Static and Dynamic Crack Propagation in Brittle Materials with XFEM Wagner Fleming Petri

2013 – 3 Fleming Petri Static and Dynamic Crack Propagation in Brittle Materials with XFEM

Static and Dynamic Crack Propagation in Brittle Materials

with XFEM

Wagner Fleming Petri

2013 – 3

Schriftenreihe

Fachgebiet Baumechanik/Baudynamik Prof. Dr.-Ing. habil. Detlef Kuhl

Universität Kassel

ISBN 978-3-86219-436-0

Schriftenreihe

Fachgebiet Baumechanik/Baudynamik 2013 – 3

Herausgegeben von

Prof. Dr.-Ing. habil. Detlef Kuhl

Universität Kassel

Wagner Fleming Petri

Static and Dynamic Crack Propagation in Brittle Materials with XFEM

kassel university

press

Die vorliegende Arbeit wurde vom Fachbereich Bauingenieur- und Umweltingenieurwesen der Universität Kassel als Dissertation zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften (Dr.-Ing.) angenommen.

Erster Gutachter: Prof. Dr.-Ing. habil. Detlef Kuhl Zweiter Gutachter: Prof. Dr.-Ing. habil. Paul Steinmann Weitere Mitglieder der Prüfungskommission:

Prof. Dr. Maria Specovius-Neugebauer Prof. Dr.-Ing. Arnd I. Urban

Tag der mündlichen Prüfung 30. November 2012

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Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.dnb.de abrufbar

Zugl.: Kassel, Univ., Diss. 2012 ISBN 978-3-86219-436-0 (print) ISBN 978-3-86219-437-7 (e-book)

URN: http://nbn-resolving.de/urn:nbn:de:0002-34371

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Der Umschlag dieser Arbeit ehrt Pers¨onlichkeiten, deren Entdeckungen und Publikationen die wesentlichen Grundsteine der Forschung und der universit¨aren Lehre des Fachgebiets Baumechanik/Baudynamik an der Universit¨at Kassel gelegt haben. Folgende Wissenschaftler sind dargestellt: Isaac Newton, Leonhard Euler, William Rowan Hamilton, Gottfried Wilhelm Leibniz, Galileo Galilei, Augustin-Louis Cauchy, Joseph-Louis Lagrange, Jean-Baptiste le Rond d’Alembert, Walter Ritz, Richard Courant, Ray William Clough, Olgierd Cecil Zienkiewicz, Boris Grigorievich Galerkin, Nathan Mortimore Newmark, John Hadji Argyris und Robert Leroy Taylor.

Die Portraits wurden in den Jahren 2012 und 2013 von Stephanie Heike, Studierende der Kunst- hochschule Kassel, als Teil des Projekts ’Pate -Permanent Assisted Teaching in Engineering’

in Grafit auf Papier kreiert. Dieses stellte einen Beitrag innerhalb des Lehrinnovationswettbe- werbes der Universit¨at Kassel dar.

