A Practical Approach to Fracture Mechanics [1° ed.] 0128230207, 9780128230206

Table of contents :
Front Cover
A Practical Approach to Fracture Mechanics
A Practical Approach to Fracture Mechanics
Copyright
Contents
Preface
1 - General concepts of mechanical behavior and fracture
1.1 Fracture mechanics field of application
1.2 Definition of stress and strain
1.3 Mechanical behavior under tension
1.4 The stress tensor
1.5 The Mohr's circle
1.6 Yield criteria
1.7 Stress concentration
1.8 Definitions and basic concepts of fracture
1.9 Object and field of application of fracture mechanics
2 - Linear elastic fracture mechanics
2.1 Cohesive strength
2.2 The Griffith criterion
2.3 The stress intensity factor (Irwin's analysis)
2.4 Solutions of the stress intensity factor
2.5 Experimental determination of the stress intensity factor
2.6 Determination of the stress intensity factor by the finite element method
2.7 The plastic zone
2.8 The crack tip opening displacement
3 - The energy criterion and fracture toughness
3.1 The energy criterion
3.2 The R-curve
3.3 Plane strain fracture toughness
3.4 Plane strain fracture toughness testing (KIC)
3.5 Effect of size on fracture toughness
3.6 Charpy impact energy fracture toughness correlations
3.7 Dynamic fracture and crack arrest
4 - Elastic-plastic fracture mechanics
4.1 Elastic-plastic fracture and the J-integral
4.2 JIC testing
4.3 Use of the J-integral as a fracture parameter
4.4 The crack-tip opening displacement as fracture parameter
4.5 The two-parameter criterion
5 - Fracture resistance of engineering materials
5.1 Remaining strength
5.2 Materials selection for fracture resistance
5.3 Material properties charts
5.4 Failure analysis using fracture mechanics
5.5 Reinforcement of cracked structures
5.6 The leak-before-break condition
6 - Fatigue and environmentally assisted crack propagation
6.1 Fatigue crack growth and Paris's law
6.2 Effect of the load ratio on the FCG rate
6.3 Fatigue crack closure
6.4 Effect of the environment on fatigue crack growth
6.5 Effect of variable loads on fatigue crack growth
6.6 Effect of a single overload on fatigue crack growth
6.7 Fatigue cracks emanating from notches and holes
6.8 Stress-corrosion cracking
6.9 Creep crack growth
6.10 Crack growth by absorbed hydrogen
7 - Structural integrity
7.1 In-service damage of structural components
7.2 General aspects of structural integrity
7.3 Remaining life of cracked components
7.4 A methodology for the estimation of remaining life
7.5 Structural integrity assessment procedure
7.6 Example of a structural integrity assessment
Index
A
B
C
D
E
F
G
H
I
J
L
M
N
O
P
Q
R
S
T
U
V
Y
Back Cover

Citation preview

A PRACTICAL APPROACH TO FRACTURE MECHANICS

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A PRACTICAL APPROACH TO FRACTURE MECHANICS

JORGE LUIS GONZÁLEZ-VELÁZQUEZ Metallurgical and Materials Engineering Deparment of Instituto Politécnico Nacional, Mexico City, Mexico

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-823020-6 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Rachel Pomery Production Project Manager: Prem Kumar Kaliamoorthi Cover Designer: Victoria Pearson

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Contents Preface

vii

1. General concepts of mechanical behavior and fracture 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Fracture mechanics field of application Definition of stress and strain Mechanical behavior under tension The stress tensor The Mohr’s circle Yield criteria Stress concentration Definitions and basic concepts of fracture Object and field of application of fracture mechanics

2. Linear elastic fracture mechanics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Cohesive strength The Griffith criterion The stress intensity factor (Irwin’s analysis) Solutions of the stress intensity factor Experimental determination of the stress intensity factor Determination of the stress intensity factor by the finite element method The plastic zone The crack tip opening displacement

3. The energy criterion and fracture toughness 3.1 3.2 3.3 3.4 3.5 3.6 3.7

The energy criterion The R-curve Plane strain fracture toughness Plane strain fracture toughness testing (KIC) Effect of size on fracture toughness Charpy impact energy fracture toughness correlations Dynamic fracture and crack arrest

1 1 3 6 13 17 21 24 27 31

35 35 37 40 48 56 62 66 72

75 75 81 86 87 94 96 98

v

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Contents

4. Elastic-plastic fracture mechanics 4.1 4.2 4.3 4.4 4.5

Elastic-plastic fracture and the J-integral JIC testing Use of the J-integral as a fracture parameter The crack-tip opening displacement as fracture parameter The two-parameter criterion

5. Fracture resistance of engineering materials 5.1 5.2 5.3 5.4 5.5 5.6

Remaining strength Materials selection for fracture resistance Material properties charts Failure analysis using fracture mechanics Reinforcement of cracked structures The leak-before-break condition

6. Fatigue and environmentally assisted crack propagation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Fatigue crack growth and Paris’s law Effect of the load ratio on the FCG rate Fatigue crack closure Effect of the environment on fatigue crack growth Effect of variable loads on fatigue crack growth Effect of a single overload on fatigue crack growth Fatigue cracks emanating from notches and holes Stress-corrosion cracking Creep crack growth Crack growth by absorbed hydrogen

7. Structural integrity 7.1 7.2 7.3 7.4 7.5 7.6 Index

In-service damage of structural components General aspects of structural integrity Remaining life of cracked components A methodology for the estimation of remaining life Structural integrity assessment procedure Example of a structural integrity assessment

107 107 113 118 125 133

145 145 153 163 165 168 172

177 177 183 184 190 192 195 200 204 208 213

219 219 227 235 241 250 256 267

Preface Right after finishing my PhD in metallurgy at the University of Connecticut in 1990, under the guidance of Professor Arthur J. McEvily, a renowned pioneer on the study of fatigue crack propagation under the fracture mechanics approach, I became a full-time professor at the Metallurgy Department of the Instituto Politecnico Nacional (IPN), the secondlargest higher education institution in Mexico, with more than 200,000 students. As a young teacher and researcher, I had the full intention of applying my knowledge to the solution of the technological challenges of my country and perhaps make a significant contribution to the fields of metallurgy and fracture mechanics. As a result, I introduced the very first graduate-level course on Fracture Mechanics in Mexico in 1990, and 2 years later, I persuaded Pemex, Mexico’s state-owned oil company, to finance an academy-industry research project to study pipeline fractures. After 4 years of intense work both at field and laboratory I funded the Grupo de Análisis de Integridad de Ductos (Pipeline Integrity Assessment Group, GAID) at IPN, an organization composed by professors, undergraduate, and graduate engineering students and professionals intended to perform fracture mechanics research, technical assistance on maintenance and failure analysis, and most important, structural assessment of pipelines and hydrocarbon processing and storage facilities. By the 2000s GAID had more than 500 collaborators to provide fracture mechanics related services in more than 60,000 km of pipelines, more than 144 marine platforms, six large oil refineries, and more than 50 hydrocarbon storage and distribution plants. The most significant contribution of GAID was to introduce the structural integrity approach into the management of maintenance at the oil and gas industry in Mexico, culminating with the issuing of the national standards on pipeline integrity management, nondestructive inspection, and many other technical specifications and recommended practices, all of them aimed to the practical applications of fracture mechanics. The topic of fracture mechanics was a rarity in the 1990s Mexico, since there were only a few professionals working on it by that time. Even after submitting a project proposal to the industry, some high-level executives told me that “fracture mechanics is a theoretical curiosity without practical application beyond explaining some failures.” This experience encouraged me to write my first book, back in 1998. The book was written in Spanish vii

viii

Preface

because I realized that many students and industry professionals did not have enough proficiency in English to fully understand and more importantly to apply the theoretical foundations of Fracture Mechanics to the solution of real-life problems. My Spanish book on fracture mechanics had a second edition in 2004, and in between I wrote a book on mechanical metallurgy, also in Spanish. Years later, encouraged by Ashok Saxena and other brilliant colleagues I published my books on Fractography and Failure Analysis and Mechanical Behavior and Fracture of Engineering Materials in English. Both books had a good reception among engineering students and professionals, but I still hesitated to publish a book on fracture mechanics in English, until Elsevier kindly invited me to do so. After considering the excellentdbut highly theoretical and full of complex mathematicsd existing books on fracture mechanics, I decided to take a different approach and to write a text intended to make this subject accessible to students that enter for the first time into this topic and professionals searching for quick and practical answers to the fracture problems they face in their field of practice. That is how A Practical Approach to Fracture Mechanics became a reality. This book places emphasis on the practical applications of fracture mechanics, avoiding heavy mathematical demonstrations, with a few exceptions, and focusing instead on making the physical concepts clear and simple, providing examples from my real-life experience, but yet at a level that can be easily understood and applied by both engineering students and practicing engineers that have the need to learn about it but have neither the time nor the background to understand a high-level textbook or research paper. My intention is to introduce the reader into the fracture mechanics field in a logical and chronological sequence, as the fracture mechanics concepts were developed along the history; therefore, the book begins with brief introduction of the importance of the study of fracture mechanics and presents the basic definitions of stress and strain its significance for mechanical design and materials selection. Chapters 2 and 3 introduce Griffith’s analysis as the background for Irwin’s analysis that led to the introduction of linear elastic fracture mechanics. The analytical and experimental methods to determine the stress intensity factor and the fracture toughness of engineering materials are described, along with the implications of linear elastic fracture mechanics and the energy criterion on the behavior of a loaded body containing cracks.

Preface

ix

Chapter 4 is dedicated to elastic-plastic fracture by Rice’s J integral and the two-parameter criterion which are nowadays the most widely used method to assess crack like flaws in structural integrity. In Chapter 5, the concepts of linear-elastic and elastic-plastic fracture mechanics are applied to the determination of remaining strength, materials selection, and to describe the use of fracture mechanics in failure analysis and the reinforcement of cracked structures. Once the static fracture problem is understood, the next development is the application of fracture mechanics concept to the understanding of the gradual or slow crack propagation phenomena in engineering materials under service conditions. This is discussed in Chapter 6, beginning with fatigue crack growth and covering stress-corrosion-induced crack propagation, creep crack growth, and crack growth by absorbed hydrogen. The book finishes with the latest and most important application of fracture mechanics today, which is the structural integrity assessment of cracked components. It presents a brief introduction to the main damage mechanisms of in-service components to then present the general concepts of structural integrity, including the estimation of the remaining life. Since the main attempt of this text is to serve as a practical guidance to practicing engineers, a complete example of the assessment of a real-life component is presented. This book is considered suitable for engineering students, design engineers, inspection and maintenance engineers performing Fitness-For-Service (FFS) assessments in all industries (oil and gas, power generation, construction, chemical and petrochemical, transportation, etc.) as well as for professionals working at research laboratories, engineering firms, insurance, and as loss adjusters. The book could be used in fracture mechanics courses that are being taught in most major universities and higher education institutions, at both undergraduate and graduate levels. I am indebted to my colleagues at the IPN, whom I do not list here to avoid involuntary omissions, and to many Pemex engineers, specially to Ing. Francisco Fernandez Lagos ( ), Ing. Carlos Morales Gil, Ing. Javier Hinojosa Puebla, and Ing. Miguel Tame Dominguez for their support and encouragement to allow me to fulfill my life project of applying scientific knowledge into the solution of strategic problems in the industry. I wish to dedicate this book to the memory of Professor Arthur J. McEvily, not only for his guidance and support when I was a doctoral student, but also for his

x

Preface

friendship and inspiration for more than 30 years. Finally, I am grateful to my daughter Carolina for her many suggestions for the improvement of this text, and to my brother Juan Manuel for his help in the preparation of the manuscript in English language. April 2020 Jorge Luis González-Velázquez

CHAPTER 1

General concepts of mechanical behavior and fracture 1.1 Fracture mechanics field of application Fracture is a phenomenon that has received constant attention, practically since machines and structures found use in both wartime and peace time. Particularly, the use of mechanical and structural components such as beams, columns, shafts, pressure vessels, cables, gears, and so forth have always come along with the risk of fracture. Frequently, the fracture of a structural component is accompanied with great material, economic and human losses. It is also common that, although failures may occur once in a lifetime, a single failure can mean a great catastrophe, which is the case for airplane crashes, explosions of gas pipelines, or nuclear reactor failures. The losses are not only limited to human and economic ones, but also there are additional losses, such as delays in production, environmental damage, and the detriment of the company’s public perception and image. Premature fracture of small components, such as screws and bolts, is also an insidious problem, since consumers associate it with poor quality, which results in sales reduction. In summary, it would be impossible to quantify the magnitude of losses caused by fracture-related failures, but what is certain is that fracture may be the limiting factor for the success of industries and entire economies. Fracture mechanics is the discipline that provides the basis and methodology for the design and assessment of cracked components in order to determine the effect of the presence of a crack. It is also applied to develop structures and materials more resistant to fracture. Through time, it has been demonstrated that the traditional criteria of structure design under the assumption of absence of flaws, and further compensating its effect by means of safety factors are risky and often lack any technical foundation. The fact is that flaws, especially cracks, inevitably appear in both mechanical and structural components due to poor manufacturing, inadequate A Practical Approach to Fracture Mechanics ISBN 978-0-12-823020-6 https://doi.org/10.1016/B978-0-12-823020-6.00001-3

© 2021 Elsevier Inc. All rights reserved.

1

2

A Practical Approach to Fracture Mechanics

construction, or introduced during service, so the engineers have to deal with them, and the best way is by analyzing their effect on the mechanical behavior. The problem of fracture has kept scientists and engineers busy since the eras of the great Leonardo DaVinci and Galileo. However, it was until the beginning of the twentieth century that Griffith was able to calculate the fracture strength of brittle materials, but even so, theoretical and experimental difficulties hindered the development of fracture mechanics until 1956, when George R. Irwin introduced the concept of stress intensity factor and fracture toughness, giving birth to modern fracture mechanics. Nowadays, the study of fracture mechanics is a fundamental part of mechanical, materials and metallurgical engineering. A significant fact is that over 40% of the articles published in engineering and materials science journals are related, directly or indirectly to mechanical behavior and fracture. Within the industry, fracture mechanics is extensively used in the aeronautic, aerospace, chemical processing, oil refining, and nuclear industries, and it has begun to be used more frequently in the automobile industry, pipeline hydrocarbon transport, and construction industries. Although the economical and safe operation of engineering components and industrial facilities requires a design resistant to cracking and fracture, fracture mechanics is of great usefulness in components that have already been in service. It helps to set the criteria for the acceptance or rejection of flaws, establishing the frequency of inspections and safe operational limits of process equipment and machinery; these studies are known as Structural Integrity or Fitness-For-Service. Fracture mechanics is also useful in failed components, since it provides the analytical tools to determine if the cause of failure was an overload or the component had defects out of the acceptance limits. The application of fracture mechanics in all stages of the life cycle of structural and mechanical components yields great wins in safety and economy since it reduces the frequency of failures and extends the life span. All of which allow engineers to pay more attention to other fundamental issues such as the development of new materials and the improvement of designs which result in further technological advance. The field of fracture mechanics is divided into two broad fields, as illustrated in Fig. 1.1. At the microscopic scale, fracture mechanics is part of the materials science field. The aim is to study the relationship between microstructure and fracture mechanisms, including plasticity and

General concepts of mechanical behavior and fracture

3

Figure 1.1 Field of study of fracture mechanics, according to the scale size.

fractographic examination. At the macroscopic scale, fracture mechanics is a branch of materials and mechanical engineering, focused on applications, such as laboratory testing to determine the fracture properties, and solving practical problems, such as defect assessment, crack arrest, materials selection, and failure analysis, among others.

1.2 Definition of stress and strain The French mathematician and scientist Augustin Louis Cauch^y introduced the concept of stress in 1833. He used the movement laws by Euler, and Newton’s mechanics, to determine the displacements produced on a static solid body subject to surface loads. Based on Newton’s second law, which states, “to every action, there is a corresponding reaction,” Cauch^y figured out that when an external force is applied on a static body, an internal reaction force balancing the external force is instantaneously produced. The magnitude of such reaction is directly proportional to the magnitude of the applied force and inversely proportional to the size of the cross-section area, this reaction being stress. Under Cauch^y’s principle, the mechanical behavior of solid materials may be summarized as follows: Loads produce stresses, the stresses cause strain, and strain leads to fracture; therefore the aim is to determine the stresses and strains produced in a loaded solid body and determine the material’s strength to withstand such stresses without neither excessively straining nor fracturing. To facilitate the analysis of the mechanical behavior of solids, it is necessary to simplify the system, because materials are complex arrays of atoms, crystalline defects, second phases, and microstructural heterogeneities.

4

A Practical Approach to Fracture Mechanics

The basic assumptions for the initial approach of mechanical behavior include the following: • The material is a continuum: This means that matter fills in the total volume and there are no voids nor interruptions. Under this assumption, it can be established that there will be an infinitesimal volume (a volume that tends to zero, but it never is zero), where the forces and areas be defined, so the stress exists in a point. This assumption is also the reason why the study of mechanics of materials is called continuum mechanics. • The material is homogeneous: The whole volume is filled in with the same type of matter. • The material is isotropic: The properties are the same in any direction. Based on these assumptions, a static solid under the action of an applied external force FA remains static, if and only if this force is balanced by an internal force Fi, of the same magnitude and opposite direction to FA. The force causes an internal reaction in the solid that is directly proportional to the magnitude of the applied force and the number of particles resisting the action of such force, being proportional to the cross-section area A. The magnitude of the internal reaction is the stress, represented by the symbol s, and can be defined as: s ¼ F=A The internal force is a vector, since it has magnitude and direction, therefore it can be split into two components: One perpendicular to the cross-section area (Fn) and the other parallel (or tangential) to the crosssection area (Ft), as shown in Fig. 1.2. The stress produced by Fn is called normal stress and is expressed by sn ¼

Fn A

Fn Internal area,

A

Reaction internal force, Fi

External applied force, FA

Ft

Figure 1.2 Internal reaction force to an applied force that gives origin to the stress concept and its decomposition into normal (Fn) and tangential (Ft) components.

General concepts of mechanical behavior and fracture

5

Normal stresses are divided into two types; when the internal forces tend to elongate the body, they are called tension stresses and its sign is positive (þ) and when they shorten the body, are called compression stresses and are of negative sign (). On the other hand, the tangential forces will produce shear stress, represented by the symbol s, being calculated as s¼

Ft A

The physical effects of the normal and shear stresses on the body are quite different and therefore, they have to be treated separately. The typical stress units are given in Table 1.1. Cauch^y also included the concept of strain in his analysis of the mechanical behavior of solid materials. In simple terms, strain is the change of shape in a body due to the action of stresses, and it is divided into two types: Elongation strain, identified by the symbol ε, and defined as the change of length (lf  l0) over the initial length (l0) ε¼

Dl lf  l0 ¼ l0 l0

Shear strain, identified by the symbol g, and defined as the change of straight angle q of a cubic volume element, so it is calculated as 1 g ¼ ðtan qij þ tan qji Þ 2 Where i is the direction of displacement and j is the original direction of the displaced side. Fig. 1.3 schematically illustrates the two types of strain. The stress and strain are related by the so-called constitutive equations, which appear in every book of mechanical behavior or strength of materials. The reader is encouraged to become acquainted with them, since they are fundamental for the analysis of the mechanical behavior of engineering materials. Table 1.1 Typical stress units. System

Units

Common multiple

International

Pascal (Pa ¼ Nw/m )

MPa ¼ 106 Pa

English

psi (psi ¼ lbf/plg2)

ksi ¼ 1000 psi

Mks Conversion factors:

2

kg/mm

2

kg/cm2 ¼ 100 kg/mm2 1 ksi ¼ 6.895 MPa 1 kg/cm2 ¼ 14.23 psi 1 MPa ¼ 10.5 kg/cm2

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A Practical Approach to Fracture Mechanics

Figure 1.3 Schematic illustration of engineering strain.

1.3 Mechanical behavior under tension The simplest way to observe the mechanical behavior of a solid is by applying a tension force on a body of regular cross-section and record the load and elongation produced in the test specimen. The typical Load versus Elongation record of an engineering material is shown in Fig. 1.4. By using the definition of stress as s ¼ F/A0, where A0 is the initial cross-section area of the test specimen, and defining the elongation strain as ε ¼ Dl/l0, where Dl is the elongation and l0 is the initial length, the stress-strain curve in tension can be reconstructed from the original Load-Elongation curve. Since A0 and l0 are constants, the shape of the curve does not change, only the scale is modified, therefore the stress-strain curve in uniaxial tension has the same shape of the Load-Elongation curve. As it can be observed on the stress-strain curve, at the onset of loading, the elongation is proportional to the stress, but if the load is removed, the body recovers its initial shape and length; this behavior is termed as elastic. In most materials, the strain is linearly proportional to the strain and the constant of proportionality is called the Young’s modulus, represented by the

Figure 1.4 Typical Load-Elongation or Stress-Strain curve in uniaxial tension of an engineering material.

General concepts of mechanical behavior and fracture

7

symbol E. When the stress surpasses a limit value the deformation becomes permanent, a condition termed as plastic strain, the stress at this point is termed as the yield strength, and it usually represented by the symbol s0. On further loading, the load has to be increased to sustain the plastic deformation, this behavior is called strain hardening and it makes the curve to take a parabolic-like shape, which maximum is referred as ultimate tensile strength, represented by the symbol suts. Just after the ultimate tensile strength is reached, the material experiences a local contraction known as necking, so the cross-section area is rapidly reduced and the load drops, leading to the final rupture. The maximum elongation in tension loading is referred as ductility, identified by the symbol εf. These essential features of the stressstrain curve in uniaxial tension allow determining the fundamental mechanical properties of engineering materials, which are: • Young’s modulus (E): Proportionality constant between strain and stress in the elastic regime. • Yield strength (s0): Tension stress at the onset of plastic deformation. • Ultimate tensile strength (suts): Maximum tension stress that a material can withstand before failure. • Rupture elongation or ductility (εf): Maximum plastic elongation measured after rupture in tension. At this point, it is important to discuss about the hardness, which is generally regarded as a mechanical property. Hardness is the resistance of a material to be scratched or indented, and its value, regardless the testing method, involves several mechanical processes such as elastic strain, plastic strain, and strain hardening. Additionally, it is affected by other factors such as friction, thickness and surface finish; therefore, hardness is not a fundamental mechanical property. However, since its testing is simple, fast and economical, it has become very popular as a parameter to characterize the mechanical behavior of engineering materials. The basic rule is that the harder a material is, the higher load bearing capability, but the disadvantage is that hard materials are usually brittle. Hardness is measured in several scales, being the most common in engineering materials: Brinell, Vickers, Rockwell, and Shore (for plastics). The uniaxial tension test not only provides the fundamental mechanical properties of materials, but also provides information on the overall performance of materials under stresses. This information is quite useful in the analysis of the mechanical and fracture behavior of materials, allowing comparisons for materials selection, materials development, design and quality control, among others. Typical material behavior categories according to the stressstrain curve in uniaxial tension is schematically depicted in Fig. 1.5.

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A Practical Approach to Fracture Mechanics

Figure 1.5 Typical material behavior categories according to the Stress-Strain curve in uniaxial tension.

The material categories according to this criterion are: Hard and brittle: High yield strength, low strain hardening and poor ductility. The Young modulus may be very high. These materials have high hardness and stiffness, but they are brittle, so they do not resist impacts, strain and thermal shock. They are used in cutting tools and parts requiring high abrasive wear and erosion resistance. Soft and ductile: Low yield and tensile strength, low strain hardening and high ductility, so they are ductile and easy to shape by mechanical processing. The Young modulus may be low, but not necessarily. They are used to fabricate components of intricate forms, but limited low load bearing applications. High strength: High yield and tensile strength, high strain hardening and moderate to high ductility. The Young modulus is usually high. These materials resist heavy loads, strong impacts and require high energy input to cause fracture, so they are widely used for high load and impact conditions, such as buildings, bridges, machinery, process equipment, all kinds of transport vehicles. Weak: They feature low mechanical strength, so they are used in low load bearing, wear, and impact conditions and low durability items, such as disposable cups and packing stuffing. The typical tensile properties of common engineering materials are shown in Table 1.2. It is observed that the strongest materials are the metallic materials, which makes them an excellent choice for structural and mechanical applications. Polymers have the lowest mechanical strength levels, although their low density, chemical stability and easy of manufacture makes them ideal for low load requirements. Ceramics and glass are highly resistant to compression and exhibit high hardness, but they are

Table 1.2 Mechanical properties of common engineering materials. Material

s0, MPa

suts, MPa

%εf

Brinell hardness

Carbon steel

200e212

250e1200

340e1600

120e80

100e1100

Low alloy steel

200e212

400e1100

460e1200

14e100

120e350

Stainless steel

200e212

170e1000

480e2240

62e180

140e650

Cast iron

200e212

215e790

350e1000

10e35

100e290

Aluminum alloys

70.3

30e500

60e550

22e35

18e160

Cooper alloys

130

30e500

100e550

30e90

29e160

Nickel alloys

200

70e1100

340e1200

80e110

100e350

Titanium alloys

90e110

250e1250

300e1600

14e120

86e460

Zinc alloys

70e80

80e450

130e520

10e100

38e150

Elastomers

0.01e0.6

2e90

25e50

0.2e0.4

7e15

Polymers

8e100

18e70

20e70

1e6

6e20

0.02e0.4

0.01e12

Foams Alumina Carbides Glass Brick Concrete

400e800

690e5500

0.005e1

N/A

350e665

3.3e4.8

600e1600

a

370e680

2.5e5

600e1000

1000-6800 a

70e80

30e1600

22e150

0.5e0.7

10e470

15e20

a

7e14

1e2

N/A

a

2e6

0.35e0.45

N/A

a

5e17

0.7e1.5

N/A

15e30

50-140 32e60

Rocks

7e20

34-298

Wood

5e20

30e70

60e100

5e9

N/A

Leather

0.1e0.5

5e10

20e26

3e5

N/A

Composites

10e200

N/A

100e1000

5e15

N/A

Reported as compressive strength.

9

a

260e300

0.01e12 a

General concepts of mechanical behavior and fracture

E, GPa

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A Practical Approach to Fracture Mechanics

brittle, so its application is limited to compressive loading and nonimpact uses. According to the continuum mechanics principles and taking into consideration the stress-strain behavior in uniaxial tension, the mechanical behavior analysis of loaded bodies is typically carried out by the following procedure: (1) Definition of the component geometry. (2) Definition of loads, magnitude, direction, and point of application. (3) Calculation of stresses. (4) Calculation of strains and displacements resulting of the stresses. The design based on continuum mechanics aims to determine the loads and/or the size of the component and to select a material of specific mechanical properties capable of withstand and transmit the applied loads and/ or displacements. To do this, the cross-section area and the applied loads should be such that the stresses are lower than the material’s strength. This process can be explained by the mathematical definition of stress: s¼

F A

Where s is the stress, F is the applied load or external force and A is the size of the cross-section area. The general design criterion is usually: If s  Limit stress, the component fails. The limit stress is usually the yield strength multiplied by a safety factor (SF). Based on the earlier-mentioned equation, the stress-based mechanical design is carried out under the procedure shown in Fig. 1.6. First, the shape and size of the component are defined, to determine the cross-section area size (in many design codes the cross-section area size is determined by the thickness), then, the loads are assigned and a material with enough mechanical strength to withstand the calculated stresses is selected. In engineering practice, the safety factor is an additional material strength, additional thickness and/or a limit load, all of them established in order to compensate the presence of additional or higher stresses, either by unexpected accidental loads or by abuse. The safety factor is also introduced to compensate for lower strength caused by the presence of flaws and defects in the material. In most mechanical design codes, the maximum allowable stress is the calculated stress multiplied by the safety factor, defining the design stress, in such case FS is more than one, but if the safety factor is applied to the material’s strength, then FS is less than one. The more uncertainty of in-service loads, the poorer quality of material, and the more severe failure consequences, the higher safety factor.

General concepts of mechanical behavior and fracture

11

Figure 1.6 Flow chart of structural components design by continuum mechanics.

Regardless to what variable the safety factor is applied, it should give a reasonably large gap between the material’s strength and the reaction stresses ad schematically illustrated in Fig. 1.7. As it may be observed FS gives the design safety margin, which is the difference between the minimum specified material’s strength and the limit design load. Because most vendors supply materials with strength somewhat greater than the minimum specified, the design safety margin becomes a minimum expected value; however, this should not be considered as a warranty of a safer design and the users should not relay on the free excess strength. Furthermore, most structural or mechanical components are not regularly operated at their design limit, and in many cases, the operational loads are far below the design limit, such as happens with automobiles, hand tools, cranes, and many others, and additional operational safety margin is introduced, allowing the implementation of damage tolerance strategies, as it will be further discussed in this book. For most engineering materials, the elastic part of the stress-strain curve linear, therefore Young’s modulus is defined as:
s E¼ ε Elastic

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A Practical Approach to Fracture Mechanics

Figure 1.7 Stress levels in the design of structural components resulting of the use of the safety factor.

Bearing in mind, that ε ¼ Dl/l0 and s ¼ F/A0, it can be stated that   l F ¼ l0 A Therefore F¼

  AE Dl l0

The term (AE/l0) is known as the elastic coefficient, high values of the elastic coefficient result in rigid structures, that is less likely to elastically deform under load; whereas low values of the elastic coefficient indicate that the structure is flexible, and it will easily deform under load. The elastic coefficient is widely applied in the design of mechanical and structural components, since it allows to set the physical dimensions and select the appropriate material to control the elastic strain. The first case of elastic design are flexible or elastic components: those are that should feature fairly large elastic strains or flexions, but not as much as to reach yield, as it is the case of helical and leaf springs. In other words, flexible components are designed to have large controlled elastic strains under the applied loads. According to the elastic coefficient, this can be achieved by long lengths, small cross-section areas and materials with a low Young’s modulus. The first two conditions make flexible components long and slender. The other type of elastic design is rigid components, which are those whose excessive elastic strain is adverse for their performance, such as building

General concepts of mechanical behavior and fracture

13

structures, supports, gears, and machine parts. In these components, the magnitude of the elastic strain must be limited to a minimum, so the elastic coefficient must be high. This can be attained by widening the cross-section, shortening the length and by selecting high Young modulus materials. The first two characteristics make rigid components short and thick. The following example illustrates the use of the elastic coefficient. A hollow rectangle bar of 5 cm external height and 4 cm external width, 1 mm thickness, and 50 cm long, to be used as part of a frame is required to support a load F ¼ 5000 N. The maximum allowable tensile strain is Dl/l0 ¼ 2.0  104. The original design uses steel, but a novel engineer suggests substituting steel by aluminum to save weight and cost. Is this a good idea? Assume that the cost of aluminum is three times the cost of steel. ESteel ¼ 200  109 N/m2; EAluminum ¼ 70.3  109 N/m2. Solution: According to the force-elongation elastic formula: (Dl/l0) ¼ F/(AE) A ¼ 1.96 cm2 ¼ 0.000,196 m2. Substituting values for steel: (Dl/l0)Steel ¼ (5000 N)/(0.000196 m2  200  109 N/m2) ¼ 1.276  104 Substituting values for aluminum: (Dl/lo)Aluminum ¼ (5000 N)/(0.000196 m2  70.3  109 N/m2) ¼ 3.629  104 Notice that the aluminum bar does not meet the specification for allowable strain. If the frame has to be made of aluminum, in order to meet the allowable strain specification, its thickness need to be changed as A ¼ F/[(Dl/l0)E] ¼ 5000 N /[2.0  104  70.3  109 N/m2] ¼ 3.556  104 m2 ¼ 3.556 cm2 If the external dimensions have to be the same, the thickness of the aluminum frame shall be 0.184 cm ¼ 1.84 mm, but the important factor is the weight. WSteel ¼ (Alor)Steel ¼ 1.96 cm2  50 cm  7.85 g/cm3 ¼ 769.3 g WAluminum ¼ (Alor)Aluminum ¼ 3.556 cm2  50 cm  2.7 g/cm3 ¼ 480.06 g The aluminum frame weights 1.6 times less than the steel frame, but it will cost 1.872 times more than steel, so in terms of weight saving, aluminum is a better choice, but in terms of cost steel is better.

1.4 The stress tensor According to Cauch^y’s stress theory, the state of stress at one point is described by the stress tensor which is determined by the stress components

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A Practical Approach to Fracture Mechanics

Figure 1.8 Stress components in a unitary volume element of a solid body.

acting on the faces of an element of a differential volume (a cube) located at the origin of a Cartesian coordinate system of x, y, and z axes, as shown in Fig. 1.8. This results in nine stress components, three normal and six shear. An index notation identifies each stress component: sij Where i is the cube’s face where the stress is acting, and j is the direction of the stress. The nine stress components written in matrix form, become the stress tensor, since they are vectors, and it has the form: 0 1 sxx sxy sxz B C s ¼ @ syx syy syz A szx szy szz Due to balance of momentum, it can be easily demonstrated that the shear components are symmetric, that means that sij ¼ sji, so that reduces the stress tensor to six independent components, three normal and three of shear. The other than zero components of the stress tensor define the state of stresses and represent the reactions that occur inside of a static solid body subject to external loads. The most common states of stress are presented in Table 1.3. The methodology to determine the state of stresses (qualitatively) of a solid of regular shape under a few known loads is the following: 1. Draw a free body diagram of the component, indicating the applied loads in the form of vectors (arrows with magnitude, direction, and application point) as well as the points of support and/or restrictions to displacement.

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Table 1.3 Common states of stress. Name

Uniaxial tension

Simple compression

Biaxial tension

Pure torsion

Tensioncompression

Plane stress

Triaxial stress

Stress tensor

0

Example

1

sxx 0 0 C B s ¼ @ 0 0 0A 0 0 0 1 0 sxx 0 0 C B 0 0A s ¼@ 0 0 0 0 1 0 sxx 0 0 C B s ¼ @ 0 syy 0 A 0 0 0 1 0 0 sxy 0 C B s ¼ @ sxy 0 0 A 0 0 0 0 1 sxx 0 0 B C s ¼ @ 0 syy 0 A 0 0 0 1 0 sxx sxy 0 C B s ¼ @ sxy syy 0 A 0

0

0

sxx

sxy

B s ¼ @ sxy sxz

syy syz

0 sxz

1

C syz A szz

Dead weight hanging of a bar

Open forge

Thin wall cylinder under internal pressure

Transmission shaft

Wire drawing

Thin sheet and free surfaces

General case

Note: The common practice is to eliminate rows and columns which values are zero.

2. Place an orthogonal coordinate system x, y, z, with its center on the body’s center of gravity or the geometrical centroid, preferably with one of the axes aligned with the body’s symmetry axis or parallel to the direction of the main applied load. 3. Draw a cube at the origin of the coordinate system, with its sides aligned parallel to the coordinate system axes. 4. Identify the internal reaction loads on each of the cube’s faces, as vectors of the same magnitude and in opposite direction to the external forces. 5. Identify normal and shear stress components generated on the cube faces by applying the index notation, bearing in mind that the first subindex is the face where the force is acting on, and the second subindex is the direction of the reaction force. Use the rule of signs to determine the

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A Practical Approach to Fracture Mechanics

stress sign, remembering that the positive faces and directions are front, right and upwards, and vice versa. The rule of signs is (þ) (þ) or () () ¼ (þ); and (þ) () or (þ) () ¼ (). 6. Write down the identified nonzero components matrix, the resulting matrix is the state of stresses. The following example illustrates this procedure. Determine the state of stress of a screwdriver of 0.5 cm diameter that is under a compressive load of 100 kgf and a torque of 6.2 kgf-cm.

z

y x

The identified reactions are: 1. Force along the z axis of positive direction (þ) acting on the face under the -z cube’s face results into a negative normal stress (compression) and corresponds to the -szz component. 2. Force along the y axis and positive direction (þ) acting on the þx face of the cube results in a positive shear stress and corresponds to the þsxy component. Therefore, the state of stress is0 1 0 sxy 0 B C s ¼ @ sxy 0 0 A 0 0 szz Performing the corresponding calculations F 100kgf szz ¼ 2 ¼ ¼ 509 kgf =cm2 pr pð0:25Þ2 cm2 16T 16ð6:2Þkgf  cm sxy ¼ 3 ¼ ¼ 253 kgf =cm2 pd pð0:5Þ3 cm3

General concepts of mechanical behavior and fracture

0

0

B s ¼ @ 253 0

253 0 0

0

17

1

C 0 Akgf =cm2 509

1.5 The Mohr’s circle As seen before, the state of stresses depends on the loads, body geometry, and the coordinate system orientation. Since the coordinate system orientation is arbitrarily chosen, the state of stresses may have an endless number of equivalent components, one for any possible orientation. This means that for the same body, with the same loads, the state of stresses changes if the orientation of the coordinate system is changed, but the stress tensor must be equivalent. Fig. 1.9 intuitively shows this idea; notice that the body is the same, so is the load, but the stress tensor changes when the orientation of the coordinate system is changed. In the left figure, the x axis is parallel to the applied force, so the stress tensor has only the component sxx, which corresponds to uniaxial tension. But, if the coordinate system is rotated around the z axis, as in the right figure, the reaction forces resulting of P on the cube’s faces produce four stress components, two of tension and two of shear on the x, y plane, thus giving a plane stress state. The procedure to calculate the state of stresses when the orientation of the coordinate system is rotated is called stress transformation. The stress transformation equations in two dimensions are

Figure 1.9 Variation of stress components after rotation of the reference coordinate system.

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A Practical Approach to Fracture Mechanics

  sxx þ syy sxx  syy sx0 x0 ¼ þ Cos2q þ sxy Sin2q 2 2   sxx þ syy sxx  syy sy0 y0 ¼  Cos2q  sxy Sin2q 2 2 syy  sxx sx0y0 ¼ Sin2q þ sxy Cos2q 2 Where the symbol (0 ) indicates the stress component after rotation. Notice that: sxx þ syy ¼ sx0 x0 þ sy0 y0 This is known as first stress invariant, which can be extended to three dimensions and states that the sum of the normal stress components is constant. The German engineer Otto Mohr showed in 1882 that the twodimensional stress transformation equations can be represented as a circle, where the shear stresses are plotted in the ordinates and the normal stress is plotted in the abscises. Such circle allows visualizing the transformation of a state of stress when changing the coordinate system orientation. The circle’s center is the average normal stress whereas the radius is the maximum shear stress. The points located at the circle’s perimeter indicate the values of the stress components after making a rotation of the coordinate system by 2q degrees, where q is the rotation angle in the real space. The rules for the construction of the Mohr’s circle in two dimensions are 1. The abscises are the normal stresses and the ordinates are the shear stresses. Strictly speaking the positive values of the shear stress are plotted below the origin, but this is only required if the proper direction of the shear stresses has to be determined. 2. The first point in the circle is the pair (sxx, sxy) and the second point in the circle is (syy, sxy). Notice that the negative value of the shear stress is used. 3. A straight line joins the two previous points, which is the diameter of the circle. 4. The intersection of the diameter with the horizontal axis is the center of the circle that is the average normal stress. 5. The circle radius is the maximum shear stress.

General concepts of mechanical behavior and fracture

19

Figure 1.10 Mohr’s circle in two dimensions.

6. The normal stress values when the shear stress is zero, that is, the points where the circle intersects the abscises axis are the principal stresses. 7. Following the diameter extreme points touching the circle’s perimeter after a rotation of 2q degrees, the new stress values are obtained. This procedure is schematically shown in Fig. 1.10. Mohr’s circle allows identifying the following properties of the stress transformation. 1. The average normal stress is the center and the maximum shear stress is the radius of Mohr’s circle. 2. An orientation on which the shear stress is zero always exists. The directions and normal stresses on those directions are called principal stresses, identified from the highest to the lowest, as: s1 > s2 > s3. 3. An orientation on which the shear stress is maximum always exists, and it is located halfway between the maximum and minimum principal stresses (90 degrees in the Mohr circle and 45 degrees in the real space).   Example: 10 4 Be the stress tensor: s ¼ 4 2 The original stress tensor points on Mohr’s circle have coordinates (10,4) and (2,4). The straight line joining these points is the circle’s diameter. The center is located at: sxx þ syy 10 þ 2 save ¼ ¼6 ¼ 2 2

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A Practical Approach to Fracture Mechanics

The radius of the circle can be easily calculated by the Pythagoras theorem, choosing, and appropriated triangle.

r¼

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 42 þ 84 ¼ 8:944

The maximum shear stress is: smax ¼ 8.944. The direction of q is: q ¼ 1/2 tan1(sxy/(sxx  save) ¼ 1/2 tan1(8/4) ¼ 31.7 Mohr’s circle is:

The maximum principal stress is: smax ¼ save þ r ¼ 6 þ 8.944 ¼ 14.944 And the minimum principal stress is: smin ¼ save  r ¼ 6  8.944 ¼ 2.944 The following figures show the physical orientation of the original and transformed stresses:

General concepts of mechanical behavior and fracture

21

Figure 1.11 Mohr’s circle for three-dimensional stresses.

