Astm stp 1207 1994

ASTM Committee on E-8 on Fatigue and Fracture formerly E-24 on Fracture Mechanics sponsored the symposium in cooperation with the University of Tennessee and the Oak Ridge National Labor

Fracture Mechanics"

Twenty-Fourth Volume

John D Landes, Donald E McCabe, and J A M Boulet, Editors

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Printed in Ann Arbor, MI December 1994

Foreword

The 24th National Symposium on Fracture Mechanics was presented at Gatlinburg, Ten- nessee on 30 June-2 July 1992 ASTM Committee on E-8 on Fatigue and Fracture (formerly E-24 on Fracture Mechanics) sponsored the symposium in cooperation with the University of Tennessee and the Oak Ridge National Laboratory John D Landes, University of Tennessee, and Donald E McCabe, Oak Ridge National Laboratory, served as chairmen of the symposium and editors of the resulting publication J A M Boulet, University of Tennessee, also served

as an editor of the publication

Contents

THIRD ANNUAL JERRY L SWEDLOW MEMORIAL LECTURE

Reflections o n P r o g r e s s i n F r a c t u r e M e c h a n i c s Research PAUL c PARIS 5

A n A p p r o x i m a t e T e c h n i q u e for P r e d i c t i n g Size Effects o n C l e a v a g e F r a c t u r e

T o u g h n e s s (Je) U s i n g the Elastic T StresS MARK T KIRK, ROBERT H DODDS, JR.,

I n t e r i m Results f r o m the H e a v y Section Steel T e c h n o l o g y (HSST) S h a l l o w - C r a c k

F r a c t u r e T o u g h n e s s Program TIMOTHY J THEISS, DAVID K M SHUM, AND

A p p l i c a t i o n of J-Q F r a c t u r e M e t h o d o l o g y to the Analysis of P r e s s u r i z e d T h e r m a l

S h o c k i n R e a c t o r P r e s s u r e Vessels -DAVID K M SHUM, TIMOTHY J THEISS,

DUCTILE TO BR1TrLE TRANSITION

S i n g l e - S p e c i m e n T e s t Analysis to D e t e r m i n e L o w e r - B o u n d T o u g h n e s s in t h e

T r a n s i t i o n - - J O H N D LANDES, UWE ZERBST, JURGEN HEERENS,

Region TED L ANDERSON, DAVID STIENSTRA, AND ROBERT H /)ODDS, JR

A P e r s p e c t i v e on T r a n s i t i o n T e m p e r a t u r e a n d Kjc D a t a C h a r a c t e r i z a t i o t r - -

DONALD E MCCABE, J G MERKLE, AND R K NANSTAD

E v a l u a t i o n of Elastic-Plastic F r a c t u r e T o u c h n e s s Testing in the T r a n s i t i o n Region

T h r o u g h J a p a n e s e I n t e r l a b o r a t o r y Tests TADAO IWADATE AND

Results of M P C / J S P S C o o p e r a t i v e Testing P r o g r a m in t h e Brittle-to-Ductile

T r a n s i t i o n Region WILLIAM A VAN DER SLURS AND MARIE T MIGLIN

Effect of S t r a i n R a t e o n S m a l l S p e c i m e n F r a c t u r e T o u g h n e s s in t h e T r a n s i t i o n

Region TADAO IWADATE, MIKIO KUSUHASHI, AND YASUHIKO TANAKA

A n a l y s i s of Results f r o m t h e M P C / J S P S R o u n d R o b i n T e s t i n g P r o g r a m in the

Ductile-to-Brittle T r a n s i t i o n Region MARIE T MIGUN, LILLIAN A OBERJOHN,

AND WILLIAM A VAN DER SLUYS

D e t e r m i n a t i o n of L o w e r - B o u n d F r a c t u r e T o u g h n e s s for H e a v y - S e c t i o n Ductile C a s t

I r o n (DCI) a n d E s t i m a t i o n b y S m a l l S p e c i m e n Tests -TAKU ARAI,

TOSHIARI SAEGUSA, GENKI YAGAWA, NAMIO URABE, AND ROBERT E NICKELL

S i m p l e r Jic Test a n d D a t a A n a l y s i s P r o c e d u r e s f o r H i g h - S t r e n g t h Steels -

J H UNDERWOOD, E J TROIANO, AND R T ABBOTT

389

410

A New Application of Normalization: Developing J-R Curves from Displacement

Versus Crack Length and from Displacement AIone -I<ANG LEE AND

JOHN D LANDES

Amounts of Ductile Crack Growth in Bending CEDRiC E TURNER

Dislocation Emission and Dynamics Under the Stress Singularity at the Crack Tip

and Its Application to the Dynamic Loading Effect on Fracture

Toughness -A TOSHIMITSU YOKOBORI, JR., TADAO IWADATE, AND

Crack Growth Under Small Scale and Transition Creep Conditions in Creep-

Ductile Materials -aSHOK SAXENA, KOICHI YAGI, AND MASAKI TABUCHI

Effects of Mean Load on Creep and Fatigue Crack Growth at Elevated

Temperature -KAi-YOUARN HOUR AND JAMES F STUBBINS

Evaluation of the Relationship Between C*, ~h, and ~h during Creep Crack

Growth ASHOK SAXENA, B DOGAN, AND KARL-HEINZ SCHWALBE

Stress Intensity Factor Solutions for Surface Cracks in Flat Plates Subjected to

Nonuniform S t r e s s e s - - - I V A T U R Y S RAJU, SAMBI R METTU, AND V SHIVAKUMAR

Weight Functions for Eccentric Cracks -XIAOGUANG CHEN AND PEDRO ALBRECHT

Fracture Criteria for Surface Cracks in Brittle Materials -WALTER G REtrrEa,

JAMES C NEWMAN, JR., BRUCE D MACDONALD, AND STEVE R POWELL

The Crack Tip Opening Displacement and J Integral Under Strain Control and

Fully Plastic Conditions Estimated by the Engineering Treatment Model for

Plane Stress Tension KARL-HEINZ SCHWALBE

617

636

ROBERT E NICKELL AND DAVID F QUIIZlONES

Fracture Capacity of High Flux Isotope Reactor (HFIR) Vessel with Random

Crack Size and Toughness -SHIH-JUNG CHANG

652

672

FATIGUE

S i m u l a t i o n of Fatigue Crack G r o w t h of an Inclined Elliptically S h a p e d S u b s u r f a c e

Crack in Residual Stress Fields -K MAYRHO~R, F D FISCHER, AND

E PARTEDER

Surface Crack Growth in Inconel 718 During Large Unload-Reload Cycles -

Effects of Cyclic Loading on the Deformation and Elastic-Plastic Fracture Behavior

of a C a s t Stainless Steel JAMES A JOYCE, EDWIN M HACKETT, AND

T h e A p p l i c a t i o n of a Ductile Fracture M e t h o d to P o l y m e r Materials -ZnEN ZHOU

AND JOHN D LANDES

Methodology for Predicting Canopy Fracturing Patterns During Ejection

R O C K Y R ARNOLD, PATRICK S COLLINS, PETER S AYOUB, AND R TUNG

Calculation of Stress Intensity Factors for Interface Cracks Under Mixed-Mode

L o a d i n g - - R A J I V A NAIK AND JOHN H CREWS, JR

I m p a c t Testing of AI203 and SiCw/AI203 Ceramics -LYLE R DEOBALD AND

STP1207-EB/Dec.1994

Overview

The 24th National Symposium on Fracture Mechanics was held 30 June to 2 July, 1992, in Gatlinburg, Tennessee, on the doorstep of the Greater Smoky Mountains National Park In addition to the fine technical program and the evening social activities, the park bears gave unusual entertainment to the symposium attendees with their apparent free access to the hotel garbage facilities The Symposium was sponsored by A S T M Committee E 24 (now E 08), with support from the University of Tennessee and the Oak Ridge National Laboratories

The Symposium had an international flavor Nine different countries were represented on the technical program with nearly one third of the presentations coming from authors outside

of the United States There was a very large participation from Japan; eight papers were pre- sented by Japanese authors This international participation added an important dimension to the symposium allowing the attendees to gain insight on the fracture work that has been going

on around the world

The book has been divided into nine topical sections; Jerry L Swedlow Memorial Lecture, Constraint Issues, Ductile to Brittle Transition, Elastic-Plastic Fracture, High Temperature Effects, K Analysis, Applications, Fatigue, and Nonmetallic Materials The Swedlow Memorial Lecture, presented by Professor Paul C Paris of Washington University, St Louis, looked at the impact that this symposium series has had on the progress in fracture mechanics research

As originator of this important series, Professor Paris is uniquely qualified to judge its merit