Contents

1 Introduction 1

1.1 Motivation . . . 1

1.2 State of the Art . . . 2

1.3 Aim of the Work . . . 6

1.4 Outline . . . 7

2 Basic Concepts of Static Linear Elastic Fracture Mechanics 9 2.1 Reference Problem . . . 9

2.2 Asymptotic Solution . . . 10

2.3 SIFs for a Kinked Crack . . . 14

2.3.1 Infinitesimal Straight Crack Extension . . . 16

2.4 SIFs for a Symmetrically Branched Crack . . . 22

2.5 GRIFFITH’s Energy Balance . . . 25

2.6 Relationship Between the SIFs and the Energy Release Rate . . . 27

2.7 TheJ-Integral . . . 27

2.8 ESHELBYTensor . . . 28

2.9 Extraction of the SIFs . . . 30

2.10 Extraction of theT-Stress . . . 33

2.11 Fracture Criteria Based on LEFM . . . 34

2.11.1 Maximum Circumferential Stress Criterion . . . 36

2.11.2 Minimum Strain Energy Density Criterion . . . 38

2.11.3 Maximum Energy Release Rate Criterion . . . 40

2.11.4 Local Symmetry Criterion,LS. . . 42

3 Basic Concepts of Dynamic Linear Elastic Fracture Mechanics 47 3.1 Reference Problem . . . 47

3.2 Elastic Waves . . . 49

3.2.1 Body Waves . . . 49

3.2.2 LAMÉRepresentation . . . 50

3.2.3 Two-Dimensional Problems . . . 50

3.2.4 Surface Waves . . . 51

3.3 Stationary Cracks . . . 52

3.3.1 Asymptotic Solution . . . 52

3.3.2 Some Basic Solutions . . . 57

3.4 Moving Cracks . . . 62

3.4.1 Steady Dynamic Crack Growth . . . 62

3.4.2 Nonsteady Crack Growth . . . 64

3.4.3 Dynamic Stress Intensity Factors for Moving Cracks . . . 65

3.5 Energy Balance . . . 66

3.6 Extraction of the Dynamic SIFs . . . 68

3.7 Dynamic Fracture Criteria . . . 70

3.7.1 Crack Initiation . . . 71

3.7.2 Crack Growth . . . 71

3.7.3 Crack Arrest . . . 73

3.8 Maximum Circumferential Stress Criterion . . . 74

3.9 Minimum Strain Energy Density Criterion . . . 75

3.10 GRIFFITHEnergy Criterion and Principle of Local Symmetry . . . 78

3.10.1 Crack Kinking . . . 78

3.10.2 Crack Branching . . . 81

4 XFEM for Crack Modelling 85 4.1 Weak Form . . . 85

4.2 Finite Element Method . . . 86

4.2.1 Partition of Unity . . . 86

4.2.2 Finite Element Approximation . . . 86

4.2.3 Standard Error Estimator . . . 88

4.2.4 Error Estimator in the Presence of Singularities . . . 88

4.2.5 Shape Functions . . . 89

4.2.5.1 Quadrilateral Elements . . . 91

4.2.5.2 Triangular Elements . . . 94

4.3 Extended Finite Element Method XFEM . . . 96

4.3.1 Discontinuous Enrichment . . . 97

4.3.2 HANSBO& HANSBOApproach . . . 99

4.3.3 Near-Tip Enrichment . . . 100

4.3.4 Intersection Enrichment . . . 103

4.3.5 Branching Enrichment . . . 103

4.3.6 High Order Elements . . . 105

4.3.7 Near-Tip Enrichment on a Fixed Area . . . 106

4.3.8 DOF Gathering . . . 107

4.3.9 Cut-off Function . . . 108

4.4 Numerical Aspects . . . 109

4.4.1 Numerical Integration . . . 109

4.4.2 Node Selection for Enrichment in Order to Avoid Linear De- pendency . . . 112

4.4.3 Transformation for a kinked crack . . . 113

4.4.4 Determination of theq-field . . . 121

4.5 Examples . . . 124

4.5.1 Plate Under Uniaxial Tension . . . 124

4.5.2 Crack Propagation in a T-Beam . . . 125

4.5.3 Convergence Analysis . . . 128

4.5.4 Kinked Crack . . . 132

5 Dynamic Fracture 139 5.1 Weak Form of the Initial Boundary Value Problem . . . 139

5.2 Semi-discrete equation of motion . . . 141

5.3 Generalized-αMethod . . . 142

5.3.1 Numerical Properties . . . 143

5.3.1.1 Stability . . . 143

5.3.1.2 Accuracy . . . 146

5.3.1.3 Numerical Dissipation . . . 147

5.4 Continuous and Discontinuous GALERKINMethods . . . 152

5.4.1 Continuous GALERKINMethod . . . 152

5.4.1.1 Stability and Convergence Properties . . . 155

5.4.2 Discontinuous GALERKINMethod . . . 155

5.4.2.1 Acceleration Determination . . . 158

5.4.2.2 Stability and Convergence Properties . . . 158

5.5 Examples for Dynamically Loaded Stationary Cracks . . . 162

5.5.1 Rectangular Plate with a Central Crack . . . 162

5.5.2 Rectangular Plate with an Inclined Crack . . . 166

5.5.3 DENG’s Solution . . . 170

5.6 Examples for Propagating Cracks . . . 170

5.6.1 Semi-infinite Crack in an Infinite Plane Subjected to a Stress Pulse . . . 173

5.6.2 Crack Propagation in a T-Beam . . . 179

6 Summary, Conclusions and Future Work 187 6.1 Summary and Conclusions . . . 187

6.2 Future Work . . . 189

References 189

List of Figures

2.1 Cracked body and boundary conditions . . . 10

2.2 Definition of polar coordinates with pole at the crack tip. . . 11

2.3 Fracture modes. . . 13

2.4 Original crack and extension of lengths. . . 15

2.5 Particular case considered to determineFandG. . . 16

2.6 ξ−Plane and contoursC⁺andC⁻. . . 17

2.7 F-functions for different kinked angles θ using the TAYLOR series (2.30) and the numerical procedure described in this section. . . 21

2.8 G-functions for different kinked angles θ using the TAYLOR series (2.31) and the numerical procedure described in this section. . . 21

2.9 Particular case considered to determineFandG. . . 22

2.10ξ−Plane and contoursC₁⁺,2andC₁⁻,2. . . 23

2.11 Components of theFmatrix for the symmetrically branched crack (solid lines) and for the kinked crack (dashed lines) as function of the anglem=θ/π. . . 25

2.12 Components of theGvector for the symmetrically branched crack (solid lines) and for the kinked crack (dashed lines) as function of the anglem=θ/π. . . 26

2.13 Arbitrary contourΓsurrounding the crack tip. . . 28

2.14 Arbitrary contoursΓ₁andΓ₂. . . 30

2.15 Vectorial fieldq: a) fixed orientation; b) orientation tangent to the crack. 32 2.16 Concentrated load per unit thickness acting at the crack tip. . . 33

2.17 Possible fracture envelopes. . . 35

2.18 Fracture toughness as a function of the specimen thickness. . . 36

2.19 Stresses in the vicinity of the crack tip in polar coordinates. . . 37

2.20 Crack with infinitesimal extensionΔs. . . 41

2.21 Crack propagation angleθcaccording to the criteria of: (a) Maximum Circumferential Stress Criterion (σ), (b) Minimum Strain Energy Den- sity, plane stress andν= 0.3(S₁), (c) Minimum Strain Energy Density, plane strain andν= 0.3(S₂), (d) Maximum Energy Release Rate (G) and (e) Local Symmetry (LS). . . 44 2.22 Fracture envelopes according to the criteria of: (a) Maximum Circum-

ferential Stress (σ), (b) Minimum Strain Energy Density, plane stress andν= 0.3(S₁), (c) Minimum Strain Energy Density, plane strain and ν = 0.3(S₂), (d) Maximum Energy Release Rate (G) and (e) Local Symmetry (LS). . . 45 3.1 Initial and current configuration. . . 48 3.2 RAYLEIGHwave speedCR. . . 52 3.3 a) Uniform pressure applied to the crack, b) Step surface pressure. . . 57 3.4 Stressσ₂₂ahead of the crack tip for a semi-infinite crack under a sud-

denly applied, spatially uniform crack surface pressure. The models of plane stress and plane strain withν= 0.3are considered. . . 60 3.5 a) Concentrated loads applied at a distanceLbehind the crack tip, b)

Step concentrated load. . . 60 3.6 Stress intensity factor history,KI(t), for a semi-infinite crack under a

suddenly applied concentrated loads, located a distanceLbehind the crack tip. The models of plane stress and plane strain withν= 0.3are considered. . . 61 3.7 ContourΓmoving with the crack tip. . . 67 3.8 Dynamic crack initiation toughness for Homalite-100. Schematic rep-

resentation of the results obtained by RAVI-CHANDARand KNAUSS

[Ravi-Chandar and Knauss, 1984]. . . 72 3.9 Schematic behaviour of the dynamic crack growth toughnessK_I^Dwith

respect to the crack speed in a nominally brittle material. . . 73 3.10 Dynamic crack propagation angleθcaccording to the criteria of Maxi-

mum Circumferential Stress forv/CR= 0,0.1,0.2,0.3,0.4,0.5and0.6. . 76 3.11 Difference between the dynamic crack propagation angle θc and

θ_c⁰according to the criterion of Maximum Circumferential Stress for v/CR= 0.1,0.2,0.3,0.4,0.5and0.6. . . 76 3.12 RatioKeq/K_eq⁰ for different crack speedsv/CR= 0.1,0.2,0.3,0.4,0.5

and0.6, according to the criterion of Maximum Circumferential Stress. 77 3.13 Circumferential stress σθθ for Mode I loading and different crack

speedsv/CR= 0.5,0.6,0.7,0.75and0.8. . . 77 3.14 Dynamic crack propagation angleθcaccording to the criterion of Min-

imum Strain Energy Density (S) forv/CR= 0, 0.4and0.5. As a ref- erence, the results for theσ-criterion corresponding tov/CR= 0and 0.5are also shown in dotted lines. . . 78

3.15 RatioSmin/S_min⁰ for different crack speedsv/CR= 0.1,0.2,0.3,0.4and

0.5, according to the criterion of Minimum Energy Release Rate. . . . 79

3.16 Strain Energy Density FactorSfor ModeIloading and different crack speedsv/CR= 0.5,0.55,0.57,0.58and0.6. . . 79

3.17 Graph of theg(v)function defined in (3.122) for different values of the KOLOSOV’s constantκ(2.6). . . 84

3.18 Critical speed for the branching phenomenon as function of the KOLOSOV’s constantκ(2.6). . . 84

4.1 Finite Element discretization of a continuum and shape function. . . . 87

4.2 Piecewise linear global shape functionφⁱ. . . 91

4.3 Linear, quadratic and cubic shape functions for equally spaced nodes and the approximation functionu^h₁. . . 92

4.4 Bilinear LAGRANGEelement in physical and natural coordinates. . . . 93

4.5 Shape functions of the bilinear LAGRANGE element and approxima- tion of the displacement componentui. . . 93

4.6 LAGRANGEfamily of quadrilateral elements. . . 95

4.7 Natural domain (left) and nodal distribution (right) for triangular ele- ments. . . 95

4.8 a) linear, b) quadratic and c) cubic triangular elements. . . 96

4.9 Finite element mesh and crack Γd. The nodes marked with red squares are enriched with the generalized HEAVISIDEfunctionH. . . . 97

