, the Heaviside (step) functions are usually employed to capture the jump in displacement, while the so-called branch functions model the singularities at crack tip The tip-enriched functions are usually chosen based on the asymptotic solutions of displacement and stress fields when crack exists Since the introduction, XFEM has been investigated and applied in various types of problems, such as crack propagation in different kinds of material like isotropic materials [11], orthotropic materials [12] and functionally graded materials [13, 14]; crack propagation in different types of structures like solid structures [15] shell/plate structures [16, 17]; dynamic crack propagation analysis [18-20]

In XFEM model, a propagating crack is often described by a series line segments The crack topology is realized by the aid of level set method [21] Usually, two types of level set functions are needed: a normal level set, which is defined by the signed-distance from a given point to the crack segment, and a tangent level set, which is defined by the signed-distance to the line perpendicular to the crack segment at crack tip One tangent level set must be defined for each crack tip As crack propagates, the level set functions must be updated In order to determine the direction of propagating crack, appropriate criterion have to be taken Usually, this criterion is calculated from the stress intensity factors, which in turn can be computed by the so-called J-integral

1.3.2 Extended Meshfree approach

Although FEM is powerful and convenient, it is not without shortcomings The existence of the finite element mesh is actually a disadvantage in many cases One example is when large deformation cannot be ignored The standard element has to be convexed in shape in order to provide good approximated solution When large deformations occur, some elements could be distorted such that convexity is lost and numerical errors add up considerably Another example is the case where the mesh has

to be updated, as already mentioned above Hence, the meshfree method has been proposed to alleviate the difficulties related to finite element mesh [22] As the term suggests, the class of “meshfree” methods do not require discretizing the problem domain into finite elements Instead, the problem domain is represented only by nodes, including nodes on boundaries and nodes inside the domain When there is a change in domain geometry, nodes could be added or removed without difficulties

In terms of XFEM, re-meshing is not required However, one has to track the position

of crack relatively to the finite element mesh One of three scenarios may happen for each element: i) the element is entirely cut into two parts by the crack, ii) the element is partially cut by the crack and thus contains the crack tip, and iii) the element is not cut

by the crack Determination which case happens could be tricky due to the complicated

of the domain geometry and the arbitrary of crack path [23] As mention above, mesh is not be required in meshfree methods but scattered nodes are used for discretion of the problem domain These nodes play the key roll of approximation or interpolation without any connection between them as "element" in mesh-based method So meshfree methods are good at dealing with problems with discontinuities such as crack propagation problems On the other hand, applying the enrichment functions into a meshfree approximation scheme results in an approach which do not need update in domain geometry and the challenge of tracking the relative position of crack with respect

to the finite element mesh can be avoided The use of meshfree methods to these fracture problems, however, lags a little behind

There are several meshfree methods, based on several types of basis functions used in meshfree approximation such as the Element-Free Galerkin method (EFG) [24], the

meshless local Petrov-Galerkin method (MLPG) [25], the Reproducing Kernel Particle method (RKPM) [26], the Radial Point Interpolation method (RPIM) [27] and the Moving Kriging method (MK) [28, 29] Most of the meshfree basis functions (e.g EFG, MLPG, and RKPM) do not possess Kronecker-delta property as in standard Finite Element basis functions Hence, essential boundary conditions cannot be enforced directly In such situation, further techniques like Lagrange multipliers or penalty method are usually employed to impose the boundary conditions The RPIM and MK basis functions are in contrast, satisfy the Kronecker-delta property, thus they allow direct imposition of essential boundary conditions Interestingly, these two meshfree approaches share similarities in formulation, although being developed from different sources, and can be classified into one category

To apply meshless methods to fracture problems, there are two types of enrichment techniques In the first type, the basics functions are augmented to include functions similar to the near tip asymptotic fields (intrinsic enrichment) as per [29-31] The second type is more efficient that the displacement field is enriched by using the Heaviside This approach is known as an extrinsic enrichment meshfree method and only the nodes surrounding the crack are taken into account As a consequence, meshfree methods using the extrinsic enrichment are thus suitable for crack growth simulation The new approaches are formed in a way that making use of not only the advantages of the RPIM and MK shape functions [32, 33] and but also the versatility of the vector level set method [34] Additionally, as compared with lower-order finite elements that are commonly applied and can be captured only the linear crack opening, the meshfree methods however are dominated and have the great advantage to capture more realistic crack openings due to the higher-order continuity and non-local interpolation character [35]

1.4 Fundamental of Fracture Mechanics

1.4.1 Crack behavior in isotropic material

In linear elastic fracture mechanics, a macro cracked problem is usually considered in which the model contents an initial crack The model can be subjected to various kinds

of load such as static, harmonic and dynamic The main tasks of such problems are to

answer these following questions:

- Does the crack propagate under the given load?

