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FPML - Fracture and

Photomechanics

LaboratoryInstitute of Mechanics, Vienna

University of Technology

The Struggle for Recognition of Engineering Fracture Mechanics

by H.P. Rossmanith

XV. EXTENSIONS OF LINEAR-ELASTIC FRACTURE MECHANICS

xtensions of the linear-elastic fracture mechanics (LEFM) concepts began toppear in the literature during the period 1960-1965 with the five most notable

xtensions related to:

1. Fatigue crack growth,

2. Stress-corrosion cracking,

3. Plasticity effects at the crack tip,

4. Dynamic fracture mechanics,5. Creep and visco-elastic fracture.

ach of these topics will be discussed in turn. Research in these fields was

imulated through Irwin's appointment as a Boeing Professor of Mechanics at

ehigh University during the period 1967-1972. The Lehigh connection is

ghlighted in the contributions by F. Erdogan et al. and R.W. Hertzberg in this

V.

XV.1 Fracture mechanics based approach to fatigue

ossibilities for using fracture mechanics approach for studies of fatigue crackin

ere recognised at NRL, but this topic was viewed as less important than the

haracterization of conditions governing abrupt crack initiation. Two members o

he engineering staff at Boeing-Seattle, W.E. Anderson and P.C. Paris (see the

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he interrelation between university, government institution and industry with

spect to fatigue research is exposed in the contribution by L.P. Pook in this AV

XV.2 Environmental assisted cracking

uring the 1930's it was noted that moisture assisted slow crack extension in gla

nd that moisture greatly reduced the force necessary for splitting of micaObreimoff 1930). These effects were explained as due to the action of the water

molecule in reducing cohesive type attractive forces between atomic size

omponents of the solid. Thus, there was a close relationship of these explanatio

the Griffith crack theory. In the Soviet Union, Rehbinder and his associates

udied and developed liquids which assisted the drilling and crushing of rock. T

Rehbinder effect' was interpreted in a similar manner and attributed to the

nfluence of water on fracture of glass. But this work was not very well knownutside of the Soviet Union until after World War II.

he post WW II work which led to the development of modern fracture mechani

egan with a modification of the Griffith theory idea in which the resistance to

rack extension was assumed to be plastic deformation at the leading edge of the

rack rather than solid state surface energy (Irwin 1948, Orowan 1949). It is of

nterest that Dr Arthur Ruark, one of the principal advisors of G.R. Irwin advoca

-programming of the existing NRL fracture studies toward use of the Griffith

heory in conjunction with the Rehbinder effect. Irwin's rejection (at that time) o

his advice was based on the fact, that at that time it was not clear that recognitio

f the dominating importance of plastic strain at the leading edge of a crack was

ompatible with the surface energy concepts of Rehbinder.

rior to 1957 the fracture control techniques for stress corrosion cracking were

eveloped entirely without reference to the synergistic effect of fabrication flawscting like initial cracks. Ships moored in sea water and underground steel piping

re usually protected by the system termed anodic protection which is based upo

ectro-chemical effects. In other cases, surface coatings which are inert to

orrosion may be applied by plating or oxidation. In the case of bridges, ship

uperstructures, and other outdoor metal structures, painting is regarded as

urnishing adequate protection of the metal surface from chemical attack. The



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tractiveness for the use of austenitic 'stainless' steels rests mainly on the well

nown inertness of these metal alloys to chemical attack. Normally they do not

quire protection from corrosion.

n the case of rotor coil support rings for turbine generator rotors the use of

ustenitic steel, primarily for its non-magnetic characteristics, appeared to provid

dditional benefits against stress corrosion cracking. Under normal conditions th

material does not show any evidence of chemical attack in the form of rusting or

tting, however, rings fabricated without cold expansion were shown to be

usceptible to stress corrosion cracking in moist conditions due to chlorides and

trides. The idea that stress corrosion cracking was a relative thing and would

ccur in a non-susceptible material due to a large enough starting crack was

nknown prior to 1960 and did not become widely known until about 1965.