4.10 Normal and tangential vectors for: a) Smooth crack b) Kinked crack. . 98

4.11 Comparison of the HANSBO& HANSBOand XFEM basis functions. . . 100

4.12 Near-tip functions. . . 101

4.13 Enrichment strategy for a boundary crack. . . 102

4.14 Enrichment strategy for an interior crack. . . 102

4.15 Enrichment strategy for intersecting cracks. . . 104

4.16 Enrichment strategy for a branched crack. . . 105

4.17 Modified near-tip enrichment: all nodes within a radiusamax are en- riched by the near-tip functions. . . 107

4.18 Enrichment with a cut-off function. . . 109

4.19 Triangulation and GAUSS points for the elements influenced by the near-tip and HEAVISIDEenrichment. . . 111

4.20 a) Failed DELAUNAYtriangulation, b) Corrected DELAUNAYtriangulation.111 4.21 Transformations for the integration of a element cut by a crack. . . 112

4.22 Mapping of a square into a triangle. . . 112

4.23 Node selection strategy to avoid linear dependency: HEAVISIDE en-

richment. . . 113

4.24 Strategy to avoid linear dependency: Intersection enrichment. . . 114

4.25 Strategy to avoid linear dependency: Branching enrichment. . . 114

4.26 Mapping for a kinked crack. . . 115

4.27 Comparison of different mappings. . . 118

4.28 Mappings atX₂=−1.0. . . 118

4.29 Partial derivative ∂F₁ ∂X₁for different mappings atX₂=−1.0. . . 119

4.30 Successive mappings. . . 120

4.31 Variables for the calculation ofvq. . . 121

4.32 Plate under uniaxial tension with a crack of length2ainclined an angle β. . . . 124

4.33 Finite element mesh and crack forβ= 45^◦. . . 125

4.34 Stress intensity factorsKIandKIIfor a plate under uniaxial tension with a crack inclined an angleβ. . . 125

4.35 Crack propagation in a T-beam. Geometry, load, boundary conditions and finite element mesh. Units:[cm]. . . 126

4.36 Crack path for different boundary conditions. a) Vertical displacement fixed at both ends (bottom crack in red) b) Vertical displacement fixed along the lower border (top crack in black). . . 127

4.37 Load displacement curves for different boundary conditions. a) Ver- tical displacement fixed at both ends (red) b) Vertical displacement fixed along the lower border (black). . . 127

4.38 Load-displacement curves for different mesh-Δacombinations. . . 128

4.39 Crack paths for different mesh-Δacombinations. . . 128

4.40 Zoom-in of the crack paths for different mesh-Δacombinations. . . 129

4.41 Zoom-in of the crack paths for theσ−criterion and theS−criterion, considering the second (fine) mesh andΔa= 0.5 cm. . . 129

4.42 Load-displacement curves for the σ−criterion and theS−criterion, considering the second (fine) mesh andΔa= 0.5 cm. . . 130

4.43 Example mesh used for the analysis. . . 132

4.44 Energy-norm in the entire domain for the FEM and the classical XFEM.134 4.45 Convergence analysis for the stress intensity factorKIIand the dif- ferent enrichment strategies. . . 135

4.46 Condition numberκof the stiffness matrix for the different enrichment strategies. . . 136

4.47 Stress intensity factorKI(s)(N/mm^3/2) for a kinked crack as a func- tion of

(s)(√mm). . . 137 5.1 Generalized mid-point approximations . . . 143 5.2 Spectral radius for different time integration schemes andξ= 0. . . 149 5.3 Spectral radius for different time integration schemes andξ = 0.1.

(^∗) In the case of the Generalized-αmethod, the spurious rootλ₃=−1 was not considered. . . 150 5.4 Algorithmic dampingξ¯for different time integration schemes andξ= 0. 150 5.5 Algorithmic dampingξ¯for different time integration schemes andξ=

0.1. (^∗) In the case of the Generalized-αmethod, the spurious root λ₃=−1was not considered. . . 151 5.6 Approximation of the primary variabley, defintion of the jump

yn

and illustration of the physical timetfor GALERKIN time integration schemes. . . 156 5.7 Spectral radius for different time integration schemes andξ= 0. . . 159 5.8 Spectral radius for different time integration schemes andξ= 0.1. (^∗)

In the case of the Generalized-αmethod, the spurious rootλ₃=−1 was not considered. . . 160 5.9 Algorithmic dampingξ¯for different time integration schemes andξ=

0.0. . . 160 5.10 Algorithmic dampingξ¯for different time integration schemes andξ=

0.1. (^∗) In the case of the Generalized-αmethod, the spurious root λ₃=−1was not considered. . . 161 5.11 Rectangular plate with a crack. . . 163 5.12 Normalized dynamic stress intensity factorKI/KI0for the rectangular

plate with a central crack. Coarse mesh and Generalized-αmethod withρ_∞= 1. . . 163 5.13 Normalized dynamic stress intensity factorKI/KI0for the rectangu-

lar plate with a central crack. Coarse mesh and the discontinuous GALERKINMethod withp= 1. . . 164 5.14 Normalized dynamic stress intensity factorKI/KI0for the rectangular

plate with a central crack. Coarse mesh and Generalized-αmethod withΔt= 0.2μs. . . 164 5.15 Normalized dynamic stress intensity factorKI/KI0for the rectangular

plate with a central crack. Fine mesh andΔt= 0.1μs. . . 165 5.16 Normalized dynamic stress intensity factorKI/KI0for the rectangular

plate with an inclined central crack,Δt= 0.1μs. . . 167 5.17 Normalized dynamic stress intensity factorKII/KI0for the rectangu-

lar plate with an inclined central crack,Δt= 0.1μs. . . 167

5.18 Normalized dynamic stress intensity factorKI/KI0for the rectangular plate with an inclined central crack,Δt= 0.1μs. . . 168 5.19 Rectangular plate with an inclined central crack, finer mesh near the

crack. . . 168 5.20 Results of KII/KI0using the dG method and the original and finer

mesh.Δt= 0.1μs. . . 169 5.21KI/K_I^max using the Generalized-α method (ρ_∞ = 1) and three

meshes.t^∗= 50μsandΔt=t^∗/1280. . . 171 5.22KI/K_I^max using the Generalized-α method (ρ_∞ = 1) and three

meshes.t^∗= 200μsandΔt=t^∗/1280. . . 171 5.23 Geometry to simulate a semi-infinite crack in a infinite plane subjected

to a stress pulseσ₀. . . 174 5.24 Results ofKI/σ₀√

H using the Generalized-αmethod withρ_∞ = 1 and 200 time steps. . . 175 5.25 Results ofKI/σ₀√

