Development of new finite elements based on consecutive interpolation for 2d and 3d thermal mechanical problems

iii ABSTRACT This dissertation presents a new group of finite elements based on the integration of consecutive-interpolation procedure into the traditional Finite Element Method.. The c

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

Independent Examiner 1: Associate Prof Dr NGUYỄN VĂN HIẾU

Independent Examiner 2: Associate Prof Dr NGUYỄN QUỐC HƯNG

Examiner 1: Dr TRỊNH ANH NGỌC

Examiner 2: Associate Prof Dr CHÂU ĐÌNH THÀNH

Examiner 3: Associate Prof Dr LƯƠNG VĂN HẢI

SCIENTIFIC SUPERVISORS:

1 Assoc Prof Dr TRƯƠNG TÍCH THIỆN

2 Assoc Prof Dr BÙI QUỐC TÍNH

i

LỜI CAM ĐOAN / DECLARATION

Tác giả xin cam đoan đây là công trình nghiên cứu của bản thân tác giả Các kết quả nghiên cứu và các kết luận trong luận án này là trung thực, và không sao chép từ bất kỳ một nguồn nào và dưới bất kỳ hình thức nào Việc tham khảo các nguồn tài liệu (nếu có) đã được thực hiện trích dẫn và ghi nguồn tài liệu tham khảo đúng quy định

This doctoral dissertation is the outcome of my original research, conducted at the Ho Chi Minh city University of Technology, VNU – HCM, Viet Nam I declare that this document is my own work and has not submitted for any other degree or qualification except as specified All the references are cited in the document

Tác giả luận án / Author

Chữ ký / Signature

ii

TÓM TẮT LUẬN ÁN

Luận án trình bày một nhóm phần tử hữu hạn mới dựa trên sự tích hợp kỹ thuật nội suy liên tiếp vào phương pháp phần tử hữu hạn truyền thống Với kỹ thuật này, không chỉ giá trị tại nút mà cả giá trị đạo hàm trung bình tại nút cũng được sử dụng trong quá trình xấp xỉ Nhờ đó trường đạo hàm thu được từ nhóm phần tử hữu hạn mới là một trường liên tục, thay vì bất liên tục tại nút (không hợp lý về mặt vật lý) như phương pháp truyền thống Sự cải thiện về tính liên tục cũng mang đến độ chính xác cao hơn của lời giải xấp

xỉ Tuy nhiên khác với các phương pháp có độ liên tục bậc cao hiện hành như phương pháp Đẳng hình học hay phương pháp không lưới, phương pháp phần tử hữu hạn nội suy liên tiếp vẫn duy trì thuộc tính Kronecker quan trọng trong tính toán số Thêm vào

đó, phương pháp đề xuất sử dụng cùng một lưới phần tử với phương pháp phần tử hữu hạn truyền thống và không làm tăng số lượng bậc tự do

Kỹ thuật nội suy liên tiếp ban đầu được giới thiệu riêng lẻ cho phần tử tam giác 3 nút và phần tử tứ giác 4 nút và áp dụng với bài toán đàn hồi tuyến tính hai chiều Trong luận

án này, phương pháp được hệ thống hóa và phát triển nâng cao để tạo ra một nhóm phần

tử hữu hạn mới phù hợp với nhiều miền bài toán từ một chiều đến ba chiều, và được áp dụng để phân tích bài toán tương tác cơ-nhiệt Phương pháp đề xuất tiếp tục được mở rộng để khảo sát ứng xử của miền chứa dạng bất liên tục như vết nứt, với vật liệu đẳng hướng và vật liệu trực hướng

iii

ABSTRACT

This dissertation presents a new group of finite elements based on the integration of consecutive-interpolation procedure into the traditional Finite Element Method With this technique, not only the nodal values but also the averaged nodal gradients are included in approximation process As a result, the gradient fields obtained by the new group of finite elements are smooth, unlike the fields which are (non-physically) discontinuous at nodes delivered by traditional Finite Element Method The improvement on continuity results in higher accuracy of approximated solution as well

On the other hand, unlike the other higher-order methods such as the Isogeometric Analysis and the Meshfree methods, the novel Consecutive-interpolation Finite Element Method possesses the important Kronecker-delta property Furthermore, the proposed method employs the same discretization mesh with the traditional Finite Element Method and does not increase the number of degrees of freedom

The consecutive-interpolation procedure was initially introduced separately for the node triangular element and the 4-node quadrilateral element to be used in analysis of two-dimensional linear elastic problems In this dissertation, the method is further developed to form a new class of finite elements which is suitable for domains from 1D

3-to 3D The new group of finite elements (being integrated with consecutive-interpolation procedure) is applied to analyze the thermo-mechanical problems The proposed method

is also extended to study behaviors of bodies containing discontinuities such as cracks, for both isotropic and orthotropic materials

iv

CONTENTS

LỜI CAM ĐOAN / DECLARATION i

TÓM TẮT LUẬN ÁN ii

ABSTRACT iii

CONTENTS iv

LIST OF FIGURES viii

LIST OF TABLES xii

NOMENCLATURE xiii

CHAPTER 1 INTRODUCTION 1

Heat transfer and thermo-mechanical problems 1

Finite element method (FEM) and its issues 2

Trends in development of numerical methods 5

Original contributions of the dissertation 7

Scientific and practical meaning of the contributions by the dissertation 8

Methodology 8

Scope of the dissertation 8

Outline of the dissertation 9

CHAPTER 2 LINEAR THERMO-ELASTIC PROBLEMS 11

Formulation 11

Discrete form 13

Time integration scheme 15

2.3.1 Backward Euler scheme 15

2.3.2 Newmark scheme 16

CHAPTER 3 CONSECUTIVE-INTERPOLATION PROCEDURE FOR 1D AND 2D PROBLEMS 17

Issue of non-physically discontinuous nodal gradient in Finite Element Method (FEM): An example of two-node bar element (L2 element) 17