Mohr’s circle in three dimensions can be constructed from the three principal stresses that appear in three dimensions, identified as s1 > s3 > s3, which are determined by the roots of the cubic equation
 s3  ðsxx þ syy þ szz Þs2 þ sxx syy þ syy szz þ sxx szz  s2xy  s2yz s  sxx syy szz þ sxy sxz syz  sxx s2yz  syy s2xz  szz s2xy ¼ 0 Once the three roots of the above cubic equation are determined, the Mohr’s circle in three dimensions is constructed by drawing one circle between each pair of principal stress, as shown in Fig. 1.11. The difference from Mohr’s circle in two dimensions is that the stress components after a rotation of the reference axes are located at some point within the shaded area, so their value cannot be determined by graphic methods. One interesting application of Mohr’s circle is to visualize the effect of the addition of a new stress component into the stress state. For example, a pipe under internal pressure is in biaxial tension, but if a shear stress is introduced, the stress state will change to a triaxial state of stress, as shown in Fig. 1.12. The effect of the addition of a shear stress is that a new compression stress appears, and the maximum principal stress significatively increases, thus the pipe will behave as if the pressure was higher and if the shear stress keeps increasing, the pipe will collapse, as if it were crushed.

1.6 Yield criteria In uniaxial tension, it is known that when a material reaches its yield strength, it will start to plastically strain. However, in practical situations, it

22

A Practical Approach to Fracture Mechanics

Figure 1.12 Effect of the introduction of a large shear stress on a thin wall pipe under internal pressure.

is common to find a combined state of stresses that make plastic strain initiate at a stress different to the yield strength. The way to calculate whether there is yielding under a combined state of stresses is called Yield Criterion. The two most well-known yield criteria are Tresca’s and Von Mises’s, which are described next: Tresca’s criterion: Tresca’s criterion, known also as the maximum shear stress criterion, establishes that the plastic strain will initiate when the maximum shear stress surpasses a critical value. According to Mohr’s circle, the maximum shear stress is the difference between the maximum and minimum principal stresses, thus, Tresca’s criterion is expressed by the following equation: s1  s3 ¼ s0 Von Mises’s criterion: This criterion states that yielding starts when the effective stress reaches a critical value, and is expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s0 ¼ pffiffiffi ðs1  s2 Þ2 þ ðs2  s3 Þ2 þ ðs1  s3 Þ2 2 For many years, Tresca’s criterion was the most widely used because it is simpler and it predicts that yielding will occur at stresses lower than the actual ones, providing an additional safety window. Von Mises’s criterion, on the other hand, is more precise and has the advantage that it does not require the calculation of principal stresses when its general equation, shown below, is used. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
1 2 2 2 s0 ¼ pffiffiffi ðsxx  syy Þ þðsxx  szz Þ þðszz  syy Þ þ 6 s2xy þ s2yz þ s2xz 2

General concepts of mechanical behavior and fracture

23

Figure 1.13 Yield map in two dimensions.

The yield criteria in two dimensions may be graphically represented by a plot of the nonzero principal stresses in each direction of the coordinate system, as shown in Fig. 1.13. These graphs are the yield maps and provide an easy way to identify the combination of principal stresses that exhibit the greatest or least yield resistance. The arrows on the yield map indicate the load path. It is evident that the tension-tension and compression-compression combinations have the greatest resistance to yielding, while the tension-compression combinations have the least yielding resistance, with the stresses to cause yield being less than the yield strength. The two-dimension yield map also show that Tresca’s criterion is more conservative, since its map is inside of Von Mises’s map, even though they coincide at uniaxial and biaxial stress states. The following examples illustrate the use of the yield criteria. Determine the minimum yield strength of the material to fabricate a cylindrical pressure vessel with a design pressure p ¼ 50 kg/cm2. If the diameter is D ¼ 100 cm and the wall thickness is t ¼ 5 mm. Solution: The hoop stress is the maximum principal stress, given by Barlow’s equation: s1 ¼ pD/2t s2 ¼ 1/2 s1 s3 ¼ 0 Applying the Tresca criterion: s1 ¼ s0, therefore: s1 ¼ (50 kg/cm2  100 cm)/(2  0.5 cm) ¼ 5000 kg/cm2

24

A Practical Approach to Fracture Mechanics

By Von Mises’s criterion: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s0 ¼ 2 s21 þ s22  s1 s2 However, because s2 ¼ 1/2 s1: s0 ¼ (O3/2) s1 ¼ 0.866  5000 kg/cm2 ¼ 4330 kg/cm2 Notice that according to Von Mises’s criterion a lower yield strength is required. A transmission circular shaft bar of diameter D ¼ 1 in has to bear a tensile load F ¼ 15,000 lbf and a torque of 1900 lbf-in. Determine the minimum required yield strength of the fabrication material by Von Mises’s criterion. Assuming that the x axis is parallel to the longitudinal axis of the shaft: sxx ¼ F/A ¼ 4F/p D2 ¼ 4  15,000 lbf/[p  (1 in)2] ¼ 19,100 psi syz ¼ 5.1 T/D3 ¼ 5.1  1900 lbf-in/(1 in)3 ¼ 9690 psi By using Von Mises’s formula in terms of the stress components 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s0 ¼ pffiffiffi 2s2xx þ 6s2yz 2 Substituting values: s0 ¼ 25,426 psi. If there is an available material of s0 ¼ 50,000 psi, how much is the reduction of diameter that can be achieved: Now Von Mises’s formula has to be written in terms of the diameter 32 F 2 T2 þ 156:06 ¼ 2s20 p D4 D6 Substituting values 2.2918  109/D4 þ 5.634  108/D6 ¼ 5  109 Performing calculations for varying diameters, the approximate solution is D ¼ 0.885 in. It is worth to mention that the approximation should be so as to obtain a slightly smaller number in the left side of the above equation, so the result is fairly conservative.

1.7 Stress concentration According to the definition of stress, a reduction in the cross-section area means an increment of the stress, in an amount proportional to the magnitude of the area reduction. However, the sharp changes of crosssection, as well as discontinuities, such as holes, grooves, gouges, etcetera, increase the stress beyond the magnitude given by the sole reduction of the cross-section area. This phenomenon is called stress concentration and is of

General concepts of mechanical behavior and fracture

25

great technological importance, because it increases the local stress above nominal applied stresses. The magnitude of the stress increment due to the stress concentration is called stress concentration factor, defined by the symbol KT, therefore smax KT ¼ sapp Where smax is the maximum stress value at the root of the stress concentrator, and sapp is the applied stress. One of the first stress concentration factors was obtained by Inglis in 1913, for an infinite thin panel with an elliptic hole in the center. The expression for KT is
a KT ¼ 1 þ 2 b where a is the long diameter length and b is the short diameter length of the ellipse. In a circular hole a ¼ b, therefore KT ¼ 3, so the stress at the edges of a circular hole may easily overcome the yield strength of the material and it will plastically deform around the hole as schematically depicted in Fig. 1.14. Furthermore, if the ellipse becomes too sharp (a [ b) KT increases to infinite, but that actually does not happen because, the stress concentration phenomenon loses physical meaning because the crosssection area at the tip of the approaches to zero, which invalidates the stress calculation. At first a crack may be conceptualized as a very sharp stress

Figure 1.14 Schematic illustration of stress concentration at a circular hole in a uniformly tension loaded plate.

26

A Practical Approach to Fracture Mechanics

Figure 1.15 Graph of Kt for a plate of unitary thickness with two U notches in tension.

concentrator, but in this case the stress values around the crack tip have to be calculated under a different approach, which is precisely what became modern fracture mechanics. In practice, the KT values are found in charts like the one shown in Fig. 1.15, which are published in numerous books, mechanical design manuals, as well as in low-cost applications for mobile devices that provide the stress concentration factors for a wide variety of geometries and load conditions. In general, the factor that most greatly determines the value of KT is the sharpness of the stress concentrator, that is the ratio between the stress concentrator root radius (r) and the depth of the notch (h); for instance, in the graph of Fig. 1.14 at a radius-to-depth ratio of r/h ¼ 1.0, a shallow notch (h/W ¼ 0.1) has a KT ¼ 1.48, and if the notch is deeper h/W ¼ 0.3, KT z 2.42, that is 1.64 fold increment. Now, if the notch is sharper r/ h ¼ 0.1, for h/W ¼ 0.1, KT ¼ 3.29, and for h/W ¼ 0.3, KT ¼ 5.73, a 1.74 fold increment, but KT increases up to 2.37 times, so not only the stress concentration factor is higher, but the effect is more pronounced. Furthermore, the effect of stress concentrators is more severe in brittle materials than in ductile ones. This effect is used in glass cutting, where a sharp and hard blade makes a deep score on the surface of the glass, then the glass plate is slightly bent in the direction transverse to the score, so the stress concentration overcomes the tensile strength and the glass breaks in a brittle way. Nonetheless, this does not mean that stress concentration may not be a problem in ductile materials, it simply means that it is less dangerous.

General concepts of mechanical behavior and fracture

27

1.8 Definitions and basic concepts of fracture Fracture is defined as the separation or fragmentation of a solid under the action of stresses through a process of creation of new surfaces that are the fracture surfaces. Normally, to fracture a material after yielding, it is necessary to increase the stress until a crack nucleation mechanism takes place, followed by crack growth and final separation. Depending on the load conditions, body geometry and the mechanical properties of the material, it may be necessary to increase the load after the crack initiation has started to further cause fracture, whereas in other cases, it will be sufficient to reach the point of crack initiation to have the crack to growing spontaneously. A very important circumstance is that the fracture can initiate from a preexisting crack or a severe stress concentrator, so the nucleation stage is suppressed, and the fracture process consists only of the crack propagation and final separation stages. It is important to point out that in order to break a solid component, it is not necessary that the initiation and crack propagation conditions be present throughout the entire body’s volume, but since the fracture process occurs on a plane, it is enough to meet the fracture conditions in that single plane. This is similar to the chain principle that says, “To break a chain, it is enough to break a single link.” The fact that a fracture may initiate on a localized plane, or more precisely in a narrow region, where there may be stress concentrators or preexisting flaws, suggests that it may take place at stresses lower than the design stresses, all of this provide the fracture characteristics of being sudden, unexpected and, very often, catastrophic. Depending on the amount of plastic strain before failure, two types of fractures are recognized, as shown in Fig. 1.16. Brittle fracture: The strain in most of the body’s volume is elastic and the extent of plastic strain is negligible; so, if after the fracture the fragments are reassembled, the component will have the same initial geometry. Ductile fracture: It is the fracture after an appreciable amount of plastic strain in most of the body’s volume and consequently, there is a permanent change of geometry. In components under tensile stresses, the plastic deformation close to the fracture surface will show a lateral contraction called neck. Quasibrittle fracture: The plastic deformation zone is concentrated close to the fracture surface, so the fracture component displays a brittle appearance, but the fracture mechanism is essentially ductile.

28

A Practical Approach to Fracture Mechanics

Figure 1.16 Classification of fractures according to the amount of plastic deformation before final fracture.

The classification of fractures as brittle and ductile is conventional, because in some fractures with brittle appearance there can be an intense plastic deformation, but located in a small region around the crack, so the majority of the material’s volume remains within elastic deformation. Traditionally, brittle fracture has received most of the attention for being sudden and catastrophic, but ductile fracture is widely studied too, because of the development of high toughness materials. In practice, both types of fracture can appear in the same component and for that reason, it is important to study the mechanical behavior in both types of fracture. To understand the fracture phenomenon, it is necessary to understand the following basic terms, schematically illustrated in Fig. 1.17. Fracture: Separation or fragmentation of a solid resulting of loads or stresses, and with the creation of at least two new surfaces, namely the fracture surfaces. Fracture plane: Macroscopic plane where fracture takes place. It is also the plane where the crack propagates. Fracture propagation direction: Direction of growth of a crack on the main fracture plane. There may be more than one propagation directions on the same fracture plane. Crack: Intersection of a partial fracture with a free surface. Since the separation between the two surfaces of a partial fracture is very narrow, the crack looks like a line to the bare eye. A crack is also a partial fracture that extends less than the entire width of the body that contains it.

General concepts of mechanical behavior and fracture

29

Figure 1.17 Schematic illustration of the basic fracture terms.

Crack tip: It is the extreme point of a crack in the direction of propagation. Physically is the point where the two fracture surfaces of a partial fracture connect. Crack front: It is the line forming the apex of two partial fracture planes and it is the boundary between the fractured and the nonfractured plane. Crack opening displacement: Separation of the fracture surfaces at the initiation edge of the crack, usually measured along the loading line. Load line: Imaginary line parallel to the direction of the applied load. The main feature of the mechanical behavior of cracked bodies is that the stresses and strains will be greater ahead of the crack tip region, thus causing greater displacements (and strains) as compared to the uncracked body. Additionally, the crack provides a favorable path for aggressive substances from the environment to penetrate into the material, contributing to the fracture process. This will bring about the following consequences: • Reduction of the capacity to bear loads, which weakens the structure or mechanical component. • Alteration of response to loads and displacements causing abnormal behavior. • Reduction of the service life. In addition to the aforementioned consequences, which are quite bad themselves, experience has demonstrated that considering the presence of cracks on a structure is very important, because there is a great variety of conditions and mechanisms that lead to the formation of cracks. These cracks may not be detected during routine quality control inspections, so it

30

A Practical Approach to Fracture Mechanics

is preferable to consider their presence and possible effects, rather than assuming that everything is as it should be, and the component is flawless. Fracture mechanics is the part of material mechanics that relates the size of a crack and the geometry of a body with the loads (magnitude and direction) leading to fracture. For this, fracture mechanics relies on the calculation of stresses and strains around a crack and on the energy balance during crack extension. The classification of fracture mechanics also depends on the amount and extension of plastic strain preceding fracture, in the following way: • Linear-elastic: The plastic zone is contained within a small region around the crack, so the strain in the majority of the body’s volume is linear-elastic. • Elastic-plastic: The plastic zone extends across the entire cross-section of the body, but stays within a relatively narrow strip around the crack plane. • Plastic collapse: The fracture occurs after generalized plastic strain. Fig. 1.18 shows schematically these three categories and Fig. 1.19 shows physical examples of these fracture categories. Fracture mechanics is also classified in terms of the time dependence of crack propagation as follows: • Static fracture: It refers to the condition where a body with a static preexisting crack just meets the conditions for rapid and unstable crack propagation under loading. This type of fracture includes brittle, ductile and plastic collapse fractures. • Slow, delayed or stable fracture: The fracture occurs by the propagation of a crack at measurable crack growth rates, and necessarily slower than the unstable crack growth rates of static fracture. The term stable also

Figure 1.18 Fracture mechanics categories according to plastic zone extension.

General concepts of mechanical behavior and fracture

31

Figure 1.19 Physical examples of the fracture mechanics categories. (A) Linear-elastic; (B) elastic-plastic; and (C) plastic collapse.

•

means that the crack can be accelerated, decelerated or arrested, in a controlled way. The typical slow fracture mechanisms are fatigue, creep crack growth, stress corrosion cracking and hydrogen induced cracking. Dynamic fracture: It is the condition where a crack is propagating under unstable conditions, usually at rates being a fraction of the speed of sound of the material.

1.9 Object and field of application of fracture mechanics As mentioned earlier, the immediate consequence of loading a solid body is the appearance of stresses. Stress produces strains, which are first elastic and after yielding, become elastic-plastic, and when the material cannot strain anymore, fracture occurs. Technically speaking, fracture is the absolute end of the service life, while in-service (the most common case) and either during the fabrication, transport, or installation. Since every structural or

32

A Practical Approach to Fracture Mechanics

mechanical component has to support or transmit loads, the fabrication material must resist the imposed stresses and strains, without fracturing. In the traditional design, the usual goal is to prevent plastic strain, because a plastically strained component will not perform as desired. To achieve a plastic-safe design, continuum mechanics seeks to fabricate the component of a material whose mechanical strength is higher than the service stresses, by applying the yield criteria, and whenever this is not possible, the crosssection area is increased, or the applied loads are limited. To compensate for unexpected loads or manufacturing defects, the calculated stress is increased by a safety factor. Since fracture is a postyield phenomenon, it is assumed that a design against plasticity is automatically a fracture proof design, but the experience shows that this does not always happen, especially when the component has cracks or defects that behave as cracks. Since a crack is a discontinuity, the continuum mechanics methods are not applicable for the design, nor assessment, and for that reason, fracture mechanics was developed. Under this perspective, fracture mechanics has three fundamental objectives: (1) To determine the load bearing capacity of a cracked body, termed as remaining strength. (2) To determine the largest tolerable flaw size, termed as critical size. (3) To determine the crack growth rate, which allows the estimation of the remaining life. The first successful engineering approach to determine remaining strength was introduced by Irwin, through the stress intensity factor, which is the parameter that characterizes the stress magnitude in the region surrounding the crack tip, which in its general form is expressed as pffiffiffiffiffi K ¼ Y s pa Where Y is a geometrical parameter, a is the crack size, and s is the applied stress. The basic criterion is: when K reaches a critical value Kc, referred as fracture toughness, the component fails. The value of Kc is a material property determined by laboratory testing. The straightforward application of the stress intensity factor is as follows: being a and Kc known, the K equation can be solved to know the fracture stress, which is the remaining strength of a cracked component. Kc sf ¼ pffiffiffiffiffi Y pa

General concepts of mechanical behavior and fracture

33

Furthermore, if the applied stress and the fracture toughness are known, then the equation of K can be solved for the crack size, obtaining the critical crack size.  2 1 Kc ac ¼ p Ys Thus, the first two fundamental objectives of fracture mechanics are reached. The stress intensity factor can be also used to analyze fatigue crack growth and stress corrosion cracking, among other slow crack growth mechanisms, because usually the components that experience these cracking mechanisms are under elastic strains, so the crack growth rate (da/ dT) is controlled by K, typically by an equation known as Paris law: da ¼ CK m dT Where C and m are empirical constants depending on the materialenvironment system. The integration of the K-based equations of crack growth rate allow determining the time between the detection of a crack by nondestructive inspection and the final fracture, that is when the critical crack size is reached, and in this way the third fundamental objective of fracture mechanics is achieved. In conclusion, fracture mechanics allows determining the loading limits at which a structural component may safely operate while containing a crack. But the power of fracture mechanics does not limits to predicting fracture loads and critical crack sizes, since it introduces additional variables to the analysis of structural components, fracture mechanics increases the number of interrelations that can be considered in the analysis of the mechanical behavior and the design, as shown in Fig. 1.20. The increased number of interrelations that fracture mechanics allows, enables it to establish whether a structural or mechanical component containing cracks of crack-like flaws can continue in service under given specific load operating conditions and for how long. These capabilities led to the introduction of the concept structural integrity, which is the capability of a component to perform the function for which it was designed, in terms of the defect content, mechanical properties and service conditions. The structural integrity assessment is a procedure that begins with a nondestructive inspection to detect and dimension the defects. Followed by

34

A Practical Approach to Fracture Mechanics

Figure 1.20 Design interrelations allowed by fracture mechanics analysis.

gathering and analyzing the design, operating conditions, and maintenance data, to determine the failure load, critical flaw size and/or remaining life, all of this to determine if the component may remain in service without any corrective action, such as repair, replace, reinforce, or reduce the service loads to ensure a safe operation. The great importance of structural integrity today is one of the reasons that make fracture mechanics a very important topic for design, operation and maintenance engineers and mechanical and material science researchers. Furthermore, every year fracture mechanics is included into more and more course programs in undergraduate and graduate schools around the world, emphasizing the increasing reach of this topic.

CHAPTER 2

Linear elastic fracture mechanics

2.1 Cohesive strength The simplest model of fracture is the direct separation of atomic planes, a mechanism known as cleavage. The cleavage fracture stress is the necessary stress to split and break apart the atomic bonds along the fracture plane, which is termed cohesive strength. Cohesive strength varies with the separation distance perpendicular to the fracture plane (x) as shown in Fig. 2.1. When the separation between atoms in the fracture plane equals l/2, the cohesive strength reaches a maximum, so a further separation will reduce the required stress to separate the atoms; when x ¼ l the interatomic attraction force is almost null and the surfaces will be completely separated. Based on this model, the cohesive strength is the maximum of the stress versus atomic separation curve, and it can be estimated as follows. The bond strength as a function of the atomic plane separation in the interval (a0 < x < l/2) can be approximated by the sine function:
 2px s ¼ s* sen a0 where s* is the cohesive strength. If x is very small, the following approximation is valid:
 * 2px s¼s a0 Assuming that the separation of planes produces elastic strain (ε), the applied stress can be calculated. According to Hooke’s law, s ¼ εE, where E is the Young’s modulus and the strain is ε ¼ x/a0, thus s ¼ (x/a0). Equaling this formula with the previous equation:
 x * 2px s ¼E a0 a0 A Practical Approach to Fracture Mechanics ISBN 978-0-12-823020-6 https://doi.org/10.1016/B978-0-12-823020-6.00002-5

© 2021 Elsevier Inc. All rights reserved.

35

36

A Practical Approach to Fracture Mechanics

V  Cohesive strength

(+) x Fracture plane

V

O

Equilibrium atomic separation, a0

x

a0

()

Figure 2.1 Cohesive strength model.

Solving for the cohesive strength: s* ¼

E 2p

Substituting the typical values of E in the previous equation, the resulting cohesive strength is much higher than the experimentally measured fracture stress. For example, the Young’s modulus for steel is E ¼ 30  106 psi, thus the cohesive strength is s* ¼ 4.77  106 psi (4770 ksi). The strongest steel has a maximum strength of about 500 ksi, therefore the logical conclusion is that there must be defects that reduce the fracture strength of the materials. Inglis,1 in 1913, analyzed the effect of the presence of an elliptical hole in the strength of metallic plates according to the following procedure. The maximum stress at the edge of an elliptical hole in a plate of infinite dimensions is given by: rffiffi   c smax ¼ s 1 þ 2 r where a is the major radius length, r is the radius at the tip, and the applied stress is s. If r / 0, the previous equation is reduced to rffiffi c smax ¼ 2s r 1

C. E. Inglis, “Stress in a plate due to the presence of cracks and sharp corners.” Transactions of the Institute of Naval Architects, Vol. 55 (1913), pp. 219e241.

Linear elastic fracture mechanics

37

To verify the implications of the above equation, consider the following example: Assuming a microscopic sharp hole, of length c ¼ 106 m (one micron) and radius at the tip ¼ 3  1010 m (typical value of an inter-atomic space in iron), determine the stress concentration. The stress concentration is smax/s ¼ 2O(c/r) ¼ 2O(106/3  1010 ) ¼ 115.47. In the previous example, if the applied stress is about 50 ksi (typical strength of a mid-steel), the local stress at the hole tip is more than 6900 ksi, enough to surpass the cohesive strength. Such reasoning encouraged the idea that the fracture stress was somehow related to the presence of defects in the material, further becoming the field of fracture mechanics.

2.2 The Griffith criterion The first significant attempt to mathematically analyze the fracture phenomenon was carried out by Alan Arnold Griffith, who was an English engineer, graduated from the University of Liverpool, who worked for the Royal Aircraft Factory and later for Rolls Royce. In addition to his contributions to the study of fracture, Griffith also designed axial flow turbo engines for airplanes and was a pioneer in vertical takeoff and landing technology of fixed wing airplanes. He developed an expression to determine the fracture stress in very brittle materials such as glass, based on the balance between the rate of conversion of stored potential energy and the energy demand for the crack extension. Griffith2 proposed that as the crack grows, the elastic energy is converted into surface energy and since the elastic energy is supplied by the applied stress, he was able to determine the fracture stress, by the following analysis: Consider a plate with a central crack that is elastically strained as it is shown in Fig. 2.2. According to Inglis,3 the elastic energy stored is U¼

ps2 a2 E

A. A. Griffith, “The phenomena of rupture and flow in solids.” Philosophical Transactions Series A, Vol. 221 (1920), pp. 163e168. 3 Inglis, C. E. “Stresses in a plate due to the presence of cracks and sharp corners.” Transactions of the Institute of Naval Architects, Vol. 55 (1913), pp. 219e230. 2

38

A Practical Approach to Fracture Mechanics

V V

U o Js

Stored energy

U

2a

Elongation

a

da

Crack extension creating two new surfaces

Figure 2.2 Griffith model of brittle fracture. Left: thin plate with central crack. Center: stress-strain record. Right: conversion of the stored energy U into surface energy gs.

where s is the applied stress, a is the crack size, and E is the Young’s modulus. In a completely brittle fracture process (without plastic strain), the stored energy is fully converted into the surface energy (gs) of the newly created fracture surfaces, one for each crack face. Therefore, the energy change is DU ¼ U þ 4gs a As the crack grows, the stored energy conversion rate has to be at least equal to the surface energy generation rate. Mathematically, this is expressed as dDU=da ¼ 0 By substituting terms and solving for the stress, the Griffith’s equation for fracture stress is obtained. rffiffiffiffiffiffiffiffiffiffi 2Egs s¼ pa To verify his theory, Griffith measured the surface tension of silica glass fibers, two inches long and 0.01 inches diameter (E ¼ 9010 ksi, v ¼ 0.251, suts ¼ 24.9 ksi). The surface energy was determined from tension tests at temperatures between 745 and 1100 C, obtaining gs ¼ 0.0031 lb-in. Further, he made light bulbs of the same material, making cracks with a cutter and pressurizing them until rupture. His results varied  10% with respect to his predictions.

Linear elastic fracture mechanics

39

The following examples show the fracture stresses calculated by the Griffith criterion: Determine the fracture stress of a glass; surface energy equal to 200 erg/cm2 (0.2 Pa/m) and Young’s modulus E ¼ 80.1 GPa, containing a 2 mm long internal crack. Compare the result with the theoretical cohesive strength. Solution: The Griffith fracture stress is: s ¼ [2(80.1  109 Pa) (0.2 Pa/m)/(p(2  106 m)]1/2 ¼ 71,409,554 Pa ¼ 74.4 MPa The cohesive strength is: s* ¼ E/2p ¼ 80,100 MPa/2p ¼ 25,497 MPa Notice that the cohesive strength is 357 times higher than the Griffith’s fracture, which confirms that indeed glass fracture strength is controlled by the presence of cracklike defects. A cast iron component with a crack of 15 mm long, fractures at 1000 MPa, determine the fracture energy according to Griffith. Assume that E ¼ 106 MPa. Solution: Solving the fracture surface energy from the Griffith’s equations: gs ¼

s2 pa 2E

gs ¼ (1000 MPa)2 p (0.015 m)/2(100,000 MPa) ¼ 0.02356 MPa m The change of energy is: DUtot ¼ 4ags 

pa2 s2 E

DUtot ¼ 4(0.015 m) (235,600 Pa m)  p(0.015 m)2(109 Pa)2/1011 Pa) The fracture energy is: DUtot ¼ 7067.4 J/m. Years after Griffith published his research, Orowan and Irving recognized that in ductile materials there should be a contribution of plastic strain work in order to calculate the fracture stress. They modified the Griffith’s equation and included the term εp, which stands for plastic strain, so the fracture stress becomes: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Eðgs þ εp Þ s¼ pa

40

A Practical Approach to Fracture Mechanics

Despite his success in predicting with fairly good accuracy the fracture stress of brittle materials, the Griffith criterion was not often used in practical applications, mainly because of the great difficulty to experimentally determine the fracture surface energy. Nonetheless, the Griffith criterion provided two fundamental concepts to understanding the fracture phenomenon. They are: 1. The fracture is the result of an energy conversion process that not only depends on the applied stress, but also on the crack size. Such a concept would be taken up 30 years later by Irwin to set down the energy criterion, which is the theoretical foundation of modern fracture mechanics. For this reason, Griffith is considered the “father of fracture mechanics.” 2. The Griffith equation predicts that fracture stress is inversely proportional to the square root of the crack size. This was repeatedly proven in laboratory and failure analysis, thus leading to the introduction of semiempirical equations to predict the fracture stress of engineering materials, just like the Griffith-Tetelman criterion. Later on, the introduction of the stress intensity factor as a parameter to characterize the fracture strength gave a rationale for this correlation, since it frequently shows a a dependency on the square root of the crack length.

2.3 The stress intensity factor (Irwin’s analysis) George Rankine Irwin, born in El Paso, Texas on February 26, 1907, was a notable physicist graduated from the University of Illinois, USA. His career started in 1937 at the US Naval Research Lab working in the ballistics area, where, among several developments, he designed the first bulletproof vest. By 1946, he was appointed to study the fracture problem of the Liberty ships.4 Irwin took over Griffith’s ideas and developed the linear elastic fracture mechanics (honoring Griffith, Irwin used the symbol G to identify the energy release rate of a cracked body, a key parameter in fracture analysis). In 1967, he was appointed Boeing University professor at Lehigh University and in 1972, joined the University of Maryland. Irwin was the object of numerous awards and recognitions, among which stands 4

The Liberty ships were the first ones built by fully welded hull and were key for the triumph of the United States of America in World War II. Of 2700 Liberty ships fabricated, 400 presented cracks, 90 had severe cracks, and over 20 broke apart while in service.

Linear elastic fracture mechanics

41

out the Gold medal from The American Society for Metals. He passed away on October 1998. His fracture analysis is described as follows. Upon loading a cracked body, the fracture surfaces undergo a displacement that can be classified into three basic modes, as shown in Fig. 2.3. Mode I involves the crack opening in a direction perpendicular to the fracture plane, and is termed the tensile opening. Mode II refers to the displacement of fracture surfaces on crack plane, but in opposite directions; it is referred to as sliding shear. Mode III involves the lateral displacement of the fracture planes and is termed tearing shear. The fracture modes can occur in single or combined ways, and even can feature the three modes altogether, which turns out to be too complicated to analyze. Fortunately, most practical cases correspond to Mode I, whereas Modes II and III are typical of torsional and shear loading. The first step of Irwin’s analysis is to determine the stress field around a crack,5 because such stresses are responsible for stretching the atom bonds and creating the new fracture surfaces. This problem was solved in the following way: assume a plate of uniform width, with a crack in Mode I under a uniform tension stress s. At any point located in a position given by the (r, q) coordinates with origin at the crack tip, there is a state of stresses sij, as shown in Fig. 2.4. If the plate is very thin, there will not be enough material in the transverse direction (z) to transmit forces, and thus the stress in this direction is zero, leaving as the only stress components sxx, syy and sxy, setting a plane

MODE I Tension opening

MODE II Sliding shear

MODE III Tearing shear

Figure 2.3 Displacement modes of fracture surfaces in a cracked body under load.

5

Published in: G. R. Irwin, “Analysis of stresses and strain near the end of a crack transversing a plate.” J. of Applied Mechanics, Vol. 24 (1957), pp. 361e364.

42

A Practical Approach to Fracture Mechanics

y

σyy τxy σxx r

Crack length (a)

θ x

Figure 2.4 Stress state around a crack in plane stress condition.

stress condition. On the contrary, if the plate is very thick, the material in the transverse direction will withstand the lateral contraction in the z direction and a plane strain condition will be set. For plane strain condition, the equilibrium equations according to the classic elasticity theory are the following: vsxx vsxy þ ¼0 vx vy vsyy vsxy þ ¼0 vy vx The strains are: εxx ¼

vu vv vv vu ; εyy ¼ ; gxy ¼ þ vx vy dx vy

where u and v are displacements in directions x and y, respectively. The solution of this differential equation system consists of finding a stress function able to satisfy simultaneously the previous equations. This is solved by using a function J(s) so that sxx ¼

v2 J v2 J v2 J ; s ¼ ; s ¼ yy xy vy2 vx2 vyvx

The combination of these equations leads to an equation known as an Airy stress function, which is expressed as V4 J ¼ 0

Linear elastic fracture mechanics

43

The boundary conditions to solve the Airy stress function in a cracked body are when (x, y) / N: sxx ¼ syy ¼ sapplied; sxy ¼ 0. On the crack surface: syy ¼ sxy ¼ 0. The solution to Airy’s function for an infinite plate with a central crack was obtained by Westergard6 in 1939, for a linear-elastic behavior of the material, so the stresses around the crack in plane stress condition are rffiffiffiffi   a q q 3q sxx ¼ s cos 1  sin sin 2r 2 2 2 rffiffiffiffi   a q q 3q syy ¼ s cos 1 þ sin sin 2r 2 2 2 rffiffiffiffi a q q 3q sxy ¼ s sin cos cos 2r 2 2 2 whereas for plane strain, szz ¼ yðsxx þ syy Þ Obviously, the rest of the stress components are zero. The previous equations can be written in general form as rffiffiffiffi a sij ¼ s fij ðqÞ 2r where i and j can take any of the x, y, or z values at the same time. Irwin7 introduced the term KI ¼ sO(pa), thus, for a crack of any length, the stress’s magnitude in a given position (r, q) from the crack tip, only depends on K, so the previous equations can be written as KI sij ¼ pffiffiffiffiffiffiffi fij ðqÞ 2pr pffiffiffiffiffi KI ¼ s pa where KI is the stress intensity factor (SIF), whose units are: K [¼] (stress) (length)1/2 Westergaard, H.M., “Bearing pressures and cracks.” Journal of Applied Mechanics, Vol. 61 (1939), pp. A49eA53. 7 The experimental measurement of the SIF was done by J. A. Kies. Irwin acknowledged him by using the symbol K to identify the stress intensity factor. 6

44

A Practical Approach to Fracture Mechanics

The commonly used units are MPaOm for the International System and ksiOin for the US Customary. The conversion factor is 1 MPaOm ¼ 1.098 ksiOin. Therefore, the equations of the stress components become   KI q q 3q sxx ¼ pffiffiffiffiffiffiffi cos 1  sin sin 2 2 2 2pr   KI q q 3q syy ¼ pffiffiffiffiffiffiffi cos 1 þ sin sin 2 2 2 2pr KI q q 3q sxy ¼ pffiffiffiffiffiffiffi sin cos cos 2 2 2 2pr The SIF is the fundamental parameter of linear-elastic fracture mechanics (LEFM), since it defines the stress magnitude around a crack in an elastically loaded body. In other words, once the SIF is known, the stress state around a crack is completely defined. The fact that the SIF is the only parameter determining the stress magnitude in a cracked body establishes a similitude principle: If two different bodies, with different loads and different cracks have the same value of the stress intensity factor, their behavior is identical. This is so because they both have the same stress field. The only restrictions to the previous principle are that the crack displacement mode stays the same and the stressestrain behavior must be predominantly linearelastic. The next examples show the application of the SIF principle of similitude: Calculate the stress intensity factor in an infinite plate with a central crack 10 cm long if the plate is under a uniform tension stress of 100 MPa. Solution: Since it is a central crack 2a ¼ 10 cm, thus: a ¼ 5 cm ¼ 0.05 m K ¼ sO(pa) ¼ 100 MPaO(p(0.05 m) ¼ 39.6 MPaOm If the crack size in the previous problem grows twice as much, what should be the applied stress so that the plate has the same SIF? Solution: s ¼ K/[O(pa)] ¼ 39.6 MPaOm/[O(p(0.1 m)] ¼ 70.7 MPa

Linear elastic fracture mechanics

45

Two infinite plates A and B are under an applied stress of 200 and 300 MPa, respectively. Calculate the crack size so that both plates have the same stress intensity factor, and therefore the same behavior. Solution: KA ¼ KB, so substituting: 200 MPaO(p aA) ¼ 300 MPaO(paB) O(paA)/O(paB) ¼ 300 MPa/200 MPa ¼ 1.5 aA/aB ¼ 1.52 ¼ 2.25. The crack is A should be 2.25 times larger than crack in B. Frequently it is important to know the principal stresses (normal stresses at a direction where the shear stresses are zero); for example, for the solution of plasticity problems and application of the yield criteria. The principal stresses ahead of the crack tip are
 KI q q s1 ¼ pffiffiffiffiffiffiffi cos 1 þ sin 2 2 2pr
 KI q q 1  sin s2 ¼ pffiffiffiffiffiffiffi cos 2 2 2pr For plane strain: 2vKI q s3 ¼ vðs1 þ s2 Þ ¼ pffiffiffiffiffiffiffi cos 2 2pr For plane stress: s3 ¼ 0 The two-dimension displacements of a volume element in a point of (r, q) coordinates are: u in the x direction and v in the y direction, as shown in Fig. 2.5. y

K>0 v

K=0 K>0 K=0

Crack

r

θ

u x

Figure 2.5 Displacements of a unit volume (point) in a cracked body under load.

46

A Practical Approach to Fracture Mechanics

The values of the displacements for plane strain are calculated by rffiffiffiffiffiffi   KI r q 2q u ¼ 2ð1 þ vÞ cos 1  2v þ sin 2 E 2p 2 rffiffiffiffiffiffi   KI r q 2q v ¼ 2ð1 þ vÞ sin 2  2v þ cos 2 E 2p 2 And for plane stress:

rffiffiffiffiffiffi   r q 2q 2q cos 1 þ sin  y cos 2p 2 2 2 rffiffiffiffiffiffi   2KI r q 2q 2q v¼ sin 1 þ sin  y cos 2p 2 2 2 E

2KI u¼ E

The formulas to calculate the stress components for Modes II and III are   KII q q 3q sxx ¼ pffiffiffiffiffiffiffi sin 2 þ cos cos 2 2 2pr 2 KII q q 3q syy ¼ pffiffiffiffiffiffiffi sin cos cos 2 2 2pr 2   KII q q 3q sxy ¼ pffiffiffiffiffiffiffi cos 1  sin sin 2 2 2 2pr KIII q sxz ¼ pffiffiffiffiffiffiffi sin 2pr 2 KIII q syz ¼ pffiffiffiffiffiffiffi cos 2 2pr For torsional loading:

pffiffiffiffiffi KII ¼ s pa pffiffiffiffiffi KIII ¼ s pa

Now, it is interesting to analyze the variation of the normal stress in the horizontal crack plane ahead of the crack tip (q ¼ 0). For an infinite plate with a central crack in Mode I, the stress variation in the crack plane is as illustrated in Fig. 2.6. Notice that very close to the crack tip (r / 0) the normal stresses go to infinite. Likewise, for high values of r, s / 0, but that is not possible

Linear elastic fracture mechanics

47

500

σ (MPa)

400 σyy = σxx = KI / (2π r)1/2

300 200

σzz = ν (σxx + σyy) plane strain σzz = 0 plane stress

100 0 0.0

2.0

4.0

6.0

r (mm)

8.0

10.0

Figure 2.6 Variation of stresses along the fracture plane (q ¼ 0) ahead of the crack tip in Mode I, KI ¼ 10 MPaOm, v ¼ 0.3.

because far from the crack tip the stress shall be equal to the applied stress. Furthermore, it is observed that in the direction perpendicular to the crack plane, syy has the greatest magnitude, whereas in the other directions, sxx and szz are lower in magnitude. The immediate consequence the high stress levels close to the crack tip is the formation of a plastic zone at the crack tip, because at some point close to the crack tip, the yield strength is surpassed, and the material will undergo to plastic deformation. The next example shows the magnitudes of the maximum principal stress and the maximum shear stress, to discuss the effect of the introduction of a crack in a uniformly loaded component. Calculate the maximum principal stress and the maximum shear stress at r ¼ 1, 10, 20, and 100 mm and q ¼ 0 , 45 y, 90 from the crack tip of an infinite plate in plane stress if KI ¼ 20 MPaOm. Solution: The corresponding equations to calculate s1 and smax are: s1 ¼ [KI/O(2pr)][1 þ sin(q/2)] smax ¼ [KI/O(2pr)] sin(q/2)cos(q/2)

48

A Practical Approach to Fracture Mechanics

The results are tabulated in the next table: r (mm) q

Stress MPa

1

20

100

0

s1

252

80

56

25

0

0

0

0

322

102

72

32

89

28

20

9

s1

305

96

68

31

smax

126

40

28

13

smax 45

s1 smax

90



10

It is observed that along the crack plane (q ¼ 0 ), smax ¼ 0, and s1 falls down rapidly (more than three times) at r ¼ 10 mm, but from r ¼ 20 mm to r ¼ 100 mm, it only drops about one half. At q ¼ 45 , s1 shows its maximum values, but decreases at ratio similar to that observed for q ¼ 0 . As r increases from 10 mm to ¼ 100 mm, s1 drops about one half. On the other hand, smax is about 3.6 times smaller than s1 and follows a tendency similar to s1. At q ¼ 90 , which is the vertical direction with respect to the crack plane, s1 is about 5% smaller that at q ¼ 45 , but follows the same tendency and r increases. Regarding smax it shows the maximum values, being about 40% of s1, therefore this direction will have the highest effective stresses. This result is very interesting because it demonstrates that the strongest effect of the introduction of a crack in a loaded body occurs in the vertical direction with respect to crack plane.

2.4 Solutions of the stress intensity factor There are several methods to determine the SIF, which can be classified as: 1. Known solutions 2. Approximated solutions 3. Experimental methods 4. Numerical methods The selection of the method for determining the SIF depends on time availability, technological resources, and the required precision level. Fig. 2.7 presents a schematic of the elapsed time to determine the SIF by the most common methods. Notice that the time spent may last from a few minutes (search in handbooks) up to hundreds of hours of hard work. The analytical methods for the determination of the SIF use advanced mathematics to solve the Airy stress function and the resulting differential

Linear elastic fracture mechanics

STRESS MEASUREMENT

HANDBOOKS

MOBILE DEVICE APP.