Husing the dG method withp= 1and 200 time steps. . . 175 5.26 Results ofKI/σ₀√

Husing the cG method withp= 1,2and3, and 20 time steps. . . 176 5.27 Results ofKI/σ₀√

Husing the dG method withp= 1,2and3, and 20 time steps. . . 176 5.28 Results of KI/σ₀√

H using the Gen.-αmethod and 20 time steps, v= 1500 m/saftert=td. . . 177 5.29 Results of KI/σ₀√

Husing the dG method withp = 1and 20 time steps,v= 1500 m/saftert=td. . . 177 5.30 Fine mesh for simulating an infinite plate subjected to a stress pulse. . 178 5.31 Results ofKI/σ₀√

Husing the mapping scheme and the dG method withp= 1and 40 time steps,v= 1500 m/saftert=td. . . 178 5.32 Dynamic crack paths obtained using the mapping scheme. . . 180 5.33 Crack propagation velocityvforT₀= 0 msand the mapping scheme. . 180 5.34 Crack propagation velocityvforT₀= 1 msand the mapping scheme. . 181 5.35 Crack length evolution forT₀= 0 msand the mapping scheme. . . 181 5.36 Crack length evolution forT₀= 1 msand the mapping scheme. . . 182 5.37 Relation between crack velocity and crack length, mapping scheme. . 182 5.38 Dynamic crack paths obtained using the dG method, mapping

scheme and different meshes and crack length incrementsΔa. . . 183 5.39 Relation between crack velocity and crack length obtained using the

dG method, mapping scheme and different meshes and crack length incrementsΔa. . . 184

5.40 Comparison of dynamic crack paths obtained using the mapping scheme (MS) and the energy-conserving scheme (ECS). Original mesh, Gen.-αmethod andΔa= 0.5 cm. . . 184 5.41 Comparison of crack velocity-crack length relation obtained using the

mapping scheme (MS) and the energy-conserving scheme (ECS).

Original mesh, Gen.-αmethod andΔa= 0.5 cm. . . 185

List of Tables

4.1 Linear, quadratic and cubic shape functions and their first derivatives for equally spaced nodes. . . 91 4.2 Convergence rater in terms of the mesh sizeh(side length of the

elements). . . 133 5.1 Time integration parameters of the NEWMARKfamily as a function of

the spectral radiusρ_∞ . . . 149

Nomenclature

δij KRONECKER’s delta

Γ Boundary

Γ⁺_d^,− Crack surfaces Γu DIRICHLETboundary Γσ NEUMANNboundary κ KOLOSOV’s constant λ,ν LAMEparameters C Constitutive tensor

Ω Domain

Φ AIRYstress function

φⁱ Shape function associated to nodei ρ Density

σ Stress tensor

E GREEN-LAGRANGEstrain tensor F Material deformation gradient

P Elastic energy-momentum tensor, ESHELBYtensor ε Linear strain tensor

q Auxiliary vector field u Displacement vector X Position vector Cd Dilatational wave speed CR RAYLEIGHwave speed

Cs Shear wave speed Fi(r,θ) Near tip function G Energy release rate Gf Fracture energy h Mesh size H(·) Unit step function J J-Integral K Kinetic energy

KI Stress intensity factor for mode I fracture Keq Equivalent stress intensity factor

KIc Critical stress intensity factor for mode I fracture KIIc Critical stress intensity factor for mode II fracture KIIIc Critical stress intensity factor for mode III fracture KIII Stress intensity factor for mode III fracture KII Stress intensity factor for mode II fracture K_I^A Dynamic crack arrest toughness K_I^D Dynamic crack growth toughness K_I^d Dynamic crack initiation toughness r Convergence rate

S Strain energy density U Internal energy

wⁱ Support of the shape functionφⁱ WDyn Work of inertial forces

WExt Work of external forces WInt Work of internal forces

WS Work requiered to create new crack surfaces

Acknowledgments

I started my PhD at the Ruhr-Universität Bochum and worked there for two years under the guidance of Dr.-Ing. habil. Detlef Kuhl and the director of the institute for statics and dynamics Prof. Dr. techn. Günther Meschke, to whom I am very grateful for his hospitality. After these two years Detlef became a Professor at the Kassel University and offered me the possibility to continue my work in Kassel, which I accepted delighted. That said, I would like to express my gratitude to my PhD thesis advisor Professor Dr.-Ing. habil. Detlef Kuhl, for his patience and helpful guidance during the development of my thesis. He encouraged me to start a PhD during my time in Bochum as DAAD scholarship holder and I am indebted to him not only for that opportunity but also for supporting me in every possible way since the day we met. The life lessons that he taught me changed me forever.

I want to especially thank my friends and colleges Sandra Krimpmann and Sönke Carstens, my stay in Germany would not have been the same without their friendship and support.

I am very grateful to Professor Dr.-Ing. habil. Paul Steinmann for agreeing to be the second advisor of my thesis, and the committe members Professor Dr. Maria Specovius-Neugebauer and Professor Dr.-Ing Arnd I. Urban. I also thank Esther Norton and Patricio Tapia for reading and correcting the text and for their valuable suggestions.

I thank my family and my girlfriend, Paola Agüero, for their love, patience and all the sacrifices over the past few years. I dedicate this work to my mother who died in July 2009.

Finally, I would like to thank the Chilean Government and the University Católica del Norte for the funding that made this research possible.

Chapter 1 Introduction

This introductory Chapter provides an overview of this thesis, beginning with the mo- tivation of studying the fracture phenomenon in brittle materials, Section 1.1. Next, the state of the art is reviewed in Section 1.2, while Section 1.3 is concerned to the aim of the present work. Finally, the organisation of the thesis is outlined in Section 1.4.

1.1 Motivation

The fracture behaviour of materials is characterized by the concentration of micro- cracks in a localized process zone. In the cracking process, the microscopic cracks grow and coalesce to form an open macro-crack. Many of these cracks can be con- sidered harmless, meaning that they will not lead to the failure of the structure.

However, those cracks subjected to high stresses and larger than a certain size, can compromise the structure and jeopardize people safety. Therefore, the realistic model of crack opening and propagation is a prerequisite for reliable forecasts of security and durability of structures.

The classical analysis of structures is normally restricted to undamaged structural models. However, a large number of failures in structural components have occurred due to crack initiation and propagation, leading to dramatic consequences in the fields of civil, military, aeronautical, astronautical and maritime engineering. Maybe the most emblematic cases are the brittle fracture of the Liberty ships and the fuse- lage fatigue failure of the Boeing 737. The purpose of this work is contributing to close the gap between the classical analysis of structures and real damage mecha- nisms. In this context, the realistic modelling of dynamic crack propagation is sought, in order to improve the damage prognosis of engineering systems.

