Astm stp 1409 2002

Four themes were encompassed in the presentations: Implications for Design and Testing; Fracture Toughness Standardization; Crack Growth Resistance; and Unique Materials and Environmenta

Fracture Resistance Testing

of Monolithic and Composite Brittle Materials

J A Salem, G D Quinn, and M G Jenkins, editors

ASTM Stock Number: STPI409

Library of Congress Cataloging-in-Publication Data

Fracture resistance testing of monolithic and composite brittle materials / J.A Salem, G.D Quinn, and M.G Jenkins, editors

p cm

"ASTM stock number: STP1409."

Includes bibliographical references and index

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Peer Review Policy

Each paper published in this volume was evaluated by two peer reviewers and at least one editor The authors addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the ASTM Committee on Publications

To make technical information available as quickly as possible, the peer-reviewed papers in this publication were prepared "camera-ready" as submitted by the authors

The quaii~ of the papers in this publication reflects not only the obvious efforts of the authors and the technical editor(s), but also the work of the peer reviewers In keeping with long-standing publication practices, ASTM maintains the anonymity of the peer reviewers The ASTM Committee

on Publications acknowledges with appreciation their dedication and contribution of time and effort on behalf of ASTM International

Pdnted in Brk~geport, NJ January 20~

This publication, Fracture Resistance Testing of Monolithic and Composite Brittle Materials, con- tains papers presented at the symposium of the same name held in Orlando, Florida, on 14 November

2000 The symposium was sponsored by ASTM Committee C28 on Advanced Ceramics The sym- posium chairman was Jonathan A Salem, NASA Glenn Research Center at Lewis Field, and the sym- posium co-chairmen were George D Quinn, National Institute for Standard and Technology, and Michael G Jenkins, University of Washington

IMPLICATIONS FOR DESIGN AND TESTING

Failure from Large Grains in PolycrystaBine Ceramics: Transitions in Fracture

Toughness -s FREIMAN

Stresses in Ceramic Plates Subjected to Loading Between Concentric R i n g s - -

L.M POWERS, J.A SALEM, AND A.S WEAVER

17

30

FRACTURE TOUGHNESS STANDARDIZATION

Development, Verification, and Implementation o f a National Full-Consensns

Fracture Toughness Test Method Standard for Advanced Ceramics

M.G mNKINS, J.A SALEM, O.D QUINN, AND L BAR-GN

Does Anyone Know the Real Fracture Toughness? SRM 2100: The World's First

Ceramic Fracture Toughness Reference Material -O.D QUtNN, K •

R J GETI'INGS, J A.SALEM, AND J J SWAB

Fracture Toughness of Ceramics using the SEVNB Method: From a Preliminary

Study to a Standard Test Method J J KOBLER

The Fracture Toughness Round Robins in VAMAS: What We Have L e a r n e d - -

Application of Quantitative Fractography to the Characterization of R-Curve

Behavior J J MECHOLSKY, JR., T J HILL, AND Z CHEN

F r a c t u r e Testing of a Layered Functionally G r a d e d Material M.d roLL,

R.D CARPENTER, G.H PAULINO, Z.A MUNIR, AND J.C GIBELING

152

169

UNIQUE MATERIALS AND ENVIRONMENTAL EFFF_~S Environmental and Thermal Effects on the Toughness of TiCN-Based M a t e r i a l s - -

S GUICCIARDI, C MELANDRI, F MONTEVERDE, A BELLOS1, AND G DE PORTU

Fracture Toughness Studies on Ceramics and Ceramic Particulate Composites

at Different Temperatures G.A C, OGOTSI

The Effect of Stress Rate on Slow Crack Growth Parameter Esfimates J.A SALEM

AND M.G JENKINS

187

199

213

Overview

During the past decade, ASTM Committee C28 on Advanced Ceramics, along with its European and Japanese counterparts, has made great progress in the development of new standards for the mea- surement of fracture toughness, slow crack growth, and biaxial strength These standards are de- signed to result in quality measurements for engineering, research, and general characterization pur- poses They strive to strike a balance between accuracy and reasonable convenience Although work continues on improvement of existing standards and the development of new standards, members of Committee C28 felt that the time had come to review and summarize the recent efforts by sponsor- ing a symposium on fracture resistance testing of brittle materials The symposium was held in Orlando, Florida during November of 2000 Participants came from Europe, Asia, and the Americas Four themes were encompassed in the presentations: Implications for Design and Testing; Fracture Toughness Standardization; Crack Growth Resistance; and Unique Materials and Environmental Effects Although three of the themes are relatively broad and papers representing a variety of subtopics were presented, the session on Fracture Toughness Standardization was rela- tively focused The emphasis on this topic merits some explanation Fracture toughness is a funda- mental measure of a ceramic's ability to tolerate flaws, or conversely, its brittleness; however, there are conflicting views on the importance of that property for design Some design methodologies cur- rently in use for ceramics (i.e., those based on strength statistics) do not employ fracture toughness, despite its overwhelming importance in classical deterministic design techniques However, it is the fracture toughness and the flaws inherent in a ceramic material that control the strength measurements used as the basis for such reliability methodologies Furthermore, deterministic design methods that employ fracture toughness are being used for ceramic component design

Thus, quality fracture toughness measurements using a reasonable degree of similitude are needed Furthermore, since research on toughening of ceramics exists and continues, the techniques need to

be efficient in terms of the material used and the time required, Committee C28 chose three tech- niques for development and standardization in the new fracture toughness standard C 1421-99 These techniques are detailed in this special technical publication and, importantly, the three techniques show convergence when good metrology is employed In addition to standardized techniques, the section Fracture Toughness Standardization discusses the single edged V-notched beam method that

is on a fast track for standardization in Europe

The section on Implications for Design and Testing contains papers on the analysis of plates for bi- axial strength testing and the transition in measured fracture toughness from a value associated with the properties of a single grain to the polycrystalline value A number of papers presented in the other sessions also had implications for design and testing These included papers on the effect of stress rate

on slow crack growth parameters for design usage, and the application of quantitative fraetography

to the characterization of the R-curve behavior of a silicon nitride for bearing applications

The section on Unique Materials and Environmental Effects includes papers on elevated tempera- ture fracture toughness testing of particulate reinforced ceramic composites, thermal and environ- mental effects on the fracture toughness of titanium carbonitrides for machining, and environmental interactions that lead to rate effects in "dynamic fatigue" (i.e., stress corrosion) testing

The section on Crack Growth Resistance includes papers on testing of functionally graded materi-

als, elevated temperature R-curve testing, and the study of a toughening mechanism Although most researchers applied classical mechanical techniques for the measurement of fracture toughness or crack growth resistance, both theoretical and fractographic methods were also presented

The papers are relevant in that they supply the background that determined many of the guidelines used in current Committee C28 standards This volume not only summarizes the latest standard meth- ods for the measurement of fracture toughness, slow crack growth, and biaxial strength, but also in- dicates new areas for fracture toughness test method development and standardization: testing of complex materials, elevated temperature measurement, and R-curve measurement Indications were also given that test method development is needed in the areas of elevated temperature biaxial strength measurement and slow crack growth by static loading It is hoped that in coming years these areas will be pursued fruitfully by Committee C28 on Advanced Ceramics

Jonathan A Salem

Life Prediction Branch NASA Glenn Research Center at Lewis Field Cleveland, Ohio 44135

Symposium co-chair and co-editor

George D Quinn

Ceramics Division National Institute for Standards and Technology Gaithersburg, MD 20899

Symposium co-chair and co-editor

Michael G Jenkins

Department of Mechanical Engineering University of Washington

Seattle, WA 98195 Symposium co-chair and co-editor

Plenary Session

Fracture Mechanics of Brittle Ceramics - 30 Years of Progress

Reference: Bradt, R C., ~Fracture Mechanics of Brittle Ceramics - 30 Years of

Progress, ~ Fracture Resistance Testing of Monolithic and Composite Brittle Materials, ASTMSTP 1409, J A Salem, G D Qninn and M G Jenkins, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2002