The consecutive-interpolation procedure (CIP) for two-node bar element: CL2 element 19

3.2.1 Calculation of CIP-based shape functions 21

3.2.2 First order derivative of CIP-based shape functions 23

3.2.3 Modification to retain the C0-continuity 24

v

3.2.4 Numerical example 25

The consecutive-interpolation procedure (CIP) for three-node triangular element (CT3) and four-node quadrilateral element (CQ4) 27

3.3.1 CIP-enhanced formulation for 2D domain 27

3.3.2 The CT3 element 28

3.3.3 The CQ4 element 28

3.3.4 Numerical examples 31

Conclusion 37

CHAPTER 4 TWO-DIMENSIONAL DYNAMIC AND QUASI-STATIC THERMOELASTIC FRACTURE PROBLEMS IN ISOTROPIC MATERIALS 38

Introduction 38

Numerical modeling of cracks 39

The extended consecutive-interpolation four-node quadrilateral element (XCQ4) 40

4.3.1 Enriched formulation for displacement field 41

4.3.2 Enriched formulation for temperature field 43

Computation of (dynamic) stress intensity factors (DSIFs) for thermo-elastic fracture problems 45

Crack growth modeling 48

Numerical results and discussion 49

4.6.1 Edge crack under constant flux (mode-I) 49

4.6.2 Static SIFs analysis: Square plate with a center crack 55

4.6.3 Static SIFs analysis: Rectangular plate with a slant center crack (mixed -mode) 59

4.6.4 Quasi-static crack propagation simulation of a slant edge crack in a cruciform panel 63

4.6.5 Dynamic SIFs analysis: Edge crack under quasi-static thermal shock 66

4.6.6 Dynamic SIFs analysis: Center crack under quasi-static thermal shock 68

4.6.7 Dynamic SIFs analysis: Curved crack under dynamic thermal shock 70

Conclusion 76

CHAPTER 5 TWO-DIMENSIONAL QUASI-STATIC THERMO-ELASTIC FRACTURE PROBLEMS IN ORTHOTROPIC MATERIALS 78

Introduction 78

vi

Formulation of XCQ4 element for linear thermo-elastic fracture problems in

orthotropic media 79

5.2.1 The characteristic equation of an arbitrary orthotropic material 80

5.2.2 Enriched formulation for displacement 81

5.2.3 Enriched formulation for temperature 83

Evaluation of SIFs by Interaction integral 83

Crack growth modeling 84

Numerical examples 86

5.5.1 Static SIFs analysis: Single edge notched specimen under mechanical tensile load 86

5.5.2 Static SIFs analysis: Rectangular epoxy/glass plate with a horizontal edge crack under constant flux 89

5.5.3 Static SIFs analysis: Rectangular epoxy/glass plate with a slant center crack 92

5.5.4 Static SIFs analysis: An anisotropic square plate with two parallel isothermal cracks 94

5.5.5 Quasi static crack propagation in an anisotropic cracked disc 97

5.5.6 Quasi static crack propagation of an edge crack in a rectangular plate under constant flux 100

5.5.7 Quasi static crack propagation in a perforated panel with a circular hole under constant heat flux 102

Conclusions 106

CHAPTER 6 DEVELOPMENT OF 3D CIP-BASED FINITE ELEMENTS 107

Generalized formulation to determine auxiliary functions 107

CIP-enhanced FEM for 3D linear heat transfer problems 110

6.2.1 Steady-state heat convection in a 3D complicated domain 110

6.2.2 Transient heat transfer in a plate with cylindrical hole 113

CIP-enhanced FEM for 3D linear elastic problems 116

6.3.1 Static analysis of a cantilever beam with T-shaped cross-section 116

6.3.2 Free vibration analysis of a hollow cylinder 120

6.3.3 Free vibration analysis of a composite sandwich beam 122

Conclusions 124

CHAPTER 7 CONCLUSIONS AND OUTLOOKS 125

vii

Conclusions 125

7.1.1 XCQ4 element for analysis of linear thermo-elastic fracture problems 125 7.1.2 General formulation of CIP-enhanced elements 126

Outlooks 127

PUBLICATIONS 128

BIBLIOGRAPHY 129

APPENDIX A 140

APPENDIX B 144

viii

LIST OF FIGURES

Figure 1.1 Thermal crack in an asphalt pavement [7] 2

Figure 1.2 Illustration of a finite element mesh for a two-dimensional heat transfer problem 3

Figure 1.3 Convergence of thermal energy corresponding to mesh size 4

Figure 3.1 Illustration of a two-node bar element (L2) in global system of coordinates (left) and in natural coordinates (right) 17

Figure 3.2 A one-dimensional domain being discretized by two L2 elements 18

Figure 3.3 Shape function associated with node 2 (global) of the Example provided in Figure 3.2, computed by traditional FEM and by CIP-enhanced FEM 22

Figure 3.4 CIP-based shape functions R1, R2 and R3 associated with the nodes given in Figure 3.2 23

Figure 3.5 First-order derivative of the shape functions associated with node 2 of the Example given in Figure 3.2, computed by traditional FEM and CIP-enhanced FEM 24

Figure 3.6 Example 3.2.4: Sketch of 1D bar subjected to body load The bar is uniformly discretized by 5 eleements 25

Figure 3.7 Example 3.2.4: Comparison of stress calculated by standard FEM and the proposed CIP-enhanced FEM 26

Figure 3.8 Example 3.2.4: Convergence rate with respect to number of degrees of freedom (DOFs) of the standard linear FEM, quadratic FEM and CIP approach 27

