The Fracture of Brittle Materials

The Fracture of Brittle Materials

Testing and Analysis

Second Edition

Stephen W. FreimanJohn J. Mecholsky, Jr.

Copyright © 2019 by The American Ceramic Society. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging‐in‐Publication Data

Names: Freiman, S. W., author. | Mecholsky, John J., Jr., author.Title: The fracture of brittle materials : testing and analysis / Stephen W. Freiman and

John J. Mecholsky, Jr.Description: Second edition. | Hoboken, New Jersey : John Wiley & Sons, Inc., [2018] |

Includes bibliographical references and index. | Identifiers: LCCN 2018036245 (print) | LCCN 2018036557 (ebook) |

ISBN 9781118769669 (Adobe PDF) | ISBN 9781118769775 (ePub) | ISBN 9781118769706 (hardcover)

Subjects: LCSH: Fracture mechanics. | Brittleness.Classification: LCC TA409 (ebook) | LCC TA409 .F765 2018 (print) | DDC 620.1/126–dc23LC record available at https://lccn.loc.gov/2018036245

Cover design by WileyCover image: Courtesy of Gina and Jason Blume

We want to thank Jason and Gina Blume for the picture of the natural fracture origin on the cover of this book.

Set in 11/13.5pts Times by SPi Global, Pondicherry, India

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

We want to dedicate this book to all the people who inspired us, guided us, and collaborated with us over the years. We especially want to acknowledge the influence of Roy Rice,

George. D Quinn, and Shelly Wiederhorn in formulating many of our ideas. In many cases, they provided the necessary

catalyst to further our knowledge through discussions and collaborative research.

vii

Preface ix

Acknowledgements xi

1. Introduction 1

2. Fracture Mechanics Background 7

3. Environmentally Enhanced Crack Growth 23

4. Fracture Mechanics Tests 37

5. Strength Testing 79

6. Thermally Induced Fracture 129

7. Modeling of Brittle Fracture 145

8. Quantitative Fractography 167

9. Microstructural Effects 207

10. Reliability and Time-dependent Fracture 223

11. Concluding Remarks 235

Subject Index 239

Contents

ix

The purpose of this book is to bring together the background, testing procedures, and analysis methods needed to design and use materials that fail in a brittle manner, primarily ceramics. In this context we define ceramics quite broadly as any inorganic nonmetal. Such a definition includes diverse materials such as semiconductors (e.g. Si, GaAs, InP), other single crystals (ZnSe, CaF, etc.), cements and concrete, and of course the oxides, carbides, nitrides, and others that we normally think of as ceramics. Ceramics are also used in composite form, either by dispersing one phase in another or by crystallizing phases from a glassy matrix. Most test procedures designed for monolithic bodies can be used here as well. However, continuous fiber‐reinforced composites behave quite differently and will not be discussed herein. Ceramics are also increasingly used in films and coatings, but determining the mechanical properties of materials in these forms is more complex and will not be addressed in this book.

This book addresses testing and analysis at temperatures for which the material behaves in a brittle manner. At elevated temperatures other modes of failure often are important. These include creep as well as general plas-tic deformation. Both of these topics are outside the scope of this book.

In this book we provide the reader some of the background needed to understand the brittle fracture process as well as a basis for choosing the proper test procedures. The mathematical development of the expres-sions used to calculate the various properties will be kept to a minimum; the reader will be referred to fundamental references. We intend to pro-vide examples to allow the reader unfamiliar with the tests to be able to perform the test procedures properly. However, the reader is strongly encouraged to consult formal national and international standards for more extensive test procedures. Questions to test comprehension for self‐evaluation are given at the conclusion of each chapter.

