Basic Fracture Mechanics and its Applications

343 Pages·2022·21.823 MB·English

by  Ashok Saxena

Basic Fracture Mechanics
and its Applications
This textbook provides a comprehensive guide to fracture mechanics and its appli-
cations, providing an in-depth discussion of linear elastic fracture mechanics and a
brief introduction to nonlinear fracture mechanics. It is an essential companion to
the study of several disciplines such as aerospace, biomedical, civil, materials and
mechanical engineering. This interdisciplinary textbook is also useful for profes-
sionals in several industries dealing with design and manufacturing of engineering
materials and structures.
Beginning with four foundational chapters, discussing the theory in depth, the
book also presents specific aspects of how fracture mechanics is used to address
fatigue crack growth, environment assisted cracking, and creep and creep-fatigue
crack growth. Other topics include mixed-mode fracture and materials testing and
selection for damage tolerant design, alongside in-depth discussions of ensuring
structural integrity of components through real-world examples. There is a strong
focus throughout the book on the practical applications of fracture mechanics. It
provides a clear description of the theoretical aspects of fracture mechanics and also
its limitations. Appendices provide additional background to ensure a comprehensive
understanding and every chapter includes solved example problems and unsolved
end of chapter problems. Additional instructor support materials are also available.

Basic Fracture Mechanics
and its Applications
Ashok Saxena
First edition published 2023
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of Taylor & Francis Group, LLC
© 2023 Ashok Saxena
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of
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reproduced in this publication and apologize to copyright holders if permission to publish in this
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Library of Congress Cataloging‑in‑Publication Data
Names: Saxena, A. (Ashok), author.
Title: Basic fracture mechanics and its applications / Ashok Saxena.
Description: First edition. | Boca Raton : CRC Press, [2023] | Includes
bibliographical references and index. |
Identifiers: LCCN 2022034759 (print) | LCCN 2022034760 (ebook) |
ISBN 9781032267197 (hardback) | ISBN 9781032273259 (paperback) |
ISBN 9781003292296 (ebook)
Subjects: LCSH: Fracture mechanics.
Classification: LCC TA409 .S2896 2023 (print) | LCC TA409 (ebook) |
DDC 620.1/126–dc23/eng/20221004
LC record available at https://lccn.loc.gov/2022034759
LC ebook record available at https://lccn.loc.gov/2022034760
ISBN: 9781032267197 (hbk)
ISBN: 9781032273259 (pbk)
ISBN: 9781003292296 (ebk)
DOI: 10.1201/9781003292296
Typeset in Times
by codeMantra
Access the Support Material: https://www.routledge.com/9781032267197
Contents
Preface.......................................................................................................................xi
Author ......................................................................................................................xv
Chapter 1 Fracture in Structural Components ......................................................1
1.1 Fracture in Engineering Materials and Structures:
Societal Relevance .....................................................................1
1.1.1 S afety Assessments ......................................................1
1.1.2 Environment and Health Hazards ................................2
1.1.3 Optimizing Costs (Fuel Economy, Material
Costs, Opportunity Costs) ............................................2
1.1.4 P roduct Liability ...........................................................2
1.2 Examples of Prominent Fractures and the Underlying
Causes ........................................................................................3
1.2.1 Failures in Liberty Ships ..............................................3
1.2.2 Failures of Comet Aircraft ...........................................4
1.2.3 Cracks in A380 Aircrafts .............................................5
1.2.4 Crack in a Structural Member of an Interstate
Highway Bridge ............................................................5
1.2.5 Cracks in Human Bones ...............................................6
1.2.6 Aneurysms in Human Abdominal Aortas ...................6
1.3 Degradation Phenomena and Fracture in Engineering
Materials and Structures ...........................................................8
1.3.1 Crack Initiation/Formation and Growth ......................8
1.4 History of Developments in Understanding Fatigue and
Fracture ......................................................................................9
1.4.1 Developments in Understanding of Fatigue .................9
1.4.2 Understanding Brittle and Ductile Fracture ...............11
1.4.3 Early Developments in Fracture Mechanics ..............12
1.4.4 Developments in Elastic-Plastic Fracture
Mechanics...................................................................15
1.4.5 Environment Assisted Cracking .................................16
1.4.6 Developments in Time Dependent Fracture
Mechanics...................................................................16
1.5 Summary .................................................................................17
References ..........................................................................................18
Chapter 2 Early Theories of Fracture .................................................................19
2.1 Microscopic Aspects of Brittle Fracture .................................19
2.1.1 Intergranular and Transgranular Fracture ..................19
v
vi Contents
2.1.2 Equi-Cohesive Temperature .......................................21
2.1.3 D uctile and Brittle Fracture .......................................21
2.2 Models of Fracture at the Atomic Scale ..................................22
2.3 Stress Concentration Effects of Flaws .....................................24
2.4 Griffith’s Theory of Brittle Fracture ........................................26
2.5 Orowan’s Modification to Griffith’s Theory ............................28
2.6 The Concept of Crack Extension Force, G ..............................29