Figure 3.9 Visualization of the shape function for a CQ4 element 29

Figure 3.10 Visualization of the first order derivative of shape function for a CQ4 element 29

Figure 3.11 Sketch of the CIP-enhanced four-node quadrilateral element (CQ4) in a 2D finite element mesh [42] 30

Figure 3.12 Example 3.3.2.1: Cantilever beam being subject to parabolic shear load31 Figure 3.13 Example 3.3.2.1: (a) The triangular mesh and (b) quadrilateral mesh at coarsest level 32

Figure 3.14 Example 3.3.2.1: Convergence rate of elastic energy with respect to mesh size, presented in log-log scale 33

Figure 3.15 Example 3.3.2.1: The field of normal stress component σxx computed by four types of element: CQ4, Q4, CT3 and T3 33

Figure 3.16 Example 3.3.2.2: Quarter model of the thick cylinder pipe and boundary conditions 35

Figure 3.17 Example 3.3.2.2: Temperature distribution in the cylinder pipe 36

Figure 3.18 Example 3.3.2.2: Hoop stress distribution in the cylinder pipe 37

Figure 4.1 Signed distance function 42

Figure 4.2 The set of nodes enriched by Heaviside function (denoted by x symbol and the set of nodes enriched by branch functions (denoted by square symbol) 42

ix

Figure 4.3 Level set functions 43 Figure 4.4 The interaction integral domain 46

Figure 4.5 The weighting function q(x) 46

Figure 4.6 Example 4.6.1: (a) Geometry and boundary conditions of a strip with an

edge crack under constant flux and (b) its deformed shape obtained with XCQ4

elements 50

Figure 4.7 Example 4.6.1: Different interaction integral domain size selected to

evaluate the interaction integral: (a) domain 1, (b) domain (2), (c) domain 3 and (d) domain 4 A mesh size of 25 x 49 elements is used for both XQ4 and XCQ4 51

Figure 4.8 Example 4.6.1: Convergence of SIFs evaluated by XCQ4 and XQ4

elements with respect to number of nodes 53

Figure 4.9 Example 4.6.1: Distribution of temperature [oC] 54

Figure 4.10 Example 4.6.1: Stress component σxx [Pa] obtained by (a) XCQ4 element and (b) XQ4 element, showing in deformed shape 54

Figure 4.11 Example 4.6.2: A square plate with horizontal center crack under two

sets of boundary conditions: (a) Adiabatic crack and (b) Isothermal crack 55

Figure 4.12 Example 4.6.2.1: (a) Temperature [oC] and (b) y-component of heat flux

[W/m2] 57

Figure 4.13 Example 4.6.2.2: (a) Temperature [oC] and (b) y-component of heat flux

[W/m2] 58

Figure 4.14 Example 4.6.3.1: Geometry and boundary conditions 59

Figure 4.15 Example 4.6.3.1: Normalized SIFs with respect to a/W at the inclined

angle θ = 30o 60

Figure 4.16 Example 4.6.3.1: Normalized SIFs with respect to crack angle θ, given

a/W = 0.3 60

Figure 4.17 Example 4.6.3.2: Geometry and boundary conditions 61

Figure 4.18 Example 4.6.3.2: Normalized SIFs with respect to a/W at the inclined

x

Figure 4.24 Example 4.6.5: Geometry and boundary conditions of an edge cracked

plate under quasi-static thermal shock 67

Figure 4.25 Example 4.6.5: Variation of normalized mode-I SIF with respect to normalized time Analytical solutions are calculated by Equation (4.26) 68

Figure 4.26 Example 4.6.6: Comparsion of normalized mode-II SIF versus normalized time, between the present XCQ4 approach (symbols) and BEM (solid lines) [95] 69

Figure 4.27 Example 4.6.7: Geometry and boundary conditions of the rectangular Bismuth layer with a pre-existing curved crack 70

Figure 4.28 Example 4.6.7: Temperature distribution in the Bismuth layer, displayed in deformed shape at t = 4 μs 72

Figure 4.29 Example 4.6.7: Variation of (a) mode-I and (b) mode-II dynamic SIFs with respect to time: comparison of mesh density 73

Figure 4.30 Example 4.6.7: Variation of (a) mode-I and (b) mode-II dynamic SIFs with respect to time: comparison of number of time steps 74

Figure 4.31 Example 4.6.7: Schematic representation of elastic wave (total displacement [m]) propagating in the Bismuth layer with a pre-existing curved crack at different time steps: (a) t = 0.4 μs, (b) t = 1.2 μs, (c) t = 2.0 μs, (d) t = 2.45 μs, (e) t = 3.2 μs and (f) t = 4.0 μs 76

Figure 5.1 Sketch of coordinate systems: X-Y are the global axes; x-y are the local axes defined at crack tip; and 1-2 are the material principal axes 80

Figure 5.2 Example 5.5.1: Sketch of the rectangular plate made of Picea abies with an inclined edge notch, subject to tensile loading 86

Figure 5.3 Example 5.5.1: Vertical displacement component uy [mm] 87

Figure 5.4 Example 5.5.1: Visualization of normal stress component σyy [MPa] fields obtained by XCQ4 and XQ4 88

Figure 5.5 Example 5.5.2: Geometry and boundary conditions 89

Figure 5.6 Example 5.5.3: Geometry and boundary conditions 92

Figure 5.7 Example 5.5.4: A glass/epoxy square plate with two parallel cracks 94

Figure 5.8 Example 5.5.4: Temperature distribution 94

Figure 5.9 Example 5.5.4: The y-component of heat flux computed by XCQ4 and XQ4 elements 95

Figure 5.10 Example 5.5.5: (a) Geometry of the cracked Brazilian disc specimen and (b) finite element mesh 97