The Finite Element Method (FEM) is by far the most used numerical technique to solve engineering problems. Although the method is now in a mature age, it is not well suited to model evolving discontinuities. The classical approach consists in us- ing a remeshing technique as the discontinuity evolves, this implies not only the mesh construction, which can be the more time consuming step in the entire simu- lation, but also the projection of the data from the old to the new mesh, leading to

projection errors. For these reasons, a different approach to model evolving discon- tinuities, theeXtended Finite Element Method (XFEM), is studied in this thesis in the framework of fracture mechanics. The method allows the crack to be arbitrary aligned within the finite element mesh and therefore, remeshing is not necessary to model crack growth.

1.2 State of the Art

The numerical modelling of cracks and the analysis of cracked structures go back to the early 70’s. Until the middle of the 1990s, the focus of the research within the failure analysis field was based on the development of continuum me- chanics models, which have a strong dependence on the spatial discretization.

Since the mid 90’s, the models based on the concept of discontinuous displace- ment approximations have gained great popularity in the scientific community.

These models take into account discontinuities within the finite elements and can be categorized into an element-wise formulation Embedded Crack Models [Simo et al., 1993, Oliver, 1996a, Oliver, 1996b, Jirásek and Zimmermann, 2001a, Jirásek and Zimmermann, 2001b] and a node-wise formulationeXtended Finite El- ement Method [Moës et al., 1999]. Within the framework of the Embedded Crack Models, the discontinuities are usually incompatible at the element boundaries and, since the work of SIMO [Simo et al., 1993], these models are based on theEn- hanced Assumed Strainconcept. This means that the jump in the displacement field is not modelled directly, and only included in the form of a discontinuous strain field. The eXtended Finite Element Method is based on the partition of unity con- cept [Melenk and Babuška, 1996], and offers the possibility to model the discon- tinuities by enriching the displacement field with discontinuous shape functions. In contrast to theEmbedded Crack Models, the discontinuities are compatible at the el- ement boundaries. A comparison of these two modelling philosophies can be found in [Jirásek and Belytschko, 2002].

A third possibility to simulate the crack propagation is the so-called remesh- ing. Within this approach, the finite element mesh changes as the crack grows, in order to match the new crack geometry. Even when this technique have been used successfully in many works, see [Nishioka, 1995, Bouchard et al., 2003, Réthoré et al., 2004, Shahani and Fasakhodi, 2009], among others, it presents three main disadvantages: the relatively high numerical effort, the loss of accuracy by mapping the information between meshes and the complexity of implementation in a finite element program.

The development of meshless methods, such as theElement-Free Galerkin Method [Belytschko et al., 1994], made the remeshing process in the simulation of discon- tinuities unnecessary. This fourth method of crack simulation uses only one set of nodes and corresponding shape functions for the discretization of the continuum.

Therefore, no remeshing is needed as the crack propagates. However, this advan- tage is offset by the very high computational cost of the method. For this reason, the coupling with the finite element method for non-damaged areas of the structures

was proposed in [Belytschko et al., 1995]. For the simulation of dynamic crack prop- agation using the element free GALERKINmethod see [Krysl and Belytschko, 1999]

and references therein. An overview on meshless methods can be found in [Belytschko et al., 1996] and in the more recent work [Nguyen et al., 2008].

The eXtended Finite Element Method starts with the pioneering work of BE-

LYTSCHKO and BLACK [Belytschko and Black, 1999], where a methodology to model crack growth with minimal remeshing is presented. Next, in the work of MOËS [Moës et al., 1999], the remeshing is completely eliminated by us- ing a HEAVISIDE function to model the displacement discontinuity on the crack, and a near-tip enrichment to take into account the singularity of the stress field at the crack tip. The doctoral thesis of DOLBOW [Dolbow, 1999]

resumes these concepts of the XFEM, including also the modelling of cracked plates [Dolbow et al., 2000b, Dolbow et al., 2000a], intersecting cracks, see also [Daux et al., 2000, Budyn et al., 2004]) and frictional contact cracks [Dolbow et al., 2001]. Next, SUKUMAR extended the method to three dimensional crack modelling [Sukumar et al., 2000b] and to the modelling of holes and inclusions by level sets in the XFEM [Sukumar et al., 2000a]. Non-planar three-dimensional crack growth was analysed in [Moës et al., 2002] using XFEM and level sets.

After the above mentioned contributions, the interest of the scientific commu- nity in XFEM increased rapidly. WELLS and SLUYS [Wells and Sluys, 2001] used only the HEAVISIDE function for modelling cohesive cracks using finite elements.

Within this approach, the crack tip is always located at the element boundaries.

This technique was improved in [Zi and Belytschko, 2003], where a new crack- tip element for XFEM and its application to cohesive cracks was proposed. The method was again based only on the HEAVISIDE function, but the crack tip can be located anywhere within an element. Higher-order displacement discontinuity in a standard finite element model was introduced using the XFEM approach in [Mariani and Perego, 2003] for the simulation of quasi-static cohesive crack propa- gation. An energy-based modelling of cohesive and cohesionless cracks via XFEM was proposed in [Meschke and Dumstorff, 2007]. The method is formulated within an XFEM based analysis model, leading to a variational formulation in terms of dis- placements, crack lengths and crack kink angles. The influence of both the numerical integration and the cohesive law on the energy distribution and the crack path was studied in the subsequent work [Meschke et al., 2006].

A slightly different approach to XFEM is the HANSBO&HANSBO method [Hansbo, 2002, Hansbo, 2004]. Within this framework, the discontinuous approx- imation is accomplished by doubling the degrees of freedom in the discontin- uous elements. Inspired in this work, a geometrically non-linear finite element framework for modelling three-dimensional crack propagation was presented in [Mergheim et al., 2006].

The convergence properties of the XFEM was studied in [Bechet et al., 2005, Laborde et al., 2005, Chahine et al., 2006] inspired in the previous work of [Strang and Fix, 1973] in the context of the finite element method. In [Laborde et al., 2005], some improvements of the XFEM were proposed in or- der to obtain optimal convergence properties for high-order elements. Furthermore,

thealmost polar integrationfor the elements containing the crack tip was introduced.

This integration issue was also studied in [Bechet et al., 2005], along with a special- ized preconditioner for enriched finite elements. Next, the use of a cut-off function in a zone independent of the mesh size was studied in [Chahine et al., 2006], where also a quasi-optimal error estimate for first order LAGRANGE finite elements on a regular triangular mesh was obtained. The influence of the partially enriched elements (blending elements) on the performance of local partition of unity en- richments was studied in [Chessa et al., 2003]. In this paper, an enhanced strain formulation, capable of achieving the optimal rate of convergence, was developed.

In [Liu et al., 2004] the XFEM was improved to directly evaluate the stress intensity factors without post-processing, considering not only homogeneous materials but also bimaterials. A statically admissible stress recovery scheme was proposed in [Xiao and Karihaloo, 2006b] in order to improve the accuracy of crack tip fields.