uring the pioneering work of Paul C. Paris and coworkers at the Transport

ivision of the Boeing Company starting in about 1958 on stable crack growth d

fatigue loading, intensive studies were also made of fracture failures occurring

mainly during hydro-testing of ultra-high-strength steel solid propellant rocket

hambers. It was not realised at first, that many of these fractures were due to ve

mall prior cracks which grew to critical size for propagation during the slow

pplication of the hydro-test pressure or during the three minute hold period at thpecified testing pressure. A corrosion inhibitor had been added to the water used

or testing and there was no evidence of corrosive pitting of the steel.

he time dependent nature of these fractures was ascribed to action of hydrogen

ntroduced into the steel in high stress regions from the water. Studies of the

nfluence of testing environment on stress rupture failures of pre-cracked ultra-hi

rength steel specimens were reported by Steigerwald (1960). Studies of crack

rowth rates during stress corrosion cracking as a function of environment and o

alue were reported by Johnson & Willner (1965).

n 1965, at NRL, Brown & Beachem (1965) extended fracture mechanics concep

stress-corrosion cracking discussing a standard testing practice for rating the

ress corrosion cracking susceptibility of a metal containing a prior crack. They



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howed that a stress-corrosion cracking threshold,KIscc, exists, below which crac

xtension does not occur for a given material-environment system. Tests were

onducted employing pre-cracked cantilever beam specimens that were loaded in

mple frame. This testing arrangement simplified specimen preparation, elimina

he need for using large tensile machines, and permitted accurate detection of cra

rowth with time. The authors demonstrated that theKIsccdetermined fromantilever beam specimens were identical toKIsccdata obtained from surface-

awed tension specimens. This success prompted subsequent investigators to use

he cantilever beam specimen to studyKIscccharacteristics of steels and alloys

uring the late 1960's and 1970's.

he novelty and great potential value of these new procedures for the study ofress corrosion cracking were responsible for the establishment of an ARPA

nanced project (monitored by B.F. Brown) at the NRL, the Boeing Company,

ehigh University, Carnegie-Mellon Institute, and Georgia Institute of Technolo

n 1966. Although this project closed in 1970 with many aspects having been lef

nclear, certainly the enhancement of the danger of stress corrosion cracking by

rior crack will be better understood in the future (Irwin 1963).

n Europe, the effect of corrosive media on crack growth behavior was studied byM.O. Speidel (1971) and others. Work performed by researchers in the countries

he former USSR is published in the journalPhysico-Khimicheskaya Mechanica

Materialovsince 1965 (English translation in the USA under the title: Soviet

Materials Science).

XV.3 Plastic crack tip behavior

inear elasticity theory predicts that the stresses become infinite as a crack tip is

pproached. In real materials, however, the stresses are limited by the flow stress

nd a plastic zone develops at the end of the crack.

XV.3.1 Plastic zone size corrections

he first attempt to characterise the size of this region was presented by Irwin an



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s associates in 1958 (Irwin et al. 1958). The paper describes test methods for

measuring Gcusing laboratory specimens to simulate both plane-strain and plane

ress conditions. By simply equating the stress ahead of the crack equal to the

eld stress, Y, Irwin was able to estimate the size of the plastic zone at the cra

p as:

here rYis the plastic zone size andKis the stress intensity factor. Taking into

ccount the redistribution of the stresses in the crack tip zone, a later analysis

rovided a better estimate of the plastic zone extension by 2rYas compared to rY

While these simplified strip zone analyses give an estimate for the extent of theassumed strip-like) plastic zone, no prediction was made for the real shape of th

astic zone. In fact, Irwin's approximations for the plastic range were based on a

mit elasticity analysis without any reference to plasticity.

truly plasticity based analysis contributed by Hult & McClintock in (1957)

howed that the plastic region at the tip of a Mode-III (transverse shear) loaded

rack was circular. Irwin initially believed that the plastic zone for Mode-I loadinould be considered as a circle centered rYahead of the crack tip (as in the Mode

I solution), with Equation (4) representing the radius of the plastic region.