Preface

x Preface

Chapter  1 is a general introduction to the concept of brittle failure. Chapter 2 provides a condensed background into the basic principles of  fracture mechanics that underlies most of the test and analysis procedures. Linear elastic fracture mechanics (LEFM) is the basis for measuring the fracture toughness of materials. Chapter 3 gives some background into the theory and mechanisms of environmentally enhanced crack growth, a process that is particularly important for designing components that are intended to survive over long periods of time under stress. Chapter 4 provides extensive details on fracture mechanics tests used to determine both a material’s resistance to fast fracture and the parameters associated with environmentally enhanced crack growth. Chapter  5 addresses the test and analysis methods to determine the strength of ceramics. New in this chapter is a section describing test procedures applicable to biomaterials. Also new is a new procedure for determining the probability of failure within a set of ceramic components based on modern statistical concepts. Chapter 6 is a new addition that provides information on the causes of thermal shock failure, a common occurrence in ceramics. Also included are some of the test procedures that are used to rank the thermal shock resistance of such materials. Chapter 7 is also new; it describes attempts to model the fracture process and to provide predictions of resistance to crack growth. Chapter 8 provides a background and discusses the methods of understanding the fracture process based on quantitative measurements made on the fracture surface. Chapter 9 discusses an important topic with respect to polycrystalline materials, namely, the effect of the micro-structure of the specific material. Chapter 10 provides the background, test methods, and analytical procedures needed to confidently predict the safe lifetimes of brittle components under stress. Finally, Chapter 11 summarizes the critical issues with respect to brittle fracture.

Stephen W. FreimanJohn J. Mecholsky, Jr.

xi

We gratefully acknowledge the work of Nicholas Mecholsky in pre-paring many of the illustrations in this volume. We also appreciate the numerous technical discussions with George Quinn and Jeffrey Fong. Finally, we would like to express our gratitude to Roy Rice for introducing us to many of the topics discussed in this book, particularly those focused on the effects of microstructure.

S. W. F.J. J. M.

Acknowledgements

The Fracture of Brittle Materials: Testing and Analysis, Second Edition. Stephen W. Freiman and John J. Mecholsky, Jr. © 2019 The American Ceramic Society. Published 2019 by John Wiley & Sons, Inc.

1

CHAPTER 1

The properties of ceramics have made them extremely attractive to society in uses such as electrical and thermal insulators, high tem-perature crucibles for steel fabrication, elegant dinnerware, etc. More recently, their applications have become even more extensive and sophisticated, ranging from complex electronic devices such as multilayer capacitors and ultrasonic transducers to thermal protection for aircraft engines and applications in the dental and medical fields. However, the brittleness of ceramics, making them subject to sudden failure without prior warning, has at times limited more extensive use. Everyone knows that traditional ceramics, such as dishes and glasses, are brittle: drop a teacup or a plate, break a window, and you experience the brittleness. By brittle we mean that there are no mechanisms to relieve or alter the high stresses at crack tips, such as dislocations in metals or crazing in polymers. The lack of any stress relief mechanism results in cracks growing to failure at stresses that are significantly less than those necessary to initiate and propagate cracks in metals.

Despite their brittleness, advanced technical ceramics form the basis for a wide variety of important products. They are used in applica-tions in which they experience significant stresses imposed by not only

Introduction

2 The Fracture of Brittle Materials: Testing and Analysis

mechanical loading but also thermal, magnetic, or electronic conditions. One sees ceramics everywhere: the large electrical insulators on poles, spark plugs, and skyscraper windows that must resist high winds. Some we do not see or are not aware of. Cell phones would not operate with-out ceramics having special dielectric properties; automobiles contain hundreds of multilayer ceramic capacitors. Aircraft engines depend on ceramic coatings to reduce the temperature of the metal blades. Turbine engines for auxiliary power generation are now being constructed with rotating ceramic blades.

Another use of ceramics that requires complete reliability is aluminum or zirconium oxide hip and knee replacements in the human body. Dental ceramic prosthetic composites are routinely implanted in many patients. The hardness, inertness, and wear resistance of these materials make them ideal candidates to replace metals in such situa-tions. Particularly when the patient is young, the lesser amount of wear debris produced by the ceramic means that the component can be used in the body for a significantly longer time than one made of metal.