Within the framework of dynamic crack propagation, WELLS [Wells, 2001] pro- posed the modelling of cohesive cracks with XFEM using only the HEAVISIDE

function, again with the crack tip located always at the element boundary. A similar method was presented in [Zi et al., 2005], coupling XFEM with level sets to represent the crack. BELYTSCHKO [Belytschko et al., 2003] proposed a model based on the loss of hyperbolicity and a new discontinuous enrichment without tip enrichment, but allowing the crack tip to be located within the elements. The crack tip behaviour was modelled using a cohesive zone model. For brittle ma- terials, RÉTHORÉ [Réthoré et al., 2005b] proposed an energy conserving scheme for modelling dynamic crack growth using XFEM, an latter developed a combined space-time extended finite element method STX-FEM [Réthoré et al., 2005a] in order to improve the quality of the numerical simulation when time discontinu- ities are present. Space-time discontinuous finite element have been also anal- ysed in [Chessa and Belytschko, 2004, Chessa and Belytschko, 2006]. A singular enrichment finite element method was proposed in [Belytschko and Chen, 2004], the method uses an enriched basis that spans the first terms of the asymptotic dynamic solution, where the stress intensity factorsKI andKII are taken as ad- ditional unknowns. XFEM simulations and mixed-mode loading experiments have been compared in [Grégoire et al., 2007].

In the context of explicit dynamics, dynamic crack and shear band propagation was analysed in [Song et al., 2006] using phantom nodes, inspired in the work of HANSBO&HANSBO. The main problem of the method is that the critical time step tends to zero when the crack passes very close to a node. To alleviate this problem, mass lumping techniques for the XFEM have been analysed by MENOUILLARD [Menouillard et al., 2006, Menouillard et al., 2008] and also in the works [Rozycki et al., 2008, Elguedj et al., 2009, Gravouil et al., 2009]. The use of basis functions which are non-zero only in the elements containing the crack was introduced in [Svahn et al., 2007], together with a rate-dependent cohesive crack model based on anisotropic damage coupled to plasticity. Time integration schemes in the XFEM for moving discontinuities were studied in [Fries and Zilian, 2009], with special emphasis on the advection-difussion equation. Recently, MENOUILLARDand co-workers [Menouillard et al., 2010] proposed a time dependent crack tip enrich- ment for dynamic crack propagation.

A comparative study of several finite element methods for dynamic fracture can be found in [Song et al., 2007]. For a state of the art review on the XFEM the interested reader is referred to the articles [Karihaloo and Xiao, 2003, Belytschko et al., 2009]

and the XFEM book [Mohammadi, 2008].

After the problem is discretized in the spatial domain, one arrives to a semi-discrete equation of motion, that is, a set of differential equations in time. The most general approach to solve the problem is the direct integration of the equation of motion.

Several time integration schemes have been developed for structural dynamics ap- plications and one of the most famous methods is the NEWMARKmethod proposed in 1959 [Newmark, 1959]. The method is a one-step procedure, which means that the solution at timetn+1depends only on the solution at timetn, and can be uncondi- tionally stable for the right choice of the parameters involved in the method. Second order accuracy can be achieved forγ= 1/2, but this value ofγleads to no numerical dissipation. The WILSON-θmethod, see [Bathe and Wilson, 1973], solves this prob- lem achieving second order accuracy together with numerical dissipation. Neverthe- less, the method overshoots significantly the exact solution in the first time steps [Goudreau and Taylor, 1972]. A further improvement of the NEWMARKmethod was introduced in [Hilber et al., 1977a] and calledαmethod, also known as the HHT-α method, which shows a better behaviour with respect to overshooting compared with the WILSON-θmethod, see [Hilber and Hughes, 1978]. Several years later CHUNG

& HULBERT proposed theGeneralized-αmethod [Chung and Hulbert, 1993]. The method offers the possibility of controlling the numerical dissipation, achieving high frequency dissipation while minimizing unwanted low frequency dissipation. The method contains the HHT-α [Hilber et al., 1977b] and WBZ-α [Wood et al., 1981]

methods.

As a consequence of the DAHLQUISTtheorem [Dahlquist, 1963], the order of ac- curacy of the methods of the NEWMARKfamily described before, and in general of any A-stable linear multistep method, can not exceed two. Due to this fact, contin- uous and discontinuous GALERKINmethods were considered in structural dynam- ics. These methods can achieve an arbitrary accuracy order by incrementing the polynomial degree of the finite element basis. The first ideas of time finite element methods were presented in [Argyris and Scharpf, 1969, Fried, 1969, Oden, 1969], several years later space finite element methods were introduced, see for exam- ple [Turner et al., 1956, Argyris, 1960]. Continuous GALERKINtime integration meth- ods were proposed in [Hulme, 1972, Argyris et al., 1977] to solve first order ini- tial value problems, while in [French, 1993] the method was used to solve the Wave Equation. The time discontinuous GALERKIN method followed the same path as its continuous counterpart, and was developed after the space discon- tinuous GALERKIN method [Reed and Hill, 1973, Le Saint and Raviart, 1974]. In [Jamet, 1978] a discontinuous GALERKIN method was used to solve parabolic initial value problems. The numerical properties of the method were anal- ysed in the latter works [Johnson et al., 1984, Eriksson et al., 1985]. In the con- text of linear elastodynamics, time discontinuous GALERKIN methods were de- veloped and analysed in the works [Gellert, 1978, Hughes and Hulbert, 1988, Hulbert, 1989, Hulbert and Hughes, 1990, Hulbert, 1992, Borri and Bottasso, 1975, Li, 1996, Li and Wiberg, 1996, Li and Wiberg, 1998, Johnson, 1993, Maute, 2001,

Steeb et al., 2003, Wiberg and Li, 1999].

1.3 Aim of the Work

The aim of this thesis is the simulation of progressive damage in brittle materials due to cracking. With this aim, the mathematical crack model will be solved using the eXtended Finite Element Method for the spatial discretization and time integra- tion schemes for the numerical integration in the time domain. The time integra- tion schemes considered are the Generalized-αmethod, the continuous GALERKIN

method and the discontinuous GALERKINmethod. The main objectives of the thesis can be summarized as follows:

• Development of a finite element program in MATLAB for two-dimensional crack propagation, which is flexible and offers the possibility of extension of the standard finite element method. This extension will include the approxima- tion of the displacement field by means of arbitrary functions on the basis of the partition of unity concept. To provide a realistic simulation that takes into ac- count any number of cracks, the special cases of crack intersection and crack branching will be considered.

• Implementation of different crack propagation criteria as basis of a reliable crack modelling. The considered criteria will be: maximum circumferential stress, minimum strain energy density, maximum energy release rate and lo- cal symmetry. These criteria will be compared at the level of crack paths and load-displacement curves.