Figure 5.11 Example 5.5.5: Crack paths obtained by initial crack inclination θ = 45o and various material orientations 99

Figure 5.12 Example 5.5.6: Geometry and boundary conditions 100

Figure 5.13 Example 5.5.6: Crack paths obtained by materials with various E1/E2 ratios, when material oriention β = 60o 101

xi

Figure 5.14 Example 5.5.7: Geometry and boundary conditions of a perforated panel

with a circular hole under constant heat flux 103

Figure 5.15 Example 5.5.7: Crack path obtained by the present approach for isotropic material 104

Figure 5.16 Example 5.5.7: Crack path obtained by the present approach for orthotropic glass/epoxy material with various material angle β 105

Figure 5.17 Example 5.5.7: Distribution of shear stress component σxy obtained for glass/epoxy material with material angle β = 0o 105

Figure 6.1 Illustration of four-node tetrahedral element defined in global coordinates (left) and natural coordinates (right) 109

Figure 6.2 Example 6.2.1: Geometry and Boundary conditions 110

Figure 6.3 Example 6.2.1: (a) Finite element meshes using four-node tetrahedrons and (b) eight-node hexahedrons 111

Figure 6.4 Example 6.2.1: Steady-state temperature field 111

Figure 6.5 Example 6.2.1: Convergence of maximum temperature with respect to the number of degrees of freedom the “exact” value is estimated by a very fine mesh of 71540 HH8 elements 112

Figure 6.6 Example 6.2.1: The y-component heat flux [W/m2] obtained by TH4 elements (upper) and CTH4 elements (lower) 113

Figure 6.7 Example 6.2.2: Geometry and boundary conditions 114

Figure 6.8 Example 6.2.2: A mesh of 600 hexahedral elements 114

Figure 6.9 Example 6.2.2: Distribution of steady-state temperature 115

Figure 6.10 Example 5.2.2: Variation of temperature at point A versus time 115

Figure 6.11 Example 6.3.1: Sketch of a cantilever beam with T-shaped cross-section subjected to shear traction 116

Figure 6.12 Example 6.3.1: Dimensions of the T-shaped cross-section 116

Figure 6.13 Example 6.3.1: Finite element mesh of 480 hexahedral elements 117

Figure 6.14 Example 6.3.1: Convergence of elastic strain energy with respect to DOFs obtained by various types of elements: TH4, CTH4, HH8 and CHH8 the “exact” value is estimated by a very fine mesh of 8800 HH8 elements 117

Figure 6.15 Example 6.3.1: Convergence rate of elastic strain presented in log-log graphs of four types of elements: TH4, CTH4, HH8, CHH8 118

Figure 6.16 Example 6.3.1: Normal stress component σxx calculated by HH8 elements and CHH8 elements 119

Figure 6.17 Example 6.3.2: Geometry of the cylinder structure with a cylindrical cavity 120

Figure 6.18 Example 6.3.2: Mode shapes corresponding to the first four non-zero eigenfrequencies 121

Figure 6.19 Example 6.3.3: Geometry of the sandwich composite beam 123

Figure 6.20 Example 6.3.3: The first five mode shapes of the sandwich beam 123

xii

LIST OF TABLES Table 3.1 Comparison of elapsed time and accuracy, measured by numerical error in

elastic energy as given in Equation (3.53), obtained by Q4 and CQ4 elements 34

Table 4.1 Example 4.6.1: Effect of interaction domain sizes on the normalized SIF 𝐾𝐼 evaluated by XCQ4 elements 52

Table 4.2 Example 4.6.1: Comparison of normalized SIF 𝐾𝐼 evaluated by the standard XQ4 and the developed XCQ4 elements using various meshes, with reference solutions 52

Table 4.3 Example 4.6.2.1: Comparision of the normalized SIF 𝐾𝐼𝐼 with respect to crack lengths among the present XCQ4 and reference solutions 56

Table 4.4 Example 4.6.2.2: Comparision of the normalized SIF 𝐾𝐼 with respect to crack lengths among the present XCQ4 and reference solutions 58

Table 4.5 Example 4.6.4: Different loading cases 64

Table 4.6 Example 4.6.7: Crack point coordinates of the curved crack 71

Table 4.7 Example 4.6.7: Material of Bismuth layer [74] 71

Table 5.1 Example 5.5.1: Evaluation of normalized SIFs numerically by XCQ4 and XQ4 elements, using the same mesh of 25 x 49 quadrilateral elements 89

Table 5.2 Example 5.5.2: Normalized SIFs with respect to different values of a/W obtained by the present XCQ4 with different mesh sizes 90

Table 5.3 Example 5.5.2: Normalized SIFs calculated by XCQ4 and XQ4 elements for the case a/W = 0.3 using various finite element mesh Reference results are taken from [114, 112] The data in brackets are elapsed time 91

Table 5.4 Example 5.5.3: Normalized SIFs numerically evaluated by using a uniform mesh of 49 x 99 XCQ4 elements 93

Table 5.5 Example 5.5.4: Normalized SIFs numerically evaluated by using a uniform mesh of 49 x 49 XCQ4 elements 96

Table 5.6 Example 5.5.6: Mode-I and Mode-II SIFs and critical angle θc after the first incrmental step, with respect to various material orientation β and six sets of material properties 102

Table 6.1 Example 6.3.1: Strain energy evaluated by CHH8 and HH8 elements using various finite element mesh The data in brackets are numerical error 119

Table 6.2 Example 6.3.2: Comparison of the first four non-zero dimensionless eigenfrequencies, obtained by analytical solution [116], the standard HH8 element and the CIP-enhanced CHH8 element 122

Table 6.3 Example 6.3.3: Material properties of carbon/epoxy and polyurethane [117] 123