Abstract: The application offiacUae mechanics to advanced technical ceramics was initiated nearly 30 years ago It was heralded by an international conference at the Pennsylvania State University in 1973 Unbridled optimism prevailed as many believed that the coupling of fracture mechanics with ceramics would enable high temperature advanced ceramics to find extensive structural applications Unfortunately, there were a number of barriers to the use

of ceramics as structural materials, even with the application of fracture mechanics The use of brittle ceramics in structural applications faced major challenges These included: (1) the absence of a reliable data base of materials properties, (2) the lack of appropriate standards for the determination o f the properties o f these materials, (3) a lack of any rational, fracture mechanics based design methodology for brittle ceramics, and (4) the high costs of producing and utilizing structural parts from advanced ceramics These were serious deficiencies that inhibited the achievement of successful applications of ceramics Fracture mechanics tests provided a basis for the design methodology and also provided understanding o f the basics of the fracture process in brittle ceramics It informed the engineering community that ceramic fi'acture toughnesses were low, < 10 MPa~/m and that there were serious problems with slow crack growth and fatigue The presence of rising R- curve behavior was observed and related to the crack tip's following wake region Cyclic fatigue was confirmed to exist and related to the presence of a rising R-curve

Today, the promise of structural ceramics remains mostly unfulfilled, except for a few niche applications that are nonetheless encouraging In addition, there have been significant advances in the properties and integrity of structural ceramics over the past three decades Advanced structural ceramics are better than ever This can be attributed to the use of fracture mechanics concepts to identify the microstructural processing defects and then improve the processing methodology to eliminate those defects There have also been major advances in the development of standards applied to ceramics in which ASTM has played a major role Unfortunately, we still have not learned to design with brittle ceramics and the cost of their utilization in structural applications remains prohibitively high

Keywords: brittle fracture, ceramics, design, fatigue, R-curves, wake region

1Department of Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa, AL 35487-0202

3 Copyright9 ASTM lntcrnational www.astm.org

4 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

Introduction

Progress in the fracture toughness testing o f brittle ceramics has been intimately related

to the attempt to utilize advanced ceramic materials in structural applications, primarily at elevated temperatures in heat engines The potential for the utilization of ceramics in heat engines funded much of the ceramic fracture mechanics studies during the past three decades For that reason, this assessment of the progress of the fracture toughness testing of brittle ceramics over the past three decades can be addressed within that perspective, for the two topics have been inseparably associated It may not be the "pure" fracture mechanics historical report that some expect from this article, but it necessarily couples the fracture mechanics testing of ceramics with the main driving force for that testing, the promise of utilization of advanced technical ceramics in structural heat engine applications

When the field of fracture mechanics began in earnest in the 1950s, the structural heat engine materials activity was focused toward cermets Cermets were believed to be capable

of combining the high t ~ properties o f ceramics with the toughness of metals This belief was never fulfilled Rather, the composite cermet developments reproduced the high temperature weaknesses o f metals and the brittleness o f ceramics, instead of the reverse Thus, into the 1960s and early 1970s, the technical ceramics field was receptive to any new approach to utilize the attractive high temperature properties o f ceramics Once introduced, fracture mechanics was immediately perceived to be the answer, for it appeared to address the primary obstacle to the application of ceramics in structural applications, brittle fracture That briefly summarizes the background as to how the two topical areas became coupled and why they are addressed together in this paper

Considering the situation, it is not surprising that an international conference on the fracture mechanics of ceramics was held at The Pennsylvania State University in 1973 It was the first of a series of such conferences held approximately every four years The proceedings have been published by Plenum Publishing [1] These conferences were among the most successful topical conferences ever held on a technical ceramics topic They have been instrumental in the development of the understanding of the brittle fracture of advanced ceramics for structural applicatons Although the theme of these conferences has drifted from purely ceramic fracture mechanics and now encompasses other related areas of mechanical behavior, this series of symposia has nevertheless remained a focal point for the application

of fracture mechanics applied to brittle advanced structural ceramics

Fracture of Ceramics in the 1960s

Because the field of ceramics is primarily chemistry and crystallography based, the introduction of fracture mechanics concepts involving more mathematics was accepted only with reservations at the start The leaders in the 1960s were the British and the Japanese: TattersaU, Tappin and Davidge in England and Nakayama in Japan Buresch and Pabst in Germany also made significant contributions In the United States, it was Wiederhorn at NBS with his basic studies on crack growth in glass that lead the way There were difficulties with the acceptance o f the concept of fracture toughness with its unusual units and with the relationship of strength to fracture toughness Many of those original challenges are accepted

as the basics today In spite o f the fact that the analytical aspects offi'acture mechanics had been firmly established for at least a decade, application to ceramics did not automatically

happen It seemed that some of the fundamentals had to be rediscovered, sometimes more than once, almost as if perhaps they did not apply to ceramics, when in fact with their brittle elastic character, ceramics were ideally suited to the application of fracture mechanics

In the e ~ arena, there were diificulties with the fracture mechanics tests, similar

to the ones that there are today, and always will be For fracture toughness measurements, the double cantilever beam and the single edge notched beam were the favorite geometries However, as the "crack" in the notched beam specimens was invariably introduced with a thin diamond saw, there was a distinct lack of reproducibility in the results Unlike metals, it was not poss~le to introduce fatigue cracks for the fracture measurements In 1970 there was almost a complete lack of understanding about the frontal process zone at the crack tip of brittle materials and absolutely no appreciation for the effects o f the following wake region behind the advancing crack tip Many individuals believed that dislocations were of paramount importance at the crack tips of brittle ceramics, just like in metals It was also difficult for some to accept that the conversions of the advanced technical ceramic fracture toughness values to surface energies did not yield the thermodynamic surface free energies that were expected for brittle ceramic materials, but it was a fact

While there were many researchers attracted to the field, often making it difficult to give anyone credit for being the first, one application stands out above all others It is the work

of Nakayama in Japan with his development of the work-of-fracture test and its application

to improve the thermal shock damage resistance of refractory bricks, initially low quality fireclays Many, not familiar with the Japanese literature, have attributed the work-of-fracture test to Tattersall and Tappin They undeniably were working in the same area, but Nakayama was the one who independently built his own testing machine, developed the test and applied

it to solve a major industrial problem for which Hasselman had previously advanced an energy balance concept To this day, 30 years later, no application of fracture mechanics to technical ceramics has had such a major industrial impact Even today, virtually all industrial refractory microstructures are designed based on some variation of the work-of-fracture test developed

by Nakayama The volume 0fthose products is at the level of millions of tons per year! Entry into the decade of the 1970s presented the technical community with the general belief of having found the answer for the utilization of technical structural ceramics in heat engine applications That answer was fracture mechanics However, many unanswered questions remained As this symposium is addressing the testing situation, a couple of problems specifically related to that issue are appropriate to note Several were the introduction o f sharp cracks into test specimens, measurements of crack lengths and the ability to obtain stable fracture conditions in work-of-fracture test specimens The testing of brittle ceramics was not nearly as advanced as it is today [2-4] It was also puT~lirlg that the total energy balance for the work-of-fracture test did not always agree with the derived energy values from fracture mechanics crack initiation tests In the late 1960s the flame of hope for structural ceramics had been ignited and in the 1970s was burning vigorously, but there were many very basic questions to be asked and equally as many to be answered That ceramic flame created a special era, one that is generally referred to as the ceramic fever of the 1970s, 1980s and into the early 1990s A level of unbridled optimism swept over the worldwide technical ceramics community It believed that many elevated temperature applications of structural ceramics were imminent Any day there would be ceramic engines

in all of our automobiles [5] The discovery of ceramic superconductors in the middle 1980s further fueled the optimism that ceramics are the answer Yet, now in the year 2001, many

6 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

of the potential applications o f advanced structural ceramics have not been realized It is appropriate to examine some oftbe reasons for this failure to achieve expectations Not a small portion of the reasom lies in the area of fracture mechanics testing and the development

of standards for those tests Utilization of fracture mechanics concepts in the design process for the application of brittle advanced ceramics remains a challenge to this day