• Numerical study of the convergence properties of the XFEM for polynomials of orderp= 1,2and3. This study will consider different enrichment strategies as the enrichment of a fixed area around the crack tip, the gathering of degrees of freedom associated to the near tip enrichment and the use of a cut-off func- tion. Furthermore, the convergence rate for theL₂-norm and the energy norm over the whole domain and over a sub-domain will be analysed as well as the convergence rate of the stress intensity factors andT-stress.

• Simulation of quasi-static and dynamic crack propagation. In the second case, the Generalized-α method will be implemented, along with the continuous GALERKIN method and the discontinuous GALERKIN method for an arbitrary polynomial degree. The numerical properties of the methods will be analysed and compared.

1.4 Outline

The remainder of this thesis is structured as follows:

Chapter 2 is a review of basic concepts of linear elastic fracture mechanics in a static framework. The reference problem, its asymptotic solution, the extraction of the stress intensity factors and crack propagation criteria are discussed.

Chapter 3 focusses on dynamic fracture mechanics. Dynamically loaded station- ary cracks and propagating cracks are analyzed in this Chapter. As in Chapter 2, asymptotic solutions and the extraction of the stress intensity factors are also im- portant topics of this Chapter. Furthermore, some basic solutions are presented in order to serve as a reference for the numerical solutions of the following chapters.

Finally, Dynamic crack propagation criteria are also discussed.

Chapter 4 introduces the XFEM to model cracks. Here, four types of enrichments are considered: HEAVISIDE, near-tip, junction and branching. Implementation details are given and crack growth simulations are carried out in order to show the advantages of the method. In addition, the crack paths and load-displacement curves obtained for different crack propagation criteria are compared. Finally, a numerical conver- gence analysis of the method for theL₂-norm, the energy norm, stress intensity factors andT-stress is performed.

In Chapter 5 the semi-discrete equation of motion is solved using three methods:

Generalized-αmethod, continuous GALERKINmethod and discontinuous GALERKIN

method. The properties of these methods, such as convergence ratio, numerical sta- bility and numerical damping, are analysed and compared. Dynamic stress intensity factors are calculated for stationary and propagating cracks. Finally, dynamic crack growth is simulated using different strategies.

Conclusions and future work are stated in Chapter 6.

Chapter 2

Basic Concepts of Static Linear Elastic Fracture Mechanics

Fracture mechanics is the field of mechanics concerned with the study of the for- mation and propagation of cracks in materials. Historically, GRIFFITH[Griffith, 1920]

was the first to point out that the presence of internal cracks and flaws plays an important role in the initiation and propagation of fracture.

The purpose of this chapter is to review the basic concepts of static linear elastic fracture mechanics (LEFM), a branch of fracture mechanics based in the applica- tion of the theory of elasticity to bodies containing cracks or defects under static loading. Accordingly, small displacements and strains, together with the General- ized HOOKE’s Law relating stresses and strains are assumed in LEFM.

In this chapter the asymptotic displacement and stress field in the vicinity of a crack tip are reviewed. The concepts of stress intensity factors (SIFs) and T-Stress are also introduced. Furthermore, expressions for the SIFs for kinked and branched cracks are studied. Then, classical aspects such as the energy balance of GRIFFITH, the energy release rate, the J-Integral and the ESHELBY tensor are also reviewed and related. The extraction of the SIFs and T-Stress from a numerical solution is described and finally classical fracture criteria are reviewed.

This chapter does not intend to cover the extensive field of fracture mechanics.

For further reading, the text books [Shah et al., 1995, Bažant and Planas, 1998, Broberg, 1999, Anderson, 2004, Gdoutos, 2005, Shukla, 2005] are recommended to the reader.

2.1 Reference Problem

In this section the reference problem is described in itsstrong form. A cracked body with open domainΩ∈R³and piecewise smooth boundaryΓis considered, see Fig- ure 2.1. The boundary is also considered as the junction of the disjointed partsΓu, Γσ,Γ⁺_d andΓ⁻_d, beingnits outward unit normal vector. Prescribed displacementsu^∗ are imposed onΓu(DIRICHLETboundary conditions), while prescribed stressest^∗

are imposed onΓσ(NEUMANNboundary conditions).Γ⁺_d andΓ⁻_d represent the crack (or cracks) surfaces, which are assumed to be traction-free (homogeneous NEU-

MANN). The problem consists in finding a displacement field functionu, satisfying the governing equations

Div(σ) =0

σ =C:ε ∀X∈Ω, ε=1

2

∇u+∇^Tu

(2.1)

together with the boundary conditions u(X) =u^∗(X) ∀X∈Γu, σ(X)·n=t^∗(X) ∀X∈Γσ, σ(X)·n=0 ∀X∈Γ⁺_d, σ(X)·n=0 ∀X∈Γ⁻_d,

(2.2)

Here,σis the stress tensor,εis the strain tensor andCis the constitutive tensor.

Equations (2.1) and (2.2) define the strong form of the problem.

2.2 Asymptotic Solution

The asymptotic solution for the stresses in the presence of a straight crack and, more importantly, the universal nature of ther^−1/2singularity were first obtained by WILLIAMS [Williams, 1957], extending the previous works of INGLIS [Inglis, 1913], SNEDDON[Sneddon, 1946] and WESTERGAARD[Westergaard, 1939], who obtained this result but only for specific configurations. WILLIAMSconsidered a homogeneous

t n

Γσ

Γ Ω Γu

Γ⁺_d Γ⁻_d

e3

e1

e2

t^∗

Figure 2.1: Cracked body and boundary conditions

isotropic linear elastic body containing a traction-free crack, see Figure 2.2, and proposed an AIRYstress function of the form

Φ =r^λ+1[c₁sin(λ+ 1)φ+c₂cos(λ+ 1)φ+c₃sin(λ−1)φ+c₄cos(λ−1)φ]. (2.3) Hereφ=π+θ.

P r

θ

Figure 2.2: Definition of polar coordinates with pole at the crack tip.

The asymptotic solutions for displacements and stresses are now well known (see [Liu et al., 2004]), [Xiao and Karihaloo, 2006a]) and are given by

Displacements

⎡

⎣ u₁(r, θ) u₂(r, θ)

⎤

⎦= ∞ n=0

⎡

⎣ f₁₁n(r, θ) f₁₂n(r, θ) f₂₁n(r, θ) f₂₂n(r, θ)

⎤

⎦

⎡

⎣ KIn

KIIn

⎤

⎦, (2.4)

wheref₁₁n,f₁₂n,f₂₁nyf₂₂nare given by

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

f₁₁n(r, θ) f₁₂n(r, θ) f₂₁n(r, θ) f₂₂n(r, θ)

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

= r^n/2 2G√

2π

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

κ+n

2+ (−1)ⁿ cos

nθ 2

−n 2cosn

2−2 θ

κ+n

2−(−1)ⁿ sin

nθ 2

−n 2sinn

2−2 θ

κ−n 2−(−1)ⁿ

sin

nθ 2

+n

2sin n

2−2

θ

− κ−n

2+ (−1)ⁿ cos

nθ 2

−n 2cosn

2−2 θ

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ (2.5) andκis the KOLOSOV’s constant defined as

κ=

⎧⎪

⎨

⎪⎩

3−4ν Plane strain, 3−ν

1 +ν Plane stress.