NUMERICAL METHODS ANALYTICAL

SUPERPOSITION

INDIRECT FATIGUE METHOD

APPROXIMATION METHODS

10 min

49

1 hr

10 hr

+100 hr

TIME

Figure 2.7 Time scale of common methods for determining the SIF.

equations, to derive formulas that include variables such as load, crack size, and the geometry; however, nowadays these are seldom used because of their great difficulty and the broad availability of known solutions and numerical-based methods. The mathematical formulas to calculate the SIF are commonly expressed in the form of polynomial expressions of the type: K ¼ P g(B, W ) f(a/W ) where g(B, W ) is a mathematical function of the thickness (B) and width (W ), and f(a/W ) is a function of the relative crack size, usually obtained by the weight function method. Table 2.1 shows some commonly known solutions of the SIF. Many formulas to calculate the SIF are available in handbooks, annexes of defect assessment standards, and, more recently, in mobile device applications, where the SIF formula or its value for a specific load condition, body geometry, crack configuration, and validity limits are provided. Some of these sources are: 1. Tada, Paris and Irwin: The Stress Analysis of Cracks Handbook. Univ. of St. Louis, USA (1973). 2. Sih: Handbook of Stress Intensity Factors. Institute of Fracture and Solid Mechanics. Lehigh Univ. Bethelehem, Pa, USA (1973).

50

A Practical Approach to Fracture Mechanics

Table 2.1 Stress intensity factors for common geometries. Geometry

K

Finite plate with central crack pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi pa KI ¼ s pa sec W pffiffiffiffiffi KII ¼ s pa Valid for: 0 < a/W  0.35

2a

τ

W

2a

Finite plate with edge crack KI ¼ 1.12 s Op pffiffi a, for 0 < a/W < 0.6 KI ¼ Y s a Y ¼ 1.122e0.231(a/W) þ 10.55(a/W)2 e21.71(a/W)3 þ 30.382(a/W)4 Valid for a/W  0.6

a W

Crack with pffiffiffiffiffiinternal pressure KI ¼ P pa P ¼ pressure inside the crack

P

2a

Linear elastic fracture mechanics

51

Table 2.1 Stress intensity factors for common geometries.dcont’d Geometry

K

Compact tension 2 3 4 KI ¼ pPffiffiffiffi ½0:886þ4:64a013:32a0 3=2þ14:72a0 5:6a0  ð1a0Þ

B W

a0 ¼ (a/W), B ¼ Thickness

a W P 2h

a

Single edge notch

P

P/2

barKI ¼ BWPS3=2 W

a S

Double pcantilever beam (DCB) ffiffi 2 3 Pa ffi 3=2 Plane strain KI ¼ pffiffiffiffiffiffiffi 1n p ffiffiffi2 BhPa KI ¼ 2 3 Bh3=2 Plane stress

3ða0Þ1=2 ½1:99a0ð1a0Þð2:153:93a0þ2:7a02 Þ 2ð1þ2a0Þð1a0Þ3=2

a0 ¼ (a/W), B ¼ thickness

P/2

2c a

σ

Semieliptical surface crack in an infinite plate with stress perpendicular to the crack Minor axis pffiffiffiffiffi KIðmaxÞ ¼ 1:12 Fs pa Major axis sffiffiffiffiffiffiffiffiffiffiffiffiffi
 2 s KIðminÞ ¼ 1:12 F p ac F ¼

3p 8

þ pa 8c 2

2

2c

Continued

52

A Practical Approach to Fracture Mechanics

Table 2.1 Stress intensity factors for common geometries.dcont’d Geometry

K

2c

a

Axial crack in a hollow thin wall cylinder under internal pressure sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
c2 KI ¼ PR 1 þ 1:61 t Rt pa

P

R

t

P

Round bar with circular V notch pffiffiffi 0:526 P D KI ¼ d2 Valid for: 1.2  D/d  2.1

D d

D

a

σ

Round bar with a radial crack under tension (t) or bending (b) pffiffiffiffiffi KI ¼ Ft;b st;b pa Ft ¼ G½0:752 þ1:286b þ0:37Y 3  Fb ¼ G½0:923 þ0:199Y 4 
  1=2 G ¼ 0:92 p2 sec b tanb b Y ¼1  sin  b p a b ¼ 2 D

3. Roche and Cartwright: Compendium of Stress Intensity Factors. Her Majesty’s Stationary Office, London (1976). 4. Marahams: Stress Intensity Handbook. Persuman Press, New York (1988) 5. Leham and Ainsworth: Stress Intensity Factor and Limit Load Handbook, Epd/Gen/Rep/0316/98 Issue 2, British Energy Generation Ltd., United Kingdom (1999).

Linear elastic fracture mechanics

53

6. Annex C: Compendium of Stress Intensity Factor Solutions, API579-1/ ASME. FFS-1 2007 Fitness-For-Service, American Petroleum Institute and The American Society of Mechanical Engineers, Washington, D.C. (2005). 7. Annex M (normative): Stress Intensity Factor Solutions. BS 9710:2005. BSI, ISBN 0580 45965 8. Stress Intensity Factors Pro, App. for iOS 11 by J.R. Hobbs In practical cases, the SIF can be obtained by approximation methods based on the general expression: KI ¼ bs O(pa) For short crack sizes (a / 0), the value of b can be easily determined if the known SIF function converges to a constant value. For example, KI for a radial external crack in a thick wall pipe under internal pressure is given by the curves of Fig. 2.8. For short cracks, a/(R2eR1) < 0.1, the curves converge at KI/K0 ¼ 1.12. Thus, if K0 ¼ 2 P[R12 /(R22 e R12 )]O(pa), an approximate function is: 2PR2 pffiffiffiffiffi KI ¼ 1:12 2 1 2 pa R2  R1 2.40 2.20

R1/R2 = 0.9

R2

KI / K0

2.00 a

1.80

R1/R2 = 0.8

R1

1.60

R1/R2 = 0.35

1.40 1.20 1.00

0

0.1

0.2

0.3

0.4

0.5

0.6

a / (R2 − R1) Figure 2.8 Approximation of the constant b of KI ¼ bsO(pa) for a radial crack in a thick wall pipe under internal pressure. K0 ¼ 2 P[R21 /(R22 e R21 )]O(pa).

54

A Practical Approach to Fracture Mechanics

The SIF can be estimated from the stress concentration factor. Irwin8 proposed that a stress concentrator of notch radius r has a stress intensity factor that can be approximated by pffiffiffi  p pffiffiffi lim Kt snom r KI ¼ 2 r/0 where Kt is the elastic stress concentration factor and snom is the nominal stress. Hasabe and Kutanda9 adapted this method to determine the value of b in the general formula KI ¼ bsO(pa) by the following procedure: The Irwin’s equation is rearranged as

 KI 1 pffiffiffiffiffi ¼ lim Aðr=hÞ0:5 snom pc 2 r/0 The values of A are obtained by the formula: pffiffiffiffiffiffiffiffiffiffiffi A ¼ Kt ðr=hÞ where h is the notch depth. The values of Kt are computed for small values of r/h, typically 0

a

a

P

a + 'a

P

v

P

v

v

Figure 3.3 Schematic representation of the loading process to attain constant displacement condition.

80

A Practical Approach to Fracture Mechanics

related the G value with K for a linear-elastic solid based on the following procedure: In the fracture plane (q ¼ 0), the normal stress is K syy ¼ pffiffiffiffiffiffiffi 2pr The crack opening displacement is K pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pðda
rÞ v ¼ 2a E where r is the distance from the crack tip, a ¼ (1
n2), and n is the Poisson ratio. The work by unit thickness, under a crack extension, is Z Z K dW ¼ svdr ¼ pffiffiffiffiffiffi vdr pr Substituting v, 2aK 2 dW ¼ pE

Z ððda
rÞ=rÞ1=2 dr

To solve this integral, the following boundary conditions are taken: r ¼ da sen2q, at r ¼ 0; q ¼ 0, when r ¼ da, q ¼ p/2, thus dr ¼ 2 da sen q cos q dq. Therefore, 1=2 Z
da
r p dr ¼ da r 2 Substituting into dW results in dW ¼ a

KI2 da E

Taking the limit when da / 0, dW/da becomes dW/da, which is the cracking resistance R; thus it is equivalent to the energy release rate, so dW/da ¼ G, which demonstrates that For plane stress: G¼

K2 E

The energy criterion and fracture toughness

81

And for plane strain: G¼

K2 ð1
y2 Þ E

These equations indicate that if the energy criterion is satisfied (G ¼ R), K will reach a critical value, which is stated as the fracture toughness (KC); it represents the fracture strength of a material under linear-elastic strain conditions. Based on this reasoning, the energy criterion can be expressed as If K > KC the crack will propagate. The fracture toughness KC can be determined experimentally by measuring the fracture load and critical crack size on a test specimen whose stress intensity factor solution is known. The value of KC is universally accepted as the governing fracture parameter and its applications extend beyond measuring the fracture resistance. The characteristics, testing, and applications of the fracture toughness will be further described in this chapter.

3.2 The R-curve The graphical representation of the energy criterion is known as R-curve, whose construction procedure is as follows. The G and R values are plotted in the vertical axis, and the crack length is plotted in the horizontal axis, where the initial crack size (a1) is plotted to the left and the crack extension (Da) to the right. The G, R axis intersects the crack size axis at Da ¼ 0. As mentioned in the previous section, in plane stress condition, G¼

K2 E

For an infinite plate with a central crack under plane stress, the stress intensity factor is KI ¼ sO(pa); therefore, G becomes G¼

ps2 a E

Plotting the above equation on the R-curve shows that the variation of G is a straight line with a slope equal to ps2/E. If R is constant and independent from the crack size, its plot is a horizontal straight line. The resulting R-curve is schematically shown in Fig. 3.4. The R-curve simplifies the use of the energy criterion to determine the behavior of a cracked component. For an initial crack size a1 and an applied

82

A Practical Approach to Fracture Mechanics

G, R

G = π(σ22/E) a

G = π(σ12/E) a R = Constant

C B

Excess energy for unstable crack growth

A

a

Δa

a1

Δa

ac = a1 + Δa Figure 3.4 R-curve of an infinite plate with a central crack under uniform stress and R independent of crack size (constant).

stress s1, G < R (point A) and the crack will not propagate; this condition is called stability. To fulfill the energy criterion the crack size has to extend a length Da until reaching the condition G ¼ R (point B). On further crack extension G will be increasingly greater than R, so the crack propagation will be self-accelerated since there is increasingly more available energy; this condition is called instability. The crack length a1 þ Da is the critical crack size (ac). The instability of the crack of length a1 can be also attained by increasing the stress to s2, so again, reaching the condition G ¼ R; in this case s2 is the fracture stress for crack size a1. In materials that fracture in a ductile mode, usually R is not constant, but it is a function of the crack size, so the R-curve has the form shown in Fig. 3.5. The dependence of R on the crack size has an important consequence on the fracture behavior as seen next. For the crack size a1, and stress s1, G < R and the crack is stable (point A). When the applied stress increases to s2, G > R, so the crack will extend, but after an extension Da, R < G, so the energy criterion is not satisfied, and the crack will become stable again (point B). Further increasing the stress to s3 increases the slope of the G curve so it becomes tangent to R (point C). At this point G ¼ R, so upon further extension the crack will be unstable. The crack size at this point will be the critical size. The stable crack extension is called pop-in and plays an important role in elastic-plastic fracture. Fig. 3.6 shows a schematic illustration of the physical appearance of the pop-in extension in a ductile material.

The energy criterion and fracture toughness

G, R

R = R(a)

σ =σ3

C Instability B

83

σ = σ2 G = π(σ12/E) a

A

a

a1

Δa

Stable extension “Pop in”

Δa

ac Figure 3.5 R-curve, R dependent of the crack size.

The following examples illustrate the practical use of the R-curve. A large thin panel containing a central crack breaks under a stress sf ¼ 300 MPa, the final crack length after fracturing was 2a ¼ 40 mm. Assuming that the crack resistance R is constant, construct the R curve and determine: (1) the critical crack length, if the initial crack size is 2a ¼ 20, 60, and 80 mm, and the applied stress is 200 MPa; and (2) determine the fracture stress if the initial crack size is 2a ¼ 40 mm. E ¼ 80 GPa.

Load

Crack tip blunting

Initial crack length

Pop-in

Figure 3.6 Schematic illustration of the pop-in crack extension in a ductile material plate under tension load.

84

A Practical Approach to Fracture Mechanics

Solution: The crack resistance is 2 R ¼ ps2f a=E ¼ pð300MPaÞ ð0:02mÞ=80; 000MPa ¼ 0:0707MPa-m At this point is more convenient to switch the units to MPa-mm, so R ¼ 70.7 MPa-mm. Theformula to  calculate  G is  2 G ¼ psf = E a ¼ pð200MPaÞ2 = 80; 000MPa a ¼ 1570ðaÞ½MPa-mm

G, R [MPa-mm]

The R-curve constructed with the above formula is shown below, with the answers indicated for each initial crack size. Of course, that it is easier to compute the values asked for, but the interest on performing the graphic procedure is that the effects of the initial crack size and the applied stress become evident. Additionally, it has to be remembered that in the 1950s the availability of calculators and personal computers was low and costly, so the graphic methods were fast and easy, a characteristic that remains even in present time. 140 120

σ = 200 MPa

σ = 424 MPa

σ = 200 MPa

100

σ = 200 MPa

80 60 40 20

40 0

10 30

20 20

← a0

0 10 10 30 0 50 6020 Half crack size (a) [mm]

30 70

8040

50 90

Δa →

ac = 45 mm

Determine the fracture stress and the pop-in extension of an aluminum panel under uniform stress containing a central crack of initial length 2a ¼ 20 mm. Assume that R ¼ 3.8Da0.23, [kg f/mm, mm], and E ¼ 7040 kg f/mm2. Solution: Since there is no analytical solution for Da, the problem has to be graphically solved. An approximate value of sf can be calculated by assuming a pop-in of 1 mm for the initial crack length of 2a ¼ 20 mm. At the onset of instability G ¼ R, thus G ¼ ps2 ða þ DaÞ=E ¼ 3:8Da0:23

The energy criterion and fracture toughness

85

Substituting values: G ¼ ðp = 7040kg f = mm2 Þð10 þ 1Þmms2 ¼ 3:8ð1Þ0:23 G ¼ 0:004909s2 ¼ 3:8kg f =mm The fracture stress for 2a ¼ 20 mm and Da ¼ 1 mm is s ¼ ð3:8=0:004909Þ1=2 ¼ 27:82kg f =mm2 At this stress G ¼ ðp = 7040Þð11Þð27:82Þ2 ¼ 3:80kg f =mm To plot the G curve a second point is necessary since it is a straight line, so taking s ¼ 27.82 kg f/mm and a ¼ 10 mm, G ¼ (p/7040) (10) (27.82)2 ¼ 3.45 kg f/mm The R versus Da graph is shown below, so plotting the points (10 mm, 3.45 kg f/mm) and (11 mm, 3.80 kg f/mm) gives the G1 curve for s ¼ 27.82 kg f/mm. 6 G = 4.6 kgf/ mm

5

Tangent of G, R curves

G, R, kgf/mm

G1

4 3 2 1 Δa = 2.1 mm

0

0

0.5

1

1.5

2

2.5

3

3.5

Δa, mm

From the above graph it is clear that at an initial crack size a ¼ 10 mm and applied stress 27.82 kg f/mm, the G1 curve is below R for Da > 1 mm, so the pop-in extension must be longer. Now knowing that the instability starts at the tangent point of the G and R curves, a good guess will be the dotted line in the above R-curve, so the tangent point is at the coordinates G ¼ 4.6 kg f/mm and Da ¼ 2.1 mm. Therefore, the critical crack size is approximately a þ Da ¼ 12.1 mm. Substituting this value on the G equation and solving for s gives the approximated fracture stress. sf ¼ (GE/pa)1/2 ¼ [(4.6 kg f/mm *7040 kg f/mm2)/p12.1 mm]1/2 ¼ 29.19 kg f/mm2

86

A Practical Approach to Fracture Mechanics

3.3 Plane strain fracture toughness It has been experimentally observed that fracture toughness (KC) varies with the thickness, as schematically shown in Fig. 3.7. The most important feature of this behavior is that KC is constant for thick specimens in plain strain condition, thus KC becomes material property identified as plane strain fracture toughness, represented by the symbol KIC, since it is determined in crack opening Mode I. It is relevant to remark that the plane strain KIC is also the minimum value of KC with respect to the thickness, while the maximum KC occurs under plane stress and plane strain combinations. Although the exact cause for such behavior is not clear yet, it is likely to be due to the plasticity effects combined with the plastic zone size and shape, since these characteristics induce more energy consumption at the onset of instability and crack propagation. Irwin introduced an expression to calculate the plane stress fracture toughness based on the plane strain fracture toughness that has the following form: "
4 #0:5 1:4 KIC Kc ¼ KIC 1 þ 2 B s0

KC B

B

2.5 KIC2 / σo2 B

Plane strain fracture toughness

Plane stress

Plane strain

KIC

Specimen thickness, B Figure 3.7 Schematic variation of KIC and fracture surface plane orientation with respect to the thickness.

The energy criterion and fracture toughness

87

Furthermore, it has been experimentally found that the plane strain condition occurs at thicknesses (B) that meet the condition: 2 B > 2:5KIC =s0

Another effect of the thickness on the fracture behavior is the fracture plane orientation and the fraction of the fracture surface occupied by shear lips. In Chapter 2 it was demonstrated that the orientation of the neck with respect to the main fracture plane depends on the plane stress or plane strain condition. It was also demonstrated that the plastic zone size in plane stress is larger than in plane strain. By applying these characteristics to the fracture of ductile materials, it can be foreseen that plane stress fractures will have slanted fracture planes, while plane strain fractures will be horizontal with respect to the maximum applied normal stress with two small shear lips on the edges, as schematically depicted in Fig. 3.7.

3.4 Plane strain fracture toughness testing (KIC) The test to determine the plane strain fracture toughness KIC was standardized in 1972 by the American Society for Testing Materials (ASTM) to provide the guidelines to obtain reproducible and accurate results. The recommended procedure appeared under the designation ASTM E399-72.2 The test consists of applying load until fracture on a specimen containing fatigue cracks emanating from a machined notch. The KIC is determined from the load-displacement record, where the maximum load at the onset of instability and the critical crack size, measured from the fractured specimen, are substituted into the stress intensity factor equation of the test specimen. Since the KIC is a quasi-static fracture parameter, the loading rates shall be controlled so the rate of increase of KI is between 0.55 and 2.75 MPaOm per second (30 and 150 ksiOin/min). The load and displacement gauge outputs shall be recorded for the full range of forcecrack opening displacement to have enough data to perform the calculations. The two main recommended specimens for KIC testing are the compact tension (CT) and the single notch bending specimen (SE(B)). The shape and dimensions of these are depicted in Figs. 3.8 and 3.9. The suggested dimensions of standard specimens for determining KIC are 2

ASTM E399-72. Standard test method for linear-elastic plane strain fracture toughness KIC of metallic materials. American Society for Testing and Materials (1972).

88

A Practical Approach to Fracture Mechanics

1.2 W ± 0.01W

Diameter = 0.25 W ±0.005W

0.275W ±0.005W an

0.6W ±0.005W

ao

W ± 0.005W 1.25 W ± 0.01W

B = 0.5W

Figure 3.8 Standard CT specimen for KIC testing per the ASTM E 1820 standard.

Wr 0.005W

a 0.2W S = 2.25W min

S = 2.25W min

B = 0.5 W

Figure 3.9 Standard SE(B) specimen for KIC testing per the ASTM E 1820 standard.

W/20  B  W/4, where W is the width and B is the thickness. The crack notch starter configurations are chevron, straight through, and slot ending in drilled hole, which size shall be an ¼ 0.2 W, while the total size of notch crack starter plus fatigue crack shall be between 0.45 W and 0.55 W. To assure the linear elasticity and plane strain conditions, W and B have to be large enough to have the plastic zone fully contained within the specimen, so the effects of plastic strain and thickness on the fracture behavior are negligible, to achieve this the specimen size should be sized to meet the following criteria.
2 KIC B  2:5 s0 Additionally, W has to be greater than 2B and B > 6.35 (KIC/s0)2. Since KIC is not yet known, the specimen dimensions are determined by an estimated KIC value. To validate the test, first a provisional fracture

The energy criterion and fracture toughness

89

toughness KQ shall be calculated using the stress intensity factor solution of the test specimen:

KQ ¼ PQ *gðB; W Þ*f ac=W where PQ is obtained from the load-displacement curve and ac is the critical crack size determined by fractographic examination of the tested specimen. Fig. 3.10 illustrates the main types of load-displacement records and the corresponding value of PQ to calculate KQ. The ideal case is TYPE III, which corresponds to brittle fracture, where PQ is the maximum recorded load Pmax. The nonlinear behavior corresponds to TYPE I, where PQ is obtained from the intersection of the load curve and a straight line with a 5% reduced slope with respect to the linear portion of the load-displacement record, referred as P5%. If a pop-in crack extension occurs, there is a local maximum in the load-displacement curve that corresponds to TYPE II; PQ will be maximum load recorded before P5% as long as P5% remains lower than PQ. To validate the results, the following criteria shall be satisfied: Pmax/PQ  1.10 B, (W
a)  2.5(KQ/s0)2 If these conditions are fulfilled, then KQ ¼ KIC and the test is valid. If the test is rejected, a thicker specimen will be required; otherwise such specimen size will be good to assess the fracture toughness of more brittle materials.

Pmax PQ = P5% Load, P

Pmax

PQ

Pmax = PQ

P5%

5% reduced slope line TYPE I

TYPE II

TYPE III

Displacement, v

Figure 3.10 Typical load-displacement records in KIC testing.

90

A Practical Approach to Fracture Mechanics

Examples of the typical calculations required to perform KIC testing are presented next. A metallic alloy, s0 ¼ 1000 MPa and estimated KIC ¼ 100 MPaOm is subject to KIC testing. Determine the CT specimen dimensions to perform a valid test. Solution: To meet the plane strain requirement: B > 2.5 (KIC/s0)2 ¼ 2.5 (100 MPaOm/1000 MPa)2 ¼ 0.025 m ¼ 2.5 mm For a CT specimen: W > 4B ¼ 4(0.025 m) ¼ 0.1 m ¼ 100 mm. Therefore, the CT specimen dimensions for a valid test are B ¼ 2.5 mm and W ¼ 100 mm. The load-displacement curve shown below was obtained from a CT specimen having B ¼ 50 mm and W ¼ 100 mm, and the material is low alloy steel in quench and tempered condition: s0 ¼ 1000 MPa. The measured critical crack size was 50 mm. Determine KIC and validate the test. Solution: Verification of the linearity condition: Pmax/PQ ¼ 0.125 MPa/0.119 MPa ¼ 1.05 < 1.1, the test is valid as per linearity. Calculation of KQ by the CT specimen K1 equation: a

P KI ¼ pffiffiffiffiffiffi f B W W For (a/W) ¼ 0.5, f (a/W) ¼ 9.66 KQ ¼ [0.119 MPa/(0.05O0.1 m)] * 9.66 ¼ 72.7 MPaOm

PQ

P5% = 0.119 MN

Load

Pmax =0.125MN

ac=50 mm B=50 mm W=100 mm

Displacement

The energy criterion and fracture toughness

91

Verifying the thickness criterion: Breq ¼ 2.5(KQ/s0)2 ¼ 2.5(72.7 MPaOm/1000 MPa)2 ¼ 0.013 m ¼ 13 mm Since Bactual > Breq the test is valid and plane strain conditions are fulfilled, therefore KIC ¼ 72.7 MPaOm. Determine the fracture toughness of the material of the above problem, but of reduced thickness, so the test plate is in a plane straineplane stress combination. Solution: Using the Irwin equation: KC ¼ KIC[1þ(1.4/B2)(KIC/s0)4] 0.5

Fracture toughness KC, MPa—m

KC ¼ 72.7[1 þ (0.00004/B2), MPaOm 180 160 140 120 100 80 60 40 20 0

0

10

20

30

40

50

Specimen thickness B, mm

Notice that Irwin estimation gives reasonable values of KC for thickness as low as 6 mm (1/400 ), but for lower thicknesses, it gives unrealistic high values. The strict thickness and linearity requirements of ASTM E399 standard limit its application to brittle materials. For example, a forged steel with s0 ¼ 500 MPa and KIC ¼ 180 MPaOm would require a test specimen of B ¼ 34.2 cm. If the specimen is SE(B), W ¼ 2B ¼ 68.4 cm, S ¼ 4.5 W ¼ 307.8 cm, so the test specimen would weigh about 2.8 metric tons. Besides, the capacity of the test machine should be around 3500 kN (360 tons), which is among the highest commercially available. Table 3.1 shows the s0 and KIC values of common metal alloys along with the required thickness for valid plane strain fracture toughness testing. In summary, the plane strain fracture toughness testing by the ASTM E399 standard is only applicable to brittle or minimum plasticity conditions, which usually occur under any or several of the following conditions: 1. Brittle materials (high s0 and low ductility). 2. Thick test specimens that behave in plane strain condition.

92

A Practical Approach to Fracture Mechanics

Table 3.1 Typical values of yield strength and fracture toughness of common engineering materials and minimum required thickness for valid KIC testing. KIC B Minimum (MPaOm) (mm) Material s0 (MPa)

High strength steel Quenched and tempered steel Stainless steel Low carbon steel Titanium alloys Aluminum heat treated Aluminum structural Cooper alloys Nickel alloys Carbides Polymers Stone and bricks Wood

680e1965 1000e1830 150e1000 240e400 500e1100 300e550 100e200 50e300 100e1000 400e700 10e100 5e20 30e70

50e150 47e70 80e200 140e220 38e80 30e50 15e30 40e80 70e100 2e5 0.3e8 0.6e2 5e10

2.1 1.65 816 750 3.0 7.5 25 44 25 0.1 1600 2.25 12

3. Low temperatures in materials with ductile-brittle transition. 4. High loading rates that restrain plasticity (impact). To make fracture toughness testing accessible for high toughness materials, the ASTM issued the standard E 1820e05a Standard Test Method for Measurement of Fracture Toughness, which describes methods to determine the plane strain fracture toughness from parameters not limited by plasticity, such as the J-Integral and CTODc, including the KIC. Another important aspect of fracture toughness testing is that KIC is an anisotropic property for most engineering materials, so the correct record of the crack orientation in the specimen is necessary; hence, the ASTM E399 standard provides a nomenclature to define the crack orientation, shown in Fig. 3.11. Another important aspect of fracture toughness testing is precracking. In theory it is necessary to have a crack tip radius very close to zero to meet the singularity condition; that is, the condition to have the stress and strain distributions as predicted by linear elastic fracture mechanics. This condition shall be independent of the specimen dimensions and the material properties. Most machined notches do not meet this criterion, since even the finest cutting tools produce a tip radius around 0.5 mm (0.02 in.). This led to the requirement of generating the initial crack by fatigue loading, since

The energy criterion and fracture toughness

T-

L L TH INA ION ON NG D CT TI LE ITU IRE REC G N G D I N D GI LO LING ON OR L SI F F ROTRU S O I EX AX

S L-

S

S-

L T-

L-

C-

R R

93

-C

T

R-L T

S

LO

-L

NG WI T TR DTH AN SV E

S SE S ES ER KN V IC ANS H T TR RT HO

RS

E

C-L L -R L -C

S

Figure 3.11 ASTM E399-09 notation for crack plane identification for fracture toughness testing. Left: rectangular sections. Right: cylindrical bars and tubes.

fatigue cracks are sharp, narrow, and follow fairly straight paths, so they provide a satisfactory geometry for fracture toughness testing. In practice, the specimens should have a starter notch that facilitates the initiation of the fatigue crack. The typical shapes of the fatigue crack starters are shown in Fig. 3.12. To use low load amplitudes in fatigue cracking, the ASTM E1829 standard requires that the root radius of the straight-trough slot should be a maximum 0.08 mm (0.003 in.), whereas for the chevron notch it should be 0.25 mm (0.010 in.). During the cyclic loading, the fatigue crack growth is carefully monitored to ensure a uniform crack length across the thickness and that the crack does not deviate from the specimen’s fracture plane until the crack reaches the desired initial size. The cyclic load amplitude shall be carefully controlled, so at the end of the precracking, the plastic zone size is smaller than the expected plastic zone size at instability. Most fracture toughness testing standards suggest that the maximum stress intensity factor at the end of the fatigue precracking shall be less than 60% of the expected KIC, so the initial load amplitude

Chevron

Straight through V

Drilled hole

U ending slot

Figure 3.12 Common fatigue crack starter notch configurations.

94

A Practical Approach to Fracture Mechanics

should be calculated to be sufficient to initiate a fatigue crack in a reasonable number of loading cycles, but the stress intensity amplitude does not increase rapidly, to induce fracture or heavy cyclic strain in the specimen. To prevent these errors, the loading cycle amplitude can be periodically reduced during the precracking process to maintain a fairly constant DK range and consequently achieve a constant and predictable crack growth rate, so the final precrack length is accurately controlled.

3.5 Effect of size on fracture toughness Although KIC is a regarded as a material property, and therefore independent of the specimen’s dimensions, the effect of the size on the fracture resistance of actual in-service structural components has strongly drawn the attention of researchers, beyond the requirements for valid testing, as previously discussed. The reasons to investigate the fracture toughness behavior of large components can be summarized as follows: Material: It is normal that large size components contain inhomogeneity in chemical composition and microstructure that cause variations in the fracture toughness across the transverse section. A common example are large turbine rotor shafts, which are manufactured by forging and heattreating, which develop different microstructures that a propagating crack will encounter across the diameter. Additionally, in large components is common to find defects such as cracks, shrinks, and pores as compared to smaller pieces. Such combination of defects and microstructural variation necessarily causes variations of KIC at different locations across the component’s volume. Since all of this is difficult to analyze theoretically, engineers had opted to carry out fracture testing in specimens of dimensions similar to the actual component, at least in thickness. Residual stresses: The residual stresses interact with the stress field around the crack tip, thus affecting the KIC values. Large size components have a strong tendency to generate residual stresses during their manufacture, construction, and in-service. Due to the difficulty to predict their magnitude and distribution of residual stresses, their effect on fracture toughness should be experimentally evaluated. Geometry and size: The size, orientation, and length/width ratio of the crack and the plastic zone greatly influence the values of KIC. If the fracture toughness is determined in a test specimen that is much smaller than the actual component, the plastic zone size/width ratio of each one will be quite different, so the state of stress may significantly vary, as will KIC.

The energy criterion and fracture toughness

95

In general, practical experience indicates that large components exhibit lower fracture toughness. In an attempt to rationalize this statement, the dimensionless number designated as the Irwin (Ir), honoring the founder of modern fracture mechanics George R. Irwin, has been introduced3 by the equation pffi s0 l Ir ¼ KIC where s0 is the yield strength, KIC is the fracture toughness, and l is the characteristic length of the component, typically the width. The parameter Ol is also referred as the scaling factor. The similitude principle of the Irwin number allows determining the effect of the size on fracture strength. For instance, if the fracture strength is determined in a small specimen, and the same geometry and loading conditions will be used to fabricate a larger size component, the (s0/KIC) ratio has to be increased in a proportion equal to the increment of Ol in order to maintain the Ir number constant, and thus have the same performance. This means that the larger component should have lower s0 and higher KIC, which may be easily achieved, since yield strength and fracture toughness usually vary in inverse proportion. However, such change of properties may be detrimental for the design, because a lower yield strength implies higher thickness, increasing the weight and fabrication costs, so the correct choice might be to select high toughness materials that have high KIC, but retain high yield strength, as the following example demonstrates. A large pressure vessel will be fabricated with a steel whose performance was determined in a small-scale vessel whose characteristic length is l ¼ 4, using a steel having a s0 ¼ 50 ksi and KIC ¼ 100 ksiOin. Select the combination of s0 and KIC to fabricate the large vessel, if its characteristic length is l ¼ 16. Solution: For the small-scale vessel: Ir ¼ (s0/KIC)Ol ¼ (50 ksi/100 ksiOin)O 4Oin ¼ 1.0. The large-size (s0/KIC) ratio should be (s0/KIC) ¼ IrOl ¼ 1.0/ O16 ¼ 0.25 in
1=2 If an ASTM A285 Grade C steel is used, s0 ¼ 30 ksi, the required KIC is KIC ¼ s0/0.25 ¼ 30/0.25 ¼ 120 ksiOin 3

Barenblatt, G. I. Dimensional Analysis, Gordon and Breach Science Publishers, Amsterdam (1987), pp. 58.

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A Practical Approach to Fracture Mechanics

which is very likely with ASTM A285 Grade C steel. If the design requires a material having s0 ¼ 50 ksi, the required KIC shall be KIC ¼ 50/0.25 ¼ 200 ksiOin which will require the use of a ASTM A529 Grade 50 steel, which meets the requirement but is more expensive.

3.6 Charpy impact energy fracture toughness correlations The Charpy impact energy is probably the most frequently used test to characterize the fracture strength, even though the Charpy absorbed impact energy is not a material property. The Charpy test consists of impacting a 10 mm  10 mm  50 mm square section bar containing a V-shaped notch with a hitter mounted on a pendulum bar, and measuring the energy absorbed by the test specimen after the impact. The absorbed energy Cv is proportional to the difference in heights of the pendulum, before and after the impact, as schematically depicted in Fig. 3.13. The typical application of the Charpy test is for quality control to verify whether the material meets a minimal specified Cv and for the determination of the ductile-brittle transition temperature, which is the temperature where the Cv drastically falls to brittle fracture ranges. However, the simplicity and economy of the test have encouraged many efforts to use the Charpy impact energy values to estimate KIC. The main limitation to obtain general Cv versus KIC correlations is because Charpy impact tests are carried out in uncracked specimens; thus, Cv includes both crack initiation and crack propagation components, whereas KIC is only related to crack Indicator dial Hitter

Notched bar 60°

Specimen

10 mm

h1

10 mm

h2 Anvil 50 mm

Figure 3.13 Pendulum and test specimen geometry and dimensions of the Charpy impact test (drawing not to scale).

The energy criterion and fracture toughness

97

propagation. The other difference is that the Charpy impact test is a high loading rate test whereas the KIC test is quasi-static. One of the most widely used Cv and KIC correlation is the RolfeNovak-Barson presented in Appendix F of the API 579-1/ASME FFS-1 2007 standard, applicable in variety of structural steels at temperatures above the ductile-brittle transition, designated as “upper shelf,” which has the following form: (KIC/s0)2 ¼ 0.64 [(Cv/s0) e 0.01], (MPaOm, MPa, J) (KIC/s0)2 ¼ 5 [(Cv/s0) e 0.05], (ksiOin, ksi, ft-lb) This correlation is valid for Cv values from 3 to 121 J (2e89 lb-ft) at room temperature (close to 21 C). For the ductile-brittle transition zone, Sailors-Corlton correlation is suggested: KIC ¼ 14.6(Cv)0.5, (MPaOm, J) KIC ¼ 15.5(Cv)0.5, (ksiOin, lb-ft)

Force

An advanced version of the Charpy test is the instrumented Charpy test introduced in the late 1950s,4 consisting of instrumenting the hitter with strain gauges in order to obtain the load-time record during the impact fracture, which is schematically depicted in Fig. 3.14. Maximum force

Yield

Crack initiation energy

Crack propagation energy

Time

Figure 3.14 Load-time diagram of an instrumented Charpy test. 4

Tanaka (J. Japan Inst. Metals, 21, 1957) and Sakui (Japan Inst. Iron and Steel, 46, 1960) made the first successful reports of Charpy impact tests instrumented with electric resistance strain sensors.

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The main advantage of the load-time diagram of the instrumented Charpy impact test is that it not only indicates the maximum impact force, but more important, the absorbed energy as a function of time, corresponding to the elastic-plastic strain energy in the first stage and the fracture energy in the subsequent stage. Both energies can be used to estimate the fracture toughness, with the advantage that the percent of plastic strain energy and fracture energy are clearly distinguished; therefore, a better assessment of the fracture mechanism can be done. Additionally, the instrumented Charpy test allows a more precise determination of the ductile-brittle transition temperature, as well as better correlations between the absorbed impact energy and fracture toughness, dynamic fracture toughness and the R-curves, all of which makes it a powerful fracture-testing method.

3.7 Dynamic fracture and crack arrest

Stress

So far, the fracture behavior has been analyzed assuming a stationary crack (not propagating) to determine the energetic conditions that initiate the propagation of the crack, namely the crack instability. However, the behavior of a growing crack is different from that of a stable crack, mainly because the plastic zone of a propagating crack is at a high strain rate, which has a stress-strain behavior quite different from the quasi-static strain conditions from which the mechanical behavior of the plastic zone and the fracture process are analyzed. Fig. 3.15 schematically shows the effect of the strain rate on the stressstrain curve in uniaxial tension of a typical engineering metal alloy. As it can

Strain rate increment

Strain

Figure 3.15 Effect of the strain rate of the stressestrain behavior in uniaxial tension in engineering metal alloys.

The energy criterion and fracture toughness

99

be seen, high strain rates increase strain hardening and reduce the ductility, but they also exert an influence on the fracture mechanism. Brittle materials that fracture by cleavage mechanism are very sensitive to the strain rate, reducing the fracture strength as the strain rate increases, whereas materials having ductile fracture mechanisms, such as microvoid coalescence, the increment of strain hardening, along with higher yield and tensile strength, at high strain rates result in higher fracture toughness. However, if the strain rate is high enough to inhibit dislocation slip-induced plastic strain, the effect is similar to that on brittle materials. Roberts and Wells5 derived an equation to calculate the velocity of a propagating sharp crack that satisfactorily fits experimental data: rffiffiffiffi E a0

V ¼ 0:38 1
a r where E is the young modulus, r is the density, and a is the instantaneous crack size. It is interesting to observe that the term O(E/r) is the speed of one-dimensional wave sound, indicating that the dynamic fracture mechanism is related to the shear wave speed. Substituting typical values for steel and taking a0 R (area A), equals the area between the R-curve and the G-curve, for G < R (area B). The fracture toughness at arrest KIA is determined by the value of G where A ¼ B. Clearly, KIA is lower than the quasi-static KIC. The following equation relates KID and KIA with the crack tip speed: KID ¼

KIA 1
ðV =VLIM Þm

The energy criterion and fracture toughness

101

where VLIM is the limit crack rate when a0/a / 0 and m is an empiric constant (typically m ¼ 2). Due to the simplicity and economy of the Charpy impact energy test, several empirical correlations between KID and Cv have been published, which are very useful to solve practical problems. One of the best known is the Salilors and Corten6 relation, expressed as pffiffiffiffi KID ¼ 15:5Cv 0:375 ; MPa m; J pffiffiffiffi KID ¼ 15:873Cv 0:375 ; ksi in; lb-ft where Cv is the absorbed Charpy impact energy. The following example gives an indication of the order of magnitude of crack tip velocity in an engineering material. Determine the crack tip velocity of a propagating crack in a large panel under uniform stress if the fabrication material is: Structural steel, E ¼ 200 GPa, r ¼ 8.05 g/cm3, Cv ¼ 50 J, KIA ¼ 40 MPaOm Aluminum alloy, E ¼ 70 GPa, r ¼ 2.7 g/cm3, Cv ¼ 26 J, KIA ¼ 15 MPaOm Solution: For the steel: VLIM ¼ 0.38(E/r)1/2 ¼ 0.38(200 GPa/8050 kg/m3)1/2 ¼ 1894 m/s KID ¼ 15.5Cv0.375 ¼ 15.5(50)0.375 ¼ 67.2 MPaOm KID ¼ KIA [1 e (V/VLIM)m], taking m ¼ 2 and solving for V:

/

V ¼ VLIM[1 e (KIA/KID)]1/2 ¼ 1894 * [1 e (40/67.2)]1/2 ¼ 1204 m/s For the aluminum alloy: VLIM ¼ 0.38(E/r)1/2 ¼ 0.38(70 GPa/2700 kg/m3)1/2 ¼ 1934 m/s KID ¼ 15.5Cv0.375 ¼ 15.5(26)0.375 ¼ 52.6 MPaOm KID ¼ KIA [1 e (V/VLIM)m], taking m ¼ 2 and solving for V: V ¼ VLIM[1 e (KIA KID)]1/2 ¼ 1934*[1 e (15/52.6)]1/2 ¼ 1245 m/s

/

6

Sailors, R.H. and Corten, H.T. ASTM STP 514, Part II (1972), p. 164e191.