Shortcomings of Structural Ceramics from the Perspective of Hindsight

Now that the 21st century is upon us and we do not have ceramic engines in all of our automobiles, it is perbaps appropriate to examine why this enthusiastic structural ceramics venture failed, although there have been ceramic successes in other venues It is the opinion

of this author that there are four major factors that were not adequate, or did not measure up

to the technical level which was required for the success of structural ceramics in heat engine applications over the past couple of decades Not necessarily in their order of importance, these include: (1) the absence of a qualified data base for those materials properties that are required by design engineers, (2) the lack of appropriate test standards for the determination

o f those properties, (3) a weakness in the knowledge of fracture mechanics based design concepts for brittle ceramic materials, and (4) the economic factor, the prohibitive costs of producing structural parts from advanced ceramics These four may not be the only reasons for the lack of success However, each of the four certainly assumed a paramount role in delaying the utilization of advanced technical ceramics in structural applications during the late 20th century Each merits individual consideration

The Absence of a Qualified Data Base of Materials Properties

Inthe 1960s and the 1970s there did not exist even a modest data base o f the properties

of advanced ceramic materials Mechanical engineers need a reliable data base for design with structural ceramics Yes, some properties were known, but only for the products of the individual ceramic manufacturers The values were the ones published by those producers in their sales literature Many of the producers only published their best results in this data, but even the best were not very satisfactory by present today standards For example, typical bend strengths were in the range of 100 to 200 MPa (15000-30000 psi) and reliable Weihnll moduli were not even available Often only a few specimens were actually tested We know today that those Weibull moduli were probably only about 5 or 6, values hardly suitable for high reliability designs! Fracture toughness values simply were not available and slow crack growth and fatigue data were non existent A tensile strength value was practically in the category of a dream It was only the continuing efforts of the government and industrial structural ceramic programs directed towards the development of ceramics for heat engines that lead to the eventual production of sufficient data bases suitable for design purposes Even now, the data bases for ceramics are not widely publicized and are very limited compared with those existing for metals Often these ceramic data bases are not generated using consensus standards consisting o f statistically significant sample sizes

There was yet another impediment to the establishment of qualified material data bases for designers to draw upon It was the continuing development of new varieties of ceramic materials by the ceramic producers In an attempt to be the provider of the best material for the anticipated structural applications that were just ahead, producers kept releasing new

ceramics with different properties An interesting example o f this activity was silicon nitride, which many engineers considered to be the primary candidate for the ceramic materials in heat engine structural applications This author did not keep a running count o f the number o f different silicon nitrides that were developed, but there were numerous reaction bonded silicon nltrides with properties that were inferior to the equally numerous densely sintered bodies and hot pressed silicon nitride formulations As it was expensive to develop a data base for any one o f these materials, the need for many data bases resulted in the production

o f few data bases with significant reliability

Lack of Appropriate Standards for Property Measurements

The title o f this section is a misnomer There were no accepted standards for the measurements o f the fracture mechanics parameters o f structural ceramics, or for other mechanical properties o f technical ceramics 30 years ago During the 1960s and 1970s, fracture mechanics measurements were being made in universities and a few industrial research laboratories by only a few individuals Often, new entries into the field visited with those scientists/engineers who were already making the measurements to learn how to apply the experimental techniques Without standards, it is not surprising that the data bases were inadequate for engineers to produce rational designs As late as 1984, Quinn [6] discussed this at length in an excellent paper, which, unfortunately was not the wakeup call that it should have been for manufacturers It is only during the past decade that the national and intemational technical communities have addressed the standards issue for advanced technical ceramics It is gratifying to see that entire symposia, such as this, are being devoted to the presentation and development o f fracture mechanics test standards for brittle ceramics

Absence of a Design Methodology

The design process for incorporating brittle ceramics into structural applications as it existed in the 1960s and 1970s left much to be desired It was essentially nonexistent Engineers knew how to design ceramic microstructures, but they didn't know how to design with those ceramic microstructures [7] The first activities in this area were the one-for-one substitutions The candidate advanced ceramics were patterned after the metals that were currently in use and then directly substituted into the turbine, or other engine Catastrophic failures occurred Successes were nmastaed in minutes and hours o f operation before failure The message was loud and clear Designing with brittle ceramics was not the same as that with ductile metals Unfortunately, this obvious lesson was not learned by the engineering community That shortcoming continued to impede progress with advanced ceramics Although practically unexplainable, instead o f refocusing and intensifying efforts to develop and advance the design process, there was a redirection o f ceramic development towards ceramic composites that would fail gracefully It has always been a puzzle to this author as to why anyone wants to use a material that fails gracefully Isn't it more natural to want to use a material that doesn't fail at all? Unfortunately, with the redirection o f some attention to ceramic composites, the advanced ceramics technical community failed to step back, recognize and accept the fact that the real problem was the design methodology, not the material failure mode Many o f the foolish mistakes that were made with monolithic ceramics were destined to be repeated and have been repeated with ceramic composites

8 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

Fortunately, a few individuals recogniTed the need for a fundamental brittle ceramic design methodology The design group at NASA Lewis Research Center (presently NASA Glenn Research Center) with their enthusiastic proponent, Gyekenyesi, have made major advances

in the design methodology They have incorporated fracture statistics and finite element stress analysis into probabalistic design concepts These applications were aided significantly

by the rapid development of computer systems over the same time period These allowed the engineers to make major advances in design methodology In the opinion o f this author, designing with advanced structural ceramics requires a distinctly different approach than metal designs A major reason why structural ceramics have not been as successful as many anticipated is that the design process has still not been perfected after nearly halfa century Perhaps some radically innovative ideas relative to design with ceramics are necessary? Hindsight is always "20/20" and it is appropriate to apply a little o f that vision at this time Although I am a materials engineer, it is obvious to me that from the very beginning there was not a sufficient level of effort devoted to the development o f design criteria and processes for the use o f brittle ceramics in structural applications Perhaps this should not be surprising,

as most of the advanced structural ceramics programs were spearheaded by materials engineers However, the best materials in the world will not perform unless incorporated into

a fundamentally sound design Needless to say, the first half of the 30 years of progress reported here lacked sufficient design activity An exception was a NASA sponsored program led by Mueller at the University of Washington in Seattle He tried to develop an academic design program combining the talents of mechanical and materials engineers, but

it did not have the longevity to be successful It is in some ways disappointing that this lesson still has not been learned nearly 20 years later The next generation of designers are now in the undergraduate classrooms They are the ones who will be expected to finally solve the brittle structural ceramic design problems with creative, innovative, radically new advances Unfortunately they are not being introduced to structural design with brittle ceramics early enough in their careers

It was noted earlier in this section that there has been a redirection o f activities to the use

of ceramic composites as a replacement for monolithic ceramics in structural applications [8] This class of materials will not substitute directly for metals either and obviously will have its own design criteria when they are eventually developed Furthermore, ceramic matrix composites have several serious problems of their own The thermal expansion and elastic modulus mismatch between the fibers and matrix will generate high internal thermal stresses

on cycling, perhaps in the GPa range These stresses will create internal cracking during cycling, eventually leading to disintegration The logic of graceful failure is also a bit

o f a mystery Once the elastic limit is exceeded in most structural applications, failure has occurred, as dimensions are not maintained and continued loading or thermal cycling is not possible Many present ceramic cong~ositcs exhibit matrix cracking at lower stress levels than comparable monolithic ceramics In addition, they bave problems with thermodynamic stability and oxidation as well Of course, a one time application is a different situation The above noted, there is no doubt in the mind of this author that the design criteria for brittle ceramics in structural applications must be a fracture mechanics based one, one involving fracture statistics and crack growth phenomena It will be limited by the brittle fracture process For that reason alone, an integral part o f the overall design process must

be the development of suitable test standards for the fracture o f brittle ceramics The results

of those standard tests will define the lifetime and the limit o f the design

Economic Factor, Prohibitive Costs

There are costs associated with everything and the costs o f generating a data base for brittle structural ceramics, developing fracture mechanics standards and perfecting the design methodology have proven to be much greater than originally anticipated in the 1960s The funding for these items would never have become available if it weren't for the potential of the utilization of ceramics in heat engines Unfortunately, these factors do not include the cost of manufacturing the ceramic components for the anticipated applications [9,10] In the final analysis, ceramics will never be used in structural applications if metals continue to perform satisfactorily and the metals are more economical to produce This is not an insignificant consideration It is a major barrier It is the bottom line for the introduction of technical ceramics into all structural applications, not only as components for heat engines