The list of ceramic applications is extensive, including materials that we do not normally think of as ceramics, e.g. semiconducting mate-rials, such as silicon, gallium arsenide, etc., and oxide films crucial for the operation of electronic devices. Because of the brittleness of these materials and their similarity in mechanical behavior to conventional ceramics, we refer to each of these materials as ceramics. Figure 1.1 shows some prime examples of advanced technical ceramics.

In each of these examples and in the myriad other applications, the brittleness of ceramics necessitates that special care must be taken in determining the mechanical properties of the material and discovering the stresses imposed on the final product during operation. The fact is that unseen, and probably undetectable, defects can lead to catastrophic failure. We will call these defects flaws. By a flaw we do not necessarily mean that errors were made in production. While improper processing can lead to pores or inclusions, component failures caused by these are relatively rare. For the vast majority of the time, brittle failure begins at the surface of a component from small cracks that are produced dur-ing the machining, finishing, or handling processes. All ceramics con-tain such flaws; there is no perfect brittle material. Even the strongest ceramic, pristine glass fibers, contains small flaws in its surface despite

CHAPTER 1 Introduction 3

the care taken to avoid any surface damage. It is the size and shape of such flaws, i.e. the flaw severity, and their location with respect to the tensile stresses that determine the strength of a component.

Brittle fracture is a statistical process. We usually think of such fail-ure in terms of a “weakest link” model. That is, failure begins from the

Metalacetabular cup

Pelvis

Femur

BIOLOX® deltafemoral head (pink)

BIOLOX® deltaliner (pink)

Femoral stem

A 16-blade silicon nitride turbine wheelfor use in small turbogenerators

Figure 1.1 Examples of advanced technical ceramics. From the left to right are an example of a ceramic hip replacement, barium titanate capacitors, various silicon nitride components, and a silicon nitride turbine wheel.

4 The Fracture of Brittle Materials: Testing and Analysis

most severe flaw located in the region of highest tensile stress. Also, the size of flaws in real components, 10–200 μm, means that detection of such defects by some nondestructive means prior to putting the part into service is extremely unlikely.

Another important aspect of most ceramic materials is that even if their strength when placed into service is sufficiently high that failure should not occur, in the presence of certain environments, e.g. water or water vapor, surface cracks will grow under the operational stresses, and failure can occur after a period of days, weeks, or even months. Fortunately, we have sufficient knowledge of this behavior, so that with proper testing and analysis, excellent predictions of the safe operating envelope, stress, and time can be given. Nonetheless, the user of ceramic components should recognize that such analysis only pertains to flaws that existed prior to putting the component in service. Other defects can be created during operation, e.g. from dust or rain, which may limit useful service life.

Knowledge of the brittle fracture process, most of which has been acquired over the past 30–40 years, has played a major part in our ability to design and use these materials, even in situations where the component is subject to significant tensile stresses. Two developments, which at the time were outside the field of materials science, were of major importance in contributing to our ability to safely use these mate-rials. One was the development of the field of linear elastic fracture mechanics. Fracture mechanics provides the framework by which the effect of the stresses imposed on a body can be translated into pre-dictions of the propensity of any cracks or flaws within the body to grow. This has led to the development of test methods and data analysis that permit designers to choose a material, machine it to shape without producing damage that could lead to premature failures, and carry out quality control procedures that provide confidence in the reliability of the part under operating conditions. A second important advancement, allowing us to design with brittle materials, was the development of statistical techniques that account for the uncertainties in the experi-mental measurements of the various parameters needed to make predic-tions of reliability.

A third factor that has greatly benefited the use of brittle ceramics in a wide variety of applications is the agreed‐upon use of a common test

CHAPTER 1 Introduction 5

methodology through national, regional, and international standards. Most of these standards have been developed by consensus by private standards development organizations such as ASTM International and the International Organization for Standardization (ISO). The details of the standards coming out of the deliberation process are based on years of data obtained in laboratories throughout the world.