It is now more than a quarter of a century since the first symposium was held at Lehigh University in June of 1967 The hundreds of authors from past symposia form a Who's Who

of fracture mechanics Many of the important new advances in the subject were first published

in the STPs that resulted from these symposia Some of these papers have been cited hundreds

of times in the literature

The central themes of this symposium were constraint issues and nonlinear fracture mechan- ics These are covered in the next four sections The number of papers dealing with the two- parameter fracture mechanics approach to constraint and its impact on transition fracture tough- ness show that it is currently the most active topic of study Much of the work on transition fracture toughness came from a round robin program sponsored by the Materials Properties Council (MPC) and the Japan Society for Promotion of Science (JSPS) Dr Martin Prager of MPC assisted with the organization and review of these sessions The sections on elastic-plastic fracture and high-temperature effects mark a continuing interest in the nonlinear fracture mechanics areas

The topics on K analysis, applications, and fatigue come from a more traditional interest area

in fracture The K analysis forms the very core of the fracture mechanics approach Applications are the ultimate goal of the fracture mechanics research The renewed interest in fatigue at the National Symposium is perhaps in anticipation of the cooperation between these two areas with the recent merger of the ASTM Committees E 09 on fatigue and E 24 on fracture into E 08

on fracture and fatigue Finally, the section on nonmetallic materials indicates that this area is one for which much of the future work on fracture and fatigue will be directed Interest in

Copyright9 by ASTM International

1 www.astm.org

fracture of nonmetallic materials is broadly based including work on polymers, ceramics and composite materials

A high point in the symposium was the Awards Banquet at which the Irwin metal was presented to Professor Ashok Saxena of the Georgia Institute of Technology and the ASTM Award of Merit was presented to Professor James A Joyce of the Naval Academy These were presented by Mike Hudson, then Chairman of ASTM Committee E 24 The members of the organizing committee should be acknowledged These include John Landes and Don McCabe, Cochairmen, Toby Boulet, Rick Link, Janis Keeney, Karl-Heinz Schwalbe, Ed Wessel, Ashok Saxena, Ted Anderson, and A1 Van Der Sluys Finally, the staff of ASTM helped with the conduct of the symposium and the development of this STP These include Dorothy Savini, Pat Barr, Kathy Dernoga, Lynn Hanson, Therese Pravitz and Kathleen Peters

John D Landes

University of Tennessee Knoxville, TN; symposium cochairman coeditor

and

Third Annual Jerry L Swedlow Memorial

Lecture

Reflections on Progress in Fracture

Mechanics Research*

REFERENCE: Paris, P C., "Reflections on Progress in Fracture Mechanics Research,"

Fracture Mechanics: Twenty-Fourth Volume, ASTM STP 1207, John D Landes, Donald E

McCabe, and J A M Boulet, Eds., American Society for Testing and Materials, Philadelphia,

to his colleagues in transmitting their work in the field, which included his enormous and

effective efforts in editorial work for the International Journal o f Fracture Mechanics and his

guidance as Chairman of the National Symposium Organizing Committee for many years, leaves us with a great debt of gratitude to him It is a distinct pleasure and honor to present the

"Third Annual Jerry L Swedlow Memorial Lecture" as a tribute not only to " J e r r y " but to his special friends, such as the late Dr John Srawley, and many others

When Symposium Cochairmen Landes and McCabe suggested a theme for this discussion, they clearly indicated an inclination toward " s t o r i e s " from "the good old days," thut is, a view of fracture mechanics history in the 1950s This was agreed upon as background to the main topic of discussing the history of the symposium itself and its contributions as a forum for fracture mechanics research as documented by the series of Special Technical Publications (STPs) books published by the ASTM The picture is most complete with the sister symposia

on elastic-plastic fracture mechanics, and its STPs, included in the discussion, as well as the retrospective volume, RPS-1, compiled by Dr John Barson (last year's Swedlow Lecturer) for

ASTM Moreover, the initial departure point of this series of A S T M STPs was really STP 381

(1965) resulting from a conference in 1964 in Chicago which chronicled the state of the art for the then infant ASTM Committee on Fracture Testing E 24 This series of books is listed as references (1 through 27) herein Currently A S T M Committee E 24 has produced more than twice this number of STPs (76 of the over 1000 now in print from ASTM), but in this author's opinion, the 27 included here are restricted to those of the greatest relevance and relationship

to the National Symposium on Fracture Mechanics

The progress in fracture mechanics research, as represented by these books [1-27], can be

viewed in terms of gross numbers There are 852 separate research papers covering 16,708 pages of print That sounds enormous, but actually it is about one average shelf of books (see Fig 1) These figures show bulk, but quality, that more illusive characteristic, is still to be demonstrated here However, first the historical background shall be developed

* Third Annual Jerry L: Swedlow Memorial Lecture

Professor of Mechanics, Washington University, Department of Mechanical Engineering, Campus Box 1185, One Brookings Dr., St Louis, MO 63130

5

www.astm.org

6 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

FIG l The Symposium related STPs

History has a strong tendancy to be one man's personal recollection of important events and

is often presented by an elder in a fashion to make the youthful in awe of exciting past times With that warning, the following " h i s t o r y " is presented

Recollections of a Graduate Student the "Good Old Days"

Some time during the 1953-1954 academic year, Professor E Orowan of Massachusetts

Institute of Technology (MIT) presented a lecture at Lehigh University As a part of that discussion, an energy analysis of progressive crack propagation was discussed It was presented

as a topic obviously needing further study (see the Introduction in RPS 1 [22]) During the same year, a Symposium on Plasticity Theory convened at Brown University At that sympo-

sium, Wendel P Roop, a representative of the Ship Structure Committee spoke of the Navy's

ship fracture problems Professor Roop indicated that progressive fracturing probably has some-

thing to do with structural stored energy fed into the process, and readily admitted he did not

understand any details There was not a bit of further fracture discussion at that most elegant Symposium on Plasticity with all the leading researchers there! The impression was left to

students of mechanics that the subject was not well understood So-called "brittle fracture"

was left for metallurgical discussions of Charpy test transition temperatures and the like

So, when this graduate student arrived at the Boeing Company in Seattle for a summer

position in 1955, it was a great shock to be assigned to study the British Comet airliner fracture

failures and to work on assuring that the Boeing 707 did not have such a problem Lacking the

courage to request an alternate assignment, the approach was to read everything which seemed vaguely related After plowing through over 100 papers and so forth in a few weeks, the clear impression began to emerge that no one really understood the problem with the single exception that the papers of G R Irwin and coworkers did make some sense One of the test engineers

at the Boeing Company, E Zapel, responded to ideas based on Irwin's analyses by working

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

overtime to do some crude tests on candidate pressure cabin skin materials, 2024 and 7075

aluminum alloy The thickness tested ranged from about 0.050 to about 0.200 in (1.3 to 5 mm)

for each with interesting results The Irwin (plastic) energy dissipation rate per unit new crack

area (or Gc) was about constant for a given thickness of a given material 2024 clearly was

much tougher than 7075 (the British Comet had an alloy very similar to 7075) That was good

news since Boeing had already selected 2024 for the 707 pressure cabin by fatigue criteria (but

they also hoped to use 7075 in the KC-135 Air Force tanker version but with a much shorter

design life and less complicated windows) The othernews with those test results was that the

toughness (Go) varied with thickness For the gages tested, 2024 had toughness increasing

slightly with thickness where, with 7075, it significantly decreased There was then no expla-

nation of these trends The Chief of Structural Research at Boeing, A Sorenson, took these

results and presented them at the Institute of Aeronautical Sciences National Meeting in New

York in January of 1956 The presentation was well attended, but no one had an explanation

of the thickness effect

Now for the benefit of the younger readers, it must be admitted that without discussion,

Sorenson had taken " m y data" and written a paper without acknowledgement, which was

perplexing A few days before the presentation he did call and invited me to dinner after the

presentation Then at dinner he asked if I would be willing to act as a consultant to Boeing 10

to 20 hours per week while continuing studies at Lehigh and spending summers at Boeing The

perplexity disappeared instantly!

In the early spring of 1956, as a consultant to Boeing and a Lehigh student, a call to G Irwin

elicited an invitation to visit him for most of a day, without an inkling that he was a busy

Superintendent of a Division of the U.S Naval Research Laboratory (NRL) He supplied copies

of NRL reports, gladly discussed work in progress, and welcomed any questions or thoughts

on the subject The discussion resolved many issues but raised just as many unanswered ques-

tions This hospitality and endless willingness to discuss the issue has been Irwin's hallmark

over the 36 years to date Mrs Irwin once related that it was about this time that Dr Irwin

made the conscious decision that "The Message" needed to be actively spread About this

same time he gave the field its formal name, "fracture mechanics."