- If the crack grows, what is the direction of the propagation?

- What is he maximum length of the crack that cannot destroy the structure

seriously?

- What is the maximum load that the structure can work well with? And so on…

The principle difficulties in cracked problems are the discontinuity at the crack path

(2-D) or crack face (3-(2-D) and the singularity at the crack tip Generally, there are three

types of loading that can make a crack develop (see Figure 1.4) Mode I occurs when

the principle load is applied normal to the crack plane, tends to open the crack Mode II

happens when the crack faces are subjected to an in-plane shear loading and tends to

slide the crack faces on each other Mode III appears when the crack faces are subjected

to an out-of-plane shear loading and tends to tear the two crack faces

Because of the singularity of stress at the crack tip, the behavior of material at this

special location cannot be calculated by normal elasticity theorem In general, the stress,

strain and displacement fields for the zone ahead of crack tip are given in term of special

factors called stress intensity factors (SIFs) There are three SIFs (K K I, II,K III) with

respect to three modes of fracture Stress fields ahead of a crack tip in an isotropic linear

elastic material can be written as

where rand are illustrated in Figure 1.7; f is tensor of non-dimensional values, ij

function of angle  and depend on crack mode

Figure 1.7 Definition of the coordinate axis ahead of a crack tip

In linear elastic, isotropic material, stress and displacement fields ahead of a crack tip

for Mode I and Mode II are given below

I

I

I

K r K r

K r

2 2

K r u

K r

2 2

K r u

where  is shear modulus;   3 4 (plane strain);   3   / 1  (plane stress)

In the linear elastic fracture mechanics, the evaluation of the stress intensity factors

under static or dynamic loading condition is so essential The SIF values play a key role

in estimating the residual strength of cracked structures, predicting the direction of crack

growth and so on However, analytical solution for stress intensity factor is given in very

few simple cases of model and load

1.4.2 Crack behavior in orthotropic materials

Assume an orthotropic crack body subjected to arbitrary forces with general boundary

conditions as shown in Figure 1.8 The global Cartesian coordinate system is denoted

by X1, X2, local Cartesian coordinate system is denoted by x x1, 2 and local polar

coordinate system at the crack-tip is denoted by r,  A following characteristic

equation can be obtained using equilibrium and compatibility conditions [36]

Figure 1.8 2D orthotropic body with crack Lekhnitskii [36] proved that the roots of Eq (1.6) are always complex or purely

imaginary and come in conjugate pairs as  1, 1 and  2, 2, they have the form:

1Re

II

II

II

K r K r K r

1.4.3 Crack behavior in functionally graded materials

Consider a 2-D functionally graded material (FGM) body with a crack as shown in

Figure 1.9 Elastic modulus and Poisson’s ratio are functions of location The global and

local coordinate systems are defined similarly to section 1.4.1 and 1.4.2 The stress fields

ahead of the crack tip are the same with the case of isotropic homogenous material

However, the displacement fields at crack tip zone are modified in which the shear

modulus parameter  and bulk modulus  are calculated at the position of the crack tip,

denoted by TIP and TIP, respectively

Mode I

2 TIP

TIP

2 TIP

I y

2 TIP

TIP

2 TIP

II y

where TIP is shear modulus and TIP is the bulk modulus at the crack tip They are

determined as TIP ETIP / 2 1 TIP; TIP TIP

TIP

31

Figure 1.9 2-D FGM body with crack

1.5 Objective of the dissertation

Together with the innovations in computer technology, computational fracture

mechanics have been intensively investigated in recent years with many reports

available in literatures discussing a wide range of aspects However, development of

advanced numerical methods and tools is a constant requirement to satisfy higher

demand in structural safety and durability Based on the advantages and disadvantages

of various trends in crack modeling, this thesis focuses on the following points:

- Develop a computational model for fracture analysis based on meshfree approach

and enrichment functions for mathematical description of discontinuities The class of

radial basis functions is employed due to its advantage of possessing the Kronecker delta

property Currently, there are models based on enriched EFG method for isotropic

materials [29], orthotropic composites [37] and thermos-mechanical fracture problems