In this second edition of the book, we summarize the concepts behind the selection of a test procedure for fracture toughness and strength determination and go into some detail in how the statistics of fracture can be used to assure reliability. We explain the importance of the role of microstructure in these determinations and emphasize the use of fractographic analysis as an important tool in understanding why a part failed. We have included a significant quantity of material related to the fracture of biomaterials. We have also included new chapters, one devoted to thermal shock and the other to the modeling of the fracture process. In addition, the portion of the book discussing how to treat the statistics of fracture strength data to ensure reliability has been greatly expanded.

The Fracture of Brittle Materials: Testing and Analysis, Second Edition. Stephen W. Freiman and John J. Mecholsky, Jr. © 2019 The American Ceramic Society. Published 2019 by John Wiley & Sons, Inc.

7

CHAPTER 2

INTRODUCTION

At the most fundamental level, brittle fracture occurs when stresses reach the level needed to break the bonds between the atoms in the material. However, if there were no means of concentrating stress, loads necessary to cause failure would be extremely large. It is the presence of small defects that concentrate applied stresses to a magnitude suffi-cient to cause them to grow. The materials we discuss in this book are brittle because plastic flow mechanisms in them are insufficient to relieve the stress concentration at the defect.

The science of fracture mechanics allows us to calculate the forc-es needed to cause defects to grow based on knowledge of specimen geometries and the applied loads. In this chapter we provide some his-torical perspective and a summary of the basic principles of fracture mechanics. The reader is encouraged to consult Anderson (1995) and Munz and Fett (1999) for more details.

Fracture Mechanics Background

8 The Fracture of Brittle Materials: Testing and Analysis

EARLY BRITTLE FRACTURE RESEARCH

Although fracture studies of ceramics can be traced as far as back as 1867, our current understanding of brittle fracture can be traced to Inglis (1913), who initiated the concept of stress concentration at a void in a material as shown in Figure 2.1.

Assuming an elliptically shaped cavity, the concentrated stress, σc,

due to the presence of the void is given by the following expression:

c 1 2

a (2.1)

where 2a is the long axis of the ellipse and ρ is its radius of curvature. As ρ approaches zero, i.e. the elliptical cavity begins to resemble a crack whose tip radius is of atomic dimensions. The atomistic nature of materials and the nonlinearly elastic or plastic deformation that occurs in the vicinity of a crack tip avoid the problem of ρ → 0, but this approach does not yet allow us to quantify the stress state at an actual crack tip and says nothing about when the void is likely to grow.

2b

Aρ

σ

2a

Figure 2.1 Void in a plate in which stresses are concentrated at point A.

CHAPTER 2 Fracture Mechanics Background 9

The second important advance in our understanding of brittle fail-ure was made by Griffith (1920) who postulated that brittle failure in glass is a result of the growth of small cracks when the material is sub-jected to a large enough tensile stress. He put forth the hypothesis that these cracks are present in all glasses in a distribution of sizes, leading to the concept that the smaller the volume (or area) under stress, the less likelihood of finding a large flaw and therefore the higher the strength. Griffith demonstrated this concept experimentally by measuring the strength of glass fibers of varying diameter and showing that strength increased with decreasing fiber diameter. The flaw that eventually grows to failure is determined by its severity as well as its location with respect to the highest tensile stress, thereby giving rise to the statistical nature of brittle failure.

Griffith also hypothesized that a material’s resistance to the growth of a crack is determined by the energy required to create the two frac-ture surfaces produced by its extension. This approach assumes that fracture occurs in an equilibrium manner, i.e. in the absence of any kinetic effects, and that no energy is lost due to plastic flow or heat. It also neglects possible effects of the test environment in which the flaw is growing, e.g. water. Griffith’s expression for glass fracture based upon this approach is given by

f

f21 2

E

a

/

(2.2)

where σf is the fracture strength, E is Young’s modulus, γ

f is the energy

required to form the crack surfaces (i.e. the fracture energy), and a is the critical flaw size. While Griffith’s calculations were not entirely accurate because of his assumption that the fracture energy of the glass could be extrapolated from measurements of surface energy car-ried out at elevated temperatures, the form of Eq. (2.2) accurately depicts the relationship between strength, flaw size, and fracture energy. However, what was still required was a way of translating the external loads on a part into knowledge that could be used to predict its resistance to crack growth. This awaited the development of frac-ture mechanics.