At the present, structural ceramics are simply much too expensive to produce to find extensive use in most heat engine applications In some respects it is a chicken and egg situation I f ceramics are to compete with metals then they must be produced in large numbers (volumes) to benefit from the economy of scale Thus, when all of the other factors have proven to be positive, a command decision must still be made to convert to ceramic components It may not be an easy decision, for it will be one with considerable risk

Contributions of Fracture Mechanics

It is evident that fracture mechanics has made major positive contributions to the understanding of the failure of brittle ceramics These have occurred in spite o f the modest advances that structural ceramics have made in the penetration o f the heat engine market Often, the contributions have been indirect ones However, many can be addressed in terms

of direct fracture related phenomena and improved ceramics themselves

a message about the fracture toughnesses of advanced ceramics and that structural component design with ceramics would be difficult That design process might have to be different Unfortunately, the message did not register Many researchers continued to strive to make tougher ceramics [8] There were numerous reports of higher toughness ceramics, some values near to 20 MPa~/m But, without the accompanying increases in strength it was evident that most of these reports were of the salesman variety, namely "my ceramic is tougher than your ceramic." Although there have been modest increases in the toughness values for advanced ceramics through microstruetural design, it is now clear that for all practical purposes the fracture toughnesses of most advanced technical ceramics are

10 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

somewhat less than 10 MPa~/m and in many instances considerably lower These toughnesses are much less than that of cast iron Everyone knows how brittle cast iron is, as well as the challenges of designing with cast iron Yet, there is a lot of cast iron used in the world I suspect that valuable lessons could be learned from cast iron applications and their design Fracture toughness values are not the only revelations of the past 30 years of fracture mechanics applications to ceramics Much has been learned about events which occur in the crack tip region, both in front ofthe crack tip and behind the crack front in the so-called wake region In the front of the crack, it is now finally accepted that dislocations are not very important Phenomena such as microcracking, phase transitions in toughened zirconias and elastic/stress effects from higher modulus particles, or materials are most common Buresch and his student Pabst were the first to recognize the importance of these phenomena Concepts of crack deflection and in-situ toughening by elongated grains have also been advanced as important frontal process zone toughening phenomena by Evans and Faber It

is interesting that only now, decades later, it has finally been realized and accepted that dislocation processes are not very important in the fracture o f advanced technical ceramics

It is, however, behind the advancing crack tip that many of the most interesting phenomena have been discovered It is these that have generally been the cause of rising R- curve behavior From the pioneering work of Steinbrech, we know that although the crack has passed, that following wake region can still contribute significantly to the crack growth resistance of advanced ceramics Many interesting things happen behind a crack in the wake region Microstructural elements become wedged, grains bridge across the crack surfaces and there are extensive frictional effects These phenomena are active for some distance behind the crack tip, until the crack opening displacement is too great for the new fracture surfaces

to continue to interact It is these active phenomena in the following wake region that are primarily responsible for rising R-curves in most advanced ceramics, the increasing crack growth resistance with crack length This phenomenon is now known to occur in most ceramic materials except for the finest grain size ceramic bodies and glasses It has created serious problems in the accurate measurement of the fracture toughness Some researchers have inadvertently reported fracture toughnesses for the ascending portion of the R-curve,

or perhaps even the R-curve plateau in the most extreme instances It is a cause of erroneously high toughness values in the literature, particularly with the indentation toughness measurement techniques These fracture toughnesses cannot be applied for design purposes Unfortunately, they have been used indiscriminately by some researchers

There is another major effect of rising R-curves in ceramics It is the susceptibility to fatigue from cyclic stresses Unfortunately, it had been previously reported that brittle ceramics were not suscept~le to cyclic fatigue from alternating stresses However, it is now well established that indeed ceramics are susceptible to fatigue and the reason is the opening and the closing of the crack in the wake region That cyclic process damages those elements which interact across the newly formed fracture surfaces This necessitates continual growth ofthe main crack to maintain a stable overall crack tip region, both in the frontal process zone and in the following wake region

Fracture mechanics has also contributed to the understanding of slow, or environmentally assisted crack growth It has enabled that phenomenon to be incorporated within the design process Slow crack growth occurs whenever the stress intensity exceeds either the threshold, I ~ , or the fatigue limit, I~o Unlike metals, glasses and brittle advanced structural ceramics are susceptible to environmentally assisted crack extension when under a constant

load or stress This phenomenon has been referred to as static fatigue It is different than the cyclic fatigue phenomena originating from rising R-curve behavior in ceramics and dislocation phenomena in metals It is a major factor in many lifetime prediction schemes for Advanced structural ceramics

Improved Structural Ceramics

It was noted earlier that ceramics in the 1960s did not have very attractive properties in terms o f strengths, or Weibull moduli In fact, they were rather poor by today's standards Presently, off-the-shelf advanced technical ceramics are available with average strengths approaching one GPa and Weibull moduli in excess o f 20 This is nearly an' order o f magnitude improvement in each o f the parameters Fracture mechanics was a major factor

in these ceramic material improvements as the combination o f fracture toughness, fracture statistics and fractography enabled the identification o f critical flaws as processing defects and subsequently the development o f better ceramics Bowen, Aksay and Lange have been active

in this segment o f the ceramic processing field and refer to the flaws as processing related fracture origins It has often been possible to relate the first generation o f critical flaws to processing defects Once these more serious defects were eliminated by improved processing techniques, then the next level o f flaws could be similarly identified and removed These successive defect eliminations have obviously been successful as ceramic strengths have improved substantially One could make the argument, and probably make it convincingly that these processing improvements have been the major contribution o f fracture mechanics

to advanced technical ceramics

As a result o f this combination o f fracture mechanics concepts and improved ceramic processing, today's advanced ceramics are better than ever It is the opinion o f this author that they are ofmafficiently high quality for more extensive applications in heat engines, if only the designs were better and the manufacturing costs were reduced

Ceramics in Structural Applications

With the significant improvements in the properties o f technical ceramics, the advances

in design and the increased knowledge o f fracture as a result o f fracture mechanics studies over the past 30 years, there have developed some utilization o f advanced technical ceramics

in structural applications Although not dominantly in heat engines, these applications are meaningful structural uses o f advanced ceramics One o f the most remarkable o f these applications is the use o f ceramic turbochargers in passenger cars by both Nissan and Toyota

[11, 12] Subjected to overload proof testing, there have not been any failures publicized to date One can imagine a ready, but perhaps not quite one-for-one translation o f some o f these turbocharger design concept to similar applications in high temperature gas turbines It is not unreasonable for this to come o f fruition in the next decade or two Ceramic valves have also made it into some reciprocating engines Ceramic seals and ceramic ball bearings have also made significant progress and are now commercial mainstays It is evident, even obvious, that there are a number o f niche markets for structural ceramics As these continue to he identified, there will be an increase in the use o f ceramics and eventually their costs will he significantly reduced Hopefully this will occur in a timely, synergistic manner

