A Practical Approach to Fracture Mechanics

By Jorge Luis González-Velázquez

A Practical Approach to Fracture Mechanics provides a concise overview on the fundamental concepts of fracture mechanics, discussing linear elastic fracture mechanics, fracture toughness, ductile fracture, slow crack propagation, structural integrity, and more. The book outlines analytical and experimental methods for determining the fracture resistance of mechanical and structural components, also demonstrating the use of fracture mechanics in failure analysis, reinforcement of cracked structures, and remaining life estimation. The characteristics of crack propagation induced by fatigue, stress-corrosion, creep, and absorbed hydrogen are also discussed. The book concludes with a chapter on the structural integrity analysis of cracked components alongside a real integrity assessment.

This book will be especially useful for students in mechanical, civil, industrial, metallurgical, aeronautical and chemical engineering, and for professional engineers looking for a refresher on core principles.

• Concisely outlines the underlying fundamentals of fracture mechanics, making physical concepts clear and simple and providing easily-understood applied examples
• Includes solved problems of the most common calculations, along with step-by-step procedures to perform widely-used methods in fracture mechanics
• Demonstrates how to determine stress intensity factors and fracture toughness, estimate crack growth rate, calculate failure load, and other methods and techniques

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 8, 2020
• ISBN: 9780128230527

Jorge Luis González-Velázquez

Dr. González-Velázquez is a Professor of Metallurgy and Materials Engineering at the Instituto Politecnico Nacional (IPN) in Mexico and Founder and Director of the Pipeline Integrity Assessment Group, with more than 27 years of continued experience on fracture mechanics and integrity assessment of pipelines and oil processing facilities. He has published over 160 papers and authored 3 books on fracture and mechanical behavior of engineering materials. He received the IPN Lazaro Cardenas Award as outstanding professor in 2007 and the Academy of Distinguished Engineers and Hall of Fame award of The University of Connecticut in 2008.

Book preview

A Practical Approach to Fracture Mechanics

Jorge Luis González-Velázquez

Metallurgical and Materials Engineering, Deparment of Instituto Politécnico Nacional, Mexico City, Mexico

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Copyright

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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 second-largest 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 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 excellent—but highly theoretical and full of complex mathematics—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.

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.

), 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 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

Abstract

A brief introduction of the importance of the study of the mechanics of materials is provided, describing the fields of applications in terms of the scale of objects and the mechanical desgin process based on the stress concept. Further, the definition of stress and strain tensors are given. The stress transformation and its significance are explained by Mohr's circle. Tresca’s and Von Mises’s yield criteria are briefly explained, along with examples of their use for design and materials selection. The chapter finishes with an introduction of the basic concepts of fracture, provinding the fundamental definitions of fracture, crack, fracture resistance, and the classification of fracture, to conclude with a discussion of the scope and field of application of fracture mechanics.

Keywords

Fracture concepts; Fracture mechanics; Mechanical design; Mohr's circle; Strain tensor; Stress concentration; Stress tensor; Yield criteria

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 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 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.

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

1.2. Definition of stress and strain

The French mathematician and scientist Augustin Louis Cauchŷ 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, Cauchŷ 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 Cauchŷ’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. 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 F A remains static, if and only if this force is balanced by an internal force Fi, of the same magnitude and opposite direction to F A . 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 σ, and can be defined as:

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 (F n ) and the other parallel (or tangential) to the cross-section area (F t ), as shown in Fig. 1.2.

The stress produced by F n is called normal stress and is expressed by

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

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 (+) 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 τ, being calculated as

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.

Cauchŷ 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 (l f   −   l 0) over the initial length (l 0)

Shear strain, identified by the symbol γ, and defined as the change of straight angle θ of a cubic volume element, so it is calculated as

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