The unresolved issues at this point, 1956, were formidable The Griffith-Irwin-Orowan

energy rate analysis, that is, the Gc approach, was not popular It depended on the plastic

dissipation rate of work with crack growth to be constant with crack extension, but why should

it be constant for different crack sizes and crack configurations? What about the thickness

effects? Professor D C Drucker of Brown about this time was saying that an energy balance

for fracturing was a necessary condition but not sufficient conditions to form an analysis E

Orowan wrote a note saying the second derivation of energy was more important in ductile

failures Kuhn of National Aeronautics and Space Administration (NASA), Professor Sachs of

Syracuse, and Professor Neuber of Munich all held strongly that their own individual analysis

were more proper approaches Indeed, it was certainly not clear that the "Fracture Mechanics"

approach of Irwin would prevail The more the experts talked about it, the muddier the waters

became In fact, at this same time, the Superintendent of the neighboring Metallurgy Division

o f the NRL, W Pellini, paid scant attention to Irwin's approach and promoted his own views

So who was convinced that Irwin's approach would prevail?

Well, to a graduate student in applied mechanics, Irwin's approach was the only one that

was comprehensible, giving a context for some clear thinking on the subject It was also sig-

nificant consolation the Dr A A Wells of the British Welding Research Institute wrote a

perceptive discussion of the Comet aircraft failures (1955, see Ref 22) following Irwin's

approach and also began in late 1954 some extended visits to work within Irwin's division at

NRL Dr Wells wrote a report at NRL analyzing a wedge-force initiation from a notch in

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

8 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

uniformly stressed plates in which he without explanation added the energy rates for each

loading by "squaring the sum of the square roots," that is,

Gtota I = (G]/2 + G~/2) 2

It took some weeks to see that this was correct and gave the impression Wells was a very clever

man and that his visits with Irwin were producing synergistic progress

The ultimate example of this synergism developed to fruition in 1956 In 1954, D Post,

working with Irwin at NRL, examined photo-elastic fringes near a crack tip and came upon a

paper by Westergaard [28], which could be used to assist in the analysis With admitted assis:

tance from Wells, Irwin used Westergaard's methods to develop the elastic crack-tip stress

field equations that he first presented at the International Congress on Applied Mechanics in

Brussels in 1956, to be published by American Society of Mechanical Engineers (ASME) in

1957 (see Ref 22) This paper immediately related the Griffith-Irwin energy rate, G, to the

intensity of the crack-tip stress field, K, and showed that the stress field always had the same

distribution

Although Professor I Sneddon of Glasgow had in 1946 [29] obtained the elastic stress

solution for an embedded circular (disk) crack and had expanded its crack tip stress field to

obtain the singular, 1/~rr, term, as well as determining the Griffith energy rate, he did not

recognize the generality or physical significance of his results Also in 1957, M L Williams

published (see Ref 22) with ASME an alternate derivation and version of the crack-tip stress

field equations, but he omitted discussion of the key physical significance of the Griffith-Irwin

approach At least for the novice, Irwin's 1957 paper contained significantly clearer applicable

results Although Williams paper appeared first, March 1957 instead of June 1957 for Irwin's,

it is of some historical interest to note that Irwin's manuscript was the first of the two received

by ASME (see footnote on first page of each)

From the point of view of immediate usefulness, it was Irwin's 1957 paper that cleared the

muddy waters Having identical crack-tip stress fields for different sizes of cracks and differing

load application methods explained why the plastic dissipation G of the Griffith-Irwin analysis

should be roughly constant It also immediately became clear that the thickness effects were

due to lateral constraint, in particular the tendency toward plane stress or plane strain within

the crack-tip plasticity Many burning issues on static fracture phenomena were rapidly

resolved By 1958, when a U.S Naval Symposium On Structural Mechanics [30] was held at

Stanford, the large audience and stirring response to Irwin's presentation demonstrated that his

"message" was being well received Some real "progress" was being made in fracture

mechanics research

Also, with the recognition of the generality of the crack-tip stress field equations, other vistas

expanded In 1955, at the Boeing Company, the question was raised whether the Griffith-Irwin

energy analysis could be used to analyze fatigue crack growth The initial reaction was negative

considering the fact that fatigue implied cyclic plasticity at a crack tip, which was seemingly

unrelatable to the static Griffith elastic energy rate However, even before Irwin's 1957 paper

was formally in print, a memo was sent within the Boeing Company saying the crack-tip stress

field intensity parameter, K, should be able to correlate fatigue crack growth rates (see also the

discussion to Irwin's paper in Ref 30) Various internal difficulties at Boeing prevented suffi-

cient data being available until 1959 to demonstrate fatigue crack growth rate correlations, so

my own first paper on that subject did not appear until 1961 (see Ref 22) Moreover, in 1958,

Professor A J McEvily published a report at National Advisory Committee for Aeronautics

based on stress concentration theory, which is relatable to K, but it seems he has never been

given sufficient credit for that discovery within the fracture mechanics community Note that

Professor G Sinclair and his student D Martin [32] at the University of Illinois attempted

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

before 1958 to use G as a correlating parameter but without real success Therefore, it was Irwin's 1957 paper that was the key to direct progress on fatigue crack growth

Consequently, by the late 1950s there existed a great deal of interest in Irwin's "fracture mechanics." The studies of others began to relate closely to his on topics of application, analysis, and reinforcement of the theory Prime examples of such efforts are the late 1950s work of Wundt of General Electric, Bueckner of General Electric, McClintock of MIT, and

visits with Irwin with occasional visits with others were completely sufficient, as compared to today with hundreds of centers of activity just in the United States

In the summer of 1957 at the Boeing Company, Bill Gray, as technical advisor within the headquarters offices, requested weekly briefings and educational discussions, which he and W

E Anderson attended, to assure that they developed a full familiarity with the field By late

1959, the head engineering people of the Transport Division of Boeing had requested that formal courses be taught internally with at least one person from each structural and materials unit attending Dr Irwin was enticed to visit and teach one of the classes to the first group, while surely similar intensive interest was occurring within other organizations in both industry and government

This intensification of interest caused the need for better communication between the expand- ing group of leading researchers to accelerate the progress in fracture mechanics The formation

of the ASTM Special Committee to resolve the Polaris Missle engine case fracture problem at the time was exactly the stimulus needed That group, in performing its primary task, was periodically bringing together most of the leading researchers in the country, and it was quickly recognized in the group that it was advantageous to keep everyone up to date on the latest progress in the area Not only did the meetings discuss the primary business related to Polaris, but extra time was scheduled for people to present their other research in progress

The formal committee opened its doors to other individuals who could contribute and benefit from the research in progress sessions For example, Dr Irwin contacted Wessel of Westing- house at a national American Society for Metals (ASM) meeting in 1960 and recognizing his interest and potential contribution brought him into the group The group grew to 20 then 30 and perhaps 40 people attending these meetings Sometimes the meetings were held at N A S A headquarters in Washington Frequently, Dr Irwin would invite the whole group to his home

in the evening (surely without his wife's knowledge of the number who would arrive), and the technical discussions would last far into the night Perhaps that was the real birth of our national

a convivial meeting, usually following dinner, for drinking, conversation and intellectual enter- tainment." Indeed, " c o n v i v i a l " was a perfect description of that group

It was about this time that Jerry Swedlow first came upon the fracture mechanics scene Dr Irwin 2 recalls that in late 1959 or 1960, Jerry visited Irwin at NRL to inquire for his employer, Hercules Powder, about the potential use of fracture mechanics in analyzing explosive frac- turing of rock As a result of that visit, Jerry was referred to Professor J Lubahn of the Colorado School of Mines Perhaps these events were key to Jerry later going to Cal Tech for his doctorate

It was also in 1960 that this student decided to return to Lehigh to finish doctoral work W

E Anderson at that time convinced his superiors at Boeing to allow me to take a modest amount

of research funding along to begin some effort there That enticed two younger Assistant Professors, F Erdogan and G Sih, to become involved in fracture mechanics, which in my opinion triggered some significant research progress The first known university courses in fracture mechanics started then, with early attendees like J R, Rice, J D Landes, J A Begley,

2 G R Irwin, private communication

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10 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

R W Hertzberg, and more (too numerous to continue), names well known to all in the field

now It is not an unbiased view that this contributed to fracture mechanics progress, but at least

it is undeniable as leading toward the first national symposia

The early 1960s initiated a great expansion in fracture mechanics activity with many centers

of activity developing throughout the country, instead of perhaps 5 places in the 1950s, it rapidly

became at least 50 centers by the mid to late 1960s, as compared to hundreds now The end of

the " g o o d old d a y s " was rapidly approaching with this acceleration in progress