McClintock and Irwin met and became first acquainted at the IUTAM Conferenc

n Brussels in 1954 with a picture taken by I.N. Sneddon. McClintock had attend

nother conference on fatigue in London prior to coming to Brussels and he

hought that his Mode-III sophisticated model of flow near the crack tip was helpor finding a solution for the other modes too. Muskhelishvili, being also the

hairman of the session where Irwin presented his ideas, was one of the most

nteresting persons attending the conference and his complex analysis of stresses

as one of the main features of the conference.

lasticity at the crack tip was also addressed in the first report of the ASTM Spec



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ommittee (ASTM 1960). The report states that the presence of plasticity at the

rack tip causesKto be underestimated if no correction is employed. For

ngineering purpose and for contained plasticity at the crack tip (quasi-brittle

material behavior) the effect of a plastic zone in an elastic-plastic material can be

ken into account by extending the length of the crack slightly in a perfectly ela

material as:

here ais the true crack length in the elastic-plastic material, aeffis the effective

rack length in the perfectly elastic material, and rYis given by Equation (6).

quations (6) and (7) give the Irwin plastic zone correction.

XV.3.2 Crack opening displacement concepts

n 1963, Wells (1963) introduced an alternate concept for modeling the plastic zo

crack tips. The idea was based on a critical displacement at the crack tip, know

s the crack opening displacement (COD), which could be used to characterise

rack extension. Wells evaluated the COD by employing Irwin's rYestimate and

splacement equations for a center crack in an infinite elastic body to find thelationship:

hereEis Young's modulus. Wells believed that Equation (8) showed that the

OD criterion was consistent with LEFM and was applicable even beyond gener

elding, although this latter point was not proven. A later paper by Wells (1965)so based on the COD concept, demonstrated the correlation between fractures o

rge (wide-plate) and small (Charpy V-notch) test specimens, either of which

ould be related to service failures. The significance of this example was not to

evelop the precise form of the correlation between specimen and service behavi

ut rather to demonstrate that such correlations were both possible and accurate.

ince this initial attempt, many correlations between Charpy V-notch test results



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nd various fracture mechanics tests have been developed and proven useful

Barsom & Rolfe 1987). The development by Wells is now known as the crack-t

pening displacement (CTOD) criterion and is widely used to characterise elastic

astic and plastic fracture behavior.

s far as spreading of fracture mechanics in Europe is concerned, A.A. Wells an

.E. Turner have been extremely active with Wells having the backing of the

nternational Welding Society.

ndependent developments in the West by Dugdale (1960) and Panasyuk in the

ast (Leonov & Panasyuk 1959, Panasyuk 1960) introduced strip-yield models f

rack-tip plasticity. For a thin sheet loaded in tension, Dugdale as well as Panasy

nd his coworkers stated that yiedling was confined to a narrow band along the

rack line. This case is equivalent, mathematically, to placing internal stressesqual in magnitude to (Y at the end of the crack, which tend to close the crack an

move the stress singularity at the crack tip. Dugdale derived a formula for the

ngth of the plastic zone d at each side of the crack as:

xperiments with steel sheets having both internal and edge slits showed goodgreement with Equation (9). Also, the size estimate of dcompares favorably wi

he size estimate of 2rYfrom Equation (6). Despite the apparent success of the

redictions, Dugdale's approach was not immediately accepted because no

hotographic evidence of yielding was presented in the paper. An etching

chnique introduced in 1965 by Hahn & Rosenfield (1965) showed the size and

hape of local yielding at the crack tip of steel specimens containing edge cracks

he results indicated that the Dugdale model is useful for approximating the sizend shape of the plastic zone under plane stress conditions.

ther variations of the strip yield model appeared during the 1962-1965 period. I

962, Barenblatt (1962) published an extensive article concerning the equilibrium

f cracks in brittle fracture. He introduced the idea of molecular cohesive forces

cting over a small distance at the end of the crack, which also tended to remove



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he stress singularity and keep the crack closed. While this approach was similar

ugdale's model mathematically, the physical basis was on molecular cohesion

ather than macroscopic plasticity. Dislocation mechanics finally yielded a furthe

rip yield model, based on linear arrays of dislocations, which was presented by

ilby et al. (1963). The BCS model represents the distribution of elastic-plastic

rain in the yielded region by an inverted pile-up of dislocations. Subsequent

evelopment by the authors (Bilby & Swinden 1965) produced a model in which

astic relaxation occurs by dislocation pile-ups on slip planes inclined

ymmetrically from the slip plane. This modification is called the Bilby-Swinden

nclined strip yield model. Work in fracture mechanics performed in Eastern

urope at the same time period 1960-1965 is summarised and critically discusse

n the contribution by V.V. Panasyuk in this AV.