12 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

Summary and Conclusions

The connection o f the application o f fracture mechanics to brittle ceramic materials and the promise of the utilization of ceramics as structural components in heat engines was established in the 1970s Contributions of fracture mechanics to understanding the fracture

of brittle ceramics were summarized and the attempted utiliTztion of ceramics in heat engines was reviewed Failure to achieve widespread use of ceramics in heat engines was examined Fracture mechanics has made major contributions to the understanding of fracture in ceramics, including the failure criteria, slow crack growth and fatigue and their incorporation into design methodologies for the utilization of ceramics Fracture mechanics has also made significant contributions to the fundamental understanding of the fracture processes in brittle materials Particularly relevant is the recognition that the following wake region of the crack

is respons~le for rising R-curves in ceramics and the presence of cyclic fatigue However, fracture mechanics was also a contributor to the identification of processing defects in ceramics and indirectly responsible for many advances in ceramic processing that eliminated those defects Progress over the past three decades has been substantial

Unfortunately, advanced technical ceramics have not achieved widespread use in structural applications in heat engines A series of reasons for this was advanced, including the lack of data bases and appropriate standards for fracture testing, a weakness in the design methodology for brittle ceramics and the cost of ceramic components In spite of these challenges, ceramics have been incorporated into automotive turbochargers, and also used in seals and as bearings It is evident that there exist niche applications where ceramics are the materials of choice It is anticipated that these types of applications will increase in number and that ceramics will become utilized in more structural applications with time

Acknowledgments

The author hesitates to acknowledge individuals for the fear of omitting some, but within that perspective apologizes in advance to any he may omit He will forever be indebted to Dick Hasselman and Fred Lange who helped organized the first "Fracture Mechanics of Ceramics" conference nearly 30 years ago He also acknowledges his graduate students who worked in the field of fracture of ceramics and glasses He is especially grateful to Mototsugu Sakai and Ken White with whom he continues to interact on the topic of the fracture of ceramics and also to George Quinn, Mike Jenkins and Jon Salem who encouraged him to write this paper and participate in this symposium All three of them were students in his classes Finally, he would like to acknowledge the influence of Bob DeVries who introduced him to the crystallography of minerals and was instrumental in creating the author's interest

in minerals, materials on which he currently spends much of his time studying the phenomenon of single crystal cleavage

References

[1] Bradt, R C., Hasselman, D P H., and Lange, F F., Fracture Mechanics o f Ceramics

Vols I - II, Plenum Publishing Co.; NY, NY (1974) (This series has > 10 volumes.) [2] Sakai, M and Bradt, R C., "Fracture Toughness Testing of Brittle Materials",

International Materials Reviews, Vol 38, 1993, pp 53 - 78

[3] Miller, J H and Liaw, P K., "Fracture Toughness of Ceramics and Ceramic Matrix

Composites", in ASM Handbook, Volume 8, Mechanical Testing and Evaluation, ASM

International, Materials Park, Ohio, USA, 2000, pp 654 - 664

[4] Jenkins, M G and Salem, J A., "Fracture Resistance Testing of Brittle Solids', inASM Handbook, Volume 8, Mechanical Testing and Evaluation, ASM International, Materials

Park, Ohio, USA, 2000, pp 665 - 678

[5] Bradt, 1L C., "Ceramic Heat Engine Programs in the United States', in Proc of the 2nd European Symposium on Engineering Ceramics, London, 1987, edit F Riley, Elsevier

Pub Co., London, England, 1989, pp 229 - 240

[6] Quinn, G D., "Properties Testing and Materials Evaluation", Ceramic Engineering and Science Proceedings, Vol 5 No 5-6, American Ceramic Society, Westerville, Ohio, 1984

pp 298 -311

[7] McLean, A F and Hartsock, D L , "An Overview of the Ceramic Design Process',

Engineered Materials Handbook, Volume 4, Ceramics and Glasses, ASM

International, Materials Park, Ohio, USA, 1991, pp 676-690

[8] Evans, A G., "Perspective on the Development of High-Toughness Ceramics", Journal AmericanCeramic Society, Vol 73, No 2, 1990, pp 187 - 206

[9] Savitz, M., "Commercialization of Advanced Structural Ceramics, Part I", American Ceramic Society Bulletin, Vol 78, No 1, 1999, pp 53 -56

[ 10] Savitz, M., "Commercialization of Advanced Structural Ceramics, Part II", American Ceramic Society Bulletin, Vol 78, No 3, 1999, pp 52 -56

[11] Matsui, M., Ishida, Y., Soma, T and Oda, I "Ceramic Turbocharger Rotor Design

Considering Long Terra Durability', Ceramic Materials and Components for Engines,

editors W Bunk and I-I Hausner, DKG Bad Honnef, 1986, pp 1043-1062

[ 12] Katano, Y., Ando, Itoh, and Sakai, M., "Applications of Ceramics to Turbocharger

Rotors for Passenger Cars', Journal Engineering far Gas Turbines and Power, Vol

l15b, No 1, 1993, pp 9-16

Implications for Design and Testing

Failure from Large Grains in Polycrystalline Ceramics: Transitions in Fracture Toughness

Reference: Freiman, S., "Failure from Large Grains in Polycrystailine Ceramics:

Transitions in Fracture Toughness" Fracture Resistance Testing o f Monolithic and

Composite Brittle Materials, ASTM STP 1409, J A Salem, M G Jenkins, and G D

Quinn, Eds American Society for Testing and Materials, West Conshohocken, PA, 2001

Abstract: This paper addresses issues of fracture for materials in which large grains, exist in a matrix of much smaller grains Polycrystalline ZnS and ZnSe, used for optical components, frequently fall into this category For material microstructures of this character failure can occur from flaws contained within the isolated large grains so that the governing fracture toughness is that of a single crystal of the material, rather than that

of the polycrystalline matrix9 I point out that there are currently limitations and

uncertainties associated with the most popular experimental procedure available to determine the fracture toughness for small crystals As an alternative to testing, I discuss the state of our ability to predict fracture toughness based solely on knowledge of lattice parameters and elastic properties The historical background and examples of such a prediction technique are demonstrated Second, the role of microstructure in governing the transition from single crystal to polycrystalline values of toughness is shown9 I postulate that mixed mode fracture mechanics expressions can be used to explain the transition in fracture toughness9

Keywords: ceramics, fracture toughness, grain size, polycrystalline toughness, single crystal toughness

Introduction

One important characteristic of ceramic microstructures is that grains are often similar

in size to the flaws that lead to failure Therefore, depending on the depth of the surface defects and the grain size, flaw environments can range from one crystal to many surrounding grains, raising the issue that fracture toughness obtained with large cracks may not be a valid predictor of the behavior of small flaws Because typical techniques for determining fracture toughness employ artificial cracks which can be many times the size of the grains, the resistance to flaw growth in-service can be quite different than that measured in the laboratory Cracks, which encompass many grains, will be more resistant to growth because of toughening mechanisms that do not apply to smaller flaws From the point of view of the user of the ceramic, it is the strength, especially the minimum in the strength distribution of a component, not the polycrystalline fracture

i Ceramics Division, National Institute of Standards and Technology9 Gaithersburg, MD, 20899

17 Copyright9 ASTM International www.astm.org

18 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

toughness, which is important for reliability, making an understanding of the small flaw behavior particularly crucial In addition, there is data to indicate that while fracture toughness can increase with grain size, strength can actually decrease [1-5]

While, fracture initiation and crack growth in ceramics can be either transgranular or intergranular in nature, this paper will be limited to a discussion of flaws initiating wholly within a grain, and propagating transgranularly A number of polycrystalline materials, e.g ZnS and ZnSe used as optical components, have microstructures consisting of large grains imbedded in a fine grain matrix One of the implications of such a microstructure

is that the determinant of the strength of such components may be the fracture toughness

of one grain, i.e a single crystal of the material A number of experimental observations

of single-grain-controlled failure have been made [1-6]

Provided that sufficient size crystals are available, the fracture toughness, Kic, or fracture energy, (, of such a grain can be experimentally determined in a number of ways