The expansion of activity was such that by 1962 at least a one-day-long research meeting,

after a ASTM Special Committee meeting the day before, was held at NRL in which the topic

was restricted to only a subtopic within fracture mechanics It was about this time that ASTM

headquarters recognized that its special committee had quite successfully discharged its original

special task yet wished to go on and did so with headquarters blessings

However, fracture mechanics was only popular within a relatively small circle of engineers

and researchers An inquiry to a leading textbook publisher at that time brought back a response

that "fracture mechanics was never going to amount to any wide interest." In many quarters,

there was active opposition to fracture mechanics from competing approaches Many of the

leading metallurgy/materials scientists often openly stated the view that if fracture mechanics

could not be directly related to dislocation theory, it was certainly doomed Others asked " d o

you believe in fracture mechanics?" as if it were a blind faith

Approaching the middle 1960s, ASTM headquarters communicated to the Special Committee

that it wished it to become a regular E committee and to get on with the task of developing

testing standards At least one other E committee had some members opposed to a new fracture

mechanics oriented committee Their ultimate argument was that "fracture is just the final

cycle in a fatigue test," implying a new committee was unnecessary Happily, ASTM Com-

mittee E 24 was formed, and the initial peaceful coexistence between committees quickly led

to active cooperation, and after 25 + years, an appropriate merger is taking place In the mean-

time, allowing fracture mechanics to have its own forum obviously permitted more rapid prog-

ress in both groups!

Just before, but during the process of forming the A S T M E 24 Committee from the Special

Committee, it was decided to hold a Symposium on Fracture Toughness Testing and Its Appli-

cations at the ASTM National Meeting in Chicago in June of 1964 With the cooperation of

Professor J Low of Carnegie-Mellon, who ably chaired the Special Committee as well as the

new ASTM E 24 Committee, the Symposium Chairman W F Brown, Jr of N A S A elicited

N A S A support Dr Brown intensely sought to present fully the state of the art at that time and

put together a quite comprehensive program His frequent phone calls to committed authors

made it clear he wanted the best that each could produce! The results are recorded in the

landmark volume, A S T M S T P 381 [1], which recorded much of the progress through the early

1960s Well before 1970, it had become A S T M ' s all time best-selling book It represented a

departure point for the ASTM E 24 Committee to begin its efforts as a regular ASTM com-

mittee It was a definitive statement of progress up to that time

There was, of course, other key early 1960s work not mentioned here, for example, the initial

application of fracture mechanics to subcritical environmental cracking by Professor H H

Johnson of Cornell [34], with apologies to the many others whose fine work is also omitted

here However, one other special development occurred that greatly assisted progress in fracture

mechanics research It was the development of the first reliable servocontrolled electrohydraulic

test equipment by Research Inc., now known as MTS Systems Corporation It had just the right

capabilities to assist in the testing requirements of fracture mechanics, but it required a new

level of electronic control knowledge of many o f us At least throughout the eastern United

States, Mr H R Hartmann was our frequent mentor on how to do the type of testing required

In 1964 at the suggestion of W F Payne of the U.S Air Force, we initiated two-week

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summer short courses on Fundamentals of Fracture Mechanics at the University of Denver

with the sponsorship of Universal Technology Corp of Dayton, Ohio At that time, almost no

university courses existed These courses were held annually throughout the remainder of the

1960s, and in later additional one-week courses special advanced topics were discussed The

courses not only served to initiate many into this field but also served as a place in which

instructors, who were established people such as Irwin, Srawley, Wessel, and so on, met for

extended periods each year and exchanged research ideas Also, many relatively unitiated in

the field who attended have gone on to devote their careers to the field, for example, John

Barsom, Don McCabe, Howard Wood of the U.S Air Force, and so on These courses assisted

progress in the field and later spread into courses at many places

With the advent of the new A S T M E 24 Committee came the responsibility to develop

standard testing specifications The group by the end of 1965 quickly shifted to the objective

of first developing a standard for plane strain fracture toughness testing, resulting in the ASTM

Test Method for Plane-Strain Fracture Toughness of Metallic Materials (E 399) Between reg-

ular committee week meeting s smaller, more intensive, task group meetings were held that

were most often led by Brown and Srawley adhering to the topic of the plane strain standard

Moreover, the A S T M E 24 Committee grew enormously over the original Special Committee

group, and one could no longer keep up with all of the research going on by simply attending

a one-day meeting a couple of times each year Were the " g o o d old d a y s " over?

The National Symposium on Fracture Mechanics

The A S T M E 24 Committee established a Task Group on Research with John Srawley as

the initial Chairman, a function he served well until called upon to become Chairman of the

main committee The meetings provided a research forum, but they did not draw the latest

work in the field by some of the leading researchers It certainly did not produce the intensity

of the 1964 meeting in Chicago resulting in STP 381 [1]

Consequently, at the National A S T M meeting in Atlantic City in June of 19661 asked Irwin,

an attendee, if we would run a research symposium on fracture mechanics at Lehigh University,

would he support the idea and lend his name as cosponsor? He seemed enthusiastic, which was

crucial to beginning the planning for the first National Symposium on Fracture Mechanics at

Lehigh i n June of 1967 NRL and ONR lent their names as cosponsors The published pro-

Symposium with repeat performances in June of 1968 and 1969

By 1970, for the fourth symposium, it appeared that a new location would be advantageous,

so the chairmanship was entrusted to E T Wessel, who with assistance from Jerry Swedlow,

held the meetings at Carnegie-Mellon University

It was during that fourth symposium in Pittsburgh, after initial discussions with Srawley as

the representative of the ASTM E 24 Committee administration, that a meeting of about five

of us was held suggesting that the symposium become an ASTM Committee E 24 sponsored

event We adopted a written set of ground rules most of which are now lost and forgotten, but

adequately followed in spirit They included:

1 The National Symposium would remain a research symposium with some emphasis on

only advanced applications

2 The symposium was to be most often held in an academic environment with occasional

meetings held at ASTM headquarters if facilities permitted

3 The selected chairman along with the permanent organizing committee would select the

papers for presentation

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12 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

4 Papers accepted for presentation were to be judged on their full manuscripts except for

special invited lectures

5 ASTM agreed to publish the papers in the STP format subject to ASTM review

requirements

6 The symposium was to have its own special ASTM account with a perpetual carryover of

surplus funds entrusted to the next chairman (the first three symposia generated a small

surplus and the fourth was also anticipated to do so at that time.)

7, All costs except for the STP publication were to be borne by the symposium funds

8 The current chairman would be allowed to set the registration fee (with the approval only

of the permanent organizing committee) and was encouraged to solicit outside funds as

he felt appropriate

9 Others of lesser importance

For the large part, at least the spirit of these rules was followed to ensure that the symposium

remain a real research forum Its longevity is one measure of its success

Along with accepting the rules, the group invited Professor H Corten to hold the fifth

symposium at the University of Illinois in 1971 and tentatively set the sixth for ASTM head-

quarters in 1972 which eventually occurred Sometime in the mid 1970s, Jerry Swedlow became

the Chairman of the Permanent Organizing Committee

The symposia occurred annually until 1977, a year in which the less frequently held Inter-

national Congress of Fracture occurred with Canada as host country It was thought tactful to

skip a year with the national symposium; Jerry Swedlow was involved with both organizing

committees However, the fast pace of progress in elastic-plastic fracture mechanics (EPFM)

research led to the organization of a special ASTM symposium on that subject at the Atlanta

meeting in November 1977 with Landes, Begley, and Clarke as Chairmen It was an outstanding

meeting, as shall be proven later herein It led to the second and third such special EPFM

symposia in 1981 and 1986, as well as a closely related (and STP documented) conference/

workshop on EPFM testing which occurred at the Spring 1982 ASTM E 24 Committee meetingl

Meanwhile, the National Symposium itself has occurred annually since the interruption in 1977,

except for a similar interruption in 1989

A summary of all of these related symposia, including their location, dates, and hosts (Chair-

men) are given in Table 1, as well as the ASTM STP numbers for the proceedings of each

Measurement of Progress in Fracture Mechanics

The National Symposium on Fracture Mechanics series was originally motivated and con-

tinues to be a forum for progress in the field However, "progress" is a nice sounding word

meaning different things to different people The series has been self-perpetuating and self-

sustaining financially for 25 years, and it has produced STPs whose sales assist ASTM, so it

is surely a "success" (at the least a long succession) However, "progress in research" here

shall be viewed as the development of good ideas and approaches technically which lead to

further discovery and developments The immediate reaction of those who know the Content

of the National Symposium series is positive of course, this series has been outstanding in this

regard, but how can we measure it and demonstrate the progress?