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Notice that even though steel and aluminum have very different mechanical and toughness properties, their theoretical crack tip velocities are quite similar. Since in many occasions unstable crack propagation cannot be prevented, the ability of a structure to arrest a dynamically growing crack is of great interest in fracture mechanics. Such capacity depends on the KIA value, because the higher KIA, the sooner the crack arrests, thus KIA experimental testing is very important for fail-safe designs that consist of arresting a crack generated after a failure condition in such a way that the overall component maintains its structural integrity and the consequences are minimized. Examples of structural components where the fail-safe designs are feasible are gas and liquid transportation pipelines, airplane fuselage, ship hulls, reactor shells, pressure vessels, and more. The testing procedure for determining KIA was introduced in the ASTM E 1221 standard in 1988,7 which consists of wedge loading a modified CT specimen thick enough to be in plane strain condition. The wedge-shaped pin is pushed into a hole in the test specimen until a fatigueinduced precrack starts to propagate. At the onset of instability, the wedge insertion is stopped, and the specimen is set in constant displacement condition; consequently, after this point, G decreases until arrest takes place. The specimen-wedge pin arrangement is schematically shown in Fig. 3.18. P

Wedge Split Pin

ao = 0.35W 1.2W

Test Specimen

W Support Bloack

1.25W

Figure 3.18 Schematic illustration of wedge-loading arrangement and side-grooved compact crack arrest specimen for KIA testing.

7

This method was introduced by Irwin and collaborators in 1988 and presented on the 19th Symposium of Fracture Mechanics (ASTM STP 969).

The energy criterion and fracture toughness

103

A summary of the test procedure provided in the ASTM 1221 standard is: 1. The specimen is loaded up to a predetermined crack opening displacement (COD). The loading rate should be relatively slow so the load signal from the testing machine and the COD from a clip gauge mounted on the specimen mouth are properly recorded. The initiation of unstable crack propagation shall be detected by a sudden load fall in the test record system or by the loud noise produced at the onset of instability. 2. If the unstable crack extension is not produced in the first load cycle, the specimen is unloaded back to zero by retrieving the wedge. The residual COD is measured and subtracted from the total displacement. 3. The specimen is reloaded to a COD higher than that of the previous load cycle until attaining instability. At this point the crack will propagate and arrest. 4. The critical COD displacement, CODc, is obtained by subtracting the first COD and half of the subsequent residual COD, as shown in Fig. 3.19. 5. The maximum load and the final crack size aa are used to calculate the provisional crack arrest toughness Ka by using the stress intensity factor solution of the test specimen. The recommended procedure to determine the final crack length is by heat tinting at 250e350 C for 10e90 min and then breaking open the specimen. Nowadays, however, there are very precise nondestructive techniques such as the potential drop that allow determining the crack length across the test specimen without heat tinting nor breaking opening the test specimen.

COD2

COD3

Load, P

COD1

CODc = COD3 – COD1 – ½ (COD1+COD2)

½(COD1+COD2)

COD

Figure 3.19 Schematic graphic of load versus COD cycles to determine the provisional crack arrest toughness Ka.

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A Practical Approach to Fracture Mechanics

The initial stress intensity KIi and the provisional crack arrest toughness KQa are calculated by the following expression: qffiffiffiffiffiffiffiffiffiffiffi E COD B=B

a N pffiffiffiffiffiffi KI ¼ f W W ffi a rffiffiffiffiffiffiffiffiffiffiffiffiffi a 3

a 2 a a f
2:55 ¼ 1
0:748
2:176 þ 3:56 W W W W W a 4 þ 0:62 W where E is the Young’s modulus, COD is the crack mouth opening displacement, W is the specimen width, B is the full thickness, and BN is the net thickness at the side grooves. Since the previous procedure may lead to underestimations of KIA, the apparent arrest toughness, Ka, should be validated by the following criteria: W e aa  0.15 W 2
KQa W
aa  1:25 s0
2 1 KIi aa
a0  2p s0
2 KQa B s0d where aa is the crack size at arrest, a0 is the initial crack size, s0d is the dynamic yield strength (it is suggested that s0d ¼ s0 þ 205 MPa (30 ksi)), and s0 is the standard yield strength of the material. In the ideal material, KA should be independent of the final crack length and instability should occur at a constant value of COD, therefore the value of Ka is at the intersection of the constant KIA line and the K(aa/W ) values corresponding to COD ¼ CODo ¼ CODa, as illustrated in Fig. 3.20A. Nevertheless, in real materials Ka falls within a scatter band for the same COD, as shown in Fig. 3.20B, where the low values of KA correspond to longer crack arrest lengths and vice versa. The following example illustrates the practical application of the above procedure. A modified CT specimen of W ¼ 100 mm and B ¼ 50 mm with side grooves 3 mm depth was used to determine KIA in a structural steel with s0 ¼ 690 MPa.

The energy criterion and fracture toughness

(A)

(B)

COD = Cte.

105

COD = Cte. KIA max

KIA

Ka

KIA= Cte.

KIA min

aa/W

aa/W

Figure 3.20 Results of KIA in terms of arrest crack size: (A) ideal, (B) real.

The initial and final crack lengths were a0/W ¼ 0.35 and aa/W ¼ 0.8. Determine the crack arrest fracture toughness (KIA) and the validity of the test if the load-COD experimental curve per ASTM E1221 is 200

% Initial Load

COD3= 2.0

150 COD1 = 0.6

100

COD2 = 0.7

50 0

0

0.5

1

1.5

2

2.5

COD, mm

CODc ¼ COD3 e COD1 e 1/2 (COD1 þ COD2) ¼ 2.0 e 0.6 e 1/2 (0.6 þ 0.7) ¼ 0.75 mm KIi ¼ E * CODi*(B/BN)1/2 f(a0/W)/(W)1/2 (a0/W) ¼ 0.35 f(a0/W) ¼ (1 e 0.35)1/2 {0.748 e 2.176(0.35) þ 2.56(0.35)2 e 2.55(0.35)3 þ 0.62(0.35)4] ¼ 0.26

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A Practical Approach to Fracture Mechanics

pffiffiffiffi KIi ¼ 200; 000MPa* 0:0006m* ð50=44Þ1=2* ð0:26Þ =ð0:1mÞ1=2 ¼ 105MPa m KIA ¼ E * CODc * (B/BN)1/2 f(a0/W)/(W)1/2 (a0/W) ¼ 0.8 f(a0/W) ¼ (1
0.8)1/2{0.748
2.176(0.8) þ 2.56(0.8)2
2.55(0.8)3 þ 0.62(0.8)4] ¼ 0.105   pffiffiffiffi KQa ¼ 200; 000MPa* 0:00075m* 1:04* 0:105 =ð0:1mÞ1=2 ¼ 51MPa m Validation: W e aa  0.15 W; (100 e 80) mm ¼ 20 mm  0.15 (100 mm) ¼ 15 mm W e aa  1.25 (KQa/s0)2; 20 mm  1.25(51/690)2 ¼ 6.83 mm aa
a0  (1/2p) (KIi/s0)2; (80
35) mm ¼ 45 mm  (1/2p)(105/690)2 ¼ 36.4 mm B  (KQa/s0d)2; 50 mm  [51/(690 þ 205)]2 ¼ 3.24 mm pffiffiffiffi The validation criteria are satisfied, therefore KIA ¼ 51MPa m.

CHAPTER 4

Elastic-plastic fracture mechanics 4.1 Elastic-plastic fracture and the J-integral In the previous chapter it was stated that when plasticity is limited to a small zone in the crack-tip and the load-displacement record is predominantly linear, the stresses in the crack-tip region are described by the stress intensity factor (SIF), but if the Load-Displacement behavior deviates from linearity, resulting in a significative amount of plastic deformation, as shown in Fig. 4.1, the use of the SIF, and elastic-plastic fracture mechanics, is no longer valid; therefore, another fracture criterion shall be used. A simple correction to the problem of plasticity in fracture was proposed by Irwin, by assuming that the plastic zone makes the crack behave as if it was longer, and introducing the effective crack size aeff ¼ ac þ rp, where ac is the critical crack size and rp ¼ 1/p (KIC/s0)2, thus pffiffiffiffiffiffiffiffiffi KIC ¼ s paeff This plasticity correction obviously provides higher KIC values, as shown in the following example: A plate of finite width W ¼ 200 mm made of material with s0 ¼ 500 MPa fractures at an applied stress of 200 MPa. If the crack critical size is 2a ¼ 50 mm, calculate the fracture toughness by using Irwin’s plasticity correction assuming a linearelastic behavior. Solution: KIC ¼ s(p a)1/2 [sec(p a/W)]1/2 ¼ 200 (p*0.025)1/2 *[sec(p*0.025/0.2)]1/2 KIC ¼ 56.1 MPa m1/2 rp ¼ (1/p) (KIC/s0)2 ¼ (1/p) (56.1/500)2 ¼ 0.004 m aeff ¼ 25 þ 4 ¼ 29 mm

A Practical Approach to Fracture Mechanics ISBN 978-0-12-823020-6 https://doi.org/10.1016/B978-0-12-823020-6.00004-9

© 2021 Elsevier Inc. All rights reserved.

107

108

P

A Practical Approach to Fracture Mechanics

PQ

Pmax < 1.1PQ

Pmax > 1.1PQ

P PQ

rp

rp

W

W

Displacement, v

Displacement, v

Figure 4.1 Load versus crack opening displacement for linear-elastic and nonlinear behavior of a cracked component.

Applying Irwin’s correction: KIC ¼ s (p aeff)1/2 [sec(p aeff/W)]1/2 ¼ 200 (p*0.029)1/2 * [sec(p*0.029/0.2)]1/2 Corrected value of KIC ¼ 60.37 MPa m1/2 In the previous example it is observed that the KIC value calculated by the plasticity correction increases as the plastic zone becomes larger, which initially is correct since more energy is consumed by the deformation and fracture process. However, when the plastic zone size approaches the ligament size (W  a), the conditions will be no longer linear-elastic, therefore the Irwin’s correction will eventually lead to incorrect values of the toughness. In 1968, James Robert Rice, an assistant professor at Brown University in the United States, developed a method based on a path-independent integral1 to determine the fracture energy of a cracked body without requiring a linear stress-strain behavior. This method is known as the J-integral and became a fundamental parameter for characterizing the behavior of cracked bodies under nonlinear strain conditions. James R. Rice was an outstanding student of mechanical engineering at Leigh 1

J.R. Rice. “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks”. Journal of Applied Mechanics, Vol. 35 (1968) pp. 379e386.

Elastic-plastic fracture mechanics

109

University, where he took lessons from great personalities in fracture mechanics such as Fazil Erdogan, George Sih, and Paul Paris. He graduated with a BS in 1962, and later he got MS and Ph.D. degrees in 1964. After graduating, he held a position at Brown University from 1964 to 1981, and since then, is an associate professor at Harvard University. From the physical point of view, the J-integral is the energy balance around a closed path around the crack-tip. This balance is done between the work supplied by the tractions acting on surface elements of a closed path around the crack, while the deformation energy is consumed within the limits of the closed path as depicted in Fig. 4.2. Since the J-integral is independent from the path, the integration can be done close or far from the crack-tip and the change of potential energy due to crack extension can be calculated independently that the stress-strain behavior is linear or nonlinear. According to Rice, the J-integral for a cracked body is defined by the following expression:  Z  dui J¼ Wdy  Ti ds ¼ 0 dx G

Where Ti is the traction on a differential surface element along a closed path, ui is the displacement and W is the deformation energy density given by Zε W¼

sij dεij 0

y

Tractions (T)

ds

x

Arbitrary path function ( Г)

a

Available energy

=

Work done by the tractions around the path



Strain energy inside the path

Figure 4.2 Energy balance of a closed path around a crack.

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A Practical Approach to Fracture Mechanics

Where sij is the stress tensor and εij is the strain tensor. The Ti tractions are the normal stresses acting on the closed path boundary given by Ti ¼ sij nj Where nj is the normal unit vector to sij. The experimental determination of the J-integral for materials exhibiting a nonlinear behavior can be done from the Load-Displacement record, as schematically depicted in Fig. 4.3. If a precracked specimen is loaded up to get a stable crack extension Da, and then the displacement is fixed (point A in Fig. 4.3A), the load will drop an amount DP (point B in Fig. 4.3A). The change in potential energy due to the crack extension is equal to Zv DU ¼ AreaOAB ¼

DPdv 0

Note that DU is negative because DP is negative. But if the load is fixed after attaining a stable crack extension (point C in Fig. 4.3B), the crack opening displacement will increase an amount Dn (point D in Fig. 4.3B), and the potential energy change due to crack extension under constant loads is ZP DU ¼ AreaOCD ¼ 

DvdP 0

(A)

(B)

P A

D

C

P

∆P B a

a a + ∆a

a + ∆a

∆v

O

Displacement, v

O

Displacement, v

Figure 4.3 Schematic Load versus Displacement curves of a cracked body on elasticplastic deformation. The area between the curves represents the energy change due to a crack extension. (A) Constant displacement, (B) Constant load.

Elastic-plastic fracture mechanics

111

Since the J-integral is indeed the change of potential energy upon the crack extension, it can be established that DU J ¼  lim ¼ Da/0 Da

ZP   Zv   vP vv dv ¼ dP va v va P 0

0

The aforementioned equations allow to establish an experimental method to determine the J-integral upon crack extension; however, a strong limitation arises from the fact that the small crack length increments make very difficult to measure the area between the curves of Fig. 4.3. To solve this problem, Rice demonstrated that when the plastic zone is a narrow stripe in the ligament; as shown in Fig. 4.4, the J-integral can be calculated by J¼

2A BðW  aÞ

Where A is the area under the Load-Displacement curve, limited by a slant line parallel to the linear portion of the curve in order to subtract the elastic part of the work. This method is referred as the “Basic J-integral Method.” The J-integral value can be calculated for different geometries and load conditions using different methods such as compliance, finite element, and so forth, all of them being complex and laborious. Fortunately, currently there are handbooks of J-integral solutions such as

Elastic line

Load, P

Plastic zone

Crack length

A

Ligament

Displacement, v

Figure 4.4 Plastic zone condition and area for J-integral calculation according to Rice (Basic Method).

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A Practical Approach to Fracture Mechanics

1. A. Zahor: Ductile Fracture Handbook, EPRI NP 6301-D, Vols. 2 and 3, (1968). 2. V. Kumar, M.D. German and C.F. Shih: An engineering Approach for Elastic Plastic Fracture Analysis, EPRI NPe1931. July 1981. The following examples illustrate the J-integral concept and provide a perception of its range of values. The graph shown below was obtained by loading a ductile material plate with side grooves, so the plastic zone is a narrow stripe along the ligament length. A crack extension Da ¼ 2 mm was attained under constant displacement conditions. The test specimen had W ¼ 100 mm, B ¼ 5 mm, and an initial crack size ao ¼ 15 mm. Determine the value of the J-integral. E ¼ 200,000 MPa. P

P(a) = 44,200 N P(a + ∆a) = 44,140 N

Drawing not at scale Apl

Unloading line

0.95 mm 1.5 mm

v

Solution: The shaded area between the Load-Displacement curves is the energy change due to crack extension DU and it is estimated as the shaded area between the curves for a and a þ Da. DU ¼ 1.8 N-m (Joule) The J-integral by unit thickness is approximately: J ¼ (1/B) (DU/Da) ¼ (1/0.005) (1.8 J/0.002 m) ¼ 180,000 J/m2 Calculation of the J-integral by the Basic Method J ¼ 2A/[B(Wa)] The area under Load-Displacement minus the elastic contribution is: Apl ¼ 38 J J ¼ 2(38 J)/0.005 m (0.10.015) m ¼ 177,823 J/m2

Elastic-plastic fracture mechanics

113

As seen in the above example, the J-integral values determined from the DU values are imprecise due to the difficulty to measure the area between the Load-Displacement curves for the initial and extendedcrack lengths, while the Rice’s basic method provides more precise values that are more consistent with the experimental evidence.

4.2 JIC testing The J-integral can be used to determine the fracture resistance of solids that exhibit a nonlinear stress-strain behavior. The specific value of the J-integral to characterize the fracture toughness of materials is identified by the symbol JIC, which is determined prior to the onset of fracture instability, typically established at a small stable crack extension. The procedure to determine JIC was developed by Landes and Bagley in 1972 and is described in the ASTM E803 standard, published in 1981, later, in 1998, this method along with the LEFM and CTOD methods for fracture toughness testing were incorporated into the ASTM E1820 standard. The procedure for JIC testing appears in the British standard BS-7448 as well. The most popular specimen geometry is the Compact (CT) shown in Fig. 4.5, but the Single Edge Bend SE(B) and the Disk-Shaped Compact DC(T) are also frequently used. The CT specimen geometry for JIC testing is modified at the notch mouth so a clip-type extensometer can be accommodated to measure the displacement in the load line. Side grooves are machined at both sides of the fracture plane to confine the plastic zone into a narrow strip. 0.45 W < a < 0.55 W D = 0.25W

1.2W

a 0.25 to 0.4 W

W

Recommended thickness B = 0.5W

1.25W Figure 4.5 Modified CT specimen for JIC testing.

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A Practical Approach to Fracture Mechanics

The aim of the test is to obtain the Load-Crack Mouth Opening Displacement (CMOD) curve from specimens with different crack lengths so as to get different crack extensions (Da) and calculating the J-integral for each test specimen with the following equation J¼

K 2 ð1  n2 Þ þ Jpl E hApl Jpl ¼ BN b 0

Where Apl is the area under the Load-CMOD curve (area A in Fig. 4.4), BN is the net width (specimen width between grooves), b0 ¼ (W  a0) and h ¼ 2 þ 0.522(b0/W ) for the CT specimen. The calculated J-integral values for each crack extension are plotted as shown in Fig. 4.6. The toughness determined by the J-integral approach is the value corresponding to the onset of a significant stable crack extension, such value is termed as JC. If the testing is performed under Mode I crack displacement, then the toughness is expressed as JIC In the ASTM E1820 standard an interim J-integral value, identified by JQ, shall be determined by the following procedure. a. Using the least squares method determine the best fit linear regression line of the plotted data points by the formula: ln J ¼ lnC1 þ C2 ln(Da/k), where k ¼ 1 mm

J

Plasticity line J = 2 σo ∆ a

JQ

Power law regression line

0.2 mm offset line 0.15 mm exclusion line

1.5 mm exclusion line

Non valid data region

Non valid data region

0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 ∆a [mm] Figure 4.6 J versus Da graph and plasticity line for JIC testing.

Elastic-plastic fracture mechanics

115

b. Plot the plasticity line, defined by: J ¼ 2 s0 Da

General

J ¼ (s0 þ suts) Da Strain hardening materials c. The exclusion lines shall be parallel to the plasticity line intersecting the abscissa (Da axis) at 0.15, 0.2 and 1.5 mm. There should be at least five valid data points between the exclusion lines for 0.15 and 1.5 mm. d. The J-integral value located at the intersection of the regression line with the 0.2 mm offset line is JQ. Nowadays, computer-controlled servo-hydraulic machines allow to make the JIC determinations by using a single precracked specimen. The specimen is loaded to attain a measurable crack extension and then it is repeatedly unloaded at least 10% of the maximum load and reloaded as shown in Fig. 4.7. The Load versus CMOD cycles are automatically recorded and digitalized to be entered as data into a software, which, by using the specimen’s compliance, calculates the crack extension and computing the areas under the Load-CMOD curves, calculates the J-integral values. The software automatically determines the test validity. Just like in LEFM, the interim JQ is limited by a series of conditions, being more important that the plastic zone should be small compared to the overall dimensions and the component has to be thick enough to assure

Figure 4.7 Load-unload sequence applied in the single specimen method for JIC testing by computer-controlled servo-hydraulic machines.

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A Practical Approach to Fracture Mechanics

plane strain conditions. To qualify JQ as JIC the following criteria should be met: (1) Thickness B > 25( JQ/s0) (2) Initial ligament (W  a) > 25( JQ/s0) (3) Slope of the power law regression line at DaQ < s0 (4) Width W >> 2J/s0 In addition to these limitations, as mentioned before, the application of the J-integral is specific for the tested material, temperature and strain rate, so unlike the stress intensity factor, that only depends on the geometry, the analysis of fracture based on J-integral is not as widespread as the stress intensity factor. Despite the aforementioned limitations, JIC testing is a frequent laboratory practice since it allows determining the plane strain fracture toughness of ductile materials by using reasonably small specimens (typically W ¼ 5 cm, B ¼ 2.5 cm in CT specimens), as compared to the specimen size required for KIC testing. The formula used to convert JIC into plane strain fracture toughness is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E JIC KJC ¼ ð1  n2 Þ To differentiate the fracture toughness determined by KIC testing from the calculated by JIC, the former is identified to as KJC. The following example illustrates the procedure and related values obtained in a multiple specimen testing to determine JIC. A CT specimen of B ¼ 25 mm, W ¼ 50 mm, with ao ¼ 22 mm, made of a metal having: so ¼ 350 MPa, suts ¼ 550 MPa, E ¼ 200,000 MPa, v ¼ 0.3, was tested for JIC, giving the following results: Specimen

J (kJ/m2)

Da (mm)

1

50

0.08

2

100

0.30

3

175

0.38

4

185

0.80

5

225

1.20

6

250

1.60

7

275

1.80

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The J versus Da plot is shown below: The J vs ∆a plot is shown below: 300

J [kJ/m2]

250 200 150 100 50 0

0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50 1.65 1.80 1.95

∆a [mm]

The plasticity line is: J[kJ/m2] ¼ 2s0Da ¼ 2(350 MN/m2) Da [mm] ¼ 700Da [mm] The solid line between the plasticity lines displaced 0.15 mm and 1.5 mm represents the best fit power law regression line of the data points, whose intersection with the exclusion line at Da ¼ 0.2 mm is JQ ¼ 155 kJ/m2 The validity of JQ depends on the following criteria: B > 25(J/so) ¼ 25(155 MPa-mm/350 MPa) ¼ 11 mm; 25 mm > 10.7 mm; the thickness criterion is satisfied. (Wa) ¼ (5022) mm ¼ 28 mm > 25( J/so) ¼ 25(155 MPa-mm/350 MPa) ¼ 11 mm. The ligament length criterion is satisfied The slope of the power law regression line at DaQ is 133 MPa < 350 MPa. The regression line slope criterion is satisfied. Therefore, the test is valid and JIC ¼ 155 kJ/m2. The estimated KJC is: KJC ¼ {E JIC/(1  v2)}1/2 ¼ {200,000 MPa 0.155 MPa-m/(1  0.32}1/2 KJC ¼ 184.6 MPaOm

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4.3 Use of the J-integral as a fracture parameter As already mentioned, the physical meaning of the J-integral is that it represents the energy release rate upon crack extension, therefore it has to be equivalent to Irwin’s energy release rate G. Rice derived the formulas that relate the SIF and the J-integral, which are valid as long as the plastic strain is limited. For plane stress: J¼

K2 E

For plane strain: J¼

K2 ð1  n2 Þ E

The equivalence between J-integral and K imply that the J-integral is also a similitude criterion; therefore, if two cracks have the same J-integral, they have equal energy release rates, and therefore, their behavior should be the same. Additionally, the variation of the J-integral with respect to the crack extension is equivalent to the R-Curve. The advantage of the J-integral over K is that it is no restricted to linear-elastic behavior, and therefore it can be applied to the analysis of elastic-plastic fracture behavior, which is nonlinear for most engineering materials, as long as plasticity is not extensive. The only disadvantage is that the J-integral depends on the stress-strain curve, which is dependent on the material condition, temperature, and strain rate, thus the J-integral is exclusive for the specific material-loading rate-environment conditions under which the J-integral is determined. The principle of the use of the J-integral in elastic-plastic fracture analysis is schematically depicted in Fig. 4.8. On loading the J-integral increases, since the material is ductile the crack-tip stretches up producing a blunt shape (point A). Upon reaching a critical J-integral value, there is a pop-in extension of the crack (point B), which corresponds to JIC, as explained before. As the load further increases, the crack will reach the instability (point C). According to the energy criterion, unstable crack propagation will take place when the energy release rate, surpasses the demand of energy to extend the crack; mathematically, this is dJA dJR > da da

Elastic-plastic fracture mechanics

C

J

119

Propagation

B

JIC

Stable extension Pop-in A

Blunting Sharp tip

0

Crack extension, 'a

ao

Figure 4.8 Ductile fracture stages as related to the J-integral.

where JA is the J-integral applied to the component, which primarily depends on the load, the strain hardening exponent, the ductility, and the crack size, and JR is the J-integral value for a specific crack extension and represents the fracture resistance of the material, which is dependent on the crack size and the stress-strain behavior of the material at the evaluation temperature. Based on the above, the instability condition can be graphically determined by plotting JA and JR in a J versus Crack length graph, like the one shown in Fig. 4.9. The instability condition occurs at the tangent point of the JA and the JR curves, since that point represents the condition where dJA/da ¼ dJR/da, and then the energy release rate is equal to the energy rate required to attain continued crack growth. J

Applied J curve Increasing load Common tangent Material J-R curve

JC Instability point: (dJ/da) > (dJR/da)

ac

Crack size, a

Figure 4.9 Characterization of instability as a function of J and Jc.

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A Practical Approach to Fracture Mechanics

Based on the above, the practical application of the J-integral for the analysis is elastic-plastic fracture is as follows: 1. Find the solution of the applied JA, from a J-integral compendium or by numerical stress analysis. The solution for the applied JA can be written as  nþ1 P2 P JA ¼ f ðaÞ 0 þ s0 ε0 ðW  aÞhða; W ; nÞ Po E Where a is the crack size, W is the width, E0 ¼ E/(1  v2), s0, and ε0 are the strength and ductility at yielding, n is the strain hardening exponent, f(a) is a constant depending on the crack size, h(a, W, n) is a constant depending on a, W and n, P is the applied load per unit thickness and P0 is the limit load per unit thickness. 2. Determine the material J-R curve by using the method described in the Annex A9 of the ASTM E1820 standard to obtain the power law regression line of the J-Da data using the following expression:

JR ¼ C1

 C2 Da k

Where k ¼ 1.0 mm (0.0394 in.). 3. Plot JR, and JA for increasing loads to obtain the crack driving force diagram in a J versus a plot, as shown in Fig. 4.9 and find the point where dJA dJR ¼ da da Read the critical crack size and record the applied load corresponding to the JA curve that becomes tangent to the JR curve. The solution can be found analytically by solving the above differential equation. The following example illustrates the above procedure. A hollow cylinder made of low carbon steel is under internal pressure and contains an axial crack of 50% thickness depth. Determine the failure pressure and the critical crack length by the J-integral method.

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The component dimensions are as shown below: b = 0.5 in Ro = 5.5 in

Ri = 5 in

p

a = 0.25 in

The mechanical properties of the steel are: s0 ¼ 40 ksi, E ¼ 29,000 ksi, v ¼ 0.3, ε0 ¼ 0.004, and n ¼ 2. The material JR curve is given by: JR ¼ 18,316 Da0.744 [psi-in]. The elastic-plastic formula for JA in an axially cracked cylinder is  nþ1 p2 p ðb JA ¼ f1 0 þ 1:12s0 ε0  aÞh1 p0 E  2 R02 f1 ¼ 4pa 2 F2 R0  Ri2 2 ðb  aÞs0 p0 ¼ pffiffiffi 3 ðRi þ aÞ From Tables 4.1 and 4.3 of V. Kumar: An engineering Approach for Elastic Plastic Fracture Analysis, EPRI NPe1931. Jul 1981, F ¼ 2.36 and h1 ¼ 11.6. Substituting data: E0 ¼ E/(1  v2) ¼ 29,000,000 psi/(1  0.32) ¼ 31,868,132 [psi] p0 ¼ (2/O3) (0.5ea)in * 40,000 psi/(5.0 þ a)in ¼ 46,188 [(0.5 e a)/ (5.0 þ a)] [psi] f1 ¼ 4pa [5.02/(5.52  5.02)]2 * 2.362 in ¼ 333.3 a [in]

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A Practical Approach to Fracture Mechanics

JA ¼ 333.3 in (p2 psi2/31,868,132 psi) a þ 1.12*40,000 psi *0.004 (0.5 e a) in*11.6* [p psi/46,188 [(0.5 e a)/(5.0 þ a)] psi]2þ1 JA ¼ 1.0459  105 p2aþ 2078.72 (0.5  a) [p3 (5 þ a)3/ (46,1883(0.5  a)3)] JA ¼ 1.0459  105 p2aþ 2078.72 (0.5  a) [p3 (5 þ a)3/ (46,1883*(0.5  a)3)] JA ¼ 1.0459  105 p2aþ 2.11  1011 p3 [(5 þ a)3/(0.5  a)2)] [psi-in] The analytical solution is too complicated; therefore, the problem is solved graphically in the crack driving force diagram, plotting the JR curve of the steel and the JA curve for the axially cracked cylinder at varying pressures. 7000

J-integral [psi-in]

6000 p = 1700 psi

5000

p = 1600 psi

4000

p = 1000 psi

3000 2000 1000 0 0.20

0.25

0.30

0.35 0.40 Crack length [in]

0.45

0.50

0.55

From the above crack driving force diagram, it is determined that the critical crack length is 0.305 in and the failure pressure is just above 1600 psi.

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123

Paris2 proposed to make dJ/da dimensionless multiplying it by (E/s2o) in order to use it as an instability criterion. Such term is defined as the “Ductile Tearing Modulus” and is represented by the equation:    dJ E T¼ da s20 Where s0 can be either the yield strength or the flow stress 1/2 (s0 þ suts). The analysis of ductile fracture with considerable strain hardening and where there is significant pop-in based on T is very simple, as shown in Fig. 4.10, because the applied J-integral has a linear tendency with respect to T, whereas JR is decreasing. Therefore, instability occurs at the intersection point of both curves. The following example illustrates this approach. A plate under uniform stress containing an edge crack has a crack resistance curve given by JR ¼ 350(a  a0)0.52 (kJ/m2, m). Determine the instability point and the fracture stress by the Tearing Modulus approach. E ¼ 200,000 MPa, v ¼ 0.3, s0 ¼ 550 MPa, ao ¼ 20 mm. Assume plane strain conditions and KI ¼ 1.12s(pa)1/2 Solution: The applied J is determined from T ¼ (dJ/da) (E/s2o ), solving for J and integrating, J ¼ (s2o /E)T !da ¼ (s2o /E) aT ¼ (5502 MPa2/200,000 MPa)*0.02 mT J (MPa-m) ¼ 0.03T J

J applied

Material curve, JR Instability point

Tearing Modulus, T

Figure 4.10 Analysis of ductile fracture based on the tearing modulus. 2

P.C. Paris, H. Tada, A. Zahoor, H. Ernst. The theory of instability of the tearing mode of elastic-plastic crack growth. In: Landes JD, Begley JA, Clarke GA, editors ASTM STP Elastic_plast Fract. Vol. 668 (1979), pp 5e36.

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The value of JR is calculated by the following expression introduced by Ernst3: !C2 =ðC2 1Þ s20 TR JR ¼ 1=C EC2 C1 2 Having JR ¼ 350(a  a0)0.52, C1 ¼ 350, C2 ¼ 0.52 JR ¼ [(5502MPa2*TR)/(200,000 MPa *0.52*3501/0.52)]0.52/(0.521) ¼ 62,535T1.083 R Plotting the above equations, the solution can be graphically found (as well as analytically). The result is that instability occurs at T ¼ 39.2, which corresponds to Jc ¼ 1.176 MPa-m ¼ 1176 kN/m The applied J is: J ¼ (1  v2)K2I /E ¼ (1  v2)[1.12s]2(pa)/E J ¼ (1  0.32)[1.12s]2MPa2(p*0.02 m)/200,000 MPa ¼ 1.7931  105 s2 MPa-m Solving for s and substituting Jc ¼ 1.176 MPa-m gives the failure stress: sf ¼ ( Jc/1.7931  105)1/2 ¼ 256 MPa 3.5 3

J, MPa-m

2.5 2 1.5 1 0.5 0

20

40

60

80

100

T

3

H. Ernst, P. Paris, and J. Landes, “Estimations on J-Integral and Tearing Modulus T from a Single Specimen Test Record,” in Fracture Mechanics, ed. R. Roberts (West Conshohocken, PA: ASTM International, 1981), 476e502.

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4.4 The crack-tip opening displacement as fracture parameter In 1960, Alan A. Wells, a researcher from the British Welding Research Association (later the Welding Institute), after spending a sabbatical year with Irwin at the U.S. Naval Research Lab, proposed a fracture model based on the observation that fracture initiates after a critical crack-tip stretching strain. As schematically shown in Fig. 4.11. If the material is ductile enough, upon loading the crack-tip will first become blunt and further it will go through a stable crack extension to finally reach instability. Wells4 proposed that the strain elongation in the vertical direction with respect to the fracture plane, which is the Crack-Tip Opening Displacement (CTOD) can be used as fracture criterion. The validity of the CTOD approach as a fracture toughness parameter is justified by the already known equation: CTOD ¼

4 KI2 p Es0

Since G ¼ K2/E, then CTOD ¼ G/s0, therefore the CTOD is directly related to the energy criterion. The advantage of applying CTOD as fracture criterion is that it is neither limited by linearity of the stress-strain behavior as with KIC nor by restricted by plasticity as with JIC, therefore it Stable crack extension

PQ

P>0

PQ

P

CTODc

Blunting CTODc

P=0

CTOD = 0 CTOD Figure 4.11 Application of CTOD as fracture criterion.

4

A. A. Wells. “Unstable Crack Propagation in Metals: Cleavage and Fast Fracture”. Proccedings of the Crack Propagation Symposium, Vol. 1, Paper 84, Cranfield UK, 1961.

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can be applied to generalized plasticity conditions. Likewise, CTOD can be applied to short cracks and high-toughness materials, as well as an assessment parameter for fracture resistance and defect assessment. The ASTM E1820 standard provides the guidelines for characterizing the fracture toughness by CTOD testing, previous to 1998 the ASTM E1290 standard was used. These standards are equivalent to the BS7448 and ISO 15,653 standards. In CTOD testing the symbol to identify the cracktip opening displacement is d, being • dIc the value of CTOD at the onset of a predefined stable crack extension, typically Dap ¼ 0.2 mm þ 0.7dIc, and • dc the critical CTOD at the onset of unstable crack extension. The types of Load-Clip gage displacement records that may be obtained by CTOD testing are shown in Fig. 4.12. The curve (A) corresponds to brittle or limited plasticity fracture. Curve (B) indicates a significant pop-in prior to instability, and curve (C) is typical for high ductility materials with significant stable crack extension. It is worth mentioning the testing standards may include up to six different types of Load-Clip gage displacement curves, but they can be treated similarly to one of the three cases depicted in Fig. 4.12. The most common specimen for CTOD testing is the SE(B), but also CT and any other geometry with known SIF solution may be used. The specimen shall be precracked by fatigue loading to attain an initial crack of size ranging 0.45 (a/W ) to 0.55 (a/W ), where W is the specimen width. It Pc

Pop-in Pc

Load, P

Pc

(A)

(B)

Vg

(C)

Vg

Vg

Clip gage displacement, Vg Figure 4.12 Types of Load versus Clip Gage Displacement records obtained by CTOD testing.

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127

is recommended that W be as close as possible to the actual width of the component for which the analysis is done and W ¼ 2B, is preferred, although specimens having B ¼ W are allowed. Since the direct measurement of CTOD is very difficult, the value of d is calculated from the measured Vg by the formula: d¼

rðW  aÞVg rðW  aÞ þ a þ z

Where Vg clip gage displacement measured at a distance z from the load line, a is the initial crack size, and r is a plastic rotational factor, equal to 0.4 for SE(B) specimen and 0.46 for the CT specimen, as schematically depicted in Fig. 4.13. An important observation is that value of Vg comes from the rotation of the of the crack faces in addition to the crack-tip opening displacement, which may lead to erroneous estimations of dc. To avoid this Vg is separated into an elastic Ve and a plastic Vp as schematically shown in Fig. 4.14, and the elastic contribution to d is calculated by the theoretical equation: de ¼

ð1  v 2 ÞKI2 2Es0

Therefore d is calculated by:

Load line

z

Vg a

r(Wa) W

Figure 4.13 Specimen parameters to be measured for CTOD testing.

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A Practical Approach to Fracture Mechanics

Load, P

Pc

Vp

Ve

Vg Figure 4.14 Separation of Vg into elastic (Ve) and plastic (Vp) contributions.

d¼

rðW  aÞVp ð1  v 2 ÞKI2 þ 2Es0 rðW  aÞ þ a þ z

As in all other fracture toughness testing methods, the experimental result is an interim value dQc that has to be validated to be expressed as dc. The qualification criteria are: (1) Ligament size (W  a)  300dQc (2) Crack extension Dap < 0.2 mm þ 0.7dQc These requirements are established to consider dc a property insensitive to the in-plane dimensions of the specimen. However, dQc values that do not meet the above requirement can still be labeled as dc as long as they are applied for a specific geometry and size. The following example illustrates the calculations and qualification of a real-life CTOD testing. A CT specimen with the following dimensions, W ¼ 50 mm, B ¼ 20 mm, a ¼ 20 mm, made of API 5L X70 steel, was tested in laboratory obtaining the Load-Clip gage displacement curve shown below. The clip gage wedges were z ¼ 2 mm and the stable crack extension was Dap ¼ 0.25 mm. Determine the critical CTOD and check the validity of the test. Solution: From the Load versus Vg curve: Pc ¼ 32 kN, Vg ¼ 0.74 mm, Vp ¼ 0.33 mm.

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129

40 35

Load. kN

30 25 20 15 10 5 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Vg , mm

For the CT specimen is: KI ¼ Pc f(a/W)/(BOW) ¼ 0.032 MN*(7.28)/[0.02 mO(0.05 m)] ¼ 41.7 MPaOm dQc ¼ (1  v2) (KI)2/(2Eso) þ [r(W  a)Vp]/[r(W  a) þ a þ z] dQc ¼ (1  0.32) (41.7 MPaOm)2 *(1000 mm/m)/ (2*200,000 MPa*483 MPa) þ [0.46(50e20) mm*(0.33) mm]/ [0.46(50e20) þ 20 þ 2] mm dQc ¼ 0.008 mm þ 0.127 mm ¼ 0.135 mm The qualification criteria are: Ligament size (W  a)  300dQc;(W  a) ¼ 30 mm; 300dQc ¼ 300(0.135) ¼ 40.5 mm. Not met 0.2 mm þ 0.7(0.135) mm ¼ 0.295 mm; Dap ¼ 0.25 mm < 0.295 mm. The crack extension criterion is met Since the ligament size criterion was not accomplished, the dQc ¼ 0.135 mm will be valid as dc only for components of 25 mm thickness. Although the CTOD is a valid fracture toughness parameter, it cannot be directly used for defect assessment since it is not explicitly related to the crack size and the fracture load, so its most frequent practical application is for quality control, as the minimal specified crack-tip ductility that a material must have in order to perform satisfactorily in a given application. Nonetheless, the relative easiness of the CTOD testing and its application to

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A Practical Approach to Fracture Mechanics

every type of fracture, from brittle to fully plastic, has encouraged attempts to make use of the CTOD for the assessment of cracked components. For example, the ASTM E 1820 standard provides an equation to calculate the J-integral from the dc values: Jc ¼ dc ms0 Where m is a constant calculated by a  a  0 0 m ¼ 1:221 þ 0:793 þ 2:751n  1:418 n W W Where n is the strain hardening coefficient determined by the ASTM E646 standard or by the empirical relation n ¼ 1.724e6.098/Rs þ 8.326/ Rs2  3.965/Rs3, Rs ¼ suts/s0. This expression enables the use of dc for every application falling into the scope of the J-integral such as the R-Curve (energy release rate vs. crack extension), referred as the d-R curve. In 1971, Burdekin and Dawes of the Welding Institute of Great Britain introduced an assessment method for steel panels containing cracks based on the CTOD.5 The principle of the method is that if during ductile fracture a plastic zone of radius rp and length equal to the thickness is formed, the fracture will surely occur by ductile shear and therefore cleavage fracture will be prevented. Thus, if rp ¼ (K/s0)2/2p and d ¼ K2/(s0 E), s  0 dc ¼ 2prp E Since it is difficult to measure the stress after yielding, the term (s0/E) is replaced by the yield strain ε0 and if rp ¼ B, the condition to prevent brittle fracture becomes dc  2pε0 B Taking experimental data from laboratory and full scale tests, Burdekin and Dawes developed two empirical equations representing the upper

5

Burdekin, FM and Dawes, MG. “Practical use of linear elastic and yielding fracture mechanics with particular reference to pressure vessels”. Proceedings of Conference on Practical Application of Fracture Mechanics to Pressure Vessel Technology, London, 3e5 April 1971, pp 28e37.

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131

envelope of the experimental failure strain εf. To make the equations as general as possible they introduced the nondimensional CTOD.  2 εf εf dc F¼ ; for  0:5 ¼ 2pε0 a ε0 ε0 εf εf dc F¼ ¼  0:25; for > 0:5 2pε0 a ε0 ε0 The plot of this equations produces a graph like the one shown in Fig. 4.15, which is known as the “CTOD design curve.” The application of the CTOD design curve is as follows. 1. The assessment point is the applied strain divided by the yield strain (ε/ε0) that is entered in the abscissa axis, while (dc/2pε0a) is entered in the ordinate axis. If the point falls within the SAFE zone, the crack is stable, and the cracked component is accepted to continue into service. If the assessment point falls within the FAIL zone, the ductile fracture toughness has been surpassed and the cracked component is rejected. 2. If the crack size is known, the dc value determined form the design curve at the applied strain shall be the minimum required ductile fracture toughness of the material to prevent brittle fracture. 3. If the ductile fracture toughness of the material dc is known, the maximum allowable crack size for safe operation can be calculated for the applied strain. 1.80 1.60

Φ c = δc / (2ε 0a)

1.40

SAFE

1.20

Design curve

1.00

Determine the allowable crack size or required δc

0.80

FAIL

0.60 0.40

Enter the applied strain

0.20 0.00 0.00

0.50

1.00

1.50

εf / ε 0 Figure 4.15 The Welding Institute CTOD design curve.