2.6.1 Estimation of Griffith’s Crack Extension Force
for an Arbitrary Shaped Body ....................................30
2.7 Crack Growth Resistance, R ....................................................34
2.8 Predicting Instability in Cracked Structures ...........................34
2.8.1 P redicting Instability Conditions for a General Case ...40
2.9 Summary .................................................................................41
References ..........................................................................................42
Homework Problems ..........................................................................42
Appendix 2A: Review of Solid Mechanics ........................................43
2A.1 Stress ..........................................................................43
2A.2 Strain ..........................................................................47
2A.3 Elasticity .....................................................................48
2A.4 Elastic Strain Energy ..................................................49
2A.5 Stress Transformation Equations ...............................50
2A.6 Stress–Strain Behavior ...............................................51
Notes ...................................................................................................52
Chapter 3 Theoretical Basis for Linear Elastic Fracture Mechanics ..................53
3.1 Engineering Materials and Defects .........................................53
3.2 Stress Analysis of Cracks ........................................................54
3.2.1 Equations of Elasticity ...............................................55
3.2.2 C ompatibility Equations .............................................55
3.2.3 Application of Airy’s Stress Function to Crack
Problems .....................................................................57
3.3 Stress Intensity Parameter, K, for Various Crack
Geometries and Loading Configurations by the
Westergaard Method ................................................................60
3.4 Crack Tip Displacement Fields ................................................66
3.5 The Relationship between G and K .........................................66
3.6 Determining K for Other Loading and Crack Geometries......69
3.7 Use of Linear Superposition Principle for Deriving
K-Solutions ..............................................................................73
3.8 K -Solutions for 3-D Cracks .....................................................76
3.9 Summary .................................................................................81
References ..........................................................................................83
Homework Problems ..........................................................................83
Appendix 3A ......................................................................................84
3A.1 Cauchy–Reimann Equations ......................................84
3A.2 Derivation of the Crack Tip Displacement Fields ......85
Contents vii
Chapter 4 Crack Tip Plasticity ............................................................................89
4.1 Estimate of the Plastic Zone Size ............................................89
4.2 Plasticity Modified Crack Tip Stress Field for SSY ................92
4.3 P lastic Zone Shape ..................................................................95
4.4 Crack Tip Opening Displacement (CTOD) .............................97
4.5 Summary .................................................................................97
References ..........................................................................................98
Homework Problems ..........................................................................98
Appendix 4A: Plastic Yielding Under Uniaxial and
Multiaxial Conditions .........................................................................99
4A.1 Uniaxial Stress–Strain Curve.....................................99
4A.2 Von Mises Yield Criterion for Multiaxial Loading .....99
4A.3 Tresca Yield Criterion ..............................................100
Chapter 5 Fracture Toughness and its Measurement ........................................101
5.1 Similitude and the Stress Intensity Parameter, K ..................103
5.2 Fracture Toughness as a Function of Plate Thickness...........105
5.3 Ductile and Brittle Fracture and the LEFM Approach .........107
5.4 M easurement of Fracture Toughness ....................................108
5.4.1 Measurement of Plane Strain Fracture
Toughness, K ..........................................................108
Ic
5.4.2 Fracture Toughness of Thin Panels ..........................112
5.5 Correlations between Charpy Energy and Fracture
Toughness ..............................................................................118
5.5.1 Charpy Energy versus Fracture Toughness
Correlation for Lower-Shelf and Lower
Transition Region .....................................................118
5.5.2 Charpy Energy versus Fracture Toughness
Correlation in the Upper-Shelf Region .....................118
5.6 Summary ...............................................................................119
References ........................................................................................119
Homework Problems ........................................................................120
Appendix 5A: Compliance Relationships for C(T)
and M(T) Specimens ........................................................................121
5A.1 Compliance Relationships for C(T) Specimen .........121
5A.2 Compliance and K-Relationships for M(T)
Specimens ..................................................................123
Notes .................................................................................................124
Chapter 6 Fatigue Crack Growth ......................................................................125
6.1 I ntroduction ...........................................................................125
6.2 Fatigue Crack Growth (or Propagation) Rates ......................126
6.2.1 D efinitions ................................................................126
6.2.2 Mechanisms of Fatigue Crack Growth ....................129