1

x

[38] Nevertheless, there is still little available literatures on meshfree model based on point interpolation for fracture problems

- Extend the model to investigate fracture problems in different types of materials: isotropic, orthotropic and functionally graded materials (FGM) The behaviors of materials have to be incorporated into the numerical model

- Consider dynamic fracture The inertia effect in most of practical engineering problems cannot be neglected, so dynamic fracture analysis is also investigated on various types of materials

In terms of numerical implementations, nodes are selected into three groups: i) enriched nodes, ii) nodes enriched by the step function and (iii) nodes enriched by the branch functions This procedure is in general simpler than those in XFEM which involves also the geometry of element The role of enrichment functions associated with the stress singularity at crack tip, i.e “tip-enriched functions” are investigated Typically, four-fold branch functions have to be considered based on the analytical solution of stress field near the crack tip Recently, ramp function has been proposed as alternative to the branch functions [39] The interesting point is that only one ramp function needed for each tip-enriched node, instead of four branch functions as usual Thus the number of additional degrees of freedom is reduced Details on aspects related

non-to the implementation of the proposed method are discussed in the thesis

The proposed computational model is validated through many problems, in order to investigate its accuracy, efficiency and applicability Comparison is conducted between numerical results and reference results, obtained from analytical solution or from other author in form of experiments or numerical simulations

1.6 Outline of the thesis

The thesis is organized as follows:

Chapter 1 shows the overview of fracture mechanics on three types of materials, i.e isotropic materials, orthotropic materials and functionally graded materials; some related studies and objectives of the thesis

Chapter 2 is dedicated for review of meshfree methods using the class of radial basis functions, including some advanced knowledge in construction of meshfree shape functions Enrichment methods, fundamental equations, J-integral for crack tip behavior and numerical implementation procedure are also presented in this chapter

Chapter 3 shows the application of extended meshless radial interpolation method RPIM) for static stress intensity factors calculation and quasi-static crack growth simulation of 2D isotropic solid

(X-Chapter 4 presents the dynamic form of J-integral and interaction integral method Several transient dynamic crack analyses of isotropic and orthotropic composite materials, quasi-static crack growth simulation of orthotropic materials are performed

to verify the accuracy of the extended meshless approach (X-RPIM) The linear ramp function is first introduced in meshless context to model the singularity at crack tip, applied for static and dynamic crack analyses of isotropic

Chapter 5 shows the application of X-RPIM for static and dynamic stress intensity factors computation of functionally graded materials

Chapter 6 are reserved for application of the improved extended meshless moving Kriging methods (X-MK) for crack modeling of isotropic and functionally graded materials Three types of correlation functions are applied to improve the moving Kriging shape functions, two of them are first presented in this thesis

Finally, some conclusions and outlooks are extracted in Chapter 7

CHAPTER 2 EXTENDED MESHFREE GALERKIN METHODS FOR

FRACTURE MECHANICS

This chapter discusses about the interested meshfree methods based on the radial point

interpolation technique The thin plate spline (TPS) function is used as radial basis

function for the construction of RPIM shape function Enrichment methods for

discontinuity at crack faces and singularity at crack tip are first presented in conjunction

with meshfree interpolation methods

2.1 The Radial Point Interpolation method (RPIM)

The radial point interpolation method based on the Galerkin weak form was presented

in 2000 by Wang and GR Liu [27] In this method, the problem domain is represented

by a set of distributed nodes Radial basis functions (RBFs) and polynomials are used to

construct the RPIM shape functions that based only on a group of nodes arbitrarily

distributed in a local support domain Similar to the EFG method, a set of background

cells covered the domain is used to evaluate the integrals in the Galerkin weak form It

should be noted that the RPIM is based on the EFG method by replacing moving least

squared (MLS) shape functions with RPIM shape functions [22]

The RPIM shape function has been developed based on the point interpolation method

(PIM) by including the radial basis function (RPF) in the interpolation formulation The

RPIM shape functions are used frequently for both mesh free strong form and weak form

in [22, 40] The RPIM interpolation with radial basis functions and polynomials is

 

i

R r : the radial basis function

n: number of nodes in the support domain defined by point x

( )

j

p x : the monomial in the 2-D space coordinates xT x y, 

m: the number of polynomial basis functions

i

a and b j: constants yet to be determined

The common radial basis functions used to construct the RPIM shape function are

r d i

the point of interest x, a system of n linear equations are obtained, one for each node,

which can be written in the matrix form as