In recent years, there has been considerable concern about the perpetuation of research and

its funding that has not often enough produced results of future usefulness See, for example,

N e w s w e e k [36] of January 1991 and Science of December 1990 and January 1991 for reaction

in both layman and scientific sources These articles sensationalize the negative aspects of this

concern by considering the large number of research papers published which seem to be of no

further worth Their shocking conclusion is that substantially more than 50% have no worth!

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

TABLE 1 ASTM E-24 National symposia and related symposia

Resulting ASTM

Nat Symp I Lehigh Univ June 1967 P Pads and G Irwin

E.P.F.M.-I Atlanta/ASTM Nov 1977 J Landes, J Begley, and G 668

Clarke

N.S XVI Battelle/Columbus, OH Aug 1983 M Kanninen and A Hopper 868

TX E.P.F.M Ili Knoxville, TN Oct 1986 J Landes, J Merkle, A 995 I and II

Saxena and J Bassani

N.S XXI Annapolis, MD June 1988 J Gudas, J Joyce, and E 1074

Hackett

N.S XXIV Gatlinburg, TN June 1992 J Landes and D McCabe 1207

These articles are subject to the criticism that they look at " t h e hole not the d o n u t " with their

negative view They use as their measure of usefulness " c i t a t i o n s " of papers, that is, whether

a paper has been " c i t e d " as a reference in a future paper Their criteria o f no worth is no

citations within five years o f p u b l i c a t i o n

The Institute for Scientific Information (ISI) is, like A S T M , an organization located in Phil-

adelphia, which since before 1965 has compiled " c i t a t i o n s " of scientific research papers [39]

T h e y currently compile the cited references of the papers in over 5000 scientific journals

H o w e v e r , disappointingly ISI compiles the reference citations in books such as A S T M STPs

in a separate less well-known index, the Index to Scientific and Technical Proceedings (IST)

(The confusion caused by using two separate indexes and resulting disregard of these most

significant works has been pointed out to them.) H o w unfair! You all know that recent papers

in our STPs frequently reference key earlier work in the E 24 symposium proceeding series

To begin remedying I S I ' s slight, a search through the references to all 852 papers was made

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

14 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

TABLE 2 Data on symposia proceedings/subsequent internal citations

Average Subsequent Citations

1988 969 939 47 (4)

1989 995 I&II 1112 53 (9)

1989 1020 701 34 (1)

1990 1074 618 31 (0)

a { 1 } through {7} indicate papers leading in subsequent citations within the series in terms of total

citations, respectively

Table 2 shows some of the data developed for each symposium Some have two STPs or

two volumes under one STP number RPS-1, the retrospective volume compiled by Barsom,

is also included for prospective In addition to the bulk quantities, the numbers of pages and

papers in each, the number of subsequent "citations," and references in later papers within

this symposium series show the relevance of earlier work to future progress This resulted in

the column labeled "subsequent internal citations." Of course, the numbers dwindle with the

years, since later volumes have less subsequent time in which to be cited Indeed, the columns

from the fourth National Symposium, STP 513 and 514 from 1972 and the first Elastic-Plastic

Fracture Mechanics Symposium, STP 668 from 1979 have quite astounding records The papers

in these symposia received 254 and 332 later citations, an average of about 10 per paper just

within this series (without counting in I S I ' s base of 5000 plus other journals) For comparison,

a column was added to Table 2 indicating the average number of citations per paper within the

volumes It is observed that symposia with about 30 papers do the best over the period 1965

through 1983 (A dashed line has been added to the table after 1983, since subsequent volumes

may not have had enough time to accumulate a significant number of citations.)

A final column in Table 2 gives the maximum number of internal subsequent citations in

this series received by an individual paper at that symposium Some of these papers seem to

be landmark papers receiving sometimes more citations than other whole books An outstanding

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

TABLE 3 The sequence of highly cited papers

and the J-R curve

and R-curve testing

J-R curve for crack stability

single test records

beyond J- controlled growth

example is a pair of back-to-back papers in STP 513-514 from 1972 by Begley and Landes,

alternating first authorship Considering the pair to be a single work leads to a citation total of

87 plus 49 or 136 total citations within the series, that is more than half the citations for that

symposium The next symposium in S T P 536 from 1973 contains a paper by J R Rice that

has by itself 76 citations within the series Those contributions are annotated { 1 } and {2},

respectively, in Table 2 along with other similarly meritorious papers, {3 } to { 7 } in the order

of total citations The first author's last name is added in parentheses in Table 2

Those landmark papers, ignoring some others of possible equal quality with apologies to

their authors, will help to demonstrate the point of this discussion Table 3 shows some further

information on those special papers, labeled { 1 } to {7 } in Table 3 As noted (chronologically)

in Table 3, those papers { 1 } to {7} are a series on a single subtopic, that is, monotonic slow

loading elastic-plastic fracture analysis: distinct steps in the progress of elastic-plastic fracture

mechanics research Of course, many other papers within this series also made significant

contributions Finally, to add perspective to the observed internal citations, for these seven

papers their total ISI external citations [39] are noted just for the period 1986 through 1990

The obvious similarities in the two citation rate columns in Table 3 reinforce the numbers as

a measure of interest and progress Again with apologies to other authors, it was quite impos-

sible to tabulate all of the ISI data for the 852 papers in this ASTM series

The building of the research progress in elastic-plastic fracture mechanics within this series

has been clarified Other subtopics such as subcritical crack growth, dynamic fracture tough-

ness, time-dependent (slow) effects, and so on have also been documented and developed in

this series (sometimes also in other A S T M STPs as well) Perhaps including here the elastic-

plastic fracture symposia created an unfair bias toward other subtopics Nevertheless, the ulti-

mate objective is accomplished of showing that the research within this A S T M symposium

series is extremely relevant and progressive step by step in developing advanced understanding

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

16 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

Some Comments about the Authors Contributing to These Symposia

Over the 25 years of these A S T M fracture mechanics symposia, more than 500 individual first authors contributed and most often presented their papers at the meetings In the " g o o d old days," if you knew what less than 20 people were doing, your knowledge was compre- hensive Of the 500 authors, many authored or coauthored other papers in the series Indeed, there are 55 who authored or coauthored 5 papers or more in the series The group narrows quickly to 25 authors with 8 or more papers

Looking over that list of the 25 most prolific authors, it is surprising to find 2 from England (both with 10 or more) who regularly contribute At least 7 were or are associated with Lehigh University, and 6 all simultaneously worked at one time within E T Wessel's group at West- inghouse Research, including both of your Cochairman at this sypoisium It is further observed that Table 3 had a distinctive Westinghouse, Harvard, and Lehigh flavor with most names associated with two of these organizations Not far behind is the U.S Navy with 4 of these prolific authors Of course the regularity of this group adds greatly to the continuity of the series, and they are to be especially lauded for their extended contributions

With a closing bit of levity, the unofficial champion author of this symposium series is John Landes He wins both on quantity, see Table 3, and on quality, 31 papers, but only if the 10 papers he has contributed in the symposia proceedings of 1990 through 1992 are counted In this context, John Landes is recognized as truly "outstanding in his field."

Finally, this discussion has attempted to draw your attention to the value of this A S T M symposium series and its proceedings Certainly, Jerry Swedlow and John Srawley, among others, who would be with us if they could, would be equally concerned with quality and progress in fracture mechanics research at these meetings and beyond

A c k n o w l e d g m e n t s

A modest fund was set aside by Washington University to enable the computerization of information on the contents of the books [1-27] resulting from this symposium series That task was entirely performed by undergraduate assistants, J Kistler and T Wilson, whose lengthy and timely efforts are gratefully acknowledged The typing of the manuscript, and essential reminders to produce it, are the key contributions of my wife Tina, with our thanks Finally, Dency Kahn of the Washington University Libraries provided key information on the Science Citation Index searches

References

[1] Fracture Toughness Testing and Its Applications, ASTM STP 381, 1955

[2] Stress Analysis and Growth of Cracks, Proc of the 1971 National Symposium on Fracture Mechanics

[3] Fracture Toughness, Proc of 1971 NSFM~Part II, ASTMSTP 514, 1972

[4] Progress in Flaw Growth and Fracture Toughness Testing, Proc of 1972 NSFM, ASTM STP 536,

1973

[5] Fracture Toughness and Stow Stable Cracking, Proc of 1973 NSFM Part I, ASTM STP 559, 1974