10 The Fracture of Brittle Materials: Testing and Analysis

DEVELOPMENT OF FRACTURE MECHANICS

Credit for the development of the science of fracture mechanics is right-fully given to George Irwin and his colleagues at the US Naval Research Laboratory (Irwin 1958). Irwin first introduced the concept of a “strain energy release rate,” G, which has nothing to do with time dependence, but is the change in strain energy with crack extension. Equation (2.2) then becomes

f

EG

a

1 2/

(2.3)

G f2 (2.4)

There are three ways that stress can be applied to a crack (Figure 2.2).

The predominant situation with respect to the stressing of brittle materials leads to crack growth under mode I loading, i.e. pure tension across the crack face.

A schematic of the stress state at the crack tip produced by an externally applied mode I load is shown in Figure 2.3.

Mode I(opening)

Mode II(in-plane shear)

Mode III(out-of-plane shear)

Figure 2.2 Modes of fracture. Source: From Anderson (1995). Reproduced with permis-sion of Taylor & Francis.

CHAPTER 2 Fracture Mechanics Background 11

The stress intensity factor is defined through the following expression:

y

IK

rf

2 (2.5)

where σy is the stress at the crack tip, K

I is the slope of the stress-

distance plot, ϕ is the angle in the plane with respect to the crack face, and r is the distance from the crack tip. Note that Eq. (2.5) is only valid outside of the singularity‐dominated zone, within which the material is nonlinearly elastic or in which some permanent deformation has taken place. The beauty of linear elastic fracture mechanics is that it can be used to explain fracture in spite of the existence of this singularity.

The relationship between KI and applied stress can take many forms

depending on the type of loading and the location of the crack relative to the load. The simple example of a surface flaw, the predominant source of failure for brittle materials (Figure 2.4), gives rise to the relationship

K Y aI1 2/ (2.6)

where Y is a numerical factor that depends on the loading method and the crack geometry and a is the depth of the crack into the interior.

KI

r ½σ y

Figure 2.3 Schematic of the stress field at a crack tip.

aθ

2c

Figure 2.4 Surface flaw.

12 The Fracture of Brittle Materials: Testing and Analysis

The surface flaw is typically elliptical rather than semicircular in shape and is usually elongated along the tensile surface. The expression for stress intensity factor for such a flaw is given by

K M

aI 2

1 2

(2.7)

where Φ is an elliptical integral that accounts for the fact that the stress intensity factor varies around the perimeter of the flaw and is given by

0

22

22

1 22

sin cosa

cd (2.8)

The stress intensity is a maximum at the tip of the minor axis of the flaw and is a minimum at the tip of the major axis. M is a surface correction factor whose value is approximately 1.12 as determined by finite ele-ment analyses (Raju and Newman 1979).

CRITICAL FRACTURE TOUGHNESS

Crack extension takes place when the applied load rises to a level such that the stress intensity factor produces crack tip stresses large enough to rupture atomic bonds at the crack tip. Fracture toughness is defined as the value of the stress intensity factor, K

I, at the point where K

I = K

IC,

the critical stress intensity factor:

K EG EIC C f

1 2 1 22 (2.9)

Equation (2.9) provides the link between fracture mechanics and Griffith’s fracture energy.

How do we measure criticality? Griffith defined γf in thermodynamic

terms, but for most materials the point of reversible crack extension and contraction is not observed. Experimentally, we can increase the stress on a crack to the point at which the crack grows to failure. How-ever, most brittle materials are sensitive to the effects of the atmosphere in which they are tested. Subcritical crack growth, otherwise known