2.00

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A Practical Approach to Fracture Mechanics

On important consideration of the CTOD Design curve is that it represents the upper envelope of experimental data and for that reason is conservative. Kamath6 estimated the reliability of the original CTOD design curve finding that the average safety factor is 1.9, but most important its confidence level is 97.5%. Based on this result, in 1980 the CTOD design curve was incorporated into the British Standard BS PD 6493:1980. To incorporate more complex stress distributions and the effect of stresses the applied strain is calculated by the following equation: 1 ε ¼ ½Kt ðsm þ sb Þ þ ss E where Kt is the elastic stress concentration factor, sm is the primary membrane stress, sb is the primary bending stress and ss is the secondary stress due to residual stresses and thermal expansion, among others. The following example illustrates the above procedure. A steel pipe is spooled onto a reel drum for subsea pipeline installation. During the spooling-on process the pipe is bent and tensioned for total strain of 0.003. The pipe field welds have an average dc ¼ 0.03 in. at the design temperature. Determine if the toughness sufficient to tolerate circumferential crack-like defects of length  2.0 in by the CTOD design curve. s0 ¼ 60 ksi, E ¼ 30,000 ksi. Solution: ε0 ¼ s0/E ¼ 60 ksi/30,000 ksi ¼ 0.002 (εf/ε0) ¼ (0.003/0.002) ¼ 1.5 (dc/2pε0a) ¼ 0.03 in/(2p*0.002*2.0 in) ¼ 1.19 (εf/ε0) ¼ 1.5 > 0.5, then F is calculated by: F ¼ (εf/ε0)  0.25 ¼ 1.5  0.25 ¼ 1.25 Since F calculated >F actual, the assessment point falls into the FAIL envelope of the CTOD design curve, the material toughness is insufficient to tolerate defects of 2.0 in. length.

6

Kamath, M.S., The COD Desing curve: An Assessment of Validity Using Wide Plate Tests. The Welding Institute Report 71/1978/E, September 1978.

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The minimum specified dc to tolerate defects 2.0 in. length can be calculated as follows: F calculated ¼ (dc/2pε0a), solving for dc: dc minimum  F (2pε0a) ¼ 1.25 (2p*0.002*2.0 in) ¼ 0.031 in.

4.5 The two-parameter criterion A different analysis for the determination of the CTOD was proposed by Dugdale in 19607 under the assumption that the plastic zone extends over the total length of the ligament, but as a narrow strip, as shown in Fig. 4.16. Dugdale assumed that the crack has a virtual extension, similar to Irwin’s correction for plasticity, so the effective crack length is: aeff ¼ a þ r Where r is the length of the plastic zone withstanding the yielding stress, so at the crack extremes K ¼ 0, and the applied K should be compensated by the plasticity introduced after yielding. Dugdale introduced a dimensionless CTOD, given by F¼

dE 2ps0 a

Uniform stress σ σ0

σ0

2a

Plastic zone

ρ

Figure 4.16 Dugdale’s strip yield model. 7

D.S. Dugdale. “Yielding in Steel Sheets Containing Slits.” Journal of the Mechanics and Physics of Solids, Vol. 8 (1960), pp. 100-104.

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A Practical Approach to Fracture Mechanics

By solving the equilibrium equations between the remote tension stress intensity factor and the stress intensity due to the closure stress nondimensional crack-tip opening displacement becomes   4 ps F ¼ log sec p 2s0 This solution reported good agreement with experimental results of ductile materials exhibiting little strain hardening. The strip yield model has been applied to simulate fatigue crack growth under constant loading amplitude and after overloads with success, but undoubtedly its most relevant application was on the development of the two-parameter criterion that will be presented in the next section. In 1975, Dowling and Townley8 developed a fracture criterion that combines the brittle fracture assessment by LEFM and the Dugdale criterion for elastic-plastic fracture. This approach is currently known as the twoparameter criterion and it allows analyzing by using a single equation, the fracture strength of cracked components, whether the fracture is brittle, elastic-plastic or fully ductile. Dowling and Townley recognized the three following scenarios of fracture: 1. Fast brittle fracture under a nominal stress lower than the yield strength. The fracture behavior is controlled by the crack-tip stress intensity factor. 2. Elastic-plastic fracture with the plastic zone extending over the entire ligament. This case can be analyzed by J-integral and CTOD approaches. 3. Plastic collapse. Here, the failure occurs by generalized plastic deformation of the ligament and the fracture stress is determined by the yield criteria and the net-section. In the first case, the fracture stress is given by: KIC sf ¼ pffiffiffiffiffi pa Note that increasing KIC, may imply that sf becomes greater than s0, so this equation is not valid for tough and ductile materials. If the fracture 8

Dowling, A:R: and Townley, C:H:A., The Effect of Defects on Structural Failure: A Two-Cirteria Approach, International Journal of Pressure Vessel and Piping, 3,pp. 77e107 (1975).

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135

occurs by ductile mechanisms, but the plasticity is limited to a narrow zone, the Dugdale strip yield model can be used:   8s0 ps dc ¼ a ln sec 2s0 pE Since dc ¼ KIC2/Es0, substituting the above equation along with sf what is obtained is:   8s20 ps 2 a lnsec KIC ¼ 2s0 p Introducing the parameters: Kr ¼ K/KIC Sr ¼ s/s0 Where K is the applied SIF, KIC is the Mode I fracture toughness, s is the applied stress and s0 is the yield strength. Substituting KIC ¼ [sO(pa)]/Kr and s0 ¼ s/Sr into Dugdales’s equation and solving for Kr:  p 1=2 8 Kr ¼ Sr 2 ln sec Sr p 2 Plotting this expression generates an envelope curve as shown in Fig. 4.17. The Kr versus Sr graph is denominated failure assessment diagram (FAD) and it represents the strength limit of a cracked component in terms of its resistance to brittle fracture, characterized by the fracture toughness and the ductile fracture strength characterized by the yield strength. The assessment point A represents a specific load and crack size condition of a material with known KIC and s0 values at the service conditions. Since Kr and Sr depend linearly on the load, as the load increases, the point A moves along a straight trajectory called load line. If the assessment point (A) is inside the envelope, the crack is stable and the component will not fail, but if the assessment point falls on or out of the curve (point F), the component is at risk of failure because the combination crack size-load-mechanical properties have reached instability conditions. The FAD allows the identification of the failure mode as schematically depicted in Fig. 4.18. If the assessment point is located on near the Kr axis, the fracture is brittle, but if it is near the Sr axis, the fracture is ductile. Also, the FAD the effect of the variation of material properties, if KIC increases

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A Practical Approach to Fracture Mechanics

1.20

FAIL

1.00

F

0.80

Kr 0.60

Load increases

A

Assessment point

0.40

SAFE

0.20 0.00 0.00

Load line

0.20

0.40

0.60

0.80

1.00

1.20

Sr Figure 4.17 Failure assessment diagram (FAD) by the two-parameter criterion.

1.20 Brittle Fracture 1.00 0.80

Elastic-plastic Fracture KIC p , Vo n

Kr 0.60 Assessment point Plastic Collapse

0.40 KIC n , Vo n

0.20 0.00 0.00

0.20

0.40

0.60

0.80

1.00

1.20

Sr Figure 4.18 Effect of fracture toughness and yield strength variation on the failure assessment diagram (FAD) assessment point.

Elastic-plastic fracture mechanics

137

and s0 decreases, the assessment point moves right and down, and the fracture is ductile and controlled by the net stress. But if KIC decreases and s0 increases, the assessment point moves left and upwards, so the fracture is brittle and controlled by the crack-tip stress intensity factor. In 1986, Milne, Ainsworth, Dowling, and Stewart submitted the CEGB R/H/R6 report in England, known as the R6 Code. Nowadays, this method is the basis for the assessment of crack-like flaws in practically every structural integrity codes and standards all over the world. The R6 Code establishes the tendency to exhibit any of the three fracture modes, determined by the slope of the load line: Kr/Sr > 1.8 Brittle fracture 0.2 < Kr/Sr  1.8 Elastic-plastic fracture Kr/Sr  0.2 Plastic collapse Extensive experimental programs by EPRI and other institutions resulted in the modification of the FAD curve to better fit experimental and real-life results. In the 1991 version of the R6 Report, the following options were introduced: Option 1: Kr ¼ (1  0.14 Sr2)[0.3 þ 0.7 exp (0.65 Sr6)] Option 2: Kr ¼ (Eεref /Sr s0 þ Sr3 s0/2Eεref )1/2 Option 3: Kr ¼ ( Je/J )1/2 Where εref is the true strain in uniaxial tension at the applied stress, J is the applied J-integral and Je is the elastic J-integral of the structural component. The new options of the FAD produce curves with a less slanted slope in comparison to the original two-parameter curve shown in Fig. 4.18. in addition, the limit of Sr has been extended to 1.6 for structural steels and 1.8 for stainless steels, to consider the effect of strain hardening. In general, the Sr limit is defined as   1 smax Srmax ¼ 1þ 2 s0 Since its publication as recommended practice in year 2000, the API RP 579 Fitness-For-Service recommended practice included the R6 Code

138

A Practical Approach to Fracture Mechanics

based FAD, with some modifications, to assess the severity of crack-like flaws. Later, in 2007 it became the assessment method for crack-like flaws in the 2007 and 2016 editions of the API 579e1/ASME FFS-1 “Fitness-For-Service” standard, as well as the BS 7910:2005 “Guide to methods for assessing the acceptability of flaws in metallic structures.” Fig. 4.19 illustrates the Option one FAD of the R6 Two-Parameter criterion for the assessment of cracked components along with the assessment procedure of cracked components established in the fitness-for-service standards. Note that Sr has been replaced for Lr, which stands for the ratio applied load (La) over limit load (Llim) in order to avoid limitations due to complex stress fields and to extend its coverage to components where the stress is not uniform or it is uncertain. The following example illustrates the use of the two-parameter criterion: An R ¼ 167.8 mm radius and t ¼ 11.3 mm thickness pipe transports a fluid at 8.3 MPa of internal pressure. An inspection revealed the presence of a long axial surface crack of 6 mm deep. The material properties are: s0 ¼ 500 MPa and KIC ¼ 80 MPaOm. Assume that the applied K ¼ 1.12sO(pa), s ¼ PR/t. Determine if the crack can be tolerated by the two-parameter criterion, and if the result is affirmative determine the failure pressure.

Material KIC

Flaw size

1.2 1.0

Loads

Kr Enter Ka and KIC Kr = Ka/KIC

0.8

FAIL

SAFE

0.6 0.4

Flaw size Material σo

Stainless steel cut-off

Assessment point

0.2 0.0 0.0

Carbon steel cut-off

0.4

0.8

Enter Load and Limit Load Lr = L/Llim

1.2

1.6

2.0

Lr Loads

Figure 4.19 Procedure of fitness-for-service assessment of cracked components by the Option 1 of the R6 two-parameter criterion and API 579e1/ASME FFS-1 Part 9.

Elastic-plastic fracture mechanics

139

Solution: s ¼ PR/t ¼ 8.3 MPa*137.8 mm/11.3 mm ¼ 123.25 MPa K ¼ 1.12s(pa)1/2 ¼ 1.12(123.25 MPa)(p0.006 m)1/2 ¼ 18.95 MPaOm Sr ¼ s/s0 ¼ 123.25/500 ¼ 0.247 Kr¼K/KIC ¼ 18.95 MPaOm/80 MPaOm ¼ 0.237 The assessment point (A) fall within FAD envelope constructed with the original two-parameter criterion, therefore the system pipe-crack under internal pressure is acceptable and the crack can be tolerated. Kr/Sr ¼ 0.237/0.247 ¼ 0.959, since 0.2 < Kr/Sr  1.8, the assessment point is the elastic-plastic failure region. 1.20 1.00 0.80

F

Kr 0.60 0.40

A

0.20 0.00 0.00

0.20

0.40

0.60

0.80

1.00

1.20

Sr

The failure pressure can be calculated simply by measuring the length of the line segments 0A and 0F an using the relation: Pf ¼ Pop(0F/0A) To simplify the calculation the Pythagoras theorem is applied taking the coordinates of A and F from the graph: A(0.247, 0.237); F(0.64, 0.9), thus 0A ¼ (0.2472 þ 0.2372)1/2 ¼ 0.342 0F ¼ (0.642 þ 0.92)1/2 ¼ 0.576 Pf ¼ 8.3 MPa*(0.576/0.342) ¼ 13.98 MPa

140

A Practical Approach to Fracture Mechanics

Of course there is an analytical solution by substituting Kr ¼ 0.96 Sr in the FAD equation: 0.96 ¼ {8/p2 ln[sec(p/2 Sr)]}1/2, and solving for Sr, but it is fairly complicated for the average field engineer, so the graphic solution is more convenient. After an in-line inspection of a 24 in. nominal diameter pipeline an internal axial crack was detected. The crack is in the base metal and it is one quarter of the nominal thickness depth (a ¼ 1/4 t), with an aspect ratio a/c ¼ 0.05, where c is the axial half length. The pipe is standard schedule, of t ¼ 0.375 in. thickness, D ¼ 24 in. external diameter and Di ¼ 23.25 in. internal diameter. The pipe material specification is API 5L X52, s0 ¼ 52 ksi, and KIC ¼ 100 ksiOin. The internal operation pressure for assessment purposes is P ¼ 900 psi at 70 F. Assuming that the aspect ratio of the crack remains constant at a/c ¼ 0.05 during its stable growth, determine the critical crack size by the Option 1 of the EPRI R6 Two Parameter assessment method. Solution: The next figure shows the basic geometry and dimensions of the pipe-crack system (the drawing is no at scale, dimensions in inches).

t = 0.375 D/2 = 12 2c = 3.75 Ri = 11.625

a = 0.094

P = 900 psi Center line (longitudinal)

The first step is the calculation of the parameter Kr and Sr of the initial crack: Conservatively, the crack profile is assumed to be rectangular, since it gives the highest K values and the SIF solution is the simplest available: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   c2 KI ¼ sm 1 þ 1:61 pa Rt Where sm is the membrane stress given by: sm ¼

PR t

Elastic-plastic fracture mechanics

141

Substituting values: sm ¼ (900 psi * 12 in.)/(0.375 in.) ¼ 28,800 psi ¼ 28.8 ksi KI ¼ 28.8 ksi *[(1 þ 1.61(1.8752 in2/(12 * 0.375 in2)) p(0.0938 in)]0.5 ¼ 23.5 ksiOin Kr ¼

KI KIC

Kr ¼ 23.5/100 ¼ 0.235 The Sr parameter is: Sr ¼

sref s0

Where sref is the reference stress, which can be calculated by the method proposed in the Annex D5 of the API RP 579 2000 for an internally pressurized thin wall cylinder with an axial internal crack, as follows: sref ¼ Ms sm a 1 1  0:85 t Mt  Ms ¼ a 1  0:85 t

0:5 2 Mt ¼ 1 þ 0:3797l  0:001236l3 1:818c l ¼ pffiffiffiffiffiffi Ri t Substituting values: l ¼ 1.818*1.875 in/(11.625 in*0.375 in)0.5 ¼ 1.6326 Mt ¼ [1 þ 0.3797(1.6326)2 e 0.001236(1.6326)2]0.5 ¼ 1.4154 Ms ¼ [1  0.85*(0.25)(1/1.4154)]/[1  0.85*0.25] ¼ 1.079 sref ¼ 1.079*28.8 ksi ¼ 31.08 ksi Sr ¼ 31.08 ksi/52 ksi ¼ 0.598

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A Practical Approach to Fracture Mechanics

The assessment point of coordinates Sr ¼ 0.598, Kr ¼ 0.235 fall within the R6 Option 1 envelope, therefore the crack is stable at the detection time. The next step is to compute the parameter Sr and Kr for a growing crack, assuming a constant aspect ratio a/c ¼ 0.05. The results are summarized in the next table: a

c

K

in

in

ksiOin

0.25

0.0938

1.875

23.47

0.235

31.08

0.598

0.50

0.1875

3.750

54.29

0.543

40.48

0.779

0.6

0.2250

4.500

69.53

0.695

47.05

0.900

0.75

0.2813

5.625

95.02

0.950

62.93

1.21

a/t

Kr

sref ksi

Sr

At this point it is more practical to visualize the assessment point in the failure assessment diagram to estimate how close to the critical crack size is each growing crack configuration. The next figure shows the results that are analyzed as follows: 1. The point for a/t ¼ 0.75 is outside of the envelope, therefore it corresponds to a failure condition. 2. The point for a/t ¼ 0.5 is still inside, so this crack size is still stable 3. The point for a/t ¼ 0.6 lies very close to the curve of the R6 Option 1, therefore it corresponds to the critical crack size. To verify the condition 3, the exact Kr coordinate on the assessment curve can be calculated for Sr ¼ 0.9, that corresponds to a/t ¼ 0.6. If the calculated Kr value is equal or greater that Kr ¼ 0.695, the assessment point is under the curve, and vice versa. The equation of the R6 Option 1 is: Kr ¼ ð1  0:14Sr 2 Þ½0:3 þ 0:7expð0:65Sr 6 Þ Substituting values: Kr ¼ (1  0.14(0.9)2)[0.3 þ 0.7 exp(0.65(0.9)6)] ¼ 0.705 This indicates that the assessment point for a/t ¼ 0.6 is almost on the envelope curve (99% close), therefore it is reasonable to assume that the critical crack size is: a ¼ 0.6 t ¼ 0.6*0.375 in ¼ 0.225 in. 2c ¼ 2(a/0.05) ¼ 2(0.225/0.05) ¼ 4.5 in.

Elastic-plastic fracture mechanics

1.2 (1.21, 0.95), a/t = 0.75

1.0 0.8

(0.900, 0.695), a/t = 0.6

Kr 0.6

(0.778, 0.543), a/t = 0.5

0.4 (0.598, 0.235), a/t = 0.25

0.2

Initial assessment point

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Sr

1.2

1.4

1.6

1.8

143

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CHAPTER 5

Fracture resistance of engineering materials

5.1 Remaining strength The main contribution of linear-elastic fracture mechanics (LEFM) is that it allows analyzing the interaction between fracture toughness, applied load, and crack size in order to provide an answer to the first two fundamental questions of fracture mechanics: 1. What is the fracture strength? 2. What is the maximum tolerable crack size? The fracture load of a cracked component is the remaining strength, and the maximum tolerable crack size under a given load is the critical size. The third fundamental question of fracture mechanics is the remaining life, but this concept will be discussed in Chapters 6 and 7. This chapter will focus on the remaining strength of engineering components containing cracklike flaws. As explained in Chapter 1 (see Section 1.3, Fig. 1.7), every structural or mechanical component is designed for a maximum allowable stress, called the “design limit,” but since normally the vast majority of the in-service components, such as building structures, machinery, transport vehicles, load lifting equipment and so on, are normally operated at a somewhat lower level, the remaining strength shall be understood as an operational paramenter. Based on these concepts, the remaining strength can be defined as the maximum stress at which a component containing defects can be operated without fracturing; therefore the fracture stress can be regarded as the remaining strength. To calculate the remaining strength (fracture stress) and the crack critical size, the general expression for the stress intensity factor can be used, as follows.

A Practical Approach to Fracture Mechanics ISBN 978-0-12-823020-6 https://doi.org/10.1016/B978-0-12-823020-6.00005-0

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145

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A Practical Approach to Fracture Mechanics

First, consider the general equation of the stress intensity factor: pffiffiffiffiffi K ¼ Y s pa where Y is the geometric factor, s is the applied stress, and a is the crack size. The fracture load (or failure load) is obtained by replacing K by KIC and solving for s, to obtain the fracture stress: KIC sf ¼ pffiffiffiffiffi Y pa Plotting this equation as a function of the crack size gives a curve similar to that shown in Fig. 5.1, which is referred to as the remaining strength curve (RSC). The RSC is not valid over the entire range of crack sizes, because when a / 0, sf / Ndwhich obviously is not possible, because eventually sf will reach the ultimate tensile strength and the component will fail by overload. Also, when a / N, sf / 0dwhich, again, is not possible because at some point the stress in the net section will be equal to applied stress. The previous analysis makes it necessary to determine the range of flaw sizes for which the RSC calculated by fracture mechanics is valid. A first approximation to determine such limits is the net section criterion, which consists of the following. Consider a finite plate of width W, with a central crack of length 2a, under a uniform applied stress (sA), as shown in Fig. 5.2. The nominal applied stress sA is

σf (Failure stress)

sA ¼

P BW

σf → ∞, when a → 0

σf = KIC / [Y√ a]

σf → 0, when a → ∞

a (crack size)

Figure 5.1 Schematic variation of the fracture stress or remaining strength of a structural component containing a crack-like flaw, under uniform stress and linear elastic conditions.

Fracture resistance of engineering materials

147

P

Cross-section (WB)

Net section (W-2a) B

Crack

2a

B W

Figure 5.2 Finite width plate with a central crack under a uniform stress.

Obviously, the crack impedes the transmission of internal forces, so it reduces the width of the cross section by an amount (W  a), referred as the ligament. The uncracked area is called net section, and the stress on it becomes sN ¼

P BðW  2aÞ

where sN is the net section stress, B is the thickness, and a is the half crack length. Substituting the applied stress sA ¼ P/BW, and multiplying, and dividing by W, the applied stress becomes   2a sA ¼ sN 1  W Introducing the failure criterion as the condition when the net section yields, sN shall be replaced by the yield strength (sys)1, but if the failure criterion is when the net section collapses, sN shall be replaced by the ultimate tensile strength (suts); therefore, the failure stress becomes   2a sF ¼ sðys;utsÞ 1  W The plot sf versus a of the net section crtiterion is a straight line of negative slope, whereas the plot of the failure stress controlled by KIC is a hyperbole. Superimposing both plots and taking the criterion that the 1

From here, the symbol for the yield strength s0 will be replaced by sys to avoid confusions with other stress symbols used herein.

148

A Practical Approach to Fracture Mechanics

residual strength is the least value of sf in each case, an RSC corrected by the net section is obtained, which has the form shown in Fig. 5.3. This figure shows that the region of valid remaining strength estimation by linear elastic fracture mechanics is between the regions where the fracture is controlled by the net section stress. Furthermore, the condition where the remaining strength is calculated by fracture mechanics is called the crack tip control zone, whereas when the strength is calculated by the net section stress is called the net section control zone. The RSC should be completed with two more parameters. The first is the critical crack size (ac), which is determined by solving the K equation for the crack size, but substituting the applied stress by the normal operation stress (sop), thus, ac ¼

2 KIC ðpY sop Þ2

The next important parameter of the RSC is the minimum allowable crack size by design, identified as amin, which is the crack size that makes the remaining strength equal to the design stress. It is calculated by substituting the design stress sd and the fracture toughness into the remaining strength equation, and solving for the crack size, being amin ¼

2 KIC ðpY sd Þ2

Since it is assumed that the component is not going to be operated above the design stress, the presence of a crack smaller than amin does not imply any risk of failure and, therefore, it is safe to operate a component σ uts or σ ys

Failure stress, σ f

σf = (σ ys or σ uts){1 – (2a/W)} σf = KIC /{Y√ a} Net section control

Net section control

Crack tip control

0

Relative crack size (a/W)

1.0

Figure 5.3 Schematic remaining strength curve showing the limits of crack tip and net section control.

Fracture resistance of engineering materials

149

containing defects of such size or smaller. Actually, this is how the defect tolerances in new components are determined in the design and construction codes. Finally, the range between amin and ac defines a condition where a component containing crack-like defects may be safe to operate or it has to be withdrawn from service, depending on whether or not the remaining strength is above or below the operating stress level, enabling the application of fitness-for-service approaches, as discussed in Chapter 7 of this book. Fig. 5.4 shows a schematic representation of an RSC in the form adequate to assess the effect of the presence of crack-like flaws in structural and mechanical components. One approach to assess the likelihood of failure in the RSC when the calculated remaining strength is above the normal operation level and the crack-like flaw is greater than amin, but less than ac, is the margin for safe operation (MSO), which can be calculated as: sf  sop MSO ¼  100 sd  sop Under this perspective, if sf ¼ sd, MSO ¼ 100%, which means that the original design safety margin is preserved, while if sf ¼ sop, MSO ¼ 0%, and the component is at an imminent risk of failure and a corrective action σf Design stress

Failure Remaining strength

Safe (Crack tolerance)

Normal operation level

Minimum allowable crack size

Margin for safe operation

Fitness-For-Service assessment region Detected crack size

Crack size, a

Critical crack size

Figure 5.4 Schematic remaining strength curve adequate to perform the assessment of crack-like flaws in structural components.

150

A Practical Approach to Fracture Mechanics

should be taken, which may be reducing the operation stress level, or repairing or installing a reinforcement. If none of these actions can be taken, the component has to be withdrawn from service. Despite the fact that the MSO provides a practical way to quantify how far a failure event is in a fitness-for-service decision-making process, many standards and codes prefer to make such assessment based solely on the design stress; for that purpose they introduce the remaining strength factor (RSF), defined by sf RSF ¼ sd Under the RSF approach, if sf ¼ sd, RSF ¼ 1.0, meaning that the component is retaining its full design strength. Any value of RSF < 1.0 represents a loss of strength below the design level, so an allowable RSF (RSFa) has to be stablished in order to determine whether or not a corrective action should be taken. In pressure vessels and piping fitness-forservice standards and codes, an RSFa between 0.7 and 0.9 is proposed, meaning that only strength reductions of 10% to 30% are accepted. This may not look like much, but in practice may signify very important reductions of maintenance frequency that are reflected in great cost reductions. The following examples illustrate the practical use of the RSC. Determine the minimum allowable size of an embedded elliptical crack-like flaw in a weld seam of thickness B > 1.0 in. under a uniform tension stress by the RSC approach. Assume that the maximum allowable working stress by the design code is 40 ksi and the minimum specified KIC is 40 ksiOin. Solution: The geometry and dimensions of the system are as indicated in the next figure, the plate width is assumed to be infinite, and (a/c) ¼ 0.25 constant. σ

B

2a 2c

Weld fusion line

Fracture resistance of engineering materials

151

The stress intensity factor is

rffiffiffiffiffi pa KI ¼ s Q a1:65 Q ¼ 1 þ 1:464 c

Substituting a/c ¼ 0.25, Q ¼ 1.1486, and KIC ¼ 40 ksiOin KI ¼ s(pa)0.5/Q0.5 Therefore, the remaining strength is sf ¼ 42.869/(pa)0.5 Since the plate width is infinite, the net stress is equal to the applied stress, therefore the failure stress is the maximum allowable working stress. Based on the above results the RSC is Remaining strength, σ f [ksi]

120 100 80 60 40

Max. Allow. Stress = 40 ksi

20 0

2amin = 0.731 in 0.00

0.20

0.40

0.60

0.80

1.00

1.20

Crack diameter in the thickness, 2a [in]

A cracked panel 100 cm wide and 5 cm thick with the stress intensity factor given by KI ¼ 1.12sO(pa) is under a uniform stress of 100 MPa. If the yield strength is 500 MPa and the fracture toughness is 80 MPaOm, determine the following: If a crack of size 2a ¼ 20 cm may be tolerated: By crack tip control: KI ¼ 1.12 (100 MPa) (p * 0.1 m)0.5 ¼ 62.8 MPaOm By net section control: sf ¼ 500 MPa(1 e 0.2 m/1 m) ¼ 400 MPa

152

A Practical Approach to Fracture Mechanics

Since KI < KIC, and sf > sA the crack is tolerable. What is the remaining strength for the 2a ¼ 20 cm crack? sf ¼ KIC/(1.12Opa ¼ 80/(1.12Op(0.1)) ¼ 127.4 MPa Since sf calculated by net section control is 400 MPa, the remaining strength corresponds to the calculated by crack tip control, 127.4 MPa. This stress is above the normal operation stress (100 MPa), the operational safety margin (OSM) is: OSM ¼ {(127.4  100)MPa/(200  100)MPa}  100 ¼ 27.4% The RSF is RSF ¼ 127.4 MPa/200 MPa ¼ 0.637 Construct the RSC of a plate with a central crack, W ¼ 20 in, t ¼ 1 in, if KIC ¼ 100 ksiOin and sys ¼ 100 ksi. The design stress is 60 ksi and the normal operation level is 20 ksi. Solution: The remaining strength by crack tip control is calculated as sf ¼ KIC/W0.5 f(a/W) The values of f(a/W) are calculated with a commercial stress intensity factor calculator. The net section remaining strength is sf ¼ sys (1 e 2a/W). The RSC is 10 9 8

σ f /σ op

7

Crack tip control

6 5

Net section control

4 3

Operation stress

Desgin stress

2 1 0

0

0.1

Minimum allowable crack size

0.2

0.3 a/W

0.4

0.5

0.6

Critical crack size

So far, the remaining strength has been treated in a rather simplistic approach, because the calculation based on the net section criterion initially seems to be logic and reasonable, but in real components may be much more complicated. The procedure to be applied in real-life cases, especially

Fracture resistance of engineering materials

153

where there is a high risk of suffering catastrophic losses, would be to carry out a precise assessment by analytical and computer assisted tools and carefully determining the realistic values of mechanical properties and applied loads at the actual service conditions (temperature, environment, strain rate, and etcetera). Nonetheless, in practical cases under short time and financial resources, the RSC is a valuable approach to perform quick and easy estimations, validate the reasonability of prioritizations for corrective actions, and as a didactic tool to educate trainees in fracture mechanics and structural integrity.

5.2 Materials selection for fracture resistance At this point it is clear that the remaining strength of cracked bodies depends primarily on fracture toughness, but as well on the yield strength. It is known that for the same material and temperature, fracture toughness and yield strength are inversely related, as shown in Fig. 5.5. The use of high toughness material displaces the RSC upward and to the right, expanding the net section dominance zone whereas the use of hard materials (high s0 and low KIC), displaces the RSC downward and to the left, expanding the crack control zone. The behavior described above makes it necessary to establish quantitative criteria for material selection that consider the interrelations among mechanical and fracture properties, as well as other physical properties. Table 5.1 provides a general recompilation of the fundamental mechanical 200 Fracture toughness [MPa √m]

180 160

Structural steel

140 120 100 80 Heat treated aluminum alloy

60 40 20 0 0

200

400 600 Yield strength [MPa]

800

1000

Figure 5.5 Fracture toughness as a function of yield strength for structural steel and heat-treated aluminum alloys. Approximated data.

Table 5.1 Typical values of yield strength and plane strain fracture toughness of common engineering materials. Group

Material

E (GPa)

sys (MPa)

suts (MPa) KIC (MPaOm)

Ferrous alloys

Low C steel

200e215

250e400

340e580

40e80

Medium C steel

200e216

300e900

400e1200

12e92

High C steel

200e215

400e1200

550e1640

27e92

Low alloy steel

201e217

400e1100

460e1200

14e200

Stainless steel

189e210

170e1000

480e2240

62e280

Cast iron

165e180

215e790

350e1000

20e55

68e82

30e500

60e550

22e35

Copper

112e148

30e500

100e550

30e90

Nickel

190e220

70e1100

345e1200

80e110

Titanium

90e120

250e1250

300e1600

14e120

Zinc

68e95

80e450

130e520

10e100

Epoxy matrix composites

Fiberglass

15e28

110e192

130e240

7e23

Carbon fiber

69e150

550e1050

500e1050

6.1e88

Polymers

Natural rubber

0.0025 max

20e30

22e32

0.15e0.25

Neoprene

0.002 max

3e24

3e24

0.1e0.3

Elastomer (PU)

0.005e0.02

25e50

25e50

0.2e0.4

Polycarbonate

2e3

60e70

60e70

2.1e4.6

Polyethylene

0.6e0.4

18e29

21e45

1.44e1.72

2e4

35e52

40e65

1.46e5.12

Flexible foam

0.001e0.012

0.01e3

0.01e3

0.005e0.09

Rigid foams

0.02e0.5

0.3e12

0.5e12

0.002e0.91

Alumina

215e413

690e5500a

Nonferrous alloys

Aluminum

PCV

Ceramics

SiC WC Glass

Borosilicate Silica Soda glass

Rocks

Clay brick Concrete

Bio materials a

300e460 600e720 61e64

350e665

3.3e4.8

1000e5250

a

370e680

2.5e5

3350e6800

a

370e550

2e3.8

22e32

0.5e0.7

45e152

0.6e0.8

30e55

0.55e0.7

7e14

1e2

2e6

0.35e0.45

5e17

0.7e1.5

260e380

a

68e74

1100e1600

68e72

360e420

a

10e50

50e140

a

25e38

a

32e60

a

a

Stone

7e21

34e298

Wood

60e20

30e70

60e100

5e9

Leather

0.1e0.5

5e10

20e26

3e5

Note: The yield strength of ceramics, glass, and rocks is replaced by the compression strength, since these materials do not exhibit a well-defined yield limit.

Fracture resistance of engineering materials

155

properties (Young’s modulus, yield strength, ultimate tensile strength) and the fracture toughness for several common metallic alloys and engineering materials. Note that the Young’s modulus is several hundred up to thousands of times higher than the tensile strength, whereas fracture toughness varies in the range of 0.1 MPaOm for plastic foams, up to 280 MPaOm for stainless steels. As already mentioned, the analysis by fracture mechanics considers interrelated factors that can be grouped as: 1. Material properties (continuum mechanics and fracture mechanics) 2. Geometry (shape and dimensions) 3. Loads (magnitude, direction and point of application) The most common situation in structural analysis is that the design engineer sets the load and the geometry, leaving to the materials engineer the task of selecting the appropriate material for a given application. The selection of a material for a fracture proof design is based on the fracture resistance, namely KIC, JIC, CTODc, and so on. It should be remembered that these properties are related to other mechanical fundamental mechanical and physical properties, the most important being yield strength (sys) and the Young’s modulus (E), in addition to the effect of environment and temperature. In conclusion, materials selection is not a simple matter, since it obeys several interrelated factors; as illustrated in the next example, it is almost impossible to modify a mechanical material property without affecting other properties required by the design. A component made of a steel plate will be subjected to a maximum design stress of 900 MPa. The maximum allowable crack size is 3 mm if KI ¼ 1.15sOpa. Select a suitable material for this application. According to the relation KI ¼ 1.15sOpa, the minimal required toughness is KIC ¼ 1.15  900 MPaOp(0.003m) ¼ 100.5 MPaOm The AISI SAE 4340 normalized steel meets this requirement, since its KIC ¼ 110 MPaOm. However, in this condition sys ¼ 860 MPa, which is insufficient to withstand the applied stress. An engineer suggests using the same steel 4340, but in quenched and temper condition, sys ¼ 1200 MPa, and KIC ¼ 70 MPaOm, which results in a critical crack size: ac ¼ (KIC)2/[1.15p(s)] ¼ 702 MPa2 m/[1.15p(900 MPa)2] ¼ 0.0017 m (1.7 mm) This critical crack size is almost half the size of the maximum allowable crack size required (3 mm), therefore the quenched and tempered AISI SAE 4340 is not correct for the intended design and another alternative material must be sought.

156

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Another important factor on the selection of materials for fracture resistance is the temperature. In general, fracture toughness increases with temperature, which is good for high temperature applications as long as the material does not go into a creep regimen, but the real problem is going down to low temperatures referred as cryogenic service. A case of particular interest is ferritic steels, which experience a ductile-brittle transition, which is a sudden change of toughness in a short temperature interval. The fracture toughness of austenitic steels, copper, aluminum, and nickel alloys, however, is little affected by temperature. The ductile-brittle transition of steels, as well as the effect of temperature on the fracture toughness on austenitic stainless steels, is shown in Fig. 5.6. The assessment of brittle fracture likelihood is a very important topic for both new and in-service components that may be exposed to low temperatures such as cold weather, shock chilling, or low water temperature during hydrostatic testing. The brittle fracture of pressure vessels, pipes, and structures resulting in catastrophic failures is a frequent risk in many process and power generation industries, so it has been included in the ASME S. VIII pressure vessel design code as well as other important design codes, and it is one of the parts of the API 579-1/ASME FFS-1 fitness-for-service standard. Sections III and IX of the ASME S. VIII code and the Annex F of the API 579-1/ASME FFS-1, Edition 2007, provide a correlation for the 180

Stainless steel (austenitic)

Fracture toughness [MPa√m]

160 140

Ferritic

120 100 80

Ductile-Brittle transition

60 40 20 0 -100

-50

0

50

100

Temperature [˚C]

Figure 5.6 Dependency of fracture toughness on temperature of ferritic steels and austenitic stainless steel. (Data from author’s unpublished research and several other sources.)

Fracture resistance of engineering materials

157

estimation of fracture toughness as a function of temperature of ferritic steels, which has the following form: KIC ¼ 36.5 þ 3.084 exp[0.036 (T e Tref þ 56)] (MPaOm,  C) where Tref is the temperature at which the Charpy impact energy is equal to 20 J (15 lb-ft). A similar correlation is used for hydrogen-saturated steels: KIR ¼ 29.5 þ 1.344 exp[ 0.026(T e Tref þ 89)] (MPaOm,  C) It is important to mention that the fracture toughness values calculated with these expressions are a lower bound, because they are obtained with the minimum values reported in the literature, so the user must be cautious as they may result in conservative fracture resistance evaluations. The most important criteria to perform an appropriate selection of materials for fracture resistance and crack-like flaw tolerance are: 1. Minimum weight 2. Maximum load bearing capacity 3. Control of elastic strain (stiffness or flexibility) 4. Yield before fracture 5. Leak before break The development of each one of these criteria is described next. The minimum weight criterion refers to the selection of a low-density material to minimize the weight, but at the same time provide the maximum fracture resistance. The selection of a material that meets such requirements can be done by the following analysis. First, if LEFM conditions are meet, the governing equation for the design may be pffiffiffiffiffiffiffi KIC ¼ Y sd paa where sd is the design stress, which is the maximum allowable stress that can be applied to the component maintaining the safety factor; aa is the maximum allowable crack size; and KIC is fracture toughness. It is important to remember that if the component is in plane strain condition, KIC corresponds to the ASTM standard testing, or equivalent; otherwise, the KIC for the corresponding thickness should be used. If the applied stress is s ¼ P/A, where A is the cross-section area, and P is the applied load, multiplying both sides of the equation by L/w, where L is the characteristic length and w is the weight, the governing equation becomes

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KIC

 w pffiffiffiffiffiffiffi w  pap ¼ YP L AL

The term w/AL is the mass density (r), so the preceding equation can be written as w  pffiffiffiffiffiffiffi KIC ¼ YPr pap L Reordering terms,

pffiffiffiffiffiffiffi YPL pap KIC ¼ r w

According to this equation, the maximum load, largest crack size, and lowest weight depend on the KIC/r ratio, therefore if two materials are compared, the best material for high fracture resistance with minimum weight is the one having the highest KIC/r ratio. The damage tolerance criterion refers to having the longest possible crack in the component, and yet be stable under the applied stress. This criterion can be expressed as  2 1 KIC amax ¼ pY 2 sys This implies that the best material for maximum damage tolerance is the one with the highest KIC and the lowest sys. Since most engineering materials feature high KIC and low sys combinations, the previous criterion is easily fulfilled. However, the previous criterion has to be taken with caution, since a low yield strength will affect the load-bearing capacity, such as the case of loading wire rope cables used in cranes and other elevation equipment. The rope cables are fabricated of cold drawn high carbon steel wires that usually have high sys values (typically over 200 ksi), which means low KIC (less than 40 ksiOin); this wires will not tolerate surface flaws because the ratio (KIC/sys)2 is very small, and since the wires in the external strands can be easily damaged, either by mechanical damage, over flexion or corrosion, the cables should be frequently inspected and properly maintained to avoid failures caused by wire fractures. In conclusion the selection of the optimum material for load bearing uses should take into consideration both properties (sys and KIC) simultaneously. The RSC is a very useful tool to solve the above problem. As it can be seen in Fig. 5.7, the RSC of material A, which has a high sys and low KIC,

Remaining strength

Fracture resistance of engineering materials

159

Material A

Material B

Normal operation level

Critical crack size, material A

Critical crack size, material B

Figure 5.7 Schematic remaining strength curve of a high yield strength/low toughness material (A) and low yield strength/high toughness material (B), showing that material A has a high load-bearing capacity, but low crack tolerance, while material B behaves the opposite.

shows that this material has a greater load-bearing capacity, as long as the crack-like flaws contained in the component are short, like rope cables, whereas material B, which has a high KIC and low sys, exhibits a low loadbearing capability, but high crack tolerance, which makes it a better option if the component will perform under medium to low loads and cracking damage is expected, such as liquid transportation pipelines. Furthermore, many engineering designs require that the component strains plastically before fracturing, because plastic strain is readily detectable, even at plain sight, or it can cause a noticeable but still safe malfunctioning, so plastic deformation becomes a self-warning signal that may help to prevent catastrophic failures. In Fig. 5.7, it is observed that material B has a wide net section control zone, meaning that ductile and tough materials that have both high sys and KIC, are appropriate for attaining the condition of plastic strain before brittle fracture, which, additionally is a much more desirable condition than catastrophic brittle fracture. In many engineering material applications, components frequently need to have high stiffness, such as columns, beams, support legs of heavy machinery, and so on, where an excessive elastic strain or flexibility is undesirable. The opposite occurs with elastic members, such as helical and leaf springs, that require a great flexibility, but without scarifying mechanical resistance. In both cases, the resulting stress must be kept under the yielding

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condition so they behave within the elastic deformation regime. The material selection criterion for controlled elasticity (stiffness or flexibility) can be established in the following way. In an elastically strained body, the stored elastic energy U is 1 1 s2 U ¼ sε ¼ 2 2 E The fracture stress is given by KIC sf ¼ pffiffiffiffiffi Y pa Combining both equations,

 2 Y 2 KIC U¼ 2pa E

From this expression, is clear that for a given crack size, the optimum material for high stiffness will be the one having the maximum K2IC/E ratio, because such material will store the largest amount of elastic strain energy, but at the same time will have high fracture strength. However, when the component has to have sufficient elastic strain, but without breaking apart, as is the case of springs and elastic leafs the material selection criterion of elastic strain-fracture resistance can be obtained by using the Hook law: selastic ¼ Eεelastic Replacing selastic by the fracture stress sf calculated by LEFM, the Hook law can be expressed as: KIC εf E ¼ pffiffiffiffiffi Y pa Rearranging terms, 1 KIC εf ¼ pffiffiffiffiffi Y pa E In this case, a high KIC/E ratio is required to assure the maximum elastic displacement but preventing fracture. An interesting criterion suited for pressure vessels and piping is the leak before break (LBB), which refers to a crack passing through the wall of pressurized vessel before instability, so the resulting leak can be detected and

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161

corrective actions can be taken to reduce the risk. To attain this condition, a surface crack has to become a through-thickness crack before reaching instability. Based on LEFM, the critical crack size in the radial direction of the cylinder wall can be expressed as:  2 1 KIC ac ¼ p sys Substituting ac for the thickness t the criterion becomes  2 KIC > pt sys Typically pressure vessels and piping in the process and power generation industries are fabricated of structural steel, typically having KIC > 110 ksiOin, and sys > 55 ksi; this combination results in values of (KIC/ sys)3 > 4.0; this indicates that, at first, the LBB condition may be attained by components of thicknesses under (4.0 in.)/p ¼ 1.27 in., which is fairly common. However, the achievement of the LBB condition is much more complicated than this simple reasoning, as it will be seen at the end of this chapter. Table 5.2 summarizes the previous criteria for material selection considering the combinations between fracture toughness and mechanical properties aimed to achieve maximum fracture resistance and/or crack-like flaw tolerance and the most common mechanical and physical material properties used in engineering design. The following example illustrates the performance of common engineering materials according to the above criteria.