In many instances, however, one may not have access to sufficient quantities of material

in single crystal form to use most, if not all of these techniques

The one technique that is most widely applicable to small volumes of material, the indentation-crack-length method [7], is subject to significant uncertainties, as will be discussed later As an alternative, I will attempt to show that models can be employed which yield realistic predictions of the fracture resistance of single crystals of a wide variety of brittle materials, and suggest that such models could be used in lieu of

available measurements

In addition, I will discuss the mechanisms governing the transition from single crystal

to polycrystalline fracture toughness As would be expected, measured fracture

toughness increases with increasing ratio of crack size/grain size until the level of polycrystalline toughness is reached While numerous papers have been written on the subject of this so-called R-curve behavior, and its relationship to strength, what has been less extensively discussed is the transition from purely single crystal to polycrystalline failure, which also involves significant increases in toughness In this paper I will put forth a hypothesis as to a mechanism governing the shape and magnitude of the

toughness transition

Failure from Large Grains in Polycrystalline Ceramics

As mentioned previously, strength is governed by failure from small flaws,

micrometers in extent, while most fracture mechanics procedures involve the use of large cracks to determine fracture toughness That results of such large crack tests can give erroneous predictions of strength is shown by the following example

Fractographic analysis shows that failure of chemically vapor deposited ZnSe is initiated at machining flaws within large, isolated, grains of the material (Figure 1)[4] Assuming that the only stresses acting on the flaws are those applied during testing, the strength of the material is related to the flaw sizes and fracture toughness, through:

Figure 1 Fracture surface of chemically vapor deposited, polycrystalline ZnSe showing

failure from isolated large grains (after Freiman et al [4]) The arrows in (B) point to the flaw boundaries

where Y is a geometric constant, and c is the flaw depth Or in terms o f the fracture energy, ~/:

where E is the elastic modulus 1 One can then plot failure stress as a function of the inverse square root o f the measured flaw size, generating the data shown in Figure 2 The slope of this curve yields a value of Klc (or 3' ) It can be seen from Figure 2 that the governing value of Ktc/T is that for a single crystal of ZnSe (~, 0.8 J/m2), and not that for the polycrystalline material (3.4 j/m2) This difference arises because the critical flaws are contained within one grain For most materials, i.e., those in which Kic

(polycrystal) is < about 20x the Ktc (single crystal) [5] the fact that the growing flaw later encounters tougher, polycrystalline regions has no measurable effect on the strength In fact, I am unaware of experimental evidence to suggest that a crack will arrest at such a boundary during mechanical loading

1 The value of E used in the expression will depend on whether one is analyzing a random polycrystalline

or a single crystal environment, as will the relationship between ~/ and K~c

20 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

What is clear from these results is that despite the fact that the polycrystalline material surrounding the large grain can be significantly more resistant to crack growth, if the flaw achieves a critical size within the grain, this increased resistance will not prevent failure This statement appears to be true regardless of how close to the large grain boundary the initial flaw exists Any possible dynamic effects resulting from a rapidly moving crack seem to be minimal

There are two questions raised by the above discussion: 1) How does one determine the fracture toughness of the large grain (single crystal) of the material, and 2) What are the mechanisms leading to the single-to-polycrystalline transition? We will attempt to address both of these questions in the remainder of this paper

Figure 2 Fracture strength o f the ZnSe materials described in Figure 1 as a function o f

the measured flaw sizes Using Equations 1 or 2 one can calculate applicable values o f Ktc or ( ~ ) (after Freiman et al [4])

Single Crystal Fracture Toughness - Experiment

If one has access to large enough single crystals of the material of interest, there are

numerous fracture mechanics techniques available to determine Kic (or y ) However, it

is not likely that crystals of large enough size will be available for many materials In such a case one is constrained to using the indentation-crack-length_procedure This technique involves placing a Vickers indentation in a material and measuring the length

of the surface cracks emanating from the comers of the impression The expression used

to calculate fracture toughness is given by [7]:

E P

where Ktc is the fracture toughness, Hv is the Vickers hardness, P is the indentation load,

and c is the measured crack length

While this procedure is used extensively, there are numerous problems associated with

it First, there is the uncertainty over the length of the crack measured in the optical microscope; any crack opening smaller than the resolution of the microscope will not be

accounted for Second, the constant in Equation 3 is strictly empirical, determined by a

fit to data for polycrystalline ceramics [7]; uncertainties in its value approach 25%

Thirdly, there are fundamental uncertainties in the procedure because of the anisotropic nature of a crystal The expression for the driving force on the flaw was obtained

assuming an isotropic elastic field Since the actual elastic modulus in a crystal is very directional, the driving force, and therefore the accuracy of this expression becomes very uncertain All of these factors contribute to a significant concern regarding the accuracy

of this technique, particularly as it applies to single crystals

What then is the alternative?

Single Crystal Fracture Toughness - Prediction

In principle, if the relationship between bond stress and strain is known, one should be able to calculate the energy required to separate two surfaces and, thereby, to predict the fracture resistance of a material Gilman [8] was the first to develop such a model He assumed a sinusoidal function for the attractive stress, o, between the surfaces, as shown

where, T is the fracture energy, i.e., cleavage energy, and E is Young's modulus E is

obtained from the definition of Young's modulus: E = do/de*e=o, i.e., where the strain

approaches zero A key issue is the value one assigns to a, i.e., where does complete fracture of a bond occur? Gilman assumed a to have a value of the combined radii of the atoms on the cleavage planes, claiming that this represented the extent of bonding

between the surfaces

22 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

Nevertheless, Gilman used Equation 5 to obtain reasonable agreement between calculated values of fracture energy for LiF, MgO, CaF2, etc., and those determined experimentally Using Gilman's methodology, Becher and Freiman [9] also showed excellent agreement between measured fracture energies of KBr, KCI, NaCI, BaF2, SrF2, and CaF2 and those calculated using Equation 5

White etal [10] calculated the fracture energy of a number of compounds having the diamond cubic structure using Equation 5 However, rather than taking a to be the atomic radii, they arbitrarily assumed that fracture would occur at a strain of 50%, i.e., a

and calculated values of T for a range of compounds Finally, Freiman and Baker [ 11 ] showed that similar calculations could be performed for heavy metal fluoride glasses, which have no long-range order In this case do was taken as the Zr - F distance in the glass, with a ~ do~2 Again, good agreement between model and experiment was

achieved

One fundamental question is what is the proper relationship between bond stress and strain, and how will the use of alternative expressions to Equation 4 affect the prediction ofT It can be shown [12] that a general functional expression for y in terms of the force law can be obtained in terms of the crystal structure and elastic modulus by making some broad assumptions about the fracture process These assumptions are (1) the process is reversible, (2) no energy is lost to processes other than bond rupture, e.g., there is no plastic deformation or phonon emission, (3) there are no environmental effects, and (4) there are no microstructural contributions The derived expression is:

7 = 2 Jo g ' ( O )

where e is the bond strain (/8~do) a n d g(e) is a load-displacement function for a bond on the cleavage plane All bonding functions will have the same general form Namely, (1) they will have an initial slope (g'(0)) which can be equated to Young s modulus; (2) there will be a maximum value of stress, ore; (3) there will be a range over which the bonding forces extend, and; (4) there will be some area under the o-e curve, from which

we can calculate 7- For most bonding laws only 2 of the first 3 parameters are

independent The equivalence of the initial slope with E leaves the maximum value of the stress, o , , , and the stress range, a, undetermined However, each of these two parameters can be expressed in terms of the other

Equation 6 can be recast as in terms of strain by taking

Combining Equations 6 and 7, the following expression for 7 results

of crack growth resistance

1.00

r ~ 0 5 0

Comparison o f Measured a ~ c l C a l c u l a t e d

F ~ c t u r e E n e : ~ e i 1.5o

Figure 3 Plot of measured and calculated values of( The calculated ( was obtained

from Equation 8, using a value of em = 25

Single to Polycrystalline Transition

I also wish to address the question of the mechanism(s) governing the transition to the increased toughness of the polycrystalline material surrounding a large grain

Experimentally, one observes that the transition from single to polyerystalline fracture resistance can occur at flaw/grain size ratios ranging from 1 to 6, i.e., from immediately

at the boundary of the initiating grain to many times this value As an example, Figure 4 shows the change in measured fracture toughness of ZnSe for a range of flaw sizes