n contrast to the US where fracture research in the late 60's and early 1970's wasrimarily driven by the nuclear power industry (see next section), in the UK

acture research was stimulated and motivated by the exploration of the oil

serves in the North Sea.

ased on previous work by Wells (1963) on the CTOD, in 1971, Burdekin & Sto

966), Burdekin & Dawes (1971) from the Welding Institute developed the CTO

esign curve which is a semi-empirical fracture mechanics methodology for weldeel structures. It is interesting to note, that the British nuclear industry develope

heir own fracture design analysis (1980) which is based on the strip zone plastic

model developed by Dugdale (1960) and Barenblatt (1962). An appraisal of the

evelopment of fracture mechanics for brittle and ductile behaving materials is

ven in the contribution by J.F. Knott in this AV.

XV.3.3 Elasto-plastic fracture mechanics

n the 50's and early 60's plasticity mechanics was used in England in the design

nderground shelter, steel frameworks etc. to find out critical loads for buckling.

rager and Hotch published several books. Irwin, finding that plasticity ideas cou

e quite helpful in fracture mechanics, wrote to Prager for advice and the respon

tter by Prager said that Irwin's ideas were nice but too complicated and it was

bout 10 years before the technology he had could be applied to fracture. This w



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n 1958. However, with Prager at Brown University, the Navy got interested in

asticity and was putting quite some money to Brown University for studying

asticity, for it was a real big thing at that time.

orrespondence from the early 60's indicates that, while still at Brown Universit

m Rice received strong encouragement and help from G.R. Irwin, who wanted

ice to develop some better model of the plastic zone which would help to impro

he meaning of a fracture toughness test. Rice was aware of the developments an

though Irwin was not directly interested in Mode-III other than using it in the

acture mechanics short courses, Rice went further and succeeded in introducing

ork hardening (see also the second contribution by Paul C. Paris in this

nniversary Volume). By idealizing plastic deformation within the deformation

heory of plasticity as nonlinear elastic material, Rice was able to generalise the

nergy release rate concept to nonlinear materials and to show that this energylease rate was equivalent to an integral expression which he called J-integral. I

act, modeling with Mode-III, suggested to him the idea of path independent

ntegral as a method of calculating a quantity which is now called J-integral (J fo

m Rice). The results of this investigation Rice published in his famous 1968 pa

or which he received an award from ASTM. At the time of publication it becam

nown to him that Eshelby (1951) (see also Eshelby et al. 1951), a dislocation

xpert in England who was not directly working in fracture mechanics, hadreviously derived so-called conservation integrals of which the Rice-Integral w

special case. Although Rice should have referenced the work, Rice, however, d

ot need Eshelby's work when he applied his integral to represent the plasticity

ffects within the crack tip region. P. Paris, a promoter of J. Rice, conceived of th

dea that the J-integral could also be used very close to the zone of yielding. The

ok a notched bent specimen assuming that the stress across the ligament was

roportional to the area under the curve and give it a value for J (Rice et al. 1973

ests were performed at Westinghouse. The J-integral was a special feature of th

ational Symposium on Fracture Mechanics held in 1970 at the University of

linois, and it received a big boost when ASTM Committee E-24 indicated that i

ould be used as a further measure of characterizing toughness.

n two very interesting papers, Hutchinson (1968) and Rice & Rosengren (1968)

ndependently showed that J controlled crack tip conditions and related the J-



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ntegral to the crack tip stress fields in non-linear elastic materials. A new multi-

xial small strain theory based singularity, named after the three authors HRR-

ngularity, contains the same apparent anomaly as the LEFM counterpart. It was

ound, that the J-integral defines the amplitude of the HRR singularity in the sam

ay as the stress intensity factor K characterises the amplitude of the crack tip

ngularity in linear elastic materials. The HRR theory does not consider the effe

f crack tip blunting nor does it take into account the large strains which develop

he immediate crack tip vicinity. These effects were taken into account in a

umerical finite element analysis by McMeeking & Parks (1979).