Table 5.2 Material selection criteria based on fracture strength and/or crack-like flaw tolerance. Design criterion

Maximum ratio

Minimum weight

KIC/r

Cracking tolerance

(KIC/sys)2

Maximum stiffness

K2IC/E

Maximum flexibility

KIC/E

Leak before break

(1/p)(KIC/sys)2 > Thickness

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Classify the best option for the common design criterion among the following materials: Material

E [GPa]

sys [MPa]

KIC [MPaOm]

r [103 kg/m3]

Steel alloy

200

500

110

7.85

Aluminum alloy HT

72

500

20

2.7

Titanium alloy

100

1100

60

4.5

Composite

100

600

40

1.2

Polymer

20

30

3

0.9

Elastomer

0.5

30

1.2

0.9

Concrete

20

35

0.7

2.4

The above data is substituted into the material selection criteria ratios presented in Table 5.2, obtaining the following results. The resulting ratios were multiplied by 1000 (103) in some cases to obtain easyto-compare numbers. Weighttoughness

Cracking tolerance

Maximum stiffness

Maximum flexibility

KIC/r

103(KIC/sys)2

103(KIC)2/E

103KIC/E

Steel alloy

14.01

48.4

60.5

0.55

Aluminum HT

7.41

1.6

5.55

0.27

Titanium alloy

13.33

2.98

36

0.6

Composite

33.33

4.44

16

0.4

Polymer

3.33

10

0.45

0.15

Elastomer

1.33

1.6

1.08

2.4

Concrete

0.29

0.4

0.025

0.035

Material

According to the above data, the steel alloy is, overall, the best material for fracture resistance, since its selection material ratios are among the top values in the table. It is interesting to notice that the steel alloy of this example, which corresponds to a high-strength specification, has a better weight-toughness ratio than the heat-treated aluminum alloy, which is against common sense, since aluminum is a light material often selected for weight saving uses. Furthermore, the best material for weight-toughness

Fracture resistance of engineering materials

163

ratio is the composite, which in combination to its excellent strengthdensity ratio makes it ideal for aerospace applications. Titanium is considered a super-material in the popular belief, however in the above example, it is a rather modest material, outstanding only in the flexibility-toughness criterion. Nonetheless, titanium is still a valuable material in engineering because of its great corrosion resistance. Again, the best material for damage tolerance was the high-strength steel alloy. However, it is interesting to observe that most materials used in this example have a poor performance in crack tolerance, as compared to the polymer, which resembles a high-density polyethylene (HDP). That is one of the reasons why HDP is gaining a place in car body parts, pipeline and storage tank applications. Finally, it is evident that concrete is the worst material for fracture resistance, cracking tolerance, and elastic flexibility, even though is the most used engineering material for building and support structure construction in terms of tonnage per year. The reasons are its great compressive strength, low cost, availability, and ease of use. To compensate for its low fracture strength, it is used in combination with steel reinforcing bars. Nonetheless, concrete structures are carefully designed to avoid flexions and the tolerance of cracking is very strict.

5.3 Material properties charts The fracture properties (KIC, JIC, CTODc, etc.) and mechanical properties (sys, suts, E, etc.) cover a wide range of values for different engineering materials. In order to facilitate the selection of materials, as well as to visualize the relations among properties, Ashby2 introduced the charts of material properties (CMP), which are graphs of two properties with different materials grouped in balloons. Since material properties vary over a wide range, the CMP is in logarithmic scale. Fig. 5.8 shows a reproduction of the CMP of fracture toughness versus yield strength, which is of particular interest for the scope of this book. The data to construct a CMP come from handbooks, research reports, and technical articles, and are validated by comparison with more than one source. In the KIC versus sys chart, it is observed that the range of fracture toughness is from 0.01 MPaOm to more than 400 MPaOm, whereas the 2

Ashby, M., Shercliff, H., and Cebon, D. Materials Engineering, Science, Processing and Design (2007). Elsevier Ltd.

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A Practical Approach to Fracture Mechanics

1000

100 mm Stainless steels

Steels

10 mm

Fracture Toughness [MPa√ m]

1 mm Ni and Cu alloys

100

Al alloys

Metals 10

Ti alloys

Polymers

Wood

Technical ceramics

1

PP, PE, PTE, PS, Epoxic, Phenolic

Concrete

KIC2/σ 0

Elastomers

0.1 Flexible polymer foams

Rigid polymer foams

0.01 0.1

1

10

100

1000

Yield Strength [MPa] Figure 5.8 Reproduction of an Ashby’s material chart. The dotted lines show the critical crack size. Graphic constructed from data of several sources.

yield limit varies from 0.1 MPa to more than 1000 MPa. Metals are located in the upper right corner, which by far, makes them the most fracture resistant materials. In the lower left corner are the foams, which are the weakest materials. In addition to displaying the ranges of properties, the CMP can identify materials with similar performance. For example, the materials that can tolerate the same crack size (i.e., the same damage tolerance) can be identified as follows. The critical condition of fracture is pffiffiffiffiffiffiffi KIC ¼ sys Y pac Introducing logarithms pffiffiffiffiffiffiffi log KIC ¼ log sys þ log Y pac For a plot log(KIC) versus log(sys), the equation is a straight line of slope 1.0; therefore, all materials touched by this line in the CMP will have the same critical crack size ac, meaning that they will have the same tolerance to

Fracture resistance of engineering materials

165

cracking. The dotted lines in Fig. 5.8 represent this criterion for different crack sizes. From these it is observed that metals, polymers, concrete, and foams can tolerate cracks up to 10 mm long; therefore, for the same load level, therefore, they will have the same performance. Similarly, the guideline for the maximum stiffness criterion, given by (KIC)2/sys, si it has a slope of 0.5; so, like in the previous case, the materials along a line of slope equal to 0.5 will perform the same way under elastic strain conditions. Additionally, the materials located above a straight line in the CMP will have a better performance that those on or below the line. In many engineering applications the allowable stress level is about 140 MPa, and the minimum required fracture toughness is about 40 MPaOm. If the allowable crack size is 1 mm, from Fig. 5.8 it is clear that steel and few other metal alloys may be the only materials that can fulfill these requirements.

5.4 Failure analysis using fracture mechanics As mentioned in Chapter 1, one of the most important applications of fracture mechanics is the engineering analysis of fractured components; in fact, it can be asserted that the study of fracture began with the aim to explain why structural components fail at stresses below the material’s tensile strength, either ultimate or yield. A description of the basic principles of the analysis of structural component failures based on fracture mechanics is presented in the next paragraphs. The general linear elastic fracture mechanics equation is pffiffiffiffiffi KI ¼ bs pa If the specific KI solution is known, there are three possible scenarios for the application of fracture mechanics to failure analysis: 1. If the fracture toughness of the material and the crack size at the onset of failure are known, the above equation allows determining the stress level just before fracture. This would indicate whether there was an overload or if the operation was within the design limits. The crack size just before the final fracture, which is the critical size, is easy to determine by fractographic techniques, and the fracture toughness can be determined by laboratory testing of extracted samples or from published data, being cautious of taking into consideration the corresponding temperature and service conditions of the fabrication material. 2. If the acting stress at the onset of failure is known, so is fracture toughness, and the crack size can be calculated, determining if such a crack could have existed since fabrication or if it was introduced during

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service. In addition, this number may be compared with the allowable crack sizes to establish if there was a progressive cracking mechanism. 3. Finally, critical crack size and stress at the onset of failure help determine the material fracture toughness. This way, it can be known if the material did comply with the required resistance or if there was any in-service damage mechanism that reduced the fracture resistance down to levels sufficient to cause the failure. Beside the critical crack size, some other fracture parameters can be obtained by fractographic examination, as schematically depicted in Fig. 5.9. For example, fracture toughness can be estimated by measuring the plastic zone size, which is approximately twice the shear lip height of the final stage of a fracture. With hpz the shear lip height or half neck height, pffiffiffiffiffiffiffiffiffiffiffi KIC ¼ 2phpz The KIC value can also be estimated by measuring the step that forms in the transition from stable to unstable crack propagation. As Fig. 5.9 shows, before the initiation of unstable crack propagation, the crack tip goes through a blunting deformation and a pop-in crack extension, leaving a step on the fracture surface. The step height is approximately on half of the crack tip opening displacement, therefore: pffiffiffiffiffiffiffiffiffiffi KIC ¼ 1:2533 yEs0 The following examples illustrate the use of fracture mechanics in failure analysis.

Step formed by crack pop-in

K=0 Crack tip

K>0

y = 1/2 CTODC ∆a

K = KC

Fracture surface Shear lip or half neck height

(hpz) Shear lip

2y

ac

ac

Figure 5.9 Schematic appearance of an elastic-plastic fracture and surface roughness parameters related to fracture parameters.

Fracture resistance of engineering materials

167

A pressure vessel of 2.0 m diameter and 50 mm wall thickness experienced a burst failure by internal pressure. The normal operating pressure is 4826 kPa (700 psi). The fabrication material had sys ¼ 345 MPa, and fracture toughness KIC ¼ 100 MPaOm. A fractographic examination revealed the presence of a preexisting surface crack of length 2a ¼ 254 mm and 25 mm depth. Determine the possible cause of failure. Solution: The KI at normal operating conditions is: KI op ¼ (PR/t){(1 þ 1.61(c2/Rt)pa]1/2 ¼ (4.826 MPa * 1 m/0.05 m) {(1 þ 1.61(0.1272 m2/1 m/0.05m)p0.127 m}1/2 ¼ 33.34 MPaOm Since KIop < KIC, it is concluded that the vessel was overpressurized. The pressure that caused the bursting is Pf ¼ t KIC/{R(1 þ 1.61(c2/Rt)pa]1/2} ¼ (0.05 m * 100 MPaOm)/{1.0 m (1 þ 1.61(0.1272 m2/1 m / 0.05m)]1/2} ¼ 14.49 MPa To verify if the result is reasonable, it is compared to the pressure that causes yield: P0 ¼ sys t/R ¼ 345 MPa * 0.05 m/10.0 m ¼ 17.25 MPa Since P0 > Pf it is concluded the pressure vessel was still under elastic strain at the moment previous to failure, therefore the calculation by fracture mechanics was valid, and in addition, there was not visible expansion or bulging that would have given a warning. The metallic plate shown below fractured in Mode I, after fatigue loading in a hydrogen-charging environment. A failure analysis engineer suggested that the material suffered embrittlement, but a second analyst said that the fracture is very ductile, so the material was not brittle. Prove who is right.

Solution: The measured shear half neck height is hpz ¼ 10 mm, from a hardness test the yield strength is estimated as sys ¼ 250 MPa. The expected toughness at the service conditions is KIC ¼ 80 MPaOm. The estimated KIC is KIC ¼ sys(2phzp)1/2 ¼ 250(2p * 0.01 m)1/2 ¼ 62.7 MPaOm

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This result suggests that the first engineer is right, and the material suffered a hydrogen embrittlement mechanism, since the toughness has a reduction of more than 20%, which is a typical value for materials exposed to hydrogen-charging environments. An absorbed hydrogen laboratory test may confirm this observation, or if there is material in the as-new condition, by determining the KIC in the unexposed condition.

5.5 Reinforcement of cracked structures The R-curve is useful to design reinforcements of cracked structures, since it represents the exchange between the supplied work and the fracture energy; therefore, since a reinforcement is basically a crack arrestor, it can be designed by determining the additional energy that might be added to consume the supplied work. The first option for arresting an unstable crack is to change a constant load condition into a constant displacement condition, so G becomes negative, and eventually the energy supply will be less than the energy demand to sustain crack propagation. To convert a loading condition into constant displacement, a reinforcing strip shall be installed ahead of the crack tip. The reinforcement strip has to be rigid enough in order to have a much less elongation than the base material and be bolt fastened or filet welded only at the upper and lower ends, as shown in Fig. 5.10. Fully attaching the reinforcing strip will integrate it to the component’s volume, and the constant displacement condition will not be achieved.

B Bolted or filet welded at the ends A

Crack(a)

Reinforcement P

A

B Propagation path

Reinforcement inserted

COD Reinforcement insertion distance (L)

Figure 5.10 Insertion of a reinforcement strip ahead of a crack to change from constant load to constant displacement condition.

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Fracture resistance of engineering materials

G, R

B Reinforcement

C

Area G>R A Area G< R Crack arrest length

a

a2

a1

∆a

Figure 5.11 Schematic representation of the R-curve with an overlapped reinforcement strip that changes from constant load to constant displacement.

When the reinforcement is inserted, the R-curve will take the form presented in Fig. 5.11. As the crack grows, G will go down until G < R; however, the crack will not stop at this point, since G and R represent an energy exchange rate, and crack propagation is driven by the total exchanged energy. This means that, even if G < R, the crack will keep propagating as long as there is still stored energy available, and arrest will occur when all the stored energy is spent. The stored energy in the R-curve is [Area G > R], and the spent energy for crack growth is [Area G > R], as depicted in Fig. 5.11. Therefore, for a crack size equal to a1, the crack will start to propagate at point A. If the reinforcement is located at point B, the change to constant displacement will make G to go downward until reaching the condition [Area G > R] ¼ [Area G < R], indicated as point C, and the crack will be arrested. Notice that if the reinforcement is placed too far from the crack tip or the initial crack length is longer (a2), the areas under and above G may never become equal, as represented by the dotted curve in Fig. 5.11, and the reinforcement will not be effective. Another way to arrest a propagating crack is by the insertion of a section of a higher toughness material (higher R). The insertion must be done by cutting and welding a plate or by a strong bonding adhesive, so the reinforcement becomes an integral part of the component volume. The objective is to get the G curve to go under the R curve, to make the [G > R] and [G < R] areas equal, as schematically shown in Fig. 5.12. Just

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Reinforcement width

G, R

E D

Reinforcement R

C Base metal R

A B

a

a1

Crack arrest length

∆a

Figure 5.12 Schematic representation of the R-curve of a structure reinforced by the insertion of a high toughness strip.

like in the previous case, if the reinforcement is placed too distant it will not have any usefulness because the [G < R] area would never be equal or greater than the [G > R] area. Likewise, if the width of the reinforcement is not sufficient, the areas will not become equal and the reinforcement will not be effective, either. The following example illustrates this method. Determine the toughness and minimum width of a strip to be inserted in a panel under uniform stress of 200 MPa, to arrest an edge crack of initial size 10 mm long. The base material fracture toughness is KIC ¼ 80 MPaOm, the Young’s modulus is E ¼ 200,000 MPa, and the SIF equation of the panel is KI ¼ 1.25 sO(pa). Solution: Assuming that the R value of the base material (Rbm) is constant, and the panel is under plane stress condition, Rbm ¼ K2IC/E ¼ 802 MPa2-m/200,000 MPa ¼ 0.032 MPa-m ¼ 32 MPamm G ¼ K2I /E ¼ 1.252s2pa/E ¼ 1.252(200)2 MPa2pa/200,000 MPa ¼ 0.982 a The critical crack size is ac ¼ (1/p)[KIC/1.25s]2 ¼ (1/p)[80 MPaOm/1.25s MPa]2 ¼ 0.0326 m ¼ 32.6 mm The R-curve for the above conditions is

Fracture resistance of engineering materials

70

Wr

60

Rr

50

G, R. MPa-mm

171

40

Rbm

30 20 10 0

0

10

ar

ac

a0 20

30

40

50

60

70

a, mm

The simplest way to determine the width and the toughness of the reinforcing strip R (KIC ) is to postulate a distance beyond ac where the insert is going to be installed and calculate [Area G > R]. Trying with ar ¼ 45.8 mm, Gr ¼ 0.982 ar ¼ 0.982 MPa (45.8 mm) ¼ 45 MPa-mm [Area G > R] ¼ [(ar  ac)/(Gr  Rbm)]/2 ¼ [(45.8  32.6)mm  (45  32) MPa-mm]/2 ¼ 85.8 MPa-mm2 Assuming that the reinforcing strip width is Wr ¼ 13.4 mm and [Area G > R] ¼ [Area G < R], the R value of the reinforcement Rr is Rr ¼ 2[Area G > R]/Wr þ Gr ¼ 2[85.8 MPa-mm2]/13.4 mm þ 45 MPamm ¼ 57.8 MPa-mm R Therefore, KIC ¼ (RrE)1/2 ¼ (57.8 MPa-mm  200,000 MPa)1/2

K2IC ¼ 3400 MPaOmm ¼ 107.5 MPaOm Therefore, the dimensions of the reinforcement are Maximum distance from the initial crack tip to the reinforcing strip

a0  ar

35.8 mm

Minimum width of the reinforcing strip

ar  ac

13.2 mm

Minimum fracture toughness of the strip

R KIC

107.5 MPaOm

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5.6 The leak-before-break condition In internally pressurized vessels and pipes two failure scenarios may occur: One is when a crack grows in the radial direction, but it reaches its critical size before passing through the thicknes, causing a burst rupture. The second scenario is when the crack grows until breaking through the thickness, but before reaching its critical size. If this happens, the crack will be stable, and the fluid contained in the vessel or pipe will leak, allowing the detection of the risk in a safer way as compared to an explosion scenario. This condition is called leak before break (LBB) and is very important in power generation, chemical process, petrochemical, and pipeline transportation industries, since when attained, it significatively reduces the risks. The analysis of the factors that allow the occurrence of the LBB condition can be done by fracture mechanics. A simplistic LBB criterion is to assume that when the crack size in the radial direction is equal to the wall thickness, but KI is less than KIC in the longitudinal direction, the LBB condition is satisfied. Mathematically this condition is expressed as  2 KIC > pt sys where t is the wall thickness. This equation is valid for semicircular cracks, where a z t, which is not a common case. Furthermore, it neglects plasticity, bulging, and other effects, but still is a rough approach that is useful for materials selection and first-time assessments, as mentioned before in Section 5.2. A more accurate way to determine the LBB condition was suggested by Broek3 based on the analysis of the propagation of a semielliptical crack though the thickness of a thin wall cylinder under internal pressure, as shown in Fig. 5.13. If the crack reaches its critical size in the radial direction, it will become a through-thickness crack, but if the internal pressure is not enough to cause instability in the axial direction, the axial crack will be stable and the LBB condition will be attained. In terms of fracture toughness this requires that once the crack becomes a through-thickness crack, KI < KcL, where KI is the stress intensity factor in the axial direction and KcL is the fracture toughness in that same direction. When the crack is still a nonpassing semielliptical crack, 3

Broeck, D. Elementary engineering fracture mechanics, Martinus Nijhoff Publishers, Dordrecht (1986), pp. 403e406.

Fracture resistance of engineering materials

173

2c

t

Kct

KcL

KcL

a

P

Figure 5.13 Sequence of propagation of a through-wall crack in a pressurized vessel.

the condition is KI > Kct, where KI is the stress intensity factor in the radial direction and Kct is the fracture toughness in the radial direction. Broek derived the following equation to establish the feasibility of the LBB condition of thin wall cylinders under internal pressure (R > 10t, where R is the internal radius), containing surface flaws of length several times the wall thickness:  rffiffi KcL p a2 c 3þ 2 > a Kct 9Mk c where Mk is a factor equal to 1.0 for a/t z 0, and Mk ¼ 2.0 if a/t z 1.0. The plots of this equation, for very shallow cracks, a/t z 0, and through thickness cracks, a/t z 1.0, are shown in Fig. 5.14. According to this criterion, it is established that the combinations KcL/Kct, a/2c falling under the curves will not meet the LBB condition and those over it will. Due to complications on the exact determination of the fracture toughness in the different crack geometries, the shaded region between the two curves in Fig. 5.14 corresponds to uncertain conditions, therefore only the upper and lower curves are used for the assessment. In Fig. 5.14 it is observed that anisotropy favors the LBB condition; that is, when KcL > Kct, but in practice, it is seldom that KcL/Kct > 2.0, which implies that the LBB condition is hard to obtain for shallow cracks, having an aspect a/c < 0.3. Another important observation is that the two curves converge at KcL/Kct, z 1.3, for crack aspects a/c > 0.8, which means that the LBB condition is more feasible for short and deep cracks, such as cracklike weld defects and stress-corrosion cracks. However, it is a rather rare condition in pipelines, where the surface cracks tend to be shallow, and combining this with low levels of anisotropy exhibited by typical pipe and

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3.5 a/ t ≈0

3

KcL/Kct

2.5

Leak Maximum anisotropy in Steel pipes

2 1.5

a/ t ≈ 1 KcL = Kct

1

Break

0.5 0 0

0.2

0.4

0.6

0.8

1

a/c Figure 5.14 Plot of the leak before break criterion as a function of the crack aspect and fracture toughness anisotropy.

pressure vessel grade steels, the LBB condition may be very difficult to obtain in real life situations. Another difficulty for the application of the LBB criterion is that Kct corresponds to a short radial direction (CR according to the ASTM E399 nomenclature), which is experimentally complicated to determine since there is not enough material to fabricate valid specimens for fracture toughness testing. Angeles et al.4 determined the fracture toughness in the short radial direction (CR) of SAW seam welds in API 5L X52 pipeline steel by using nonstandard curved SE(B) test specimens. They found Kct values around 56.3  2.9 MPaOm, while Kct was above 110 MPaOm, which indicates that, at least in SAW seam welded pipe, the LBB criterion is satisfied for cracks in SAW seam welds having aspect ratios a/c > 0.3, according to Fig. 5.14; unfortunately there is no data published for the base material. The LBB criterion has become a required design concept for highenergy piping systems, such as nuclear plants. Up to 1973, the design criterion for piping systems in nuclear plants was based on the occurrence of a sudden double guillotine rupture at the ends of a pipe segment (a guillotine 4

Angeles-Herrera, D., Gonzalez-Velazquez, J. L., and Morales-Ramirez, A. J. Fracture-Toughness Evaluation in Submerged Arc-Welding Seam Welds in Nonstandard Curved SE(B) Specimens in the Short Radial Direction of API 5L Steel Pipe. J. of Testing and Evaluation (2012), Vol. 40, No. 6.

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175

rupture is propagation of an unstable crack in the entire circumferential direction). To protect the plant from possible impacts of guillotine broken pipes, a high number of antilash supports and antiimpact bars have to be installed, causing adverse effects in the safety, because they restrict the thermal expansion of the pipes and reduce the accessibility of the installation for inspection and maintenance, resulting in longer radiation exposure times for the personnel. The feasibility of attaining an LBB condition in piping systems allowed great savings by eliminating supports and protections along with better operational safety and reduced risks. The implementation of the LBB criterion in piping systems include several steps, as established in design codes such as ASME, Sec. XI. This particular procedure is summarized here: 1. Evaluation. Each system eligible for the LBB has to be carefully evaluated in order to ensure the absence of conditions that may hinder an accurate assessment of loads and cracks in pipes. Such conditions are (1) sudden high loads, such as hammer hit; (2) fast and continuous crack growth, by fatigue, stress-corrosion, or creep; (3) unexpected events such as impacts, fire, and explosion. 2. Data gathering. The entry data for the systems passing the initial assessment will be needed to carry out a fracture mechanics analysis: (1) loads and pressures in piping circuits; (2) diameter, thickness, operation temperature, and pipe flow conditions; (3) material mechanical properties of base metal, welding, supports, and so forth, in the current service conditions, such as temperature, radiation, and so on. The mechanical properties may include ductile fracture properties like J-curves and R-curves, because of the prevalence of ductile and high toughness materials. It is also necessary to count on information on the leak detection capability in the plant under normal operating conditions. The aforementioned is an LBB fundamental point, since the objective is to detect a leak before the crack reaches its critical size, causing a burst rupture or an explosion. A leak detection sensitivity of 1 gallon per minute is acceptable in most cases. 3. Safety margin. An essential condition of data gathering is that it has to be completely conservative so as to ensure a sufficient safety margin in each stage of the analysis. Such conservationism must expand to the leak detection capability. For example, if a plant is able to detect a 1 gpm leak, then the analysis by fracture mechanics must consider a crack leading to a larger leak. Obviously, the greater the leak detection sensitivity, the more effective the LBB implementation.

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4. Calculation of the postulated leaking crack size. It is the critical crack size of a pipe having the minimum mechanical properties and the maximum operating loads and pressures that produce a leak equal to the minimal detection capability multiplied by the safety factor. The calculation of the leaking rate in terms of crack size depends on the specific conditions of pressure and fluid flow conditions, which is usually calculated by numerical computational models. 5. Demonstration of a leaking crack stability. The stability of a crack under normal conditions plus seismic and shutdown loads must be set with a safety margin of at least 2.0 between the leaking crack size and the critical crack size. This means that if the leaking crack size is less than half the critical size, the leaking crack is stable.

CHAPTER 6

Fatigue and environmentally assisted crack propagation

6.1 Fatigue crack growth and Paris’s law Fatigue is a process of damage accumulation, crack growth and fracture of a body under fluctuating, variable or cyclic loads. In order to occur fatigue requires at least three conditions: 1. Stress variation above a minimum value called fatigue limit. 2. The load cycle should generate a tension stress component. 3. A sufficient number of load cycles, fluctuations, or variations. If one of these conditions is not present, fatigue will not occur. Since the very beginning of fatigue studies, it was observed that it starts with an internal damage accumulation stage, which accounts for 90%e95% of the life endurance until eventually one or several cracks are nucleated. After the initial damage accumulation stage, a macroscopic crack appears, which propagates gradually at a rate controlled by the stress range magnitude. Finally, as the crack length approaches the critical size, the fracture mechanism becomes a combination of cyclic loading and static modes of fracture until the crack reaches its critical size and the component fails. If the component contains preexisting cracks or severe stress concentrators the first stage is suppressed, and the macroscopic cracking starts beginning with the first load cycles. Furthermore, if during the macroscopic crack propagation stage, the loading amplitude is reduced enough or interrupted, the crack growth will stop; similarly, if the load amplitude increases, the crack will accelerate its growth. From the fracture mechanics point of view, fatigue crack growth (FCG) proceeds under subcritical conditions, that means before Irwin’s energy criterion is satisfied. Since the vast majority of mechanical and structural components operate within the elastic strain regime, the FCG is within linear elastic conditions, thus the crack propagation can be directly related to the stress intensity factor (SIF). A Practical Approach to Fracture Mechanics ISBN 978-0-12-823020-6 https://doi.org/10.1016/B978-0-12-823020-6.00006-2

© 2021 Elsevier Inc. All rights reserved.

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The previous idea was introduced by Paul C. Paris in 1961.1 Paris was one of the first researchers to notice that the FCG rate (da/dN) is directly related to the stress intensity factor amplitude (DK ), when he was studying the fatigue problem of aircraft in 1955 at Lehigh University, shortly after the tragic accidents of the de Havilland Comet jet airplanes in 1954. It is interesting to remember that, at first, Paris idea of the da/dN dependency on DK was rejected by the scientific community by the argument that an elastic parameter such as K could not truly represent the FCG, based on the observations by several researchers that fatigue cracks propagate by cyclic plastic strain, criticizing Paris’s results as fortuitous. It was until 1967 that the overwhelming experimental data proved that Paris was right. Another curious anecdote is that Arthur J. McEvily in 1958 published a paper2 describing a relationship between the FCG rate and the square root of the crack length, and thus somehow he anticipated Paris’s results since the stress intensity factor is proportional to the square root of the crack length. Later, in 1978, McEvily introduced his own equation to relate the FCG rate to the SIF. According to Paris’s ideas, the load amplitude DP is defined as DP ¼ Pmax  Pmin Where Pmax is the maximum load, and Pmin is the minimum load. Under linear elastic fracture mechanics (LEFM) conditions, the SIF may be expressed by K ¼ PbO(pa) Consequently, the maximum load will generate a maximum value of K and the same occurs at the minimum load, so it can be established that Kmax  Kmin ¼ Pmax bO(pa)  Pmin bO(pa) The stress intensity factor amplitude is DK ¼ Kmax  Kmin Therefore: DK ¼ DPbO(pa) Paris, P.C., Gomez, M.P., and Anderson, W.P., “A Rational Analytic Theory of Fatigue”. The trend in Engineering, Vol. 13 (1961), pp. 9-14. 2 A.J. McEvily, and W. Illg, “The rate of propagation in two aluminum alloys”. NACA Technical Note 4394, Sept. 1958. 1

Fatigue and environmentally assisted crack propagation

179

Once a fatigue crack begins to grow, it propagates by the effect of the stress variation at the crack tip. If the load amplitude is constant, as the crack grows, the SIF amplitude increases and consequently the crack growth rate increases. This process goes further until the maximum K is equal to the material fracture toughness (KIC) and the final fracture occurs. This sequence is schematically shown in Fig. 6.1. Paris showed that by plotting the da/dN versus DK data on a log-log scale, the data point falls into a straight line, allowing him to postulate that da ¼ CDK m dN Where C and m are empirical constants. Further experimental evidence at the extreme values of DK showed the tendency illustrated in Fig. 6.2, designated as the Paris curve. Notice that the Paris curve is divided into three regions that correspond to the three stages of fatigue: • Region I: Near threshold. Characterized by a strong dependency on DK, and a DK below which the crack growth rate is negligible, typically below 108 mm/cycle. This value is referred as DKth. • Region II: Paris region. Range of DK values where the relation da/ dN ¼ CDKm is valid. • Region III: Near final fracture. Characterized by a rapid increment of da/dN up to the point where Kmax is close to KIC. The standard ASTM E647 sets the guidelines to perform the FCG growth tests and to get the Paris curve. Within Region I the cyclic plastically strained zone is very small, so the interaction with the microstructure is strong. In Region II the strongest ' K = ' P E —(Sa) Kmax = Kc

' P = Constant

ac 'K

'P Time

ao

da/dN

No. of cycles (N)

NF

Figure 6.1 Variation of the stress intensity factor amplitude and crack size as a function of the number of cycles of a plate under constant load amplitude and linear elastic conditions.

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Region I

Region II

Region III

log da/dN

da/dN = C 'Km

m

'Kth

log 'K

KIC

Figure 6.2 General form of the Paris curve and three regions of FCG.

influence is by the mechanical properties and the environment, which are mainly reflected on the value of the exponent m. McEvily3 introduced an equation that adjust well the three regions of the Paris’s curve:
 da DK 2 ¼ ðCDK  DKLIM Þ 1 þ dN KIC  KMAX Annex 9F.5.3 of the API 579-1/ASME FFS-1 “Fitness for Service” standard, 2016, provides typical values of the Paris equation constants of several metallic alloys, some of these values along with other published values are given in Table 6.1. Several attempts have been made to rationalize the Paris equation. One proposed by McClintock4 is based on the assumption that da/dN is

3

McEvily, A.J. On the Quantitative Analysis of Fatigue Crack Propagation. Fatigue Mechanisms: Advances in Quantitative Measurement of Physical Damage, ASTM STP 811 (1983), pp. 283-312. 4 McClintock, F.A. Discussion to C. Laird’s paper “The influence of metallurgical microstructure on the mechanisms of fatigue crack propagation”. In Fatigue Crack Propagation, ASTM STP 415, American Society for Testing and Materials, Philadelphia, PA, pp. 170-174.

Fatigue and environmentally assisted crack propagation

181

Table 6.1 Paris equation constants of metallic alloys. DK in MPaOm, and da/dN in mm/cycle. Material

m

C

Low carbon steel

3.0

10e8

Forged steel

2e3

10e10

High strength structural steel

3.0

10e11

Stainless steel (austenitic)

3.8

10e12

Gray iron (cast)

4.0

8  109

Nodular iron

3.5

10e8

Aluminum 7021

2.5

10e8

Nickel base alloy

3.3

4  1012

Titanium base alloy

5.0

10e11

Data taken from several sources.

proportional to the crack tip opening displacement (CTOD), which, as seen in Chapter 2, under LEFM conditions is given by:   4 K2 CTOD ¼ p Es0 Where E is the Young’s modulus and s0 is the yield strength. Therefore: da 4 DK 2 w dN p Es0 According to this analysis the value of m should be equal or close to 2.0 for most engineering materials, but as seen in Table 6.1, the average of m is around 3.4, this behavior indicates that there are other factors that influence the FCG rates in addition to the plastic stretching at the crack tip, which is the physical cause of the CTOD. The treatment of FCG based on fracture mechanics has two main advantages as compared to the classical continuum mechanics approach (the SeN curves). As depicted in Fig. 6.3, the first advantage is that the data scattering in the da/dN versus DK plot is much less than the observed in the SeN curves (it should be remembered that the scatter of the SeN data requieres a statistical treatment). The second advantage of the LEFM approach to FCG is that the Paris’s equation can be integrated to calculate the number of load cycles to make the crack grow from an initial to a final size; if the final size is critical, then the remaining fatigue life of a cracked component can be estimated.

A Practical Approach to Fracture Mechanics

'V V0

Data scatter band

0.6 95% Prob. fail 0.4 5% Prob. fail

0.2

Nf 103 104

105

Data scatter band

10-2 da/dN, mm/cycle

182

10-4

10-8 1

106

10

100

'K (MPa—m)

Figure 6.3 Typical fatigue curves of a normalized low carbon steel. Left, SeN curve. Right, Paris’s curve.

The integration of Paris’s equation from an initial crack size detected in a nondestructive inspection (a0) to a critical size determined by 2  1 KIC ac ¼ p smax Where smax is the maximum stress of the load cycle. Taking the case of an infinite plate with a central crack, the SIF amplitude is given by pffiffiffiffiffi DK ¼ Ds pa Substituting into the Paris’s equation and solving for N Zac N¼ a0

da pffiffiffiffiffi m CðDs pa Þ

And by performing the integral N¼

h i 2 1m=2 pffiffiffi m a1m=2  a 0 c Cð2  mÞðDs p Þ

For low carbon steels the average value of m is 3.0, so the former equation can be simplified to 
2 1 1 N¼ pffiffiffi 3 pffiffiffiffi  pffiffiffiffi a0 ac CðDs p Þ

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183

The following example illustrates the practical use of the Paris equation integration. Example: Determine the number of stress cycles to failure of a low carbon steel large panel that contains an initial crack of 20 mm length, if the panel is stressed at constant amplitude of 1/3sys. Consider that sys ¼ 358 MPa psi and KIC ¼ 100 MPaOm. Solution: Assuming that the stress range is from zero to maximum load: Ds ¼ smax ¼ (1/3)sys ¼ (1/3) * 358 ¼ 119.3 MPa The critical crack size is: ac ¼ (1/p)(KIC/smax)2 ¼ (1/p)(100 MPaOm /119.3 MPa)2 ¼ 0.224 m ¼ 224 mm Taking C ¼ 1011, the number of cycles to failure is: N ¼ [2/(C(DsOp)3)] * [1/Oa0  1/Oac] ¼ [2/(1011 (119.3Op)3)] * [1/O20  1/O224] ¼ 3317 cycles

6.2 Effect of the load ratio on the FCG rate The load ratio R in a fatigue load cycle is defined as: R ¼ Pmin/Pmax ¼ Kmin/Kmax The value of R has been used in substitution of the average load to characterize the load cycle as illustrated in Fig. 6.4. Notice that the higher the R value, the higher average load. The main effect of R in the FCG behavior is on DKth, and this is of great practical importance, since most of the designs are aimed to infinite life, therefore the SIF amplitude must be less than DKth so a crack that may be present in the component do not grow. Table 6.2 show some typical values of DKth of engineering materials. Notice that the lower the R, the higher the DKth, and steels and nickel base alloys have the highest DKth values, while the nonferrous alloys have the lowest. The effect of R on the FCG behavior has been extensively tested due to its great impact on the Paris’s curves, as can be seen on the example shown in Fig. 6.5. Basically, as R increases, the da/dN versus DK curve is displaced upwards, however the effect disappears at R > 0.5 in most cases. One of the best-known equations that consider the effect of load ratio on the FCG rate is Forman’s, which has the expression da CDK m ¼ dN ð1  RÞðKIC  KMAX Þ

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A Practical Approach to Fracture Mechanics

Tension-Tension: 0 < R < 1 Pmax

P

Tension-Compression: -f < R < -1

+P

Pm 0

Pmin

0

P -P

Time

Compresion-Compresion: R > 1

Fully reversed Tension-Compression: R = -1

+P

Pmax

Time

0 Pmax

0 Pmin = - Pmax

-P

Time

Pmin

-P

Time

Figure 6.4 Types of load cycles according to the value of R.

6.3 Fatigue crack closure The physical explanation of the effect of R on the Paris curve was first reported by Elber5 in 1969 and is known as crack closure. The simplest definition of crack closure is the contact of the opposing fracture surfaces of a crack before the load reaches its minimum value in the cycle, as schematically depicted in Fig. 6.6. Under ideal linear elastic conditions, the Load-Displacement record of a cracked body should be a straight line, which slope is related to the compliance. Elber observed that some LoadDisplacement records of fatigue cracks had a change of slope, meaning a change of compliance, as schematically shown in Fig. 6.6. The reduction of compliance was attributed to a reduction of the crack length, therefore, once the crack fully opens, the compliance returns to the value corresponding to the total crack length. The SIF value at which the crack fully opens was designated as Kop. The explanation of the R effect on FCG rate based on the crack closure phenomenon is as follows. If a fatigue crack experiences closure, the crack 5

Elber, W. The Significance of Fatigue Crack Closure, Damage Tolerance in Aircraft Structures, ASTM STP 486 (1971), pp. 230-247.

Fatigue and environmentally assisted crack propagation

185

Table 6.2 DKth values as a function of R for common metallic alloys fatigued in air at room temperature, Data from several sources. Material

A533 steel

ASTM 508 steel

Titanium

Copper

Brass 60/40

Nickel

Stainless steel 300 series

Aluminum 2219 e T851 Cast aluminum A356

R

DKth Ksi Oin

MPa Om

0.1

8

7.3

0.3

5.7

5.2

0.5

4.8

4.4

0.7

3.1

2.8

0.8

3

2.75

0.1

6.7

6.1

0.5

5.6

5.1

0.7

3.1

2.8

0.2

w5.5

w5

0.4

w4.4

w4

0.9

w3.3

w3

0

2.5

2.3

0.33

1.8

1.6

0.56

1.5

1.4

0.80

1.3

1.2

0

3.5

3.2

0.33

3.1

2.8

0.51

2.6

2.4

0.72

2.6

2.4

0

7.9

7.2

0.33

6.5

5.9

0.57

5.2

4.7

0.71

3.6

3.3

0.05

8.5

7.6

0.7

3.7

3.3

0.1

3

2.7

0.5

1.7

1.5

0.1

6.1

5.5

0.8

2.4

2.1

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1.E-02

R t 0.5

da/dN [mm/cycle]

1.E-03

R = 0.4 R = 0.2

1.E-04

R = 0.1 1.E-05 1.E-06 1.E-07 1

10

100

'K [MPa—m]

Figure 6.5 Effect of R on the Paris curve of low carbon steel fatigued in air at room temperature. Estimated data for illustration purposes.