6.2.3 Fatigue Crack Growth Life Estimation ....................131
viii Contents
6.3 The Effect of Load Ratio, Temperature, and Frequency
on Fatigue Crack Growth Rate in the Paris Regime .............136
6.4 Wide Range Fatigue Crack Growth Behavior .......................137
6.5 Crack Tip Plasticity during Cyclic Loading ..........................142
6.5.1 C yclic Plastic Zone ...................................................142
6.5.2 Crack Closure during Cyclic Loading......................144
6.6 Fatigue Cycles Involving Compressive Loading ...................146
6.7 Models for Representing Load Ratio Effects on Fatigue
Crack Growth Rates ..............................................................147
6.8 Fatigue Crack Growth Measurements
(ASTM Standard E647) .........................................................150
6.9 Behavior of Small or Short Cracks ........................................159
6.9.1 Limitations of ∆K for Characterizing Small
Fatigue Crack Growth Behavior ...............................161
6.10 F atigue Crack Growth under Variable Amplitude Loading ....164
6.10.1 Effects of Single Overloads/Underloads on
Fatigue Crack Growth Behavior ...............................164
6.10.2 V ariable Amplitude Loading ....................................166
6.11 Summary ...............................................................................169
References ........................................................................................170
Homework Problems ........................................................................171
Note ..................................................................................................173
Chapter 7 Environment-Assisted Cracking.......................................................175
7.1 I ntroduction ...........................................................................175
7.2 M echanisms of EAC ..............................................................175
7.3 Relationship between EAC and K under Static Loads ..........179
7.4 Methods of Determining K ..............................................181
IEAC
7.5 Relationship between K and Yield Strength and
IEAC
Fracture Toughness ................................................................186
7.6 Environment Assisted Fatigue Crack Growth .......................191
7.7 Models for Environment Assisted Fatigue Crack Growth
Behavior .................................................................................193
7.7.1 L inear Superposition Model .....................................194
7.7.2 A Model for Predicting the Effects of Hydrogen
Pressure on the Fatigue Crack Growth Behavior .....196
7.8 Summary ...............................................................................202
References ........................................................................................203
Homework Problems ........................................................................205
Chapter 8 Fracture under Mixed-Mode Loading ..............................................207
8.1 I ntroduction ...........................................................................207
8.2 Stress Analysis of Cracks under Mixed Mode Loading ........209
Contents ix
8.3 Mixed Mode Considerations in Fracture of Isotropic
Materials ................................................................................211
8.3.1 Fracture Criterion Based on Energy Available for
Crack Extension .......................................................211
8.3.2 Maximum Circumferential Stress Fracture
Criterion ...................................................................215
8.3.3 Strain Energy Density (SED) as Mixed Mode
Fracture Criterion .....................................................218
8.4 Fracture Toughness Measurements under Mixed-Mode
Conditions ..............................................................................223
8.4.1 F racture in Bones .....................................................223
8.4.2 Measurement of Fracture Toughness under
Mode II (K ) ............................................................225
IIc
8.4.3 Measurement of Interfacial Toughness in
Laminate Composites ...............................................229
8.5 Fatigue Crack Growth under Mixed-Mode Loading.............235
8.6 Summary ...............................................................................240
References ........................................................................................241
Homework Problems ........................................................................242
Chapter 9 Fracture and Crack Growth under Elastic/Plastic Loading .............245
9.1 I ntroduction ...........................................................................245
9.2 R ice’s J-Integral .....................................................................246
9.3 J-Integral as a Fracture Parameter ........................................249
9.4 Equations for Determining J in C(T) Specimens ..................251
9.5 Fatigue Crack Growth under Gross Plasticity Conditions ....254
9.5.1 Experimental Correlation between da/dN
and ∆J ......................................................................256
9.6 Summary ...............................................................................259
References ........................................................................................259
Homework Problems ........................................................................260
Chapter 10 Creep and Creep-Fatigue Crack Growth ..........................................263
10.1 I ntroduction ...........................................................................263
10.2 Creep Crack Growth ..............................................................266
10.2.1 The C*-Integral .........................................................266
10.2.2 C (t) Integral and the C Parameter ............................268
t
10.2.3 Creep Crack Growth in Creep-Brittle Materials ......271
10.3 Crack Growth under Creep-Fatigue-Environment
Conditions ..............................................................................273
10.3.1 da/dN versus ∆K Correlations ..................................274
10.3.2 Creep-Fatigue Crack Growth Rates for Long
Cycle Times ..............................................................281