24 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

Figure 4 Transition in fracture toughness in CVD ZnSe as a function of the ratio of flaw

size to grain size (after Rice et al., [1])

Figure 5 shows transitions for a number of different materials Another example of a transition from single crystal to polycrystalline toughness is shown in Figure 6 for ZnS [6] This figure represents data from a controlled study employing indentation-induced cracks In this case the transition is very abrupt, occurring essentially at the boundary of the large grain in the material The cleavage plane(s) in grains surrounding the large grain in which fracture initiated will be misoriented with respect to it as well as to one another Another factor will be the number of cleavage planes on which the crack will grow, and their relative resistance to such growth

Figure 5 Transition in fracture energy for a number of materials as a function of flaw

size~grain size (after Rice et al., I21)

1.4

1.2

~ 1.0-

~ 0.8 g

~ 0.6

o

o 0.4, 0.2~-

0.1) 01

Figure 6 Transition in fracture toughness of ZnS as a function of flaw size, C, and grain

size G LG and SG refer to large grain size and small grain size material, respectively, (after Yoder 16]

There are a number of models explaining the increased fracture toughness of

polycrystalline ceramics in terms of mechanisms such as crack deflection [ 13, 14] or microcracking [l 5-16] However, these models do not quantitatively predict the steep rise in Ktc observed in the transition from single crystal to polycrystalline fracture such as that observed for ZnS From a crack perspective, we can view the transition

schematically as shown in Figure 7 When the crack is contained within the grain it propagates along a cleavage plane Once the growing crack reaches the large grain

Figure 7 Schematic showing the relationship between a flaw growing in a large grain

under stress, and the segments of the growing crack in the adjoining smaller grains

26 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

boundary, in order for it to continue to propagate on this plane in the polycrystalline region, segments o f the crack must alter their orientation to conform to that o f the grain over that segment The stress intensity factor needed to extend such misaligned segments can be calculated fi'om mixed-mode models o f crack growth One such expression assumes a coplanar extension o f the crack under mixed mode loading [17] While the use

o f such an expression may be questioned for isotropic materials, its use under this

condition in which the crack is constrained to the cleavage plane seems reasonable The expression governing such a crack is:

where Ktc(so is the fracture toughness o f a single crystal o f the material

Assuming that there is only one crack path, namely the single cleavage plane, then in a randomly oriented polycrystalline microstructure an average 0 will be determined by the grain size For self-similar grain shapes, tbe area o f fracture surface will be independent

o f grain size However, finer grain size material will give rise to smaller values o f 2 While other possible toughening mechanisms such as crack bridging, formation o f subsidiary cracks, grain boundary fracture, etc could come into play, Equation 11

provides the minimum extra energy that would be required over and above that needed to grow a crack through a single grain Taking plausible values o f 2, one can calculate expected values o f Ktc/K/cesc) as shown in Table 1

Table 1 Ratio of Kict Ktctsc~ as a function of misorientation angle, n, of cleavage plane segments

Summary

In light of the fact that fracture of a number of important ceramic materials initiates within one grain, the fracture toughness of a single crystal is the key parameter needed to predict strength Because of limitations to the availability of accurate experimental techniques applicable to small single crystals, I suggest that the use of predictive models might serve as guides to fracture resistance I have shown that these models follow a general form for the relationship between fracture energy and the fundamental crystal structure and bonding laws for brittle materials independent of any specific stress-strain relationship, which will allow an estimate of fracture energy to be made Namely:

Where k is a force-law-specific constant It is hypothesized that k will vary by an amount less than the uncertainties in the experimental measurements themselves I have used this relationship to show that reasonable values of fracture energy can be calculated for a number of single-crystal materials

I have also put forth a model based upon mixed -mode fracture that can account for the single crystal to polycrystalline transition in fracture resistance

28 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

References

[1] Rice, R W., Freiman, S W., Pohanka, R C., Mecholsky, J J., and Wu, C C.,

"Microstructural Dependence of Fracture Mechanics Parameters in

Ceramics," Fracture Mechanics of Ceramics, 4, Edited by R.C Bradt, D.P.H

Hasselmann, and F.F Lange, 1978, pp 849-875

[2] Rice, R W., Freiman, S W., and Mecholsky, J J "The Dependence of Strength- Controlling Fracture Energy on the Flaw-Size to Grain-Size Ratio," Journal

of the American Ceramic Society, 63, No 3-4, 1980 pp 129-136

[3] Rice, R W., Freiman, S W., and Becher, P W., "Grain-Size Dependence of Fracture Energy in Ceramics: I, Experiment," Journal of the American Ceramic Society, 64, [6] 1981, pp 345-50

[4] Freiman, S W., Mecholsky, J J., Jr., Rice, R W., and Wurst, J C., "Influence of Microstructure on Crack Propagation in ZnSe," Journal of the American Ceramic Society, 58, No 9-10, 1975, pp 406-408

[5] Singh, J P, Virkar, V., Shetty, D K, and Gordon, R S., "Strength-Grain Size Relations in Polycrystalline Ceramics," Journal of the American Ceramic Society, 62, No 3-4, 1979, pp 179-183

[6] Yoder, P L., M.S Thesis, Pennsylvania State University, 1989

[7] Anstis, G R., Chantikul, B R., Lawn, B R., and Marshall, D B., "A Critical Evaluation of Indentation Techniques for Measuring Fracture Toughness: I,

Direct Crack Measurements," Journal of the American Ceramic Society, 64

[9], 1981, pp 533-538

[8] Gilman, J J., "Direct Measurements of the Surface Energies of Crystals," Journal of Applied Physics, 31 (12) 1960, pp 2208-2218

[9] Becher, P F and Freiman, S W., "Crack Propagation in Alkaline-Earth

Fluorides," Journal of Applied Physics, 49(7), 1978, pp 3779-3783

[10] White, G S., Freiman, S W., Fuller, E R., Baker, T L "Effects of Crystal Bonding on Brittle Fracture," Journal of Materials Research, 3, No 3, 1980 pp

[13] Faber, K T and Evans, A G "Crack Deflection Processes -l Theory," Acta Metallurgica, 31, No 4, 1983, pp 565-76

[14] Faber, K T and Evans, A G "Crack Deflection Processes -ll, Experiment"

Acta Metallurgica, 31, No 4, 1983, pp 577-84

[15] Fu, Y and Evans, A G., "Microcrack Zone Formation in Single Phase

Polycrystals," Acta Metallurgica, 36 1982, pp 1619-25

[16] Evans, A G and Faber, K T., Crack-Growth Resistance of Microcracking

Brittle Materials," Journal of the American Ceramic Society, 67, [4] 1984, pp

255-60

ASTM STP 381, 1965

Lynn M Powers, 1 Jonathan A Salem, ~ and Aaron S Weaver ~

Stresses in Ceramic Plates Subjected to Loading Between Concentric Rings

of Monolithic and Composite Brittle Materials, ASTM STP 1409, J A Salem, G D Quinn and M K Jenkins, Eds., American Society for Testing and Materials, West

Conshohocken, PA, 2002

and fatigue properties o f ceramics under multiaxial stresses The necessary tolerances required for accurate testing are not well documented in published literature The stresses

in square and round plates subjected to loading between concentric rings were

investigated via the finite element analysis method In particular, the effects o f load ring concentricity and the amount o f the plate overhanging the support ring were investigated Also, the errors generated by testing square plates between circular rings were

investigated The analyses were performed in support o f an effort by the American Society for Testing and Materials Committee C28 on Advanced Ceramics to standardized multiaxial testing o f ceramic plates