Kmax

Kmax

No closure Crack closure

P Kop

Kop Kmin

Kmin

Displacement Closure Figure 6.6 Closure of a fatigue crack and Load-Displacement record showing the change of compliance due to crack closure.

will no grow until the value of Kop is surpassed, so an effective amplitude DKeff can be defined as DKeff ¼ Kmax  Kop So, as illustrated in Fig. 6.7, at high values of R, Kmin > Kop, and the full range of the SIF will drive the crack growth, while at low R values Kmin < Kop and only the DKeff portion of the load cycle will drive the crack. Elber proposed that the FCG rate can be expressed as da ¼ CðDKeff Þm dN

Fatigue and environmentally assisted crack propagation

187

R = 0.5 'K (R=0.5) = 'Keff = Kmax - Kmin Kmin > Kop

K R=0

'Keff (R = 0) = Kmax - Kop Kmin < Kop

'K (R=0)

Kop

Time Figure 6.7 Schematic illustration of the effect of R on the SIF amplitude due to the crack closure phenomenon. DK(R ¼ 0) ¼ DK(R ¼ 0.5).

Introducing the ratio U as U¼

DKeff Kmax  Kop ¼ DK Kmax  Kmin

In terms of DK and R, U can be written as U¼

Kop 1  1  R DK

At threshold, U ¼ 0, therefore DKLIM ¼ Kop(1  R) Then da ¼ CðUDKÞm dN A graphic representation of this equation is shown in Fig. 6.8. The four basic mechanisms of crack closure recognized up today are: Plasticity: The most important and is due to the compression of the expanded plastic zone in the wake of the crack tip during the unloading portion of the load cycle. Fig. 6.9 presents a schematic representation of this mechanism. Roughness: Caused by the contact of the crests of the mating fracture surfaces when there is a horizontal displacement in the crack due to a Mode II component, as shown in Fig. 6.10.

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A Practical Approach to Fracture Mechanics

102 m=3 'Keff

1 1

R=0.4 R=0.2

R=0

10-3 10

'K / Kop

0.1

Figure 6.8 Graphic representation of Elber’s model of FCG based on DKeff. Elastic strain returns to zero

Elastic strain

Kmax

K = Kop Plastic strain expansion

Plastic zone in the wake of the crack tip

Plastic zone in front of the crack tip Compressive stress

Figure 6.9 Plasticity-induced crack closure mechanism.

Kmax

Kmin

Crack fully open

Mode II

Contact before K = Kmin

Figure 6.10 Roughness-induced crack closure mechanism.

Debris: Caused by the accumulation of debris or corrosion products in the space between the mating surfaces of a cyclically loaded crack, as depicted in Fig. 6.11. As the debris build up increases, and the Kop does so, it becomes time dependent. Phase transformation: Occurs in materials that exhibit a stress or strain induced phase transformation that involves a volume expansion, such as the

Fatigue and environmentally assisted crack propagation

Kmax

189

K = Kop

Contact Debris and corrosion deposits Figure 6.11 Debris-induced crack closure mechanism.

strain induced martensitic transformation in austenitic stainless steels. This mechanism acts together with the plasticity-induced crack closure mechanism, so it produces high Kop values. The phenomenon of crack closure is very important for understanding several mechanical, metallurgical, and environmental effects on FCG behavior, such as the effect of overloads, ductility, grain size, etc. The effects of some of these factors is explained next. Effect of ductility: Usually ductile materials exhibit greater fatigue strength at low load ranges as compared to very hard or brittle materials, even though there are other reasons to explain this behavior, an important one is plasticity-induced crack closure, since ductile materials produce large plastic zones that favor this mechanism. Fig. 6.12 shows the effect of ductility on the Paris curve and the schematic variation of Kop. Effect of grain size: Materials with similar strength level but small grain size, show the tendency to have lower fatigue threshold levels than materials with coarse grain size. This behavior is attributed roughness-induced crack closure, since in the near threshold regime (Region I), the crack path

log da/dN

Maximum hardness

Annealed

log 'K

Kop

Annealed

Maximum hardness

'K

Figure 6.12 Schematic plots of the Paris curves and Kop behavior of FCG in steel as a function of the ductility.

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Kop Fine grain

log da/dN

Coarse grain

Fine grain Coarse grain

'Kth

log 'K

'K

Figure 6.13 Schematic plots of the Paris curves and Kop behavior of FCG in a metallic alloy as a function of the grain size.

follows crystalline planes and directions, which change from one grain to another, thus producing a rough fracture surface, with many peaks and Mode II crack surface displacement that favor roughness-induced crack closure. Fig. 6.13 shows the effect of grain size on the Paris curve and the schematic variation of Kop.

6.4 Effect of the environment on fatigue crack growth It is widely known that fatigue can occur in inert environments, being called pure mechanical fatigue when this happens; nonetheless the environment plays a fundamental role in fatigue, especially in metallic alloys, even if the environment is regarded as noncorrosive or the material is assumed to be corrosion-resistant. The main effect of environment is related to the corrosivity, as it increases, the Paris curve is shifted upwards and to the left, thus, increasing the crack growth rate and reducing the threshold levels, as seen in the Paris curves of stainless steel of Fig. 6.14. This behavior is referred as corrosion-fatigue, and for most engineering material is considered the normal condition of fatigue. Furthermore, in aqueous media and very corrosive environments, the Region II of the Paris curve may show a plateau, making the FCG rate independent of DK as depicted in Fig. 6.15. When this behavior is observed it is said that the mechanism is Corrosion-fatigue þ Stress-Corrosion Cracking (CF þ SCC), since it is assumed that an SCC dissolution mechanism controls the crack propagation. A schematic comparison of the Paris curves

Fatigue and environmentally assisted crack propagation

191

1.00E-02 Air at 21 qC

da/dN [mm/cycle]

1.00E-03

Air at 600 qC Vacuum at 538 qC

1.00E-04

1.00E-05 Vacuum at 21 qC 1.00E-06 1

10

100

'K [MPa—m]

Figure 6.14 Region II Paris curves of 304 stainless steel in different environments (frequency 30 Hz, R ¼ 0.05). (Adapted from J. L. Gonzalez-Velazquez, “Fatigue Crack Tip Deformation Processes as Influenced by the Environment”, PhD Thesis, University of Connecticut (1990), pp. 75.)

log da/dN

Corrosion-fatigue + SCC (Very corrosive)

Corrosion-fatigue (Mildly corrosive) Mechanic fatigue (Inert)

log 'K

Figure 6.15 Categories of FCG as influenced by the environment.

of a metallic alloy for the three categories of FCG in terms of the environment corrosivity is shown in Fig. 6.15. The effects of environmental variables, including temperature, is attributed to the interaction of the plastic deformation mechanisms at the crack tip with the corrosive and embrittlement chemical spices within the crack cavity, which, by the way, may significantly differ from the external

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environment. The main environmental interaction mechanism that contribute to increase the FCG rate are: 1. Anodic dissolution in aqueous environments. 2. Formation of brittle oxide layers, particularly at high temperature. 3. Formation and rupture of passive layers. 4. Hydrogen embrittlement by absorption from the environment. 5. Hydrogen adsorption strain localization at the crack tip stretching zone. James and Schwenk6 proposed a modification of the Paris equation to consider the effect of temperature and the activation energy Q of the chemical reaction with the environmental corrosive species, the equation is:   da Q m ¼ CDK exp  dN RT Where T is the temperature and R is the ideal gases constant. This equation correctly predicts the effect of temperature (the higher temperature, higher FCG rate), but the effect of Q depends on the sign; for negative Q (spontaneous process), as in open circuit corrosion, a greater Q, leads to a greater FCG rates; which is consistent with experimental observations. Independently of the interaction mechanism between fatigue and environment, the loading frequency is a fundamental factor for corrosionfatigue. In general, a reduction in the load cycling frequency will displace the Paris curve upwards, as shown in Fig. 6.16. This behavior is attributed to the fact that at low frequencies the time of contact between the crack and the aggressive species from the environment increases. It may be foreseen that the dependency of the FCG rate on the frequency is a continuum, but the experimental evidence indicates that there is a critical frequency above which the corrosion-fatigue crack growth rate is independent of the loading frequency, as shown in Fig. 6.17.

6.5 Effect of variable loads on fatigue crack growth The constant loading amplitude fatigue is a rather rare case, since in general, in-service components are exposed to variations of loading amplitude and overloads, which have a strong effect on the fatigue behavior. The immediate effect of load variation might be a change on the crack growth rate; however, such change is not instantaneous, but it goes through a transient 6

James, M.H., Schwenk, E.B., Fatigue crack propagation behavior of type 304 stainless steel at elevated temperatures. Metallurgical Transactions A2 (1971), pp. 491-496.

Fatigue and environmentally assisted crack propagation

1.00E-02

0.1 Hz 1 Hz

1.00E-03

da/dN [mm/cycle]

193

10 Hz 1.00E-04

100 Hz

1.00E-05

1.00E-06

1.00E-07 1

10

100

'K [MPa—m]

Figure 6.16 Effect of the frequency on the corrosion-FCG rate of carbon steel fatigued in salt water, R ¼ 0.3. (Estimated data for illustration purposes.)

da/dN [mm/cycle]

1.00E-02

1.00E-03

1.00E-04

655 qC

1.00E-05

500 qC

1.00E-06 0.01

0.1

1

10

100

Frequency [Hz]

Figure 6.17 Corrosion-FCG rate as a function of the frequency of 18%Cr ferritic steel, Kmax ¼ 16 MPaOm, R ¼ 0.1. (Adapted from: Makhlouf y J.W. James, Int. J. Fatigue 15 No. 3 (1993), pp.163e171.)

state, before stabilizing to the FCG rate corresponding to the load amplitude. So, in the case of an increment in the loading amplitude, the cracking will accelerate at a rate greater than the one corresponding to the new loading amplitude during a determined number of load cycles; but, if the loading amplitude decreases, the crack will slow down to a rate lower than

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da/dN

'K(2)

'K(1)

'K(1)

da/dN = f('K1) da/dN = f('K2)

Transient

Transient

Time

Figure 6.18 Transients of FCG rate after loading amplitude variations.

the corresponding to the new loading amplitude before stabilizing to such value. These transients are schematically depicted in Fig. 6.18. The effect of variable loading on the fatigue life, ignoring the transients, has been traditionally estimated by the Miner’s rule, also known as the linear damage rule. It is based on decomposing the loading spectrum into a series of constant amplitude loading blocks. The number of loading cycle for each block is ni, and the number of cycles to failure at the corresponding constant loading amplitude is N f i. The Miner’s rule establishes that: X ni ¼1 Nfi In the case of components containing cracks, Nfi can be calculated by integration of the Paris Law. The following example illustrates the use of the Miner’s rule. A machinery component is fabricated of a material with KIC ¼ 70 MPaOm. The normal operation stress amplitude is 80 MPa at a frequency of 30 Hz for 8 h. On the start, the component is stressed at an amplitude of 140 MPa at 50 cycles per minute and takes 5 min. In both cases R ¼ 0. During a programmed inspection an edge crack of 15 mm long was detected. The Paris’s equation constants of the material are: C ¼ 1012 and m ¼ 3 (for da/dN in mm/cycle and DK in MPaOm). The SIF amplitude for the component’s geometry is DK ¼ 1.2DsO(pa). Estimate the remaining life in terms of number of daily operations.

Fatigue and environmentally assisted crack propagation

195

Solution: The problem is divided into two loading blocks, one for the start (s) and the other of daily operation (op). The Miner’s rule is expressed as: nop ns þ op ¼ 1 s Nf Nf The critical crack sizes for normal operation and starts are: Operation:ac ¼ (1/p)[70/(1.2  80)]2 ¼ 0.1692 m Start: ac ¼ (1/p)[KIC/(1.2smax)]2 ¼ (1/p)[70/(1.2  140)]2 ¼ 0.055 m s The values of Nop f and Nf are calculated by integration of the Paris’s law: 3 3/2 1 Nop ) * (a1/2  a1/2 ) ¼ 2(1012 * 803p3/2)1 f ¼ 2(CDs p 0 f (151/2  1691/2) ¼ 127,206 cycles

Nsf ¼ 2(1012  1403  p3/2)1 (151/2  55.261/2) ¼ 16,189 cycles The number of loading cycles in each block is: no ¼ 8 h  60 min/h  30 cycles/min ¼ 14,400 cycles Solving for ns in the Miner’s rule: s ns ¼ [1  (no/Nop f )] * Nf ¼ (1  14,400/127,206)  16,189 ¼ 14,356 cycles

If the start takes 5 min at 50 cycles/minute, the remaining life in terms of daily operations is: No. Daily Ops. ¼ ns/[Start frequency * Time of start] ¼ 14,356/[10 * 1] ¼ 1435 Notice that the result is rounded up to the immediate inferior integer, since there is no sense on determining fractions of operations.

6.6 Effect of a single overload on fatigue crack growth Contrary to what may be expected, overloads cause retardations of the FCG rate so its study is of great practical interest, since retardation can increase the fatigue life, and even more arrest a propagating crack. The effect of overloads has been treated under several approaches, but the most accepted one is plasticity-induced crack closure, because the overload plastic zone surrounds the original plastic zone, so after the overload, when the loading amplitude returns to its level the crack tip has to go through a

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compressive residual stress zone. This process generates high Kop levels that reduce DKeff, thus slowing down the crack propagation. When the crack tip exits the overload plastic zone, the crack growth rate is resumed to the value corresponding to the normal loading amplitude. This behavior is depicted in Fig. 6.19. Wheeler7 proposed a mathematical model to estimate the FCG rate in the transient state after a single overload. The model is based on a retardation parameter that is a function of the plastic zone size at the normal loading amplitude and the length to go through inside the overload plastic zone. The schematic representation of this model is shown in Fig. 6.20. The sizes of the normal plastic zone (rp) and the overload plastic zone (rol) are given by the usual Irwin’s formulas: 2  1 Kmax rp ¼ bp s0  2 1 Kol rol ¼ bp s0 Where b ¼ 2 for plane stress and b ¼ 6 for plane strain. According to Wheeler, the retardation effect lasts while rp is contained within rol, but when the normal plastic zone exits the overload plastic zone, the effect disappears. 'Kol da/dN

'K

Transient

Time

Figure 6.19 Effect of a single overload on the FCG rate.

7

Wheeler, O.E., Spectrum Loading and Crack Growth, J. of Basic Engineering, Transactions ASME, Vol. 94 (1972), pp. 181-186.

Fatigue and environmentally assisted crack propagation

197

rol Propagation through the overload zone

a0

rp

ai

Overload plastic zone

O

Figure 6.20 Wheeler model of a fatigue crack propagating through a single overload plastic zone.

The FCG rate in the transient state (da/dN)R is calculated by multiplying the FCG rate at the normal loading amplitude (da/dN)I by the retardation parameter f. (da/dN)R ¼ f(da/dN)I r m p f¼ l Where m is an experimental constant, ranging between 1.3 and 3.4 for common metallic alloys and l is the remaining distance to go through the overload plastic zone; according to Fig. 6.20, it is calculated as: l ¼ rol  ðai  a0 Þ Notice that when a0 ¼ ai and l ¼ rsc, f takes its minimum value, since rp/rol < 1, and when the boundary of rp touches the boundary of rol, f ¼ 1 and the transient state ends. Obviously the computation of the number of delay cycles caused by the overload requires the integration of the (da/dN)R values obtained by step increments of l, which is a complicated task and may require a computer program to perform it. However, for quick estimates the procedure followed in the nnext example may be carried out: Example: A thin panel cyclically loaded at constant DK ¼ 5 MPaOm, and R ¼ 0.09 (Kmax ¼ 5.5 MPaOm) suffers and overload of Kol ¼ 45 MPaOm. If s0 ¼ 400 MPa,

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A Practical Approach to Fracture Mechanics

and da/dN ¼ 108DK3 (mm/cycle, MPaOm), estimate the number of retardation cycles. Solution: (da/dN)i ¼ 108(5)3 ¼ 1.256 mm/cycle rp ¼ (1/2p) (5.5 MPaOm /400 MPa)2 ¼ 0.00003 m ¼ 0.03 mm rol ¼ (1/2p) (45 MPaOm /400 MPa)2 ¼ 0.002 m ¼ 2 mm l [mm]

f ¼ (rp/l)2

(da/dN)R [mm/cycle]

1.9

0.00025

3  1010

1.6

0.00035

4.4  1010

1.2

0.00063

7.84  1010

0.8

0.0014

2  109

0.4

0.0056

7  109

Plotting the data of the previous table the number of cycles can be roughly and conservatively (calculating the least number per each crack increment) estimated by adding the area of each block in the next graph: (da/dN)R (mm/cycle)

1.00E-08

1.00E-09

1.00E-10 0

0.5

1

1.5

2

ai -a0 (mm)



The areas of each block are: #        " 0:3 0:4 0:4 0:4 mm þ þ þ ; 4:4  1010 7:8  1010 2  109 7  109 mm=cycle

The result is: 1.45  109 cycles. If the loading frequency is 60 Hz, a common number in rotating machinery, the retardation time is about 6712 h (approximately 280 days), that may be a significant number to program a monitoring program, instead of just taking the component out of service.

Fatigue and environmentally assisted crack propagation

199

A more sophisticated model of fatigue crack retardation after a single overload was developed by McEvily and Yang8 in 1990, based on the experimental observation of a second plasticity-induced crack closure event after an overload, which decreases as the crack advances through the overload plastic zone, as shown in Fig. 6.21. According to this model FCG rate is given by the following equation: da 2 ¼ AðDKeff  DKeffth Þ dN Where DKeffth is the DKeff near the fatigue threshold and DKeff ¼ Kmax e Kop. On the other hand, the overload generates an excess closure level EC, given by the equation: 8 2 !1=2 39 < = 2 Kol 5 EC ¼ 0:6Kol 1  exp4  : ; 2ps2y B

Before OL

Second closure level

Kmax

K

Kol

Variation of the second level of closure through rOL

Kop

First closure level, Kop

normal

Kmin Extension of overload plastic zone

Displacement

Figure 6.21 Load versus displacement of a fatigue cracked specimen showing the closure levels after a single overload.

8

McEvily A. J., Yang Z. On the Significance of Crack Closure in Fatigue Crack Growth. Metallurgical Transactions 26A (1990).

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A Practical Approach to Fracture Mechanics

Where B is the thickness, Kol is the overload K value, and sy is the yield strength. The number of loading cycles during the retardation Nd is calculated by the difference between the necessary cycles to cross the overload plastic zone at DKeff minus the cycles to cover the same distance at the normal DK level, given by the following equation: # " Kol2 1 1  Nd ¼ 2 psy A ðDK  EC  DKth Þ2 ðDK  DKth Þ2 Plotting this formula as Nd versus rol gives the characteristic U-shaped curve that indicates that near the fatigue threshold, a small reduction of DKeff results in large increments of Nd, while at high DKeff levels, again Nd becomes very large.

6.7 Fatigue cracks emanating from notches and holes One case of practical importance are fatigue cracks emanating from stress concentrators like holes or notches. In general, the stress concentrators can be characterized by a notch root radius represented here by the symbol r and a notch depth, represented by D (which is replaced by the diameter or the long radius in the case of circular and elliptical holes), as shown in Fig. 6.22. Smith and Miller9 proposed the following expression for DK, valid for short cracks emanating from notches, when a < 0.13(Dr)1/2: 'V

U

D

a

Figure 6.22 Parameters to characterize cracks emanating from a notch.

9

R. A. Smith and K. J. Miller, “Fatigue Cracks at Notches”. Int. J. Mech. Sci., Vol. 19, 1977, pp. 11-22.

Fatigue and environmentally assisted crack propagation

201

rffiffiffiffi !1=2 pffiffiffiffiffi D DK ¼ Ds pa 1 þ 7:69 r Based on the similitude principle of LEFM, an already existing crack emanating from a notch will propagate if: DK  DKth Where DKth is the threshold SIF for a short crack that is a crack within the notch stress field. Bazant10 proposed an equation relating the short and long DKth values, Meggiolaro11 found the best fit to experimental data of Bazant’s equation, obtaining that: h a i1=2 0 DKth ¼ DKthN 1 þ a Where DKthN is the threshold value of a long crack and a0 is a characteristic crack length, which can be estimated by: 2  1 DKthN a0 ¼ p Dsth Where Dsth is the fatigue limit of the material, determined from SeN test of unnotched bars. These equations are useful to determine if a stress concentrator such as a notch introduces a risk of fatigue failure, as illustrated by the following example: A metal sheet panel under a cyclic stress amplitude of 500 kg/cm2 contains a notch of 1.0 cm radius and 2.0 cm depth. If the material fatigue limit is 2400 kg/ cm2 and the SIF fatigue limit is 650 kg cm3/2, determine if there is a risk of having fatigue cracks emanating from the hole. Solution: The first step is to determine the short crack length: a < 0.13(Dr)1/2 ¼ 0.13 (2.0 cm * 1.0 cm)1/2 ¼ 0.184 cm Assuming a short crack size of 0.1 cm < 0.184 cm, the SIF amplitude is:

Z.P. Bazant, “Scaling of quasibrittle fracture: asymptotic analysis. Int. J. of Fracture, 1997, 83, pp. 19-40. 11 M. A. Meggiolaro, A. C. de Oliveira, J. T. Pinho, “Short crack threshold estimates to predict notch sensitivity factors in fatigue”. Int. J. of Fatigue, 29 (2007), 2022e31. 10

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A Practical Approach to Fracture Mechanics

DK ¼ DsO(pa)[1 þ 7.69O(D/r)]1/2 ¼ 500 kg/cm2O(p * 0.1 cm) [1 þ 7.69O(2 cm/1 cm)]1/2 ¼ 966 kg cm3/2 N a0 ¼ (1/p)(DKth /Dsth)2 ¼ (1/p)(650 kg cm3/2/2400 kg/cm2)2 ¼ 0.023 cm 1/2 ¼ 650 kg cm3/2 [1 þ (0.023/0.1)]1/2 DKth ¼ DKN th [1þ(a0/a)] ¼ 585 kg cm3/2

Since DK > DKth, the answer is, yes! There is a risk that a crack will grow by fatigue. Furthermore, the number of cycles at which the crack will reach a given size can be calculated by integration of the corresponding Paris equation for the materialenvironment condition. Another topic of great practical importance is to determine whether or not a crack will grow from a circular hole in a flat plate under a uniform tensile stress amplitude. Again, for a crack to grow the condition DK > DKth must be meet. It is known that the SIF amplitude of a crack emanating from a hole in an infinite plate can be estimated as12 pffiffiffiffiffi DK ¼ 1:12 Kt Ds pl Where Kt is the stress concentration factor, and l is the crack length, when the crack is within the stress field of the hole, whose value can be estimated by lt < 0:0919d This formula is obtained from the Smith and Miller expression to determine the transition length of a crack from short to macroscopic: lt < 0.13(Dr)1/2, making D ¼ d and r ¼ D/2. Now, assuming that in most of the cases the diameter of the hole (d) will be less than 10% of the width (W) an approximated expression of Kt for in the range 0 < d/W < 0.1 is   d Kt ¼ 3  2:789 W Substituting Kt, making DsO(pl) ¼ DKsc (the SIF amplitude of the short crack without the influence of the hole’s stress field), replacing DK by DKth, and rearranging terms, the condition for a fatigue crack to propagate from a circular hole is   DKth d < 3:36  3:12368 W DKsc 12

J. A. Bannantine, J. J. Comer and J. L. Hardrock, “Fundamentals of Metals Fatigue Analysis”, Prentice Hall, New Jersey (1989), Secc. 4.4.2.

Fatigue and environmentally assisted crack propagation

203

Notice that for a very large plate d/W / 0, so the above equation tells that the DKsc acting in the short crack emanating from a hole has to be at least 3.36 times smaller than DKth of the material to prevent the propagation of a fatigue crack, a situation not very common in many structural and mechanical components. Many steel structure inspection and repair manuals propose a much simpler empirical formula to determine the minimum hole size required to prevent fatigue crack initiation. The formula is r ¼ 0.01(DK2/sys), where r is in in., sys in ksi and DK in ksiOin. However, this formula is not accurate and underestimates the minimum requiered hole radius to arrest a crack, as illustrated in the next example. A 40 in wide panel with a central circular hole of 0.25 inches of diameter is cyclically stressed at one third of the yield strength at R ¼ 0. If sys ¼ 36 ksi and DKth ¼ 6.0 ksiOin, determine if a fatigue crack may emanate from the hole. Solution: l ¼ 0.0919d ¼ 0.0919 * 0.25 in ¼ 0.023 in DKsc ¼ DsO(pl) ¼ (36/3) ksi * O(p * 0.023 in) ¼ 3.23 ksiOin (DKth /DKsc) ¼ (6.0 ksiOin/3.23 ksiOin) ¼ 1.86 d/W ¼ 0.25/40 < 0.1, then: 3.36  3.12368(0.25 in/40 in) ¼ 3.34 Since 1.86 < 3.34 a fatigue crack will propagate from the hole. Making a larger hole make things worse, for example if the hole diameter is 2 in, so d/W ¼ 0.05; l ¼ 0.184 in; DKsc ¼ 12 ksi * O(p * 0.184 in) ¼ 9.12 ksiOin; (DKth/DKsc) ¼ 6.0/9.12 ¼ 0.66; 3.36  3.12,368(2 in/40 in) ¼ 3.2. By using the empirical formula: r ¼ 0.01(62/36) ¼ 0.01 in., so the hole of 0.25 in. would prevent fatigue cracking, which may not be true. In conclusion, sooner or later a hole in a structural component under cyclic loading will generate a crack, thus demonstrating that drilling a hole at the tip of a fatigue crack to arrest it may not be a good idea. The author personally verified the previous statement in a failure analysis case (not published for confidentiality), shown in the photograph of Fig. 6.23. The component was a large vessel made of A580 steel (DKth z 10 MPaOm) and it was fatigue cracked by thermal gradients, the SIF amplitude was DKapp z 20 MPaOm, so DKth/DKapp ¼ 0.5. A maintenance crew drilled holes at the crack tips in an attempt to arrest the cracks, but according to an analysis done by the previous approach, to prevent FCG, d/W had to be z 0.91. Obviously, such condition was not met and weeks after drilling the holes, the fatigue cracks grew again from the holes.

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A Practical Approach to Fracture Mechanics

Figure 6.23 Practical example of fatigue cracks emanating from holes drilled in an attempt to arrest the previously formed fatigue cracks.

6.8 Stress-corrosion cracking The stress-corrosion cracking (SCC) is a slow fracture mechanism that occurs by the interaction of a corrosive environment, a susceptible material and a sufficiently high applied stress. These three factors act in combination to nucleate and propagate a crack and make it grow up to its critical size, leading to a complete fracture. There are two general forms of SCC: • Controlled by the environment: The crack growth rate is relatively independent of the applied stress, so the crack extends by dissolution of the material at the crack tip, forming layers of corrosion products, and localized attack (pitting) of the fracture surface. • Controlled by the stress: The crack is driven by the stress field at the crack tip, thus is dependent on KI. The crack growth mechanism is by brittle fracture modes such as cleavage and intergranular cracking. The typical SCC testing consists of measuring the time of rupture as a function of the applied stress in a smooth specimen, similar to a tension test bar. By plotting the applied stress versus log(rupture time) data, the result is shown in Fig. 6.24. The main characteristics of SCC are that the rupture time depends exponentially on the applied stress, and there is a threshold stress below which the rupture time is infinite; this stress level is usually referred as sSCC and it depends on the material and the environment. If the applied stress is below the yield strength, the SCC crack will propagate through an elastically strained solid, therefore the crack behavior

Fatigue and environmentally assisted crack propagation

205

Vmax Vo Vapp Time to failure o f LEFM Conditions

Threshold stress, VSCC

log (Rupture time)

Figure 6.24 Log(time of rupture) versus applied stress plot used to assess SCC in smooth bar tension test specimens.

can be characterized by the stress intensity factor. The SCC growth rate can be measured by exposing a precracked loaded specimen to the testing environment, applying a constant load or constant displacement (the specimen is loaded in Mode I), and periodically measuring the crack advance. By recording the crack length and knowing the applied load, the value of KI can be determined during the course of the test, as schematically depicted in Fig. 6.25, corresponding to constant load condition, therefore, as the crack grows, KI increases and the crack is accelerated until it reaches the critical size and the specimen breaks. The specimen geometries that best perform in this kind of test are the single edge notched bending beam and the cantilever bar with an edge crack. In both specimens it is relatively easy to attach the environmental chamber and to monitor the crack advance. KIC

KI a Fixed end

KI0 Environmental chamber

Dead weight

a0 Time

Figure 6.25 Precracked cantilever beam specimen for SCC testing, and typical record of the crack growth indicating an increasing-KI condition.

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A Practical Approach to Fracture Mechanics

Another option for SCC testing is to place the test specimen under constant displacement conditions. In this case, a precracked specimen is loaded by a bolt to a point that assures the propagation of the crack by SCC and then the bolt is locked, so the crack opening displacement is kept constant, setting the conditions for a K-decreasing condition as the crack grows. The results of this test are the crack growth rates and the arrest KI, defined as KISCC. The typical geometry used for this test is a CT specimen modified to introduce the loading screw. A schematic of this testing method is shown in Fig. 6.26. The dependency of SCC growth rate on the stress intensity factor is explained by the fact that K determines the magnitude of the stress at the crack tip and the crack opening displacement (COD), so the higher K the higher stresses; additionally, the greater COD facilitates the income of the corrosive species into the crack cavity, while the stretching plastic deformation at the crack tip can break the passive layers and expose new fresh material to the corrosive environment. All of these provide an extraordinary energy supply, so to the crack can easily overcome the energy barriers along its path through the microstructure. Similar to the fracture mechanics treatment of FCG data, the SCC growth rates, and the corresponding applied KI values can be plotted in a log-log graph to obtain a Paris-like curve, as shown in Fig. 6.27. Just as in fatigue, in SCC an amount of time is spent on nucleating cracks, certainly in a lesser fraction of the total rupture time than fatigue, so it is of great practical interest to measure and predict the growth rates of

a, KI

Loading bolt

Constant opening displacement

K0

a

a0

KI

KISCC

Environmental chamber

Time

Figure 6.26 Precracked-modified CT specimen for SCC testing, typical record of the crack growth indicating a Decreasing-KI condition.

Fatigue and environmentally assisted crack propagation

Region I

Region III

Region II

Control by KI

207

Log(da/dt)

Control by the dissolution rate at the crack tip

da/dt = M i /zFd

da/dt = C Km’

KSCC

KIC

log (K)

Figure 6.27 SCC growth rate versus KI curve of metallic materials showing the three regions of SCC behavior.

SCC cracks. According to Fig. 6.27, the crack growth rate in Region I can be expressed as da ¼ A0K m0 dt Where A0 and m0 are experimental constants, specific to a material and environment combination. In Region II the crack growth is controlled by the anodic dissolution rate, so the rate of SCC crack propagation can be estimated by da M ¼ icorr dt zFr Where M is the molecular weight of the metallic material, z is the number of electrical charges transferred in the dissolution reaction, F is the Faraday’s constant, r is the material density and icorr is the ionic exchange density current at the crack tip. The integration of both equations gives the time to grow an SCC crack from an initial size (a0) to a critical size (ac). 1 t¼ pffiffiffi m0 Cðbs p Þ

Zac



m=2

a a0

Micorr da þ zFr

1 ðac  a0 Þ

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A Practical Approach to Fracture Mechanics

An expression for the time of crack growth in the Region III is not considered because, in general, the stressecorrosion induced cracks are very unpredictable, and fast growing, so when a component is in Region III of SCC, it is considered as failure and no further assessments are done. The Annex 9F.5.4 of the API 579-1/ASME FFS-1 2016 “Fitness for Service” standard, presents several formulas for (da/dt)SCC in 2.25Cr-0.5Mo steels exposed to 500 ppm H2S solutions. This standard suggests that the (da/ dt)SCC data for other materials and environments shall be searched in the scientific literature, making sure that the material and environment conditions are as close as possible to the actual service conditions of the component being assessed. The analytical treatment and prediction of SCC are so complicated, uncertain and random, that in practice, the testing and assessment of SCC kinetics is done merely for classification and material selection purposes, and most structural integrity standards establish that once an SCC crack is detected, the component that contains it should be scheduled for repair or replacement.

6.9 Creep crack growth In conditions of high temperature exposure, a crack can extend under constant load because the deformation is time dependent. Such phenomenon is known as creep and consists of the plastic deformation of a material under constant stress. Creep makes stresses and strains vary continuously and as a consequence, typical fracture parameters such as K, J-Integral and CTOD are dynamic, as well. Creep deformation usually features three stages, as schematically shown in Fig. 6.28. The stages of creep deformation include the following: Primary: It consists of an instantaneous deformation, product from the applied stress, followed by a deformation that decelerates as time passes. Secondary: This is the most important stage because it takes most of the time and deformation. It is characterized by its dε=dtstrain rate. It is also called steady creep. The strain rate in steady creep depends on the stress according to the Secondary Creep Power Law, expressed by: dε ¼ Csn dt

Fatigue and environmentally assisted crack propagation

Stage I

Creep strain

Rupture strain

Stage II

209

Stage III

dH/dt = C V n

Initial strain Time

Rupture time

Figure 6.28 Typical creep curve.

Tertiary: It is characterized by a sudden increase in the strain rate. In this stage the damage mechanisms such as grain boundary cavitation are extensive and end up in final fracture. If the process zone is in secondary creep, the strain rate is constant, so it might be expected that the crack propagates at a constant rate, as schematically shown in Fig. 6.29. However, the phenomenon is not that simple, and complications arise because as the crack advances the stress in the plastic zone increases, thus increasing the strain rate, as predicted by the power law of creep, all of which hinders the analysis based only on the applied stress level.

Uniform stress, V

Creep growing process zone:

G

drc/dt

Crack extension, 'a Initial crack length, a

Process zone size, rc

Figure 6.29 Crack extension by time-dependent creep strain.

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A Practical Approach to Fracture Mechanics

Landes and Begley13 introduced the C* parameter to characterize the crack growth behavior under dynamic strain condition, which is defined as: C* ¼

1 dU B da

Where U is the change of stored energy by crack growth and B is the thickness. Therefore C* is the dynamic energy release rate in the process zone due to a crack extension. Since the process zone is under timedependent strain the crack opening displacement is also dynamic. If d is the crack opening displacement, the dynamic crack opening displacement rate is D ¼ d/t. Similar to static crack loading, a dynamic creep compliance Cn can be defined, thus the dynamic crack opening displacement can be expressed as: D ¼ CnP n Where n is the exponent of the secondary creep power law. Thus, C* is defined by:   P nþ1 dCn * C ¼ ðn þ 1ÞB da The C* parameter has been calculated for simple geometries in order to perform test time-dependent fracture testing, using the following expression: PD  n  C* ¼ Y B nþ1 Where Y is a geometry function: Y ¼ 2 þ 0.52(B/W) CT specimen Y ¼ 2.0 SE(B) specimen Y ¼ 1.1 Tension bar

13

Landes, J.D and Begley, J.A. A Fracture Mechanics Approach to Creep Crack Growth, ASTM STP 590 (1976).

Fatigue and environmentally assisted crack propagation

211

Nikbin, Smith, and Webster14 demonstrated that C* can be determined from the balance of the stored energy and the work done by the stresses, given by the following expression:  Z
vui C* ¼ ðdW =dtÞdy  sij nj ds vx Which corresponds to C* ¼

dJ dt

Where J is the applied J-integral. Experimental studies have demonstrated that the C* parameter correlates well with secondary creep crack growth rate through * m AC da ¼ dt εf Where εf is the creep fracture strain under uniaxial tension and A and m are experimental constants. By performing creep crack growth testing, they found that for high temperature ferrous alloys the secondary creep crack growth rate becomes 0:85 da Q C * ¼ dt εf where Q ¼ 0.3 for plane stress, and Q ¼ 15 for plane strain, having da/dt in m/hour, C* in MPa-m/hour and εf in percentage strain. The experimental evidence indicates that this model is good only as a first approximation to the data scattering tendency, as shown in Fig. 6.30. The creep crack growth can also occur by a mechanism of growth and coalescence of grain boundary voids in the cracking plane, commonly known as cavitation damage. This mechanism is controlled by the vacancy diffusion rate and the viscous flow of the matrix around the voids.

14

K. M. Nikbin, D. J. Smith and G. A. Webster: Prediction of creep crack growth from uniaxial creep data. Proc. R. Soc. Lond., Ser. A 396 (1984), pp. 183e197.

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A Practical Approach to Fracture Mechanics

1.0E+03 1.0E+02

da/dt [mm/hour]

Plane strain 1.0E+01 1.0E+00 1.0E-01 Plane stress 1.0E-02 1.0E-03 1.0E-04 0.01

0.1

1

10

100

C* [kJ/m2/hour]

Figure 6.30 Typical values of creep crack growth rate as a function of the C* parameter by the Nikbin, Smith, and Webster model da/dt ¼ Q(C*)0.85/εf.

A theoretical model to predict the creep crack growth rate by grain boundary cavitation was introduced by Miller and Pilnkington,15 in 1978. The schematic representation of this model is shown in Fig. 6.31, where d is the grain size, r0 is the initial void radius, and l is the separation between voids. According to this model, the voids in the grain boundary grow by stressenhanced vacancy diffusion and viscous flow of the matrix until the sum of the void diameters is equal to the grain size d, thus making the crack to extend the same length, and then, the process is repeated to sustain a macroscopic crack growth rate. Miller and Pilkington derived the following expression for the grain boundary cavitation creep crack growth: da ¼ CK n dt where K is the Mode I stress intensity factor and n is the power law creep exponent. The experimental results correlated well with this model as shown in Fig. 6.32. Even though the theoretical models to predict the creep crack growth rates are still in the state of the art, they are of practical use in the assessment of creep fracture, since usually the limit state is not defined by a rupture 15

D. A. Miller and R. Pilkington: Creep Crack Growth and Cavitation Damage. Met. Trans. A, 1978, vol. 9A, pp. 489e94.

Fatigue and environmentally assisted crack propagation

a

213

d Grain boundary

Crack

2ro O

Void

Coalescence

Void growth

Crack extension

6 rf = d

Figure 6.31 Schematic representation of the mechanism of crack propagation by growth and coalescence of grain boundary voids. 103

da/dt, (Pm /hr)

da/dt = C Kn 102

10 40

K (MPa m1/2)

100

Figure 6.32 Dependency of the creep crack growth rate on the stress intensity factor of CreMo ferritic steels at temperatures ranging from 400 to 838 C. Graph constructed from data of several sources.

time or a critical crack size, but by a maximum allowable crack growth rate; therefore there is no need to integrate the creep crack growth equations and it is enough with a fairly precise estimation of the creep crack growth rate.

6.10 Crack growth by absorbed hydrogen In the scope of this book, hydrogen-induced cracking (HIC) is a phenomenon susceptible to occur in any metal-environment system where there is freshly generated hydrogen, like sour gas, or cathodic charging, in

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A Practical Approach to Fracture Mechanics

Extension of the crack when KI = KIC

Extension of the crack when KI = KIC

B

p

Process zone

2a

Process zone

H diffusion

Figure 6.33 Mechanism of hydrogen-induced cracking (HIC) by pressurization of an internal cavity.

the presence of a poison chemical spice (sulfhydric acid or fluourhydric acid) that promotes the diffusion of atomic hydrogen into the metal, and that further recombines to form molecular gas at internal sites in the metal, such as nonmetallic inclusions. The very high fugacity pressures cause the decohesion of the interfaces at the hydrogen recombination sites, forming subsurface planar cracks that grow in planes parallel to the component wall. This mechanism is schematically depicted in Fig. 6.33 and may be analyzed under the principles of LEFM as described next. At the sites inside the metal where the hydrogen recombines to form a gas, the interfaces around the edges crack in a brittle way, partly because of the sharp radius of the cavity and part by the embrittlement caused by the hydrogen dissolved into the metal. From the fracture mechanics point of view, the cavity is an internally pressurized crack contained in an elastically strained material, so its behavior is characterized by Mode I SIF (KI); as more hydrogen arrives to the cavity, KI increases until the fracture toughness in the plane of cracking is reached, and the crack experiences an extension. Upon crack extension, the volume of the cavity increases. If the hydrogen flux into the cavity is not high enough, the pressure will drop, and the crack will be arrested until the cavity is filled again with hydrogen, and the process repeats. The HIC rate by this pressure mechanism was modeled by González et. 16 al. under the following considerations: a. The crack has a disk-like shape and is located in a plane at the midthickness of the metal plate (planar crack). 16

J.L. Gonzalez, et. al. Hydrogen-Induced Crack Growth Rate in Steel Plates Exposed to Sour Environments. The Journal of Science and Engineering Corrosion, Vol. 53, No. 12, NACE International (1997), pp. 935e943.