[6] Fracture Analysis, Proc of 1973 NSFM Part II, ASTM STP 560, 1974

[7] Mechanics of Crack Growth, Proc of the 8th (1974) NSFM, ASTM STP 590, 1976

[8] Cracks and Fracture, Proc of 9th (1975) NSFM, ASTM STP 601, 1976

[9] Flaw Growth and Fracture, Proc of 10th (1976) NSFM, ASTM STP 631, 1977

[10] Elastic-Plastic Fracture, Proc of the Elasto-Plastic Fracture Symposium (1977), ASTM STP 668, J

Landes, J Begley, and G Clarke, Eds., 1979

[11] Fracture Mechanics, Proc of 1 lth (1978) NSFM Part I, ASTM STP 677, C W Smith, Ed., 1979

[12] Fracture Mechanics Applied to Brittle Materials, Proc of 1 lth (1978) NSFM Part II, ASTM STP

678, S W Friedman, Ed., 1979

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[13] Fracture Mechanics, Proc of 12th (1979) NSFM, ASTM STP 700, 1980

[14] Fracture Mechanics, Proc of 13th (1980) NSFM, ASTM STP 743, R Roberts, Ed., 1981

[15] Fracture Mechanics Theory and Analysis, Proc of 14th (1981) NSFM, ASTM STP 791, Vol I, J

Lewis and G Sines, Eds., 1983

[16] Fracture Mechanics Testing and Applications, Proc of 14th (1981) NSFM, A STM STP 791, Vol II,

J Lewis and G Sines, Eds., 1983

[17] Elastic-Plastic Fracture, Proc of 2nd (1981) EPFM Symp., Vols I and II, ASTM STP 803, F Shih

and J Gudas, Eds., 1983

[18] Fracture Mechanics, Proc of 15th (1982) NSFM, ASTM STP 833, R Sanford, Ed., 1984

[19] Fracture Mechanics, Proc of 16th (1983) NSFM, ASTM STP 868, M Kanninen and A Hopper,

Eds., 1985

[20] Elastic-Plastic Fracture Mechanics Technology, Workshop on EPFM Technology (1983), ASTM STP 896, J Newman and F Loss, Eds., 1985

[21] Fracture Mechanics, Proc 17th (1984) NSFM, ASTM STP 905, Underwood, Chait, Smith, Wilhem,

Andrews and Newman; Eds., 1986

[22] Fracture Mechanics Retrospective, Early Classic Papers, ASTM RPS 1, J Barsom, Ed., 1987 [23] Fracture Mechanics, Proc 18th (1985) NSFM, ASTM STP 945, D Reed and R Reed, Eds., 1988 [24] Fracture Mechanics, Proc of 19th (1986) NSFM, ASTM STP 969, T Cruise, Ed., 1988

[25] Non-Linear Fracture Mechanics, Proc of 3rd (1986) Syrup on EPFM Vols I and II, ASTM STP

995, Landes, Saxena, Bassani and Merkle, Eds., 1988

[26] Fracture Mechanics, Proc of 20th (1987) NSFM, ASTM STP 1020, R Wei and R Gangloff, Eds.,

[32] Martin, D E and Sinclair, G M., "Crack Propagation Under Repeated Loading," in Proceedings

of the Third U.S National Congress of Applied Mechanics, June 1958, pp 595-604

[33] The American College Dictionary, Random House, New York, 1967

[34] Johnson, H H and Willmer, A M., in Applied Materials Research, Vol 4, 1965, p 34

[35] Engineering Fracture Mechanics, Vol 1, No 1, June 1968

[36] Begley, S., "Gridlock in the Labs," Newsweek, 14 Jan 1991, p 44

[37] Hamilton, D., "Publishing by and for the Numbers," Science, 7 Dec 1990, pp 1331-1332 [38] Hamilton, D., "Research Papers: Who's Uncited Now," Science, 4 Jan 1991, p 25

[39] The Science Citations Index, The Institute for Scientific Information, Philadelphia, published annually

(for data herein) 1986, 1987, 1988, 1989, 1990

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Constraint Issues

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

Two-Parameter Fracture Mechanics" Theory and Applications

REFERENCE: O'Dowd, N P and Shih, C F., "Two-Parameter Fracture Mechanics: The- ory and Applications," Fracture Mechanics: Twenty-Fourth Volume, ASTM STP 1207, John

D Landes, Donald E McCabe, and J A M Boulet, Eds., American Society for Testing and Materials, Philadelphia, 1994, pp 21-47

the full range of high- and low-triaxiality crack tip states The two parameters, J and Q, have distinct roles: J sets the size scale of the process zone over which large stresses and strains develop, whereas Q scales the near-tip stress distribution relative to a high-triaxiality reference stress state An immediate consequence of the theory is this: it is the toughness values over a range of crack-tip constraint that fully characterize the material's fracture resistance It is shown that Q provides a common scale for interpreting cleavage fracture and ductile tearing data, thus allowing both failure modes to be incorporated in a single toughness locus

The evolution of Q, as plasticity progresses from small-scale yielding to fully yielded condi- tions, has been quantified for several crack geometries and for a wide range of material strain hardening properties An indicator of the robustness of the J-Q fields is introduced; Q as a field parameter and as a pointwise measure of stress level is discussed

KEYWORDS: constraint, stress triaxiality, elastic-plastic fracture, fracture toughness, crack initiation, cleavage, ductile tearing, J integral, finite element method

A two-parameter fracture theory can be motivated by considering the progression of plastic states as loading on a cracked body is increased At low loads, the near-tip stresses and defor- mations evolve according to a self-similar field, scaled by Rice's J integral [1] This field, characterized by a high level of stress triaxiality, also describes the evolution of the near-tip stresses and deformations in certain crack geometries as plastic flow progresses from well- contained yielding to large-scale yielding Although this high-triaxiality field is one of m a n y possible states that can exist under fully yielded conditions, it is the only field that has received careful study until recently W h e n the high-triaxiality field [2-5] prevails over distances com- parable to several crack-tip openings, J alone sets the near-tip stress level and the size scale of the zone of high stresses and deformations Considerable efforts have been directed at estab- lishing, for different crack geometries, the remote deformation levels that ensure that the near- tip behavior is uniquely measured by J [6,7] The end result is a framework, based on J and the high-triaxiality crack-tip field, for correlating crack growth over a range of plane strain yielding conditions (see review articles by Hutchinson [8], Parks [9]) and for relating critical values of the macroscopic parameter J~c to fracture mechanisms operative on the microscale (see review article by Ritchie and Thompson [10])

Arguments that a single parameter might not suffice to characterize the near-tip states of fully yielded crack geometries have been raised by McClintock [I1] He noted that nonhard-

1 Lecturer in mechanical engineering, Department of Mechanical Engineering, Imperial College of Sci- ence, Technology & Medicine, London SW7 2BX, United Kingdom

2 P~ofessor of engineering, Division of Engineering, Brown University, Providence, RI 02912

Copyright9 by ASTM International

21

www.astm.org Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

22 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

ening plane strain crack-tip fields of fully yielded bodies are not unique but exhibit levels of

stress triaxiality that depend on crack geometry Although high stress triaxiality is maintained

in geometries involving predominantly bending over the uncracked ligament, the level of crack-

tip stress triaxiality in geometries dominated by tensile loads generally decreases as yielding

progresses into the fully plastic state (see Refs 6 and 7) Indeed, experimentally measured J-

resistance curves for center-cracked panels exhibit significantly higher slopes than those for

purpose of this paper to show that this viewpoint can be properly reconciled by a two-parameter

through full-field finite element calculations, that the J-Q fields dominate over physically sig-

nificant size scales, that is, they represent the environment in which the ductile and brittle

failure mechanisms are operative An approach based on higher-order asymptotics has been

Extending the analysis in Refs 17 and 18, Xia et al [20] have obtained up to five terms in the

asymptotic series and showed that the collective behavior of the series is consistent with the

J-Q field

An alternative two-parameter approach based on J and the elastic T stress has been advocated

by Beteg6n and Hancock [21], A1-Ani and Hancock [22], Du and Hancock [23], Parks [24],

Hancock et al [25], and Wang [26,27] Under circumstances where it is applicable, the J-T

theory can be shown to be equivalent to the J-Q theory This is discussed in the section on

small-scale yielding The toughness scaling approach of Dodds et al [28] can also be shown

to be consistent with the J-Q theory (see Kirk et al [29]) Cleavage toughness data interpreted

by J-Q theory are presented in Refs 29 and 30

The J-Q Theory

Consider a cracked body of characteristic dimension L loaded remotely by a stress denoted

yield stress It can be shown from dimensional grounds that, when L >> JhYo, all near-tip fields

are members of a single family of crack-tip fields Each member field is characterized by its

level of deformation as measured by J/~o and its level of crack-tip stress triaxiality, as measured

by Q, which also identifies that field as a particular member of the family For example, the

self-similar solution of Rice and Johnson [4] and McMeeking [5] or the Hutchinson, Rice, and