he importance of this non-linear extension of J as a stress field characterizing

arameter was first fully conceived by the experts concerned about safe operatio

f nuclear power plants. Political and public awareness regarding safety of nucle

ower plants together with the active research efforts of the nuclear power indusn the USA in the early 1970's to integrate the state-of-the-art technology in react

esign, construction and maintenance, brought fracture mechanics into the field o

uclear engineering. Fracture mechanics as developed by 1971 could not easily b

pplied to nuclear pressure vessel grade steels as the enormous toughness of this

material did allow a rational description in terms of LEFM and acceptable testing

ould require large equipment. In 1971, Begley and Landes, two research

ngineers at Westinghouse, recognised the importance of Rice's 1968 paper andecided to apply the J-integral to characterise fracture toughness of these reactor

eels. Despite high level scepticism offered by their colleagues, Begley & Lande

972) succeeded with their experiments and a decade later a standard procedure

or J-testing was issued by ASTM (1981).

owever, a general relationship between toughness, flaw size and stress loading

asto-plastic material behavior, similar to the Griffith equation well established

EFM, was still missing. A fracture mechanics based design methodology becam

vailable with the theoretical framework worked out by Shih & Hutchinson (197

hich, a few years later, formed the basis of a fracture design handbook (EPRI

981) published by the Electric Power Research Institute (EPRI) in California. T

hih analysis (1981) which uses the HRR solution to evaluate displacements sho

hat there is a unique relationship between J and the CTOD for a given material,

ence, fracture toughness can be quantified in terms of a critical value of either J



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TOD.

Many materials characterised by high ductility (or toughness) do not fail in a

udden manner (often termed catastrophical). These materials exhibit a rising R-

urve where J as well as CTOD both increase with crack extension.

Whereas initiation toughness characterises and provides information about the

nset of stable cracking of ductile materials, J-controlled continuing stable crack

rowth stretches this concept to its limits when excessive plasticity or significant

rack growth occur. The conditions for application of J as a fracture characterizin

arameter beyond ductile initiation include the requirement that the HRR based

ress and strain relationships be valid in the process zone surrounding the crack

ont, where the microscopic physical processes that lead to fracture occur.

tability of ductile crack growth is determined by essentially the same condition

s in the linear-elastic case with load control being less stable than displacement

ontrol. Stability of real structures is determined by conditions varying between

ad and displacement control and the R-curve slope can be characterised by the

aring modulus concept (Paris 1979).

or elastic materials all relevant state variables and derived quantities arendependent of the loading history. In fracturing of real materials (elasto-plastic,

sco-plastic etc.), the energy absorbed during crack growth exhibits a history

ependence (incremental theory of plasticity). In addition, the J-R curve become

eometry dependent when crack growth occurs. In an effort to repair this

eficiency, Ernst (1983) proposed a modification of the deformation theory of

asticity. At present, the effect of specimen dimensions and loading on J-R curv

n ductile materials has not been quantified for ready engineering application and

ork is in progress.

he historical development of elasto-plastic fracture mechanics and the pertinent

sues associated with it are thoroughly presented and discussed in the

ontributions by C.E. Turner, J.D. Landes et al., J.M. Barsom, A.G. Atkins, and H

. Bui and A. Ehrlacher in this AV. The particular view on fracture research tak

om a national materials testing laboratory (EMPA in Switzerland) is given by R



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ieselbach in this AV.

XV.4 Dynamic fracture mechanics

he response of materials and structures to time-varying loading is different from

atics. When, in fracture processes, inertia effects become important, one is led t

he field of dynamic fracture mechanics. Dynamic fracture is intrinsically moreomplicated than static fracture mechanics because of three effects:

1. Inertia effects must be taken into account when either the load changes

abruptly or the crack movement occurs in a jump-like fashion;

2. Rate dependent material behavior may play an important role;

3. Reflected stress waves interfere with the rapidly propagating crack.

XV.4.1 Analytic and numerical work

arly work on dynamic fracture mechanics is summarised in the review papers b

rdogan et al. (1968) and Irwin & Wells (1965). The classical (though somewha

athological) problem of the constant-speed moving Griffith crack considered by

offe gave the first hint for the possibility of crack branching for running cracks

on-stationary crack problems were considered by Broberg (1960), Baker (1962

nd others (see the second contribution by B. Broberg in this AV). In hisoneering four part paper in theInternational Journal of the Mechanics and

hysics of Solids(1972-1976), L.B. Freund derived the fundamental solution for

atic and variable speed running cracks subjected to dynamic loading. Essential

ontributions to dynamic fracture were done in the field of numerical simulation

nd development of contour integrals (see the paper by T. Nishioka in this AV).