Fatigue and environmentally assisted crack propagation

215

b. The flux of hydrogen is constant, and the diffusion coefficient of hydrogen is constant. c. The pressure inside the crack depends on the crack size and the number of hydrogen moles contained in it. d. The material is linear elastic and isotropic. The crack size is 2a, the pressure inside the crack is p, and the plate thickness is B. The equation of the SIF for this geometry is rffiffiffi a KI ¼ 2p p The crack extension criterion becomes rffiffiffi KICH p p a 2 It is worth mentioning that the fracture toughness value should be the corresponding to a hydrogen saturation condition KICH. Deriving the previous equation, the variation of pressure in the crack, as the crack grows is obtained pffiffiffi dp p ¼ KICH a3=2 da 4 The pressure of hydrogen gas within the crack can be expressed by the ideal gas law modified by a constant a that corrects the deviation from ideality. p¼

anRT V

Where p is the pressure, n is the number of gaseous hydrogen moles, R is the ideal gas constant, T is the temperature, and V is the volume. Assuming that the crack is an internally pressurized disk of radius equal to a, in an elastic medium V¼

16pa3 3E

where E is the Young’s modulus. Therefore, the pressure in the crack is  n 1=2 pða; nÞ ¼ C 3 a

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A Practical Approach to Fracture Mechanics



3aRTE C¼ 16

1=2

To determine the variation of pressure with respect to the time, the total differential of the expression above is calculated     dp dp da dp dn ¼ þ dt da n dt dn a dt substituting dp 3 1 5 da 1 dn ¼  Cn2 a2 þ Ca3=2 n1=2 dt 2 dt 2 dt To establish the crack growth rate (da/dt), as a function of the variation of pressure, the chain rule is used da da dp dp ¼ x ¼ K'a3=2 dt dp dt dt 4 K 0 ¼ pffiffiffiffiffi p KICH On the other hand, the pressure causing the crack extension is given by pffiffiffi KICH p 1=2 p¼ a 2 The number of hydrogen gas moles is n¼

 p 2 C

a3

At constant temperature, n becomes pffiffiffi KICH pa2 n¼ 2C The flux of hydrogen ( J ) though the in-plane projected crack area (pa2) is dn ¼ Jpa2 dt

Fatigue and environmentally assisted crack propagation

217

Assuming steady hydrogen flux into the crack and neglecting the hydrogen flux out, the first law of Fick can be used to determine the flux J J ¼ 2DH

DCH B

Substituting J, into dn/dt dn 2pDH DCH 2 ¼ a dt B Where DCH is the hydrogen concentration gradient between the metal surface and the crack, and DH the diffusion coefficient of hydrogen in steel. Substituting the previous equations and the constants K0 y C, the HIC rate of growth is obtained
 da 3aRTEDH DCH ¼ a 2 dt 4KICH B Integrating the previous equation gives the crack size as a function of time a ¼ a0 eHt where a0 is the initial HIC crack nuclei and H is
 3aRTEDH DCH H¼ 2 4KICH B Fig. 6.34 shows the experimental results of HIC growth, obtained from cathodic charging laboratory experiments on a type API 5L X52 steel plate, 4.5 Exponential tendency

Crack length [cm]

4.0 3.5

Overall tendency

3.0 2.5 2.0

Interconnection of cracks

1.5 1.0

Growth of individual cracks

0.5 0.0 0

20

40

60

80

100

120

140

Time of exposure [days]

Figure 6.34 Comparison of experimental and predicted results of HIC growth of an API 5L X52 steel plate by cathodic charging and a synthetic sour medium.

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A Practical Approach to Fracture Mechanics

exposed to a synthetic sour solution. It is observed that the model prediction fits well with the experimental data, only in the early moments of HIC, that is when the cracks grow individually, since after some time, the crack begin to interconnect, and the growth becomes asymptotic.

CHAPTER 7

Structural integrity

7.1 In-service damage of structural components When structural or mechanical1 components are put into service, they are exposed to the action of damage mechanisms caused by the service conditions, the environment, or from external forces. The consequence of this damage is the deterioration of the physical state, leading to a reduction of the material strength, loss of mass (resulting on cross-section or thickness reduction), material alteration, and the formation of cracks and flaws. The main consequences of the presence of crack and flaws in a structural component are 1. reduction of the remaining strength; 2. increment of the probability of failure; and 3. reduction of remaining life. It is clear that, regardless of the deterioration mechanism, to reduce the remaining life of an in-service component, it is required that the effect of the damage mechanism accumulates through time or the resulting flaw continuously grows, otherwise, the formed flaw will only reduce the remaining strength, increasing the probability of failure, but without a predictable time to failure. A common practice in engineering is to test the component prior to enter into service, by the application of loads higher than the design load. These tests are meant to verify if there are undetected defects capable of reducing the strength below the design limits. They have the advantage of preventing in-service failures that are by far more dangerous and have greater consequences than a failure that may occur under the controlled testing conditions; their disadvantages are not providing quantitative information about the 1

In the context of this book, the terms “structural component” and “mechanical component” are synonyms and refer to any solid piece whose main purpose is to bear and/or transmit loads. Occasionally a mechanical component will be referred as the part of a machinery or process equipment, while the term structural component will be used for general cases.

A Practical Approach to Fracture Mechanics ISBN 978-0-12-823020-6 https://doi.org/10.1016/B978-0-12-823020-6.00007-4

© 2021 Elsevier Inc. All rights reserved.

219

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A Practical Approach to Fracture Mechanics

overall defect content, nor damage accumulation or defect growth, therefore passing the test cannot be considered as a guarantee of the absence of defects. Furthermore, the response of the component under the test conditions may be different from the actual service conditions. Some authors suggest that the overload or overpressure tests before placing into service structural components can be risky because some defects may grow to their near-critical size and later cause in-service failures. However, as predicted by fracture mechanics, overloading stable crack-like defects will produce a residual compressive stress that will increase the load range necessary to activate the defect, thus increasing the strength and arresting or retarding the growth of defects while in service. Therefore, with a few exceptions, the preservice overload tests, such as the hydrostatic test, are beneficial. The most important characteristic of the mechanisms of material deterioration is its accumulative nature, because that is the cause that the in-service components have a finite in-service life. Fig. 7.1 shows schematically the in-service deterioration rate graphic of a typical structural component, known as “the bathtub curve,” because its shape resembles the profile of this appliance. According to the bathtub curve, initial service stage corresponds to the youth, where the damage accumulation rate is initially fast due to the initial RSFa III. Data and information gathering. This stage consists in gathering data and information about the design, manufacture, construction and service conditions, necessary to carry out the FFS assessment. If any piece of information or data is not available, inspections, field

Structural integrity

253

surveys and examinations have to be carried out as necessary. The minimum required information is: 1. Name and identification tag of the component. 2. Design and material specifications data, including reports of laboratory tests and hydrostatic or preservice testing. 3. Installation date, date of operation start-up and service time. 4. Inspection records. 5. Actual operating conditions (load, pressure, temperature), including normal, peaks and any significant changes of service conditions in the past and future. 6. Safety and monitoring systems, such as relief valves, instrumentation and automatic control systems. 7. Failure analysis reports, if any. 8. Reports of repairs, modifications and alterations. 9. Responsible staff within the company. IV. In-service inspection. This stage consists of the detection, identification and determination of the significant dimensions of defects present in the component subject of the FFS assessment. The inspection engineers along with the structural integrity engineers shall elaborate the inspection plans. The plans shall consider the following: a. Nondestructive inspection techniques. b. Inspection procedures, including scope, sensitivity, minimum detection size and precision. c. Defect significant dimensions according to evaluation method. d. Criteria for grouping indications. e. Inspection date and date of report. f. Report formats. V. Defect evaluation. This is the core part of FFS assessments, consisting of the calculation of remaining strength of the defective component under the service loads. Depending on the assessment level, the criteria can be rather simple, for example, by comparing the defect size against an allowable size, or involving complex calculations by fracture mechanics and stress analysis. An important aspect of the defect evaluation is to have the correct thickness measurement data because almost all evaluation methods depend on this variable. Likewise, the loss of thickness, local or generalized, is accompanied by other damage mechanisms, so obtaining the required thickness is mandatory for the evaluation. Fig. 7.7 shows a scheme of the thicknesses to be reported for an FFS assessment.

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A Practical Approach to Fracture Mechanics

FCA LOSS tnom

tmin tc

tmm

tam

tam: average thickness of a zone of local metal loss. tc: remaining thickness of an area or uniform metal loss tmm: the minimum measured thickness in the zone of local damage. tnom: nominal thickness or thickness of new material right after construction. tmin: thickness required by the design code. It usually includes a safety factor. LOSS: uniform thickness loss. LOSS = tnom –tam, both sides (internal-external) FCA: Future Corrosion Allowance, is an additional thickness to compensate the uniform corrosion metal loss caused by the service.

Figure 7.7 Definition of thicknesses required for FFS assessments.

The Failure Analysis Diagram (FAD), described in Chapter 4, is the usual tool to evaluate crack-like defects. The FAD allows the determination of the in-service margin (iSM), which is a measure of how close if the component to the failure condition. The iSM is calculated by dividing the distance from the origin of the FAD to the failure envelope passing by the assessment point by the distance from the origin to the assessment point, as shown in Fig. 7.8. The allowable iSM is  1.0, since any iSM < 1.0 means that the assessment point is outside of the FAD envelope and therefore is prone to failure. Kr

F iSM = OF / OA A

O

Sr

Figure 7.8 Definition of the service margin (iSM) in the FAD of a cracked component. The segment OA is proportional to the evaluation load and segment OF is proportional to failure load.

Structural integrity

255

VI. Remaining life estimation. Only when a component is accepted, the remaining life must be calculated. Generally, it is advisable that the use of the remaining life is just to set an appropriate interval of inspection or to determine and appropriated schedule for repair or rehabilitation. The remaining life calculation must not be interpreted as a precise estimation of time of failure, nor a guarantee of a safe performance during the remaining life period. The remaining life estimations categorize as follows: • The remaining life can be established with reasonable certainty. In this case remaining life can be used to establish the appropriate inspection intervals. This estimation must be conservative and consider uncertainties related to material properties, stresses and the variability of service conditions affecting the damage accumulation rate. • The remaining life cannot be established with reasonable certainty. In these cases, appropriate repairs or rerating must be done in order to return the structural integrity up to acceptable levels. The inspection is limited to assure acceptability of the remediation methods. • The remaining life is minimal or null. The repair and/or shutdown is immediately required. More frequent monitoring of future operations of the repaired equipment may be applied since the damage mechanism proved to be severe. VII. Selection of repair or remediation methods. As a rule, the repair or remediation measures must be able to restore the component capability to its original design level, or at least, reach the target structural integrity level. Repair or remediation measures demand detailed analysis by specialists in accordance with the applicable standards. This must be carried out when: a. A defect is not accepted in its current condition. b. The remaining life is minimal or cannot be estimated. c. The reliability of the assessment or the knowledge of the damage mechanism is insufficient to provide an accurate estimation of the remaining strength or the remaining life. d. Establishing a reduced service load is not feasible due to process requirements and/or service demands. VIII. In-service monitoring. It applies to components returned to service, but where accumulative damage mechanism can be active generating future

256

A Practical Approach to Fracture Mechanics

damage. At this stage, based on results obtained from monitoring, the data to calculate the damage rate should be adequate to predict the type and occurrence of a failure or an unacceptable risk condition. The most important aspect of this stage is to assure the minimum precision, reproducibility and frequency of the monitoring and inspection systems. To achieve this, the following tasks must be carried out: 1. Define monitoring procedures according to the operation conditions. 2. Make monitoring reports at the appropriate frequency. 3. Define nondestructive inspection procedures and reports. 4. Assure that the data obtained is properly stored and can be retrieved and consulted at the inspector’s convenience. IX. Documentation (report making). The documentation for the FFS assessments consists of a written technical report, complemented with graphs, photos, tables and annexes, necessary and sufficient in order to make a more detailed analysis or allow a specialist engineer to repeat the assessment. The documentation of an FFS assessment is mandatory for the compliance of mechanical integrity management programs. The following information must be included as a minimum in the report. a. Design data, maintenance and operative history. b. Inspection data used in the assessment. c. Assumptions and special considerations. d. Additional data such as: B Part, edition and evaluation level of employed standard. B Service condition for the assessment. B Calculations, including maximum allowable load, reduced load and in-service margin. B Calculation of remaining life and criterion for scheduling future inspections and repairs. B Recommendations for risk mitigation and monitoring.

7.6 Example of a structural integrity assessment (1) Technical data: The component is a heat exchanger constructed as a horizontal cylindrical shell, with internal components, thermal insulation and nozzles of different diameters. The service is as reformer of hydrocarbon

Structural integrity

257

Figure 7.9 Heat exchanger subject of FFS assessment.

vapors coming from a direct fire heater. The design is in accordance with code ASME B&PV Code, Section VIII, Division 1. The design data are shown in Table 7.4. (2) Nondestructive inspection Fig. 7.9 shows a photograph of the heat exchanger in the condition as it was found at the time of performing the nondestructive inspections. The design drawing is shown in Fig. 7.10. The heat exchanger was subjected to an upset condition that caused an overheating that induced a bulging deformation along the upper length of the vessel. The peak height over the cylinder horizontal line of the bulging is 166 mm, located at the 12:00 h position and 5500 mm away from the reference weld (weld of the north head). Another bulging was found at the 6:00 h position, of peak height of 64.6 mm (2.54300 ), at 1500 mm from reference weld. The longitudinal extension of the bulging area is 1600 mm. The external perimeter of the most expanded part of the shell is 6730 mm, which corresponds to 2142.2 mm in diameter. The shell thickness was measured in 13 levels, at distances from 500 mm (19.68500 ) to 6500 mm (255.900 ) form the north head weld, at 500 mm (19.68500 ) intervals. The minimum measured thickness was 45.7 mm (1.80000 ), located at 1000 mm from the reference weld at the 12:00 h position maximum. The average thickness was 62.7 mm (2.46900 ); the standard deviation was 5 mm (0.19700 ). Neither metal loss nor internal discontinuities were found in the shell plate, therefore, it was not necessary to further continue the inspection.

258 A Practical Approach to Fracture Mechanics

Figure 7.10 Design drawing of the heat exchanger.

Structural integrity

259

Table 7.4 Component data. Material

Shell and heads

A 516 gr 70 N 



Maximum allowable stress at 290 C (554 F)

19,400 psi 3.2 mm (0.12500 )

Future corrosion allowance Dimensions

Internal diameter

1890 mm (74.400 )

Length (cylindrical shell)

7650 mm (301.2)

Nominal thickness

65 mm (2.54300 )

Design thickness per ASME B&PV Code Section VIII, Division 1, PG 27.2.2

54 mm (2.13000 )

Design and operating conditions

Operation pressure

72.5 kg/cm2 (1032 psig)

Design pressure

72.5 kg/cm2 (1032 psig)

Operation temperature

290 C (554 F)

Design temperature

290 C (554 F)

Hydrostatic test pressure Welds

109.6 kg/cm2 (1559.6 psig) RT 100%, weld joint efficiency ¼ 1.0

Empty weight

87,000 kg

Time of service

175,000 h (20 years)

(3) Identification of damage mechanisms According to the physical examination and nondestructive inspection of the heat exchanger, the damage mechanisms that have an adverse effect on its structural integrity are: (a) General metal loss caused by atmospheric corrosion and a contribution of short time creep strain. The assessment is done per API 579-1/ASME FFS-1 2007, Part 4 “Assessment of general metal loss.” (b) Shell distortion caused by the upset conditions (overpressure and overheating) that surpassed the mechanical strength of the shell causing plastic flow. The assessment is done per API 579-1/ASME FFS-1 2007, Part 8 “Assessment of weld misalignment and shell distortions.”

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A Practical Approach to Fracture Mechanics

Table 7.5 Level 1 Assessment of general metal loss. Step

1

Description

Nomenclature

Result

Assessment data Note: The nominal thickness is replaced by average measured thickness

Design temperature

290 C

MAWP

1032 psig

Nominal thickness tnom

2.54300

Average thickness tam

2.46900

Minimum thickness tmm

1.80000

Standard deviation

0.19700

Uniform metal loss LOSS

0.07400

Design thickness: tmím.

2.13000

FCA

0.12500

tc ¼ tnom. e LOSS  FCA

2.34400

2

Verification of creep condition (paragraph 4.1.1). UTS for carbon steel >414 MPa (60 ksi) at 371 C

Design temperature 290 C < 371 C

Pass

3

Applicability (paragraph 4.3.3.2). If COV  0.1 it is general metal loss

COV ¼ 0.197/ 2.469 ¼ 0.08 < 0.1

Pass

4

Level 1 assessment criteria (paragraph 4.2.2.1) MAWPr with (tam e FCA) If MAWPr  MAWP pass

MAWPr ¼ 1118 psig MAWP ¼ 1032 psig MAWPr > MAWP

Pass

5

Result: The Level 1 Assessment criteria for general metal loss are satisfied

(c) Creep damage caused by the exposure to elevated temperatures over a period of time under excessive stresses (short time creep). The assessment is done per API 579-1/ASME FFS-1 2007, Part 9 “Assessment of components operating in the creep range.” (4) Assessment of general metal loss The Level 1 Assessment of general metal loss in the heat exchanger, is performed according to the standard API 579-1/ASME FFS-1 Fitness-For-Service 2007, Part 4 “Assessment of general metal loss,” per paragraph 4.2.2.

Structural integrity

261

Table 7.5 shows the evaluation procedure and the result. Since the heat exchanger containing general metal loss is acceptable for continued operation by the Level 1 Assessment, the evaluation of remaining life for general metal loss proceeds as follows. The remaining life (Rlife) calculation is done by the thickness approach per paragraph 4.5.1, assuming a constant corrosion rate (Crate) and using Eq. (4.6) (K ¼ 1.0): tam  tmin Rlife ¼ Crate Crate is estimated by the following formula: Crate ¼

tam ðpÞ  tam ðcÞ Time between inspections

Where tam(p) is the average measured thickness in a past date, tam(c) is the average measured thickness at the current inspection. The current inspection reported tam(c) ¼ 2.46900 , an inspection performed at the installation time 20 years before reported tam(p) ¼ 2.54300 , substituting data. Crate ¼ ð2:543  2:469Þinch=20 years ¼ 0:0037 inch=year ¼ 3:7 mpy Therefore: Rlife ¼ ð2:469  2:130Þinch=0:0037 inch=year ¼ 91:6 years Since this remaining life surpasses the design life, the remaining life is assigned as the design life that is 30 years. (5) Assessment of shell distortion The Level 1 Assessment for shell distortion in the heat exchanger, is performed according to the standard API 579-1/ASME FFS-1 FitnessFor-Service 2007, Part 8 “Assessment of weld misalignment and shell distortion,” per paragraph 8.4.2. Table 7.6 shows the evaluation procedure and the result. Since the heat exchanger containing shell distortion is unacceptable for continued operation by the Level 1 Assessment, the evaluation of remaining life does not proceed per paragraph 2.5.1. (6) Assessment of creep damage The Level 1 Assessment for creep in the heat exchanger, is performed according to the standard API 579-1/ASME FFS-1 FitnessFor-Service 2007, Part 9 “Assessment of components operating in

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A Practical Approach to Fracture Mechanics

Table 7.6 Level 1 Assessment or shell distortion. Step

1

Description

Assessment data

2

Verification of the creep condition (paragraph 4.1.1)

3

Verification of cyclic service (paragraph B1.5.2)

4

Nomenclature

Result

Design temperature

290 C

MAWP

1032 psig

Nominal thickness tnom.

2.54300

Average thickness tam.

2.46900

Minimum thickness: tmm.

1.80000

Uniform metal loss: LOSS

0.07400

Design thickness: tmím.

2.13000

FCA

0.12500

tc ¼ tnom. e LOSS - FCA

2.34400

Limit temperature carbon steel UTS > 60 ksi

371 C

Design temp. < limit temp. 290 C < 371 C

Pass

Stop-start cycles < 150

Pass

Level 1 assessment criteria [(2032 e 1890)/1890]100 ¼ 7.5 > 1%

Not pass

1%(1890 mm) ¼ 18.9 mm 166 mm > 18.9 mm

Not pass

4.1

Roundness: Variation of ID < 1% D

4.2

Bulging < 1% ID

5

Result: The Level 1 Assessment criteria for shell distortion are not satisfied

the creep regime,” per paragraph 10.4.2.1. The upset caused an overheating to a maximum temperature of 650 C (1202 F) during 12 hours that induced operation in creep range, causing plastic strain and metallurgical alteration. Table 7.7 shows the evaluation procedure and the result. Since the heat exchanger containing creep damage is unacceptable for continued operation by the Level 2 Assessment, the evaluation of remaining life does not proceed per paragraph 2.5.1.

Table 7.7 Level 1 Assessment of creep damage. Step

1

Description

Assessment data

Nomenclature

Result

Material

A 516 Gr70

Design temperature

290 C 554 F

MAWP

1032 psig

Nominal stress

19.4 ksi

Expanded diameter

84.33900

Average thickness tam

2.46900

Brinell harness

Minimum hardness (paragraph 10.2.2.1 c)

3

Operation cycles (paragraph 10.2.2.2 c)

4

Maximum permissible time for operation (paragraph 10.4.2.1 d)

5

Level 1 assessment criterion Dc ¼ Rc tse  0.25 Effective stress (Eq. 10.22 of Part 10 API 579-1/ASME FFS-1 2007):

650 C (1202 F)

Exposure time tse

12 hrs

Carbon steel (UTS > 60 ksi), Brinell harness 100 132 > 100

Pass

50 cycles

Pass

From the screening curve (Fig. 10.3 of API 579-1/ASME FFS-1 2007) tse < 2.5 hrs Rc z 1.0 hrs1

Determine Dc

Dc ¼ 1.0 Hrs1  12 Hrs ¼ 12

Conduct level 2 (P. 10.5.2.3)

s1 ¼ PD/2tam ¼ 17.6 ksi s2 ¼ 1/2 s1 ¼ 8.8 ksi

Seff ¼ 17.26 ksi Continued

263

6

Assessment temperature

Structural integrity

2

132 

Step

Seff ¼ se exp 0.24

Description



J1 SS

Nomenclature

Result

s3 ¼ -P ¼ 1.0 ksi ss ¼ (s21 þ s22 þ s23)0.5 ¼ 19.7 ksi se ¼ (1/O2) [(s1-s2)2 þ (s2-s3)2 þ (s1-s3)2 ]0.5 ¼ 16.1 ksi J1 ¼ s1 þ s2 þ s3 ¼ 25.4 ksi

Larson-Miller parameter: LMP(Seff) (Table F.31 of Appendix F of API 579-1/ASME FFS-1 2007)

LMP ¼ 45.56157 e 3.9292158lnSeff

34.37

Rupture time L (Eq. 10.21 of Part 10 API 579-1/ASME FFS-1 2007) 1000 LMPðS Þ logL ¼ ðT þ460Þ eff  CLMP CLMP ¼ 20.0

Seff ¼ 17.26 ksi, T ¼ 1202 F

4.786 hrs

9

Creep assessment damage (paragraph 10.5.2.3 i) Dc ¼ tse/L

Dc ¼ 12/4.876

2.46

10

Assessment criterion (paragraph 10.5.2.3 L) Dc  0.8

2.46 > 0.8

Not pass

7

8

Result: The Level 2 Assessment criteria for creep damage are not satisfied.

A Practical Approach to Fracture Mechanics



264

Table 7.7 Level 1 Assessment of creep damage.dcont’d

Structural integrity

265

(7) Recommendations for remediation Since the general metal loss contained in the heat exchanger is acceptable for continued operation by the Level 1 Assessment, remediation actions are not required, it is only recommended monitoring thicknesses in accordance with the normal inspection program. Since the shell distortion and creep damage contained in the heat exchanger are unacceptable for continued operation, remediation actions are required. Because the bulged plates of the shell also suffered unacceptable creep damage, the recommendation is to substitute the damaged plates by new plates of the same material specification and nominal thickness as the original plates, following the guidelines of the API 510 Pressure Vessel Inspection Code: Maintenance Inspection, Rating, Repair, and Alteration, and ASME PCC-2 Repair of Pressure Equipment and Piping, presented in the articles related to mechanical repairs, replacement of pressure components. (8) In-service monitoring If the heat exchanger is repaired and rehabilitated and placed back into operation, it is recommended monitoring thicknesses in accordance with the normal inspection program to verify the general metal loss progress, and to monitor the temperature, especially during upsets. In case of detecting overheating perform the calculations of creep remaining life to determine if the component is acceptable for continued operation and perform dimensional verifications to detect possible shell distortion.

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Index Note: ‘Page numbers followed by “f ” indicate figures and “t” indicate tables.’

A Absorbed hydrogen, 213e218 Airy stress function, 42 American Petroleum Institute (API), 230 American Society of Mechanical Engineers (ASME), 230 Ashby’s material charts, 165

B Bathtub curve, 220, 220f Bazant’s equation, 201 British Standards Organization (BS), 230 Brittle fracture, 27, 89, 204 catastrophic, 159 Griffith criterion, 37, 38f two-parameter criterion, 134

C

Cauch^y’s stress theory, 13e14 Charpy impact test application, 96e97 fracture resistance, 157 load-time record, 97, 97f pendulum and test specimen geometry, 96, 96f Sailors-Corlton correlation, 97 Charts of material properties (CMP), 163e165, 164f Cleavage, 35, 204 Cohesive strength, 36f bond strength, 35 Hooke’s law, 35 stress concentration, 37 Young’s modulus, 36 Compliance, 75, 76f Compliance method, 56e57 Compression stresses, 5 Constitutive equations, 5 Continuum mechanics, 4

Crack arrest, 105f crack opening displacement (COD), 103, 103f R-curve, 99e100, 100f specimen-wedge pin arrangement, 102, 102f Crack closure, 184. See also Fatigue crack closure Crack front, 29 Crack instability, 98 Crack opening displacement (COD), 29, 72, 72f, 103, 103f, 206 Crack-tip, 29, 45 finite element method (FEM), 63, 64f J-integral, 109 plastic zone, 67 Crack tip opening displacement (CTOD), 72e74, 72f, 180e181 fracture parameter application, 125, 125f design curve, 131e132, 131f elastic (Ve) and plastic (Vp) contributions, 127, 128f load vs. clip gage displacement records, 126, 126f qualification criteria, 128e129 single notch bending specimen (SE(B)), 127 specimen parameters, 127, 127f two-parameter criterion, 133 Creep crack growth, 212f crack extension, 209, 209f creep curve, 208, 209f dependency, 212, 213f growth and coalescence, 212, 213f J-integral, 211 primary, 208 secondary, 208 tertiary, 209

267

268

Index

D Damage mechanism, 209, 219 adverse effects, 221 defects and typical affected components, 221e226, 226f mechanical and structural components, 221, 222te225t structural integrity. See Structural integrity Damage tolerance, 230 Debris-induced crack closure mechanism, 188, 189f Double cantilever beam (DCB), 57 Ductile fracture, 27 analysis, 123, 123f stages, 118e119, 119f Ductile tearing modulus, 123 Ductility, 6e7 crack-tip, 129e130 fatigue crack closure, 189 Dugdale’s strip yield model, 133, 133f Dynamic fracture, 98e106

E Elastic coefficient, 12e13 Elastic-plastic fracture mechanics, 30, 30fe31f. See also J-integral crack-tip opening displacement (CTOD), 125e133 critical crack size, 107 Irwin’s correction, 108 JIC testing, 113e117 load vs. crack opening displacement, 107, 108f plasticity correction, 107 two-parameter criterion, 133e143 Elber’s model, 187, 188f Elongation strain, 5 Energy criterion constant displacement condition, 75e79, 79f constant load condition, 75, 78 crack opening displacement, 80 energy release rate, 76e77 fracture toughness, 81 load-crack opening displacement, 75, 76f normal stress, 80

plane strain, 81 plane stress, 80 Poisson ratio, 80 stored energy exchange rate, 77

F Failure analysis critical crack size and stress, 166 elastic-plastic fracture and surface roughness parameters, 166, 166f fracture toughness, 165 Failure assessment diagram (FAD), 135, 136f Fatigue crack closure, 186fe187f debris, 188, 189f ductility, 189, 189f Elber’s model, 187, 188f grain size, 189e190, 190f phase transformation, 188e189 plasticity, 187, 188f roughness, 187, 188f Fatigue crack growth (FCG) environmental interaction mechanism categories, 190e191, 191f frequency, 192, 193f region II Paris curves, 190, 191f variables, 191e192 load ratio, 185t, 186f fatigue load cycle, 183 types, 183, 184f and Paris’s law conditions, 177 crack size, 179, 179f crack tip opening displacement (CTOD), 180e181 fatigue curves, 181, 182f linear elastic fracture mechanics (LEFM), 178 load amplitude, 178 metallic alloys, 180, 181t stress intensity factor (SIF), 177e178, 179f three regions, 179, 180f Young’s modulus, 180e181 single overload, 195e196, 196f load vs. displacement, 199, 199f normal plastic zone, 196

Index

overload plastic zone, 196 Wheeler model, 196, 197f variable loads, 192e194 Miner’s rule, 194 transients, 192e194, 194f Fatigue cracks emanating, notches/ holes, 204f Bazant’s equation, 201 parameters, 200, 200f FCG. See Fatigue crack growth (FCG) Finite element method (FEM), 62, 64f crack tip, 63, 64f displacement equations, 63e65 normal stress, 63 stiffness matrix, 65 First stress invariant, 18 Fitness-for-service (FFS), 2, 227, 230e231, 236, 250 Fracture plane, 28, 35, 80 Fracture propagation direction, 28 Fracture resistance charts of material properties (CMP), 163e165, 164f failure analysis, 165e168 leak before break (LBB), 172e176 materials selection, 162t brittle fracture, 156 Charpy impact energy, 157 critical crack size, 155 damage tolerance criterion, 158 ductile-brittle transition, 156 fracture strength/crack-like flaw tolerance, 161, 161t fracture stress, 160 fracture toughness, 153, 153f high-density polyethylene (HDP), 163 Hook law, 160 interrelated factors, 155 leak before break (LBB), 160e161 remaining strength curve (RSC), 158e159, 159f stored elastic energy, 160 temperature, 156, 156f titanium, 163

269

yield strength and plane strain fracture toughness, 153e155, 154t Young’s modulus, 153e155 reinforcement, of cracked structures, 168e171, 168fe170f remaining strength, 145e153 Fracture stress, 32, 81e82, 146 Griffith criterion, 38e39 Fracture toughness, 2, 32, 81 crack propagation rate, 99, 100f geometry and size, 94 J-integral, 113 material, 94 residual stresses, 94

G Griffith criterion brittle fracture, 37, 38f cohesive strength, 39 elastic energy, 37e38 fracture stress, 38e39 surface energy, 38

H High-density polyethylene (HDP), 163 Hook law, 160 Hydrogen-induced cracking (HIC), 217f mechanism, 213e214, 214f stress intensity factor (SIF), 215

I Indirect fatigue method, 60e61, 61f In-service failures, 219e220 International Standards Organization (ISO), 230 Irwin’s correction, 67, 108 plasticity, 133

J JIC testing, 114f, 116t compact tension (CT), 113, 113f load-crack mouth opening displacement (CMOD), 114 load-unload sequence, 115, 115f plasticity line, 115

270

Index

J-integral, 108e109, 137 closed path boundary, 109e110 cracked body, 109e110 crack extension, 111 crack-tip, 109 creep crack growth, 211 deformation energy density, 109e110 energy balance, 109, 109f fracture parameter ductile fracture, 118e119, 119f, 123f instability condition, 119, 119f plane strain, 118 plane stress, 118 tearing modulus, 123, 123f geometries and load conditions, 111 load-displacement record, 110, 110f solutions, 112

L Leak before break (LBB), 160e161, 172, 173f data gathering, 175 evaluation, 175 fracture toughness, 172e173, 174f leaking crack size, 176 leaking crack stability, 176 safety margin, 175 Linear damage rule, 194 Linear elastic fracture mechanics (LEFM), 30, 30fe31f, 145, 178 cohesive strength, 35e37 crack tip opening displacement (CTOD), 72e74 Griffith criterion, 37e40 plastic zone, 66, 66f, 69f, 72f cracked plate, 70, 70f crack tip, 67 Irwin’s correction, 67 Mohr’s circle, 69e70, 69fe70f plane strain, 68 plane stress, 68 plastic constriction, 71 shape and size, 68e69, 68f Von Mises yield criterion, 68

stress intensity factor. See Stress intensity factor (SIF) Load line, 29, 135, 137

M Material deterioration, 220 Mechanical behavior, tension, 9t, 11f ductility, 6e7 elastic coefficient, 12e13 hard and brittle, 8 hardness, 7 high strength, 8 limit stress, 10 load-elongation curve, 6, 6f necking, 6e7 plastic strain, 6e7 safety factor (SF), 10, 12f soft and ductile, 8 strain hardening, 6e7 stress-strain curve, 6, 6f, 8f ultimate tensile strength, 6e7 weak, 8 yield strength, 6e7 Young’s modulus, 6e7, 11 Mechanical integrity. See Structural integrity Miner’s rule, 194 Mohr’s circle, 17f shear stress, 21, 22f plastic zone, 69e70, 69fe70f rules for, 18e19 stress transformation, 19 three-dimensional stresses, 21, 21f two dimensions, 19, 19f

N Necking, 6e7 Newton’s second law, 3 Nondestructive testing (NDT), 242e249, 243te247t Normal stress, 4, 63 compression, 5 energy criterion, 80 physical effects, 5 tension, 5

Index

271

O

Q

Operational safety margin (OSM), 152

Quasibrittle fracture, 27

P

R

Paris’ law, 33, 60e61, 229e230 Photoelasticity, 59 Plane strain fracture toughness, 86e87, 86f Plane strain fracture toughness testing (KIC) ASTM E399-09, 92, 93f ASTM E399-72, 87 compact tension (CT), 87e88, 88f cyclic load amplitude, 93e94 fatigue crack starter notch configurations, 93, 93f linear elasticity and plane strain conditions, 88 linearity condition, 90 load-displacement records, 88e89, 89f precracking, 92e93 single notch bending specimen (SE(B)), 87e88, 88f thickness criterion, 91 yield strength and fracture toughness, 91, 92t Plastic collapse, 30, 30fe31f Plastic constriction, 71 Plasticity-induced crack closure mechanism, 187, 188f Plastic strain, 6e7 Plastic zone, 66, 66f, 69f, 72f cracked plate, 70, 70f crack tip, 67 Irwin’s correction, 67 Mohr’s circle, 69e70, 69fe70f plane strain, 68 plane stress, 68 plastic constriction, 71 shape and size, 68e69, 68f Von Mises yield criterion, 68 Poisson’s ratio, 65, 80 Pop-in crack extension, 82, 83f Principal stresses, 19 Pure mechanical fatigue, 190

R6 Code, 137 R-curve, 82f crack arrest, 99e100, 100f crack resistance, 84 critical crack size, 81e82, 83f energy criterion. See Energy criterion fracture stress, 81e82 instability, 81e82 pop-in crack extension, 82, 83f reinforcements of cracked structures, 168, 169fe170f stability, 81e82 stress intensity factor (SIF), 81 Remaining life cracked components crack propagation rate equation, 239 critical crack size, 239 damage remaining life, 238 economic life, 238 fitness-for-service (FFS), 236 load reduction and increment of resistance, 237, 237f material strength, 236e237, 236f remaining strength curve (RSC), 235e236, 235f stress intensity factor (SIF), 239 estimation, 242f defect critical size, 248e249 defect growth rate equations, 249 failure criterion, 248 material properties, 248 monitoring, 249 nondestructive inspection, 242e249 service conditions, 247e248 in-service component, 219 reduction, 219 Remaining strength, 32, 146f allowable remaining strength factor (RSFa), 150 crack tip control, 147e148, 148f finite width plate, 146, 147f

272

Index

Remaining strength (Continued) fracture stress, 146 ligament, 147 margin for safe operation (MSO), 149 net section control, 147e148, 148f, 151 net section criterion, 146 operational safety margin (OSM), 152 remaining strength factor (RSF), 150 stress intensity factor (SIF), 146, 151 Remaining strength curve (RSC), 146, 148, 149f, 235e236, 235f Remaining strength factor (RSF), 150, 252e253 Roughness-induced crack closure mechanism, 187, 188f

S Sailors-Corlton correlation, 97 SCC. See Stress-corrosion cracking (SCC) Secondary creep power law, 208 Shear strain, 5 Shear stress, 5 Sliding shear, 41 State of stresses, 14, 15t Steady creep, 208 Strain, 6f elongation strain, 5 hardening, 6e7 shear strain, 5 Stress, 3, 10 internal force, 4, 4f normal stress, 4e5 shear stress, 5 units, 5t Stress concentration, 24e26, 25fe26f Stress-corrosion cracking (SCC), 206f analytical treatment and prediction, 208 characteristics, 204 crack opening displacement (COD), 206 environments, 204 growth rate, 204e205 dependency, 206 log(time of rupture) vs. applied stress plot, 204, 205f

Paris-like curve, 206, 207f precracked cantilever beam specimen, 204e205, 205f Stress intensity factor (SIF), 2, 40e41, 47f, 107 Airy stress function, 42 crack tip, 45 experimental determination best fit equation, 58e59 compliance vs. crack size plot, 56e58 crack plane, 60 direct stress measurement, 60, 60f double cantilever beam (DCB), 57 indirect fatigue method, 60e61, 61f Paris’ law, 60e61 photoelasticity, 59 stress concentration factor, 54 finite element method (FEM), 62e66 fracture surfaces, 41, 41f hydrogen-induced cracking (HIC), 215 known solutions, 49f, 56f geometries, 49, 50te52t radial external crack, 53, 53f superposition method, 55, 55f plane strain, 45 plane stress, 45 R-curve, 81 remaining strength, 146, 151 sliding shear, 41 state of stresses, 41, 42f stress-corrosion cracking (SCC), 206 tearing shear, 41 tensile opening, 41 torsional loading, 46 two-dimension displacements, 45, 45f Stress tensor, 13e17, 14f, 15t, 16f, 19 Stress transformation, 17e18 Structural components bathtub curve, 220, 220f crack and flaws, 219 damage mechanisms, 221, 222te225t maturity stage, 220e221 old age stage, 221 overload/overpressure tests, 220 Structural integrity, 2, 33e34 assessment procedure

Index

component data, 259t creep damage, 261e262, 263te264t damage mechanism and defect type, 251e252, 259e260 data and information gathering, 253e254 defect evaluation, 254e255 documentation (report making), 256 failure analysis diagram (FAD), 254e255 fitness-for-service (FFS), 250e256, 251f general metal loss, 260e261, 260t in-service monitoring, 254, 256, 265 nondestructive inspection, 257, 257fe258f remaining life estimation, 255 remaining strength factor (RSF), 252e253 remediation recommendations, 262e265 repair/remediation methods, 255 shell distortion, 261, 262t technical data, 256e257 techniques and acceptance criteria, 252e253 damage mechanisms, 227 damage tolerance, 230 defects by service, 234 design and quality assurance, 227 external force induced, 234 fabrication and construction, 233e234 failure and safe operation load, 228 fitness-for-service (FFS), 227, 230e231 fracture mechanics engineering, 234e235 inspection engineering, 234 mechanical/structural engineering, 234 metallurgical/materials engineering, 234 process engineering, 234 input data and precision, 231e232 in-service loads, 227

273

maintenance and operational history, 227 maximum allowable defect size, 228 mechanical properties, 227 nondestructive inspection, 227 Paris’s law, 229e230 probabilistic variables, 231e232 problems, 229 remaining life, 228 the rule, 231 safe life philosophy, 228e229 standardization organizations, 230 worst-case approach, 233 Superposition method, 55, 55f

T Tearing shear, 41 Tensile opening, 41 Tension stresses, 5 Tresca’s criterion, 22 Two-parameter criterion crack tip opening displacement (CTOD), 133 Dugdale’s strip yield model, 133, 133f effective crack length, 133 elastic-plastic fracture, 134 failure assessment diagram (FAD), 135, 136f fracture toughness and yield strength variation, 135e137, 136f fast brittle fracture, 134 fitness-for-service assessment, 137e138, 138f load line, 135, 137 plastic collapse, 134 Pythagoras theorem, 139 R6 Code, 137 Sr parameter, 141

U Ultimate tensile strength, 6e7

V Von Mises yield criterion, 22, 68

274

Index

Y Yield criterion, 21e24 Yield maps, 23, 23f Yield strength, 6e7 Young’s modulus, 6e8 cohesive strength, 35

crack opening displacement (COD), 73 fatigue crack growth (FCG), 180e181 fracture resistance, 153e155 hydrogen-induced cracking (HIC), 215e216