Rosengren (HRR) field (Refs 2 and 3) can be taken as the Q = 0 member field In short, the

Q family of fields provides the proper characterizing parameters for the full range of near-tip

fracture states

The weak coupling between deformation and stress triaxiality in a plastically deforming

material provides another argument in favor of a two-parameter description of near-tip states

Because plastic flow is incompressible, the superposition of a purely hydrostatic stress state

induces only an elastic volume change Consider a plastically deforming material element in

the forward sector of a crack as depicted in Fig 1 We can superpose a hydrostatic stress Q~o

with little or no effect on the deformation state It follows that near-tip deformation and stress

triaxiality cannot be scaled by a single parameter such as J A second parameter is required to

quantify the level of crack-tip stress triaxiality Clearly this argument does not apply to the

back sector because traction-free conditions must be satisfied on the crack faces However, this

is of no physical consequence because the fracture processes occur in the forward sector, which

is therefore the region of interest

A size scale must enter into the fracture description In this paper, we focus on fields ahead

of the crack that are relevant on the scale of the crack opening displacement ~,, or J/%, rep-

resenting the environment in which the failure mechanisms are operative

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FIG 1 Schematic illustrating the necessity for a hydrostatic stress parameter and a deformation

parameter to characterize the full range of near-tip states in the forward sector

Here 8,~ is the Kronecker delta; r and 0 are polar coordinates centered at the crack tip with 0

= 0 corresponding to a line ahead of the crack as shown in the insert to Fig 2

Fields of different crack-tip stress triaxialities can be induced by applying different levels of

T/~r o From dimensional considerations, these fields can be organized into a family of crack-tip fields of the form:

where J is Rice's J integral [1] That is, the load parameter T/tr o provides a convenient means

to investigate and parameterize specimen geometry effects on near-tip stress triaxiality under conditions of well-contained yielding Indeed, such studies have been carded out by Beteg6n

and Hancock [21], Bilby et al [32], and Harlin and Willis [33] Nevertheless, the result in Eq

2 cannot have general applicability under large-scale yielding because the elastic solution (Eq 1) upon which the T-stress is defined is an asymptotic condition that is increasingly violated

as plastic flow progresses beyond well-contained yielding

Recognizing the above limitation, O ' D o w d and Shih [13,14], henceforth referred to as OS,

identified members of the plane strain family of fields by the parameter Q, which arises naturally

in the plasticity analysis OS write:

t r l j = t r ~ j , 0 ; Q , % = eoga , 0 ; Q , u i = - - h i , 0 ; Q

O" o

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2 4 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

The additional dependence o f f j , glj, and hi on dimensionless combinations of material param- eters is understood The form in Eq 3 constitutes a one-parameter family of self-similar solu- tions, or, in short, a Q family of solutions The annular zone over which Eq 3 accurately quantifies the actual field is called the J-Q annulus Representative distributions of the Q family

of fields are presented in Fig 4 of Ref 13 and Fig 1 of Ref 14

Difference F i e l d a n d N e a r - T i p Stress Triaxiality

Using the modified boundary layer formulation, and considering a piecewise power law hardening material, OS generated the full range of small-scale yielding plane strain solutions, Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015

FIG 2 (cont.) Mean stress reference fields: (c) small strain (d)finite strain

designated by (~ij)ssv OS considered the difference field defined by

where (O'Ij)HRR is the HRR field They systematically investigated the difference field within the forward sector, [01 < 7r/2, of the annulus J/tr o < r < 5J/%, because this zone encompasses the microstmcturally significant length scales for both brittle and ductile fracture (see Ref 10)

Remarkably, the difference field in the forward sector displayed minimal dependence on r Noting this behavior, OS expressed the difference field within the forward sector as

C o p y r i g h t b y A S T M I n t ' l ( a l l r i g h t s r e s e r v e d ) ; S a t D e c 1 9 2 0 : 0 1 : 4 0 E S T 2 0 1 5

26 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

where the angular functions 6ij are normalized by requiring 6"0o(0 = 0) to equal unity More-

over, the angular functions within the foward sector exhibit these features: ~rr ~ (~00 ~ constant

Thus the difference field within the sector, 101 < and J/tr o < r < 5J/~o, correspond

effectively to a spatially uniform hydrostatic stress state of adjustable magnitude (i.e., ((rlj)d~fr

= Q(ro~j ) Therefore, Q defined by

O-co - - ( O ' 0 o ) H R R

(3" o

is a natural measure of near-tip stress triaxiality, or crack-tip constraint, relative to a high-

triaxiality reference stress state In words, Q is the difference between the actual hoop stress

and the corresponding HRR stress component, the difference being normalized by (to For

definiteness OS have evaluated Q at r = 2J/Cro; however, OS point out that Q is effectively

independent of distance The distance chosen for the definition of Q lies just outside the finite

strain blunting zone so that Q from a small and finite strain analysis should be nearly the same

OS also considered the difference field whereby the standard plane strain small-scale yielding

solution (crlj)SSV;T= o, which is driven by K alone, serves as the reference solution, that is,

(O'ij)dif f = ( O ' i j ) S S Y - - ( O ' / j ) S S Y ; T _ O (7)

In this case, the difference field in the forward sector matches a spatially uniform hydrostatic

stress state even more closely Thus, an alternative definition of Q is

where tr,, is the hydrostatic stress OS have calculated Q based on the hoop stress (Eq 8) and

the mean stress (Eq 9) for the full range of T stresses and several finite width geometries OS

have found that the difference between Q and Qm is always less than 0.1 Although the values

of Q presented in this paper are calculated from the hoop stress by way of Eq 8, it is clear from

the above that these Q values can be used to" calculate the corresponding hydrostatic stress

levels

Difference Field and Higher-Order Terms o f the Asymptotic Series

The connection between the difference field and higher-order terms of the asymptotic series

can be understood in the context of the MBL formulation Here the stress field obeys the

functional form

which also should apply to finite-width crack geometries as long as the characteristic crack

dimension L is sufficiently large compared to J/{r o Now, if one assumes a product dependence

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on the first argument in Eq 10 and works within deformation plasticity theory and an elastic

power-law hardening material, then one obtains a series in r/(J/tro):

/ _ _ \ < + i ) second order t e r m + h i g h e r order terms

Difference Field where

% = a reference strain,

r = a material constant (for a piecewise power-law material r = 1), and

In = an integration constant

By definition, the asymptotic series beyond the first term is equivalent to the difference field

because (see previous section)

[~ij = (O'/j)HRR + D i f f e r e n c e F i e l d (12) The HRR field and the second-order term provides only a two-term approximation to the

solution for the MBL problem, and this point appears not always to be understood

They have obtained a five-term expansion for the series in Eq 11 for n = 3 and four-term

expansions for n = 5, 7, and 10 Furthermore, they have successfully matched the four-term

series to the radial and angular variations of the difference field given in Figs 3 and 5 in Ref

13 for an n = 10 material Indeed, in the forward sector 101 < the collective behavior of

the second-, third-, and fourth-order terms is effectively equivalent to a spatially uniform hydro-

static stress state This observation together with the discussion of the previous section supports

the following approximate form for the near-tip fields:

Furthermore, note that an admissible range of stress states for an elastic-perfectly plastic

material can be written as

O'lj = (O'ij)Prandtl + Q%aij, 101 < ax/4

Difference Field

(14)

where (~rij)pr~nd,~ designated the Prandtl slip-line solution and again the difference field corre-

sponds simply to a uniform hydrostatic stress state scaled by Q (see Refs 14 and 23)

Variation of Q with Distance

Because Q scales the difference field relative to a reference stress state, it provides a sensitive

measure of the evolution of near-tip stress triaxiality in finite width cracked bodies It also can

be used to detect changes in the stress triaxiality that deviates from the pattern that develops

under MBL loadings For this purpose, we consider Q(i:) defined by

O" o

where ? ~ - r/(J/ffo) Note that (a0o)ssY;T=O is chosen as the reference field

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28 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