Table 6.4 Example 6.3.3: Natural frequencies of the first five modes in L-t plane, compared with experimental data [117] 124

E Young’s modulus in 2-direction (orthotropic material)

12

G Shear modulus in 1 – 2 plane (orthotropic material)

xiv

XFEM Extended finite element method

CIP Consecutive-interpolation procedure

LEFM Linear elastic fracture mechanics

LTEFM Linear thermoelastic fracture mechanics

CT3 CIP-enhanced three-node triangular element

CQ4 CIP-enhanced four-node quadrilateral element

CTH4 CIP-enhanced four-node tetrahedral element

CHH8 CIP-enhanced eight-node hexahedral element

XCQ4 Extended CIP-enhanced four-node quadrilateral element SIF Stress intensity factor

1

Heat transfer and thermo-mechanical problems

Heat transfer problems constitute a large class of engineering problems, which can be encountered nearly in every activities For example, the air-conditioning exploits the convection; both heat conduction and convection can be found in cooking; and the Earth receives heat from the Sun through thermal radiation In fact, heat transfer analysis is required in wide range of industrial disciplines such as aeronautical, electrical, mechanical and civil engineering etc [1, 2, 3]

Heat transfer within a system can be viewed as the “kinetics of storage, transport, and transformation of microscale energy carriers” [4] The process obeys the law of conservation of energy Thermal energy is transported from a hotter region to a colder one Heat conduction is the transfer of thermal energy internally of a body In order for heat conduction between body A and body B to occur, the two bodies has to be in contact Heat convection indicates the thermal interaction between a body with surrounding environment Thermal radiation is the emission of electromagnetic waves from a body, though the intensity depends on the temperature of that body [5] In engineering applications, the design’s primary goal is maximization of the amount of thermal energy transport while cost has to be kept as low as possible [6] Such a goal cannot be accomplished without heat transfer analysis

The change of temperature causes thermal expansion in materials Without appropriate consideration, this phenomenon would result in extra stress on mechanical components and structures In some cases, thermal stress may lead to breakage For example, pouring hot water into an ice-cold glass would make the glass shatters Due to the poor heat conduction of glass, the inner layer is hot (because of the contact with hot water), while the outer layer is still relatively cold The differences in thermal expansion between inner and outer layer builds up large thermal stress and cause the glass to break Cracks

are also detected in Ontario roads due to temperature changing [7], see Figure 1.1

Thermal shock cracking also occurs in reactor pressure vessel [8], a key component of

2

a nuclear power plant The existence of damage and fracture reduces the reliability and safety of structures, causing heavy financial loss due to maintenance and even human casualties In addition, material properties are dependent on temperature [9] High temperature usually makes metal sheet alloys more ductile and thus the formability is increased [10, 11] In contrast, low temperature tends to make materials more brittle [12,

13, 14] Therefore, the study on thermo-mechanical behaviors and thermo-mechanical induced fracture of solids is necessary and of great importance in industries

Finite element method (FEM) and its issues

Engineering problems are usually described by a single or a set of partial differential equations, which capture the change of variables of interest both in space and time, such

as the balance of momentum equation in solid mechanics, the Navier-Stokes equation

in fluid dynamics, the energy conservation in heat transfer, the Maxwell equation in electro-dynamics, etc By solving these equations, the engineers can extract important information For example, the distribution of temperature and heat fluxes can be obtained from the heat transfer equation Based on such knowledge, engineers can analyze and make decisions on possible improvements Unfortunately, available closed-form solutions are limited to some ideal cases Problems encountered in engineering applications are more challenging, which usually involve complicated geometries and

Figure 1.1 Thermal crack in an asphalt pavement [7]

3

boundary conditions Therefore, it is difficult and time-consuming to find solution by analytical means Instead, numerical methods would be more suitable

A numerical method does not try to find the “exact” solution but instead an

“approximate” one The “true” solution is approximated by a set of calculated values using linear combination The current most popular numerical method is the Finite Element Method (FEM) The main idea of FEM is to discretize the domain into many non-overlapping sub-domains, namely elements A finite elment mesh for a two-

dimensional heat transfer problem is illustrated in Figure 1.2 (more details on the

problem, including the boundary conditions will be discussed in Chapter 3) In practice,

an element is constructed with simple geometrical shapes, for example a line dimensional problems), triangle and quadrilateral (two-dimensional – 2D – problems), tetrahedron and hexahedron (three-dimensional – 3D – problems) The partial differential equation is required to be satisfied in each element The vertices of elements are called nodes The unknowns, i.e the values that will be used to approximate the global solution, are associated with nodes and are named by the degrees of freedom (DOFs) The number of DOFs per node varies depending on problem types For instance, in heat transfer problems, the field variable to be solved is temperature, which

(one-is scalar, thus there (one-is one DOF per node For two-dimensional elasticity problems, there

Figure 1.2 Illustration of a finite element mesh for a two-dimensional

heat transfer problem

4

are two DOFs per node, i.e the horizonal and vertical components of displacement For three-dimensional elasticity problems, there are three DOFs per node to represent the three components of displacement

Obviously, a numerical solution is expected to converge to the “true” solution when more nodes, and thus more elements, are employed in approximation An example of convergence of thermal energy corresponding to mesh size, an inverse measure of

number of elements, is depicted in Figure 1.3, in which the calculated thermal energy

gets closer to the “exact” values when smaller mesh size, i.e more elements, are involved

Being originated from the 50s – 60s of the 20th century, FEM is until now the most popular numerical tool in industries for solving partial differential equations This method is implemented in various commercial softwares such as ANSYS, ABAQUS, COMSOL, NASTRAN, DIANA etc as well as open source softwares like FEAP, OOFEM, CalculiX, FeniCS, and Code Aster etc A large number of Universities and Research Institutes around the world also develop their own in-house FEM packages to serve for teaching and doing research The FEM did not gain its popularity for no reason