Key W o r d s : ceramics, multiaxial strength, stress, plates

The strength of brittle materials such as ceramics, glasses and semiconductors is a function

of the test specimen size and the state of applied stress [1] Engineering applications of such materials (e.g., ceramics as heat engine components, glasses as insulators, silicon and

germanium as semiconductors) involve components with volumes, shapes, and stresses substantially different from those of standard test specimens used to generate design data Although a variety of models [2] exist that can use conventional, uniaxial test specimen data to estimate the strength of large components subjected to multiaxial stresses, it is frequently necessary to measure the strength of a brittle material under multiaxial stresses Such strength data can be used to verify the applicability of design models to a particular material or to mimic the multiaxial stress state generated in a plate-like component during service

1Research engineer, Case Western Reserve University, NASA Glenn Research Center, MS 49-7, Cleveland, OH, 44135

2Research engineer, Life Prediction Branch, NASA Glenn Research Center, MS 49-7, Cleveland, OH,

44135

30 Copyright9 ASTM International www.astm.org

Further, the strength of brittle materials is sensitive to machining and handling o f the test specimens This can lead to spurious chips at the test specimen edges, which in turn can induce failure not representative of the flaw population distributed through the material's bulk In the case of a plate subjected to bending, the stresses developed are lowest at the edges, thereby minimizing spurious failure

For components that are subjected to multiaxial bending, three different loading assemblies, shown schematically in Figure 1, can be used to mimic biaxial stresses by flexing circular or square plates: ball-on-ring (B-O-R), ring-on-ring (R-O-R), or pressure- on-ring (P-O-R) The R-O-R and the P-O-R are

preferred because more of the test specimen

volume or surface area is subjected to larger

stresses However, significant frictional or

wedging stresses associated with the loading ring

can be developed in the highly stressed regions

of the R-O-R specimen [3,4] These stresses are

not generated in the P-O-R configuration and the

P-O-R configuration has been investigated for

testing of both isotropic [5-8] and anisotropic

materials [9,10] Although the P-O-R has been

experimentally determined to produce results in

agreement with plate theory, it is not as

temperature testing

A wide variety of both square and round

R-O-R loaded plates are used to determine

fatigue and strength properties, however, the

necessary tolerance and geometric ratios for Figure l-Multiaxial bending test

accurate testing are not well-documented in the configurations a)ball-on-ring, b)

literature The objective of this work was to ring-on-ring, c)pressure-on-ring

investigate the stresses in square and round plates

subjected to loading between concentric rings The following aspects were investigated via the finite element method: the amount of the plate overhanging beyond the support ring; the errors generated by testing square plates between circular rings; the effect of the plate thickness on the wedging stresses; and the effect of concentricity between the load and support rings The analyses were performed in support of an effort by the American Society for Testing and Materials Committee C28 on Advanced Ceramics to standardized multiaxial testing of ceramic plates

Theory

A schematic of the ring-on-

ring specimen is shown in Figure

2 The theoretical solution for the

deflection at the center of the

s

r

Figure 2-Geometry of a ring-on-ring test

32 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

plate is given by [11]

where P is the applied load, E is Y o u n g ' s modulus, v is Poisson's ratio, and the geometry

o f the disk in shown in Figure 2 The geometry o f the R-O-R configuration consists o f its

radius, Rn, thickness, t, load ring radius, RL, and the support ring radius, Rs This

geometry may also be described by the respective diameters, D, DL and Ds The radial

and tangential stresses in the center section on the tensile surface are equal and may be expressed as

For all o f the plates studied herein, the ratio o f Rs to RL is 2

Another important measure to consider when testing brittle materials is the

probability o f failure o f the test specimen Since the plates are subjected to bending, the stresses are highest at the surface Thus, surface failure analysis was used The probability

o f failure for a surface element is

where trema~ is the maximum effective stress, Ns is the crack density function, to is the length

of an angle, or, projected onto a unit radius circle in principal stress space containing all of the crack orientations for which the effective stress is greater than or equal to the critical stress and 5" is the stress state Since the stress state is equibiaxial inside the load ring, Y~ = (tr,~r), Oremax

= tr, and to/2~r = 1 Equation 4 simplifies to

'5'

where kBs is the Batdorf crack density coefficient, eros and ms are the Weibull parameters The

Weibull parameters are determined experimentally

Finite Element Analysis

One o f the key elements o f the ASTM draft standard on Monolithic Equibiaxial Flexure Strength Testing o f Advanced Ceramics at Ambient Temperatures is setting up the guidelines for specimen geometry and load The stresses and deflections generated by these tests are subject to the limitations o f linear plate theory Many o f these limitations have been determined for ceramic equibiaxial plates through trial and error or experience This paper will confirm these ideas using finite element modeling o f ceramic plates o f various geometry and loads The modeling in this study was completed using the ANSYS program The plates were assumed to have an elastic modulus of 300 GPa and a Poisson's ratio o f 0.2 The load rings for the contact analysis are steel with an elastic modulus o f

200 GPa and a Poisson's ratio of 0.3 The friction coefficient between the load and support ring and the plate was 0.05

The first issue addressed was to determine the load levels appropriate for linear plate theory To accomplish this, the contact between the load rings and the ceramic plate was modeled The mesh for this model is shown in Figure 3 An axisymmetric model was used which contained 2518 8-node quadrilateral (PLANE82), 80 3-node target surface (TARGET169) and 128 3-node contact (CONTACT172) elements The boundary conditions applied included the pressure on the load ring, the rigid support ring and the coupled displacement on the load ring As a result o f this coupling, these nodes are forced

to take the same displacement in the specified nodal coordinate direction Both the radial and axial directions are coupled individually

Figure 3-Axisymmetric mesh of the plate and ring assembly

Using the model shown in Figure 3, the load vs displacement curve was constructed for the plate The geometry of the plate was RD = 25mm, Rs = 20mm, RL = 10mm, and t = 2mm The normalized load vs displacement/thickness curve is shown in Figure 4 The dashed and solid lines represent linear plate theory and the contact analysis, respectively Linear plate theory is assumed to the applicable where these two lines are coincident

Where linear plate theory is not applicable, the stress state is no longer predictable

by equation (2) The maximum stress is not constant at the surface inside the load ring The stress is maximum under the load and has been shown to be as high as 50% larger than the stress at the center [12] Figure 5 shows the maximum stress in the plate as a function

Figure 4-Normalized load as a function o f deflection

34 FRACTURE TESTING OF MONOLITHIC/COMPOSITE MATERIALS

D e f l e c t i o n / T h i c k n e s s

Figure 5-Ratio o f the maximum to center stress in the plate as a function o f deflection

of the deflection This figure is consistent with Figure 4 The maximum-stress to center-stress ratio is a minimum and constant at deflections lower than '/, the thickness of the plate All subsequent analyses performed in this paper are done in the range of linear plate theory This

is representative of deflection/thickness of less than '/4

The mesh used for the modeling of the circular plates is shown in Figure 6 This model is axisymmetric using PLANE82 (quadrilateral with midside nodes) elements The boundary conditions applied were the pressure load and the support ring The mesh used for the square plates is shown in Figure 7 A quarter symmetry model was used with SOLID95 (bricks with midside nodes) elements Boundary conditions applied were the pressure and support as shown

in Figure 7 The number of elements used for each model varied depending on the geometry of the plate

Figure 6-Axisymmetric mesh for circular plates

Figure 7-Quarter symmetry mesh for the square plate

Results

The radial stress distribution for the round and square plates are shown in Figure g The stress on the tensile surface is maximum and constant inside the load ring and decreases toward zero at the support The radial stress outside the support is nominally zero However, the tangential stress outside the support is not zero and is a concern at the edge where flaws may cause failure These flaws can be more severe than those inherent

on the surface o f the disk As a result, failure o f the plate may occur at the edge instead o f inside the load ring as expected

To ensure that the plates ultimately fail from flaws originating from inside the rings, test specimens should be designed so that the edge stresses are minimized One o f the ways to accomplish this is to increase the amount o f material outside the support ring The effect o f overhang on the edge stresses is shown in Figure 9 for round plates As the amount of overhang increases the tangential stress at the edge decreases The stress decreases as the ratio o f the support ring to thickness increases For thin (Ds/t=-30) plates, the edge stresses are lower than for thick (Ds/r=-lO) plates