(2.6) The displacements corresponding ton= 0

u⁽⁰⁾₁ = κ+ 1 2G√

2πKI0, u⁽⁰⁾₂ =− κ+ 1 2G√

2πKII0 (2.7)

are rigid body translations, representing the displacement of the crack tip. Taking that into account, the displacements relative to the crack tip can be written as

u₁=KI

2G r

2πcos θ

2

(κ−cosθ) +KII

2G r

2πsin θ

2

(κ+ 2 + cosθ) +O(r), u₂=KI

2G r

2πsin θ

2

(κ−cosθ) +KII

2G r

2πcos θ

2

(2−κ−cosθ) +O(r) (2.8)

or in polar coordinates ur=KI

2G r

2πcos θ

2

(κ−cosθ) +KII

2G r

2πsin θ

2

(2−κ+ 3 cosθ) +O(r), uθ=−KI

2G r

2πsin θ

2

(κ−cosθ)−KII

2G r

2πcos θ

2

(2 +κ−3 cosθ) +O(r), (2.9) whereKI1andKII1in (2.4) has been replaced byKIandKIIrespectively, the usual notation for thestress intensity factors. For the case of out-of-plane deformation, the displacement field is given by

u₃=KIII

G r

2πsin θ

2

. (2.10)

Analysing the deformation state on the crack surfaces,θ=±πand from (2.8) and (2.10) yields

u₁=±KII

2G r

2π(κ+ 1), u₂=±KI

2G r

2π(κ+ 1), u₃=±KIII

G r

2π.

(2.11)

Then, any deformation of the crack surfaces (or more generally, the deformation state of a cracked body), in a neighbourhood of a crack tip, can be viewed as a superposition of three basic modes of crack deformation ([Gdoutos, 2005]):

Opening Mode,ModeI: The crack surfaces separate symmetrically with respect to the planese1e2ande1e3.

Sliding Mode,ModeII: The crack surfaces slide relative to each other symmetrically with respect to the planee1e2and skew-symmetrically with respect to the planee1e3. Tearing (orantiplane)Mode,ModeIII: The crack surfaces slide relative to each other skew-symmetrically with respect to both planese1e2ande1e3.

Stresses In the case of the stresses, the asymptotic solution can be written as

⎡

⎢⎢

⎢⎣ σ₁₁(r, θ) σ₂₂(r, θ) σ₁₂(r, θ)

⎤

⎥⎥

⎥⎦= ∞ n=1

⎡

⎢⎢

⎢⎣

g₁₁n(r, θ) g₁₂n(r, θ) g₂₁n(r, θ) g₂₂n(r, θ) g₃₁n(r, θ) g₃₂n(r, θ)

⎤

⎥⎥

⎥⎦

⎡

⎣ KIn

KIIn

⎤

⎦, (2.12)

ModeI- Opening ModeII- Sliding ModeIII- Tearing

X₁ X₂

X₃ Figure 2.3: Fracture modes.

with

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣ g₁₁n

g₁₂n

g₂₁n

g₂₂n

g₃₁n

g₃₂n

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

= nr^n/2−1 2√

2π

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

2 +n 2+ (−1)ⁿ

cos

n 2−1

θ

−n 2−1

cos

n 2−3

θ

2 +n

2−(−1)ⁿ sinn

2−1 θ

−n 2−1

sinn 2−3

θ

2−n 2−(−1)ⁿ

cos

n 2−1

θ

+

n 2−1

cos

n 2−3

θ

2−n 2+ (−1)ⁿ

sin

n 2−1

θ

+

n 2−1

sin

n 2−3

θ

−n

2+ (−1)ⁿ sinn

2−1 θ

+n 2−1

sinn 2−3

θ n

2−(−1)ⁿ cosn

2−1 θ

−n 2−1

cosn 2−3

θ

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ .

(2.13) Concentrating the attention in the leading terms of (2.12), follows

σ₁₁= KI

√2πrcos θ

2 1−sin θ

2

sin 3θ

2

+ KII

√2πrsin θ

2 −2−cos θ

2

cos 3θ

2

+T+O(r^1/2), σ₂₂= KI

√2πrcos θ

2 1 + sin θ

2

sin 3θ

2

+ KII

√2πrsin θ

2

cos θ

2

cos 3θ

2

+O(r^1/2), σ₁₂= KI

√2πrsin θ

2

cos θ

2

cos 3θ

2

+ KII

√2πrcos θ

2 1−sin θ

2

sin 3θ

2

+O(r^1/2)

(2.14)

and in polar coordinates σrr= KI

√2πr 5

4cos θ

2

−1 4cos

3θ 2

+ KII

√2πr

−5 4sin

θ 2

+3 4sin

3θ 2

+Tcos²(θ) +O(r^1/2), σθθ= KI

√2πr 3

4cos θ

2

+1 4cos

3θ 2

+ KII

√2πr

−3 4sin

θ 2

−3 4sin

3θ 2

+Tsin²(θ) +O(r^1/2), σrθ= KI

√2πr 1

4sin θ

2

+1 4sin

3θ 2

+ KII

√2πr 1

4cos θ

2

+3 4cos

3θ 2

−Tcos(θ) sin(θ) +O(r^1/2).

(2.15)

Here,T= 4KI2/√

2πis known as theT-Stressand governs the configurational sta- bility of a growing crack [Cotterell and Rice, 1980].

In the case of antiplane deformation, the stress field is σ₁₃=−KIII

√2πrsin θ

2

, σ₂₃= KIII

√2πrcos θ

2

.

(2.16)

Equations (2.14) and (2.16) show that the stresses approach infinity asr→0. Near the crack tip the stresses are proportional to the stress intensity factorsKI,KIIand KIII. Furthermore if the SIFs are known, then the stresses and displacements can be directly calculated as a function ofrandθ, near the crack tip. The utility of the SIFs depends on being able to determine them from different load conditions and geometries. Closed-form solutions are known for simple configurations, generally considering an infinite or semi-infinite body, a survey of stress intensity factors can be found in [Murakami, 1987], [Tada et al., 1985]. For complicated configurations, numerical methods are needed, this will be the subject of Chapter 4.

2.3 SIFs for a Kinked Crack

The calculation of the stress intensity factors for a kinked two-dimensional crack is an old problem in Linear Elastic Fracture Mechanics. Despite the intensive research carried out in the 70’s and 80’s in this area, see for example [Hussain et al., 1974], [Bilby and Cardew, 1975], [Chatterjee, 1975], [Cotterell and Rice, 1980], [Karihaloo et al., 1981] and [Sumi, 1986] the solutions obtained were only partial in the sense that only specific geometries and load- ings were considered or the cracks were nearly straight. It was not until the work