In fact, it possesses many advantages: stability, simplicity, fast calculation and

Figure 1.3 Convergence of thermal energy corresponding to mesh size

“Exact” solution

5

acceptable accuracy However, the method still contains inherent shortcomings, which were pointed out in available literatures [1, 15] as follows:

 A finite element mesh is pre-requisite For problems in which domains need to

be updated, such as the simulation of crack propagation and the fluid-solid interaction, mesh discretization has to be updated accordingly

 The elements have to satisfy certain geometric conditions If the elements get distorted, e.g a four-node quadrilateral element loses its convexity due to large deformation, numerical errors could add up considerably

 The gradient fields obtained by FEM, e.g strain and stress fields in solid mechanics, are non-physically discontinuous at nodes

 The elements are suffered from locking phenomena, e.g volume-locking is encountered in dealing with nearly compressible materials (the Poisson ratio is close to 0.5), while shear-locking is usually found in solving bending beams or plate/shell problems

Trends in development of numerical methods

The shortcomings of FEM motivates researchers on developing new numerical methods For example, the Isogeometric Analysis (IGA) [16] introduces the employment of the well-known basis functions in computational geometry – B-splines and NURBS – as an alternative shape function to approximate the field variables, which enables the control

of shape function degree as well as continuity IGA has been extensively investigated in various fields of application such as plate/shell analysis [17, 18, 19], unsaturated flow analysis [20, 21], elasto-plastic behavior analysis [22, 23], etc In overall, this method requires complicated computation and encounters difficulties in treatment of boundary conditions due to the lack of Kronecker-delta property Additional techniques, such as the Lagrange multipliers or the penalty method could be utilized to do the enforcement Another potential group is the class of meshless methods, such as the Element Free Galerkin (EFG) [24, 25], the Reproducing Kernel Particle Method (RKPM) [26, 27], and the Radial Point Interpolation Method (RPIM) [28, 29] The meshless methods have the advantage that the problem is represented by a set of discrete nodes within the

6

domain as well as on the boundaries Element is not required in meshless methods, thus the difficulties and constraints due to element mesh can be avoided However, the construction of meshless shape functions depends on some user-chosen parameters These parameters have influences on the accuracy of the numerical solutions but there

is no certain criterion to select them Selection of parameters relies mostly on experiences of users Also, different sets of parameters have to be chosen for different problem types, which is not favourable in practical applications Furthermore, like the IGA, a large number of current meshless methods do not satisfy the Kronecker-delta property Another famous numerical method is the Boundary Element Method [30, 31], which is relatively time-efficient since discretization is needed only on domain boundaries However, extraction of data at points inside the problem domain is usually not an easy task More importantly, BEM requires fundamental solution, which limits the application of the method in complicated problems

The second approach is improving the Finite Element Method to utilize its own strength, while the weakness is reduced A famous contribution in this direction is the smoothed finite element method [32] The numerical solution is smoothened by modification of the derivative fields (e.g strain field) The idea of smoothed FEM has been developed into four variants: cell-based [33, 34], node-based [35, 36], edge-based [37, 38] and face-based [39, 40] All the four variants of smoothing technique used in smoothed FEM involve extra points (which are not the field nodes) Recently, the Consecutive-Interpolation Procedure (CIP) has been proposed for triangular and quadrilateral elements [41, 42] CIP possesses many desirable features, such as higher accuracy, smooth gradient fields and easy to be implemented into any existing FEM codes Unlike the smoothed FEM, CIP approach extends the shape function of traditional finite elements by terms related to the averaged nodal gradients No extra point is required in the calculation Interestingly, the total degrees of freedom remains the same with FEM, given the same mesh Also, the Kronecker-delta property of FEM is preserved With all

of the above properties, CIP is quite promising and worth further investigation At the beginning of the current research, CIP has been introduced in literatures just for two-dimensional (2D) elements [41, 42] A minus point on the applicability of CIP is that it has to be formulated differently for each type of element, which is not a trivial task CIP-

7

enhanced FEM has also been extended for two-dimensional linear elastic fracture mechanics by [43, 44], in which better accuracy than the familiar extended finite element method (XFEM) was reported In 2019, when this dissertation is being prepared, it is realized by the author that the CIP approach has been further developed for plate/shell element using First-order Shear Deformation Theory [45]

Original contributions of the dissertation

The CIP approach is promising but the number of research works on this method is still limited, probably because of the necessity to develop a different formulation for each type of finite element Being inspired by the desirable properties of CIP and being aware

of its bottleneck issue, this dissertation presents the development of a new group of finite elements based on the CIP, with focus on heat transfer and thermo-mechanical analysis

of solids, including intact and cracked bodies The novel contributions are highlighted

c) Developing CIP for three-dimensional elements

d) Introducing a general formulation for CIP, which is applicable for a wide range

of element types from one-dimensional to three-dimensional elements This general formulation resolves the bottleneck issue mentioned in Section 1.3 The proposed computational model is implemented by an in-house computer code Performance of the novel group of CIP-enhanced elements is assessed by various numerical examples, through which the accuracy and efficiency are analyzed and

8

demonstrated Comparison is conducted between the results obtained by the proposed method and reference results available literatures, including analytical solutions, experimental data and numerical simulations by other authors

Scientific and practical meaning of the contributions by the dissertation

The dissertation proposes a formulation for two-dimensional linear thermoelastic fracture problems by combination of CIP-enhanced finite element and the enriched functions The current approach is an improvement of the extended finite element method (XFEM) [46] by the employment of CIP, in the sense of higher accuracy and smooth representation of gradient fields