XV.4.2 Experimental work

rom a material characterization point of view, it was recognised that the toughn

f the material would change at higher crack propagation speeds. Dynamic crack

ropagation toughness could be measured as a function of crack speed by means

hotomechanics techniques, primarily photoelasticity and the method of caustics

onjunction with high speed photography. The Fracture Mechanics Group at the

niversity of Maryland advised by Prof. Irwin and a string of other laboratories



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ver the world employed dynamic photoelasticity in dynamic fracture mechanics

n effort to characterise the dynamic relationship between the crack speed and th

ress intensity factor during rapid crack propagation (Irwin et al. 1976) (see the

aper by A.S. Kobayashi in this AV). Other labs such as the one in Freiburg in

ermany developed and extensively used the method of caustics in their work on

ynamic fracture (Kalthoff et al. 1980). Most of these studies were motivated an

artially supported by the nuclear industry (in the USA the Nuclear Regulatory

ommission) and electricity generating companies (in the USA the Electric Pow

esearch Institute) pressed by growing concerns about crack arrest in pressure

essels and piping (see papers by C.W. Smith, W.L. Fourney et al., etc. in this A

XV.4.3 Crack arrest

n the 1970's it was noted that a staticKIcmeasurement for a structural steel mayometimes yield smaller values for the toughness than the value at crack arrest,K

his effect was attributed to crack front roughening with increasing crack speed

nd explained as an R-curve effect. Methods of rapid loadingKIctesting and KIa

valuation can be regarded as measurements of the minimum resistance of

ructural steel to crack extension. During the period 1970-90 several research

roups in the US, in Europe and elsewhere were heavily involved in characterizirack arrest and developing appropriate design codes and standards primarily for

he nuclear industry where the problem was addressed first. Crack arrest

methodologies and the important role of Irwin in this field are discussed in the

apers by W.E. Anderson, and A.R. Rosenfield in this AV.

XV.4.4 Creep and visco-elastic fracture

tructural components that operate at high temperatures may fail by slow stableeformation, so-called creep. Classical creep assumes uniform conditions for cre

occur, whereas in fracture mechanics creep is localised in the immediate vicin

f a crack where several hierarchical creep controlled deformation regimes form

he crack tip is embedded or even replaced by a tertiary process zone, contained

he steady state creep zone which in turn is embedded in the primary creep zone,

hich is engulfed by elastic material. As the material fails locally the tertiary



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rocess zone is the controlling zone on a microscopic level.

fter the J-integral was accepted as a fracture characterizing parameter, various

searchers (Landes & Begley 1976, Ohji et al. 1976, Nikbin et al. 1976) propose

creep version of the J-integral for crack growth in material undergoing steady

ate creep which became known as the C*-integral which characterises the crack

p fields in a viscos material for which the time-dependent growth rate depends

nly on the C*-integral. Experimental studies (Landes & Begley 1976, Riedel

989) compare well with the theoretical predictions provided steady state creep i

he dominant deformation mechanism and crack growth occurs at speeds slow

nough that the creep zone can spread throughout the structure before fracture

ollows. If crack growth becomes larger and eventually overtakes creep growth,

hen a C* characterization becomes invalid and the classical K-approach is

dequate. The transition zone from short time elastic to long time viscous behavias analyzed by Riedel & Rice (1980) by utilizing a simplified stress-strain law

nd neglecting primary creep.

s much of the fracture mechanics as was developed for steel and other metals

annot directly be applied to polymers, visco-elastic fracture mechanics requires

he incorporation of visco-elastic material response. Familiar with the concepts o

acture mechanics as well as with the molecular structure of polymeric material. Zhurkov contributed to the elucidation of the physical phenomena which take

ace at a crack tip in a stressed polymer by pointing out that the breakage of ma

molecule chain bonds plays a fundamental role in the fracture of polymers.

undamental to the development of visco-elastic fracture mechanics is the work

chapery (1975, 1984, 1990) who assumed a nonlinear visco-elastic constitutive

quation in the form of a hereditary integral and, by means of the well known

orrespondence principle, developed a generalised J-integral (Schapery 1984).

Major contributions to the field of visco-elastic fracture mechanics may be found

he book by J.G. Williams (1984).

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