2 J h r o - - a h e a d o f the c r a c k tip

Reference Field Distributions

T a b l e 1 p r o v i d e s the r e f e r e n c e field distributions for a b r o a d r a n g e o f n values F o r com- pleteness, we h a v e i n c l u d e d the h o o p stress d i s t r i b u t i o n s a c c o r d i n g to the H R R singularity a n d the small-scale yielding solutions for small strain a n d finite strain ( p i e c e w i s e p o w e r - l a w hard-

e n i n g material; E/tr o = 500, v = 0.3) F i g u r e 2 presents the h o o p stress and m e a n stress reference fields e s t a b l i s h e d b y the M B L f o r m u l a t i o n with T = 0 T h e original studies o f OS were b a s e d o n a finite strain f o r m u l a t i o n to e n s u r e a full description o f the near-tip states Our

s u b s e q u e n t studies h a v e s h o w n that small and finite strain analyses p r o v i d e essentially identical

TABLE 1 Reference stress distributions, o-oo/o-o, from HRR field and small and finite strain boundary layer solutions

Q results over the region of interest 1 < ? < 5 This can also be seen by comparing the finite strain and small strain distributions in Fig 2

Two reference fields have been proposed, (~00)HRR and (Cr0O)SSV;T= o Our numerical investi- gations of different crack geometries show that, when the small-scale yielding solution is chosen

as the reference state, the difference fields correspond more closely to a uniform hydrostatic stress state over a greater range of plastic deformation However, the choice of reference dis- tribution used in the definition of Q remains a matter of convenience We emphasize that once

a choise is made, it must be applied consistently throughout the analysis We recommend that

Eq 8 be used as the standard definition for Q with the small strain solution as the reference field Having a standard definition facilitates the comparison of solutions obtained by different investigators and the tabulation of a handbook of Q solutions

Although we have limited our discussion to a piecewise power-law hardening material, the

J-Q theory is independent of the form of the material's constitutive relation For example,

(cr00)ssv;T=0 can be evaluated for an actual stress-strain relation Of course, for consistency, the analyses in the fracture application should also use the same stress-strain relation

Engineering Applications o f the J-Q Theory

For engineering applications, two forms of the near-tip plastic states are proposed:

% = (%)H + Q~o~ij and

(17)

where Q in Eqs 17 and 18 are defined by Eqs 6 and 8, respectively

The values of the hoop stress of the HRR field for 1 < F < 5 is given in Table 1 The other

of the small-scale yielding field with T = 0 are given in Table 1 and Fig 2 More details are found in Refs 13 and 14

The physical interpretation of Eqs 17 and 18 is this: negative (positive) Q values mean that the hydrostatic stress ahead of the crack is reduced (increased) by Qo" o from the J-dominant stress state, or the standard small-scale yielding stress state This interpretation is precise when

IO'l < 1

As stated previously, we recommend the use of Eq 18 in the J-Q fracture methodology

However, the explicit representation in Eq 17 can facilitate approximate analyses leading to predictions of constraint effects on toughness as outlined in the section on a cleavage toughness locus The Q values presented in this paper are based on the definition in Eq 8

30 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

with an additional weak dependence on E/(r o and v, where E is Young's modulus and v is

Poisson's ratio Curves of Q versus T/(r o for n = 3, 5, 10, 20, and oo are displayed in Fig 3

These Q values, based on the definition in Eq 8, were determined by small strain analyses,

using El(r o = 500 and v = 0.3; essentially identical results were obtained from finite strain

analyses It can be seen that Q increases monotonically with T/go Also note that crack-tip stress

triaxiality can be significantly lower than the reference state (the Q = 0 state) but cannot be

elevated much above it The values Q and Q ' are given in Table 2 retaining only two places

beyond the decimal point Note that the largest value of Q ' is less than 0.04 Thus Q is effec-

tively constant over the distance 1 < ? < 5 for all MBL loadings The behavior of Q ' for

finite width geometries is discussed later

The curves in Fig 3 can be closely approximated by

The values of a ], a 2, and a3, obtained by least squares fitting, are listed in Table 3 for several

n values We have explored several other values of E/(r o and v and found that the effect on the

T with near-tip hoop stress, also can be rearranged into the form of Eq 20

To facilitate the use of Eq 20 in the analysis of finite width geometries we have provided

normalized values of the stress intensity factor K, F ( a t W ) , and the T stress, hr(alW) and E(a/

and Radon [36] The tabulated values allow us to calculate Q in these geometries under con-

tained yielding

J-T a n d J-Q A p p r o a c h e s

Two approaches to specifying families of Mode I plane strain elastic-plastic crack-tip fields

have been proposed The first approach, suggested by Hancock et al [25], uses the elastic T

3 T L Sham, private communication, manuscript in preparation

TABLE 2 Values of Q and Q 'for several values ofT/o" o

conditions OS propose to quantify near-tip constraint using the J-Q theory, which has a strong

theoretical basis as discussed earlier

Within the M B L formulation a description of near-tip states by J and Q is equivalent to that phrased in terms of K and T because Q and T are related by Eq 19 and J and K are related through

section on finite width geometries, show that the J-T approach overestimates the actual stress

triaxiality for some geometries and underestimates it in other cases so that there is not a con-

sistent trend Stated in another way, a T-stress fracture methodology could be conservative for some geometries and nonconservative in others this suggests that such an approach may be impractical

TABLE 3 Polynomial expression for Q in terms of T-stress

32 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

T A B L E 4 Values of K, T and X for CCP, DECP, and TPBB

3, (d) n = 5; J normalized by remaining ligament

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34 FRACTURE MECHANICS: TWENTY-FOURTH VOLUME

FIG 6 -Center-cracked panel evolution of Q with increasing J a/W = 0.05, 0.1, 0.2, 0.3, 0.4,

and 0.5, (a) n = 10, (b) n = 20; J normalized by crack length, a/W = 0.5, 0.6, 0.7, and 0.8, (c) n

= 10, (d) n = 20; J normalized by remaining ligament

In the figures, J is normalized by the crack length a when a/W < 0.5 and by the remaining

ligament b when a/W > 0.5 Observe that the stress triaxiality decreases steadily with increasing

J and approaches a steady-state slope at fully yielded conditions

Figure 7a and 7c show the effect of strain hardening on Q for a short crack and a deep crack,

respectively For both geometries, the loss of stress triaxiality is greater in the lower hardening

materials

The variation of Q with distance is shown in Fig 7b and 7d Here, Q is evaluated at r/(J/

(to) = 1, 2, 3, 4, and 5 It can be seen that Q has only a slight dependence on r under fully

yielded conditions For the range of loading shown in Fig 7, IQ'I < 0.03 indicating that the J

and Q are accurate descriptors of the field over distance 1 < r/(J/(ro) < 5

In Fig 7b and 7d OS also provide a comparison between the actual stress triaxiality and the

prediction by the T-stress by way of Eq 20 The open circles in Fig 7b and 7d are the T-stress

predictions, and the solid lines are the actual near-tip triaxiality already noted above At low

loads, Eq 20 predicts the evolution of near-tip stress triaxiality accurately However, at fully

yielded conditions, the stress triaxiality is incorrectly predicted In the case of a l W = O 1, T

underestimates the stress triaxiality by about 0.5(to For a deep crack a / W = 0.8, T overestimates

the stress triaxiality by a similar amount

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FIG 7 ~ e n t e r - c r a c k e d panel Effect of n on the evolution of Q; (a) short crack, (c) deep crack

Q evaluated at r/O/o'o) = 1, 2, 3, 4, and 5 for n = 10; (b) short crack, (d) deep crack The open

circles are predictions based on the T stress

T h r e e - P o i n t B e n d B a r ( T P B B )

Solutions for Q for the three-point bend bar are shown in Figs 8 and 9 for 0.05 -< a / W <

0.8 The behavior of Q in shallow cracked specimens, a / W < 0.3, is similar to that seen for

the center-cracked panel, that is, the loss of stress triaxiality occurs gradually When the crack

is sufficiently deep, a / W > 0.3, high stress triaxiality is maintained for deformations charac-

terized by J/(a(ro), or J/(b(~o), less than about 0.01 At higher J levels, the global bending stress

field impinges on the near-tip region, r -~ 2J/(ro, causing a rapid loss of stress triaxiality This

occurs at about J/(b(ro) = 0.02 corresponding to a deformation level that is less than the A S T M

limit for a valid J~c test (Jl(b(ro) = 0.04)

Strain-hardening effects on the evolution of Q are displayed in Fig 10a, 10c, and 10e It

can be seen that the effect of strain hardening on Q is weak for deeply cracked bend bars, a / W

> 0.4

The actual Q values and the T-stress predictions are compared in Fig 10b, 10d, and 10f It

can be seen that T correctly estimates the stress triaxiality for the short crack geometry (a/W

= 0.1) but fails to predict the stress triaxiality under large-scale yielding in the long crack

geometries

Copyright by ASTM Int'l (all rights reserved); Sat Dec 19 20:01:40 EST 2015