For fracture problems of orthotropic materials, the necessity of taking material orientation into account is pointed out The obtained results help to explain the influence

of material orientation and the loading mode-mixity on how cracks propagate

The development of general formulation resolves the bottleneck issue of CIP, making a systematic procedure to apply CIP into a wide range of finite elements from 1D to 3D The general formulation also enables the implementation of CIP as an add-on to any existing FEM code

Methodology

The dissertation focuses on further development of existing numerical approaches, i.e Finite Element Method (FEM) and eXtended Finite Element Method (XFEM), based on the technique of Consecutive-Interpolation Procedure (CIP) It is demonstrated through numerical examples and comparison with available reference data, that the proposed formulations have smooth representation of gradient fields, higher accuracy and higher time-efficiency Also, the applicability of the CIP technique is expanded thanks to the introduction of a general formulation

Scope of the dissertation

The proposed approach is aimed to be applied in thermal-mechanical analysis of intact and cracked solid bodies Linear elasticity (Hooke’s law) and small strain/deformation (geometrical linearity) are assumed

9

For cracked solids, currently only two-dimensional problems are considered

For intact solids, the formulation is generalized and is applicable for 1D, 2D, and 3D problems

Outline of the dissertation

The dissertation is organized as follows:

Chapter 1 provides an overview on finite element method (FEM) and its issues in solving partial differential equations The shortcomings of FEM are motivations to develop alternative methods

Chapter 2 briefly presents formulation of linear thermo-elasticity as a coupled system of equilibrium and heat transfer problems The strong-form partial differential equations are transformed into weak-form, which is then discretized in both space and time Chapter 3 describes in details the consecutive-interpolation procedure (CIP) for one-dimensional and two-dimensional thermo-mechanical problems Some desirable properties of CIP are demonstrated in numerical examples Chapter 3 is written based

on Publication A and Publication B

Chapter 4 is dedicated for novel formulation of two-dimensional linear thermo-elastic fracture mechanics based on the CIP-enhanced Finite element method and the enriched functions approach The formulation is applied to analyze cracked bodies made of isotropic materials Both the adiabatic and isothermal conditions of crack are considered

As important fracture parameters, Stress intensity factors (SIFs) are evaluated and used

to estimate the possible direction of crack growth Chapter 4 is written based on Publication C

Chapter 5 presents the extension of the formulation in Chapter 4 for orthotropic materials Material orientation has to be considered in both the enriched formulations of field variables and the prediction of crack propagation Chapter 5 is written based on Publication D and Publication E

10

Chapter 6 develops a general form of CIP that is suitable to be incorporated into a wide range of finite elements to discretize from one-dimensional to three-dimensional domains This novel development resolves the dependence on element type, which is known to be a bottleneck issue of the CIP With the proposed general form, CIP can be implemented as an add-on for any available FEM codes Chapter 6 is written based on Publication A and Publication B

Finally, some conclusions and outlooks are provided in Chapter 7

Equation Chapter 2 Section 1

11

Formulation

Considering an isotropic body Ω bounded by Γ, the governing equation for a linear

elastic problem is given by

where b is the body force acting on Ω; ρ is the mass density; 𝒖̈ is the acceleration, i.e

the order derivative of displacement u with respect to time; and σ is the order Cauchy stress tensor The stress tensor is calculated from strain tensor ε by the

Here,   u ,   t and     u t   u is the prescribed displacement, t is the

prescribed traction and n is the outward normal unit vector

When thermal expansion is considered, the Hooke’s law in Equation (2.2) is rewritten

In Equation (2.7), α is the tensor of thermal expansion coefficients; and ΔT = T – Tref is

the difference between current temperature T and a reference temperature Tref The term

I denotes the second-order identity tensor Based on principle of energy conservation,

the governing equation for transient heat transfer process is generally written by

The boundary conditions associated with Equation (2.8) are given by

T T  , on Γ1: prescribed temperature (Dirichlet boundary condition) (2.10)

q n , on Γ4: heat convection (Robin boundary condition) (2.13)

In Equations (2.8-2.13), k is the thermal conductivity; Q stands for the heat sink/source;

ρ is the mass density; c is the specific heat capacitance; T denotes the first-order

derivative of temperature with respect to time; h is the coefficient of convection; and Ta

is the ambient temperature

The sets of Equations (2.1 – 2.13) describe the coupled thermo-elastic problem Let us

denote δu and δT as arbitrary test functions for displacement and temperature,

respectively In order to solve the problem numerically, the following variational form

13

is obtained by multiplying both sides of Equation (2.1) with δu and Equation (2.8) with

δT and then integrate those equations over domain Ω

Using a numerical scheme, e.g finite element method, displacement u and temperature

T as well as the corresponding test functions are approximated from nodal values by

n

TI I T I

TI y

R R

In Equation (2.32), E is the Young’s modulus and ν is the Poisson’s ratio By using

appropriate time integration scheme, i.e Backward Euler scheme for 𝑇̇ and Newmark scheme for 𝑢̈, the set of Equations (2.25 – 2.26) can then be solved for the vector of nodal values ˆT and ˆu Once the nodal values are known, the stress/strain and temperature gradients can be computed, then the elastic energy, u, and the equivalent thermal energy, T , and are calculated as follows

Time integration scheme

As already mentioned above, the Backward Euler scheme is used to approximate 𝑇̇, while the acceleration 𝑢̈ is calculated using Newmark scheme

2.3.1 Backward Euler scheme

Given that the quantity z at time t, z t = z(t), is known, the first order derivative of a quantity z with respect to time, at t   t , i.e ztt , can be approximated using Backward Euler scheme as follows

The Newmark scheme is unconditionally stable, i.e no critical time step is required, if

parameters β and γ satisfy the following conditions [48]

From conditions (2.39) and (2.40), it is reasonable to choose β=0.25 and γ = 0.5

Equation Chapter (Next) Section 1

17

FOR 1D AND 2D PROBLEMS

This chapter is written based on Publication A and Publication B

Issue of non-physically discontinuous nodal gradient in Finite Element Method (FEM): An example of two-node bar element (L2 element)

Let us consider a one-dimensional (1D) domain Ω An arbitrary function u(x) defined in

Ω is approximated by FEM as follows [49, 50]

where N I is the Lagrange basis function associated with node I (global index), uI is the

value of function u(x) at node I and n is the total number of nodes A two-node bar

element (L2) is a straight segment connecting two nodes, see Figure 3.1

Denoting the two local nodes within an L2 element as i and j, the Lagrange shape functions Ni and Nj associated with node i and node j, respectively, are written in natural

coordinates as follows

 

1

12

j

By using isoparametric concept, the geometry is approximated by the same manner as

Figure 3.1 Illustration of a two-node bar element (L2) in global system

of coordinates (left) and in natural coordinates (right)

Now, consider that the 1D domain Ω of length L = 1 is uniformly discretized into two

L2 elements, as depicted in Figure 3.2 The two elements are denoted as e1 and e2, while

the nodes are globally indexed by 1, 2 and 3

Figure 3.2 A one-dimensional domain being discretized by two L2 elements

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Node 2 is the common node of both element e1 and element e2, therefore the derivative

of function u evaluated at node 2 can be calculated by Equation (3.9) as

x

e

l

  , (consider node 2 as part of element e2) (3.11)

The nodal values u1, u2 and u3 are generally not identical, leading to different values of

u Hence, the gradient u,x is non-physically discontinuous at node 2 This is

a common issue of traditional FEM, which is known by the C0-continuity of Lagrange

bases More details on C0-continuity are discussed in [49]

The consecutive-interpolation procedure (CIP) for two-node bar element: CL2 element

The consecutive-interpolation two-node bar element (CL2) is developed by introduction

of the consecutive-interpolation procedure (CIP) into the L2 element Using the CIP [41,

42, 51, 52, 53], function u(x) in Equation (3.1) is approximated by

     [ ] [ ]

, 1

n

I Ix x I

where ne is the number of nodes being included in element e The weight function we in

Equation (3.15) is a measure of element size For one-dimensional elements, we can be

chosen as the ratio of the length le of element e and the total length of all elements in SI

I

e e

e

e S

l w

l





For two-dimensional elements, the ratio of element area can be used Similarly, the ratio

of element volume is the weight function of three-dimensional elements

Substituting Equation (3.16) into Equation (3.15) one obtains

in which N[ ],xI is the weighted averaged of nodal gradients of Lagrange shape functions

at node I With Equation (3.13) and Equation (3.18), Equation (3.12) is rewritten as

   [ ] [ ]

, 1

n

I Ix x I



Equation (3.19) has similar form with Equation (3.1), except that vector of Lagrange

shape functions N is now replaced by the vector R of CIP-based shape functions Vector

R is computed by

21

 [ ] [ ]

, 1

n

I Ix x I

local node i and local node j are constructed on top of Lagrange basis functions by

3.2.1 Calculation of CIP-based shape functions

The Example in Figure 3.2 is recalled here The vector R which contains the shape

functions R1, R2 and R3 associated with global node 1, 2, and 3 are computed following Equation (3.20) by

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[2][ ] [2][ ] [2]

02

Insert the results in Equations (3.26-3.29) into Equation (3.25), the CIP-based shape

functions are obtained For example, noting that node 2 is node j (local) of element e1,

while in element e2 it is node i, fucntion R2 is calculated as follows

Figure 3.3 Shape function associated with node 2 (global) of the Example

provided in Figure 3.2, computed by traditional FEM and by CIP-enhanced FEM

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the proposed method On the other hand, the corresponding Lagrange shape fucntion exhibits an acute peak at node 2

For more details, the three shape functions R1, R2 and R3 are plotted in Figure 3.4 All

of them are smooth curves, indicating that the order of the basis function is higher than the linear Lagrange basis function The span of the CIP-based shape function is also wider than that of traditional FEM In FEM, the Lagrange basis function associated to a specified node is defined only within elements containing that node For instance, the

Lagrange shape function associated with node 1, N1, is only non-zero within element e1

and is zero outside element e1 In contrast, the CIP-based shape function, R1, is defined

not only within element e1 but also in the adjacent element e2

3.2.2 First order derivative of CIP-based shape functions

Starting from Equation (3.25), the first order derivative of CIP-based shape functions are given as follows

24

The first order derivative of shape function R2 is demonstrated in Figure 3.5, in which

a smooth curve is observed In contrast, the first order derivative of N2 (traditional FEM)

is not smooth A jump of derivative of N2 at node 2 is clearly observed, which explains the discontinuity in nodal gradients in traditional FEM

3.2.3 Modification to retain the C 0 -continuity

The formulation of consecutive-interpolation leads to finite elements that can reproduce

continuous nodal gradients In practice, there are cases where C0-continuity at node is required, for instances on material interfaces and geometrical boundaries In such cases,

it is necessary to modify the formulation, such that the “nodal averaged gradient”, see Equation (3.18), is replaced by nodal gradient, i.e [41, 42]

By this simple modification, C0-continuity can be recovered for any given nodes In fact,

if C0-continuity is required on all the nodes in the problem domain, the CIP approach will degenerate to standard FEM

Figure 3.5 First-order derivative of the shape functions associated with

node 2 of the Example given in Figure 3.2, computed by traditional FEM

and CIP-enhanced FEM