classical fracture mechanics methods · classical fracture mechanics methods ... crack-tip opening...

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DE07FE537 GKSS fO-- ! I- 1 " H U N C>?. SNTP IJ.V in der HELMHOLTZ GEMEINSCHAFT Classical Fracture Mechanics Methods Authors: K.-H. Schwalbe J. D. Landes 1. Heerens HELMHOLTZ I GEMEINSCHAFT wissen scha/Tt nutzen GKSS 2007/14

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Page 1: Classical Fracture Mechanics Methods · Classical Fracture Mechanics Methods ... crack-tip opening displacement (CTOD) ... • If stable crack extension occurs during a test,

DE07FE537

GKSSf O - - ! I-1" H U N C > ? . S N T P I J . Vin der HELMHOLTZ GEMEINSCHAFT

Classical Fracture Mechanics Methods

Authors:K.-H. SchwalbeJ. D. Landes1. Heerens

HELMHOLTZI GEMEINSCHAFT

wissenscha/Ttnutzen

GKSS 2007/14

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Classical Fracture Mechanics Methods

K.-H. SchwalbeGKSS, Geesthacht, Germany

J. D. LandesUniversity of Tennessee, Knoxville, TN, USA

J. HeerensGKSS, Geesthacht, Germany

This article was originally written in the form of a chapter in:

Comprehensive Structural Integrity, Online UpdateFracture of Materials from Nano to Macro

ISBN: 0-08-043749-4 (Set)

New Online Volume 11ISBN 978-0-0804-3749-4

pg. 3-42.

Original Print Edition Editors: Milne, R.O. Ritchie, B. Karihaloo (Editors)

Online update available at: www.sciencedirect.com

© Copyright 2007, Elsevier Ltd., All Rights Reserved.

Comprehensive Structural Integrity is a reference work which covers all activities involved in the assurance ofstructural integrity. It provides engineers and scientists with an unparalleled depth of knowledge in thedisciplines involved. The scope covers all industries and technologies, from the massive offshore structures tothe miniscule biological structures, and includes consideration of heavy section structures, thin sheet structures,and structures at the nano scale. Volume 1 covers these issues in general, using examples and case studies to givepractical examples of how the disciplines are applied. Volumes 2 to 6 address the underlying theories andmethodologies, covering theoretical and computational methods, fatigue, environmental influences, and hightemperature effects. Volume 7 covers practical failure assessment methods, and addresses the assessment ofstructures which contain crack-like defects. Volumes 8 and 9 cover in turn, interfacial and nano-scale failure andthe treatment of structures engineered for bio-medical applications. A subject index is contained in Volume 10 ofthe print edition and the new online Volume 11 is dedicated to the mechanical characteristics of materials.

Comprehensive Structural Integrity provides a first point of entry to the literature for both the engineer andresearcher across the whole field of structural integrity.Comprehensive Structural Integrity is published by Elsevier, and this article is reprinted with the permission ofElsevier.

Information about Comprehensive Structural Integrity can be obtained fromwww.elsevier.com or www.sciencedirect.com

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11.02Classical Fracture MechanicsMethodsK.-H. SCHWALBE

GKSS, Geesthacht, GermanyJ. D. LANDESUniversity of Tennessee, Knoxville, 77V, USAJ. HEERENSGKSS, Geesthacht, Germany

11.02.1 INTRODUCTION 5

11.02.1.1 Purpose and Goals of Fracture Toughness Testing 5

11.02.1.2 Historic Development 5

11.02.2 TEST TECHNIQUES 7

11.02.2.1 General Requirements 111.02.2.2 Specimens and Fixtures 711.02.2.3 Test Machine 1111.02.2.4 Instrumentation and Requirements II

11.02.2.4.1 Instrumentation 1111.02.2.4.2 Instrumentation requirements 14

11.02.2.5 Crack-Length Measurement 1511.02.2.5.1 Determination of initial and final crack lengths 1511.02.2.5.2 Visual method 1511.02.2.5.3 Indirect methods 17

11.02.2.6 Conducting the Test 1911.02.2.6.1 Loading the specimen 1911.02.2.6.2 Recording 19

11.02.3 ANALYSIS 19

11.02.3.1 Introduction 1911.02.3.2 Linear-Elastic Fracture Toughness 20

11.02.3.2.1 Expressions for the stress intensity factor 2011.02.3.2.2 Limits of the applicability of the stress intensity factor 21

11.02.3.3 Elastic-Plastic Fracture Toughness 2111.02.3.3.1 ./-integral 2111.02.3.3.2 Crack-tip opening displacement 2211.02.3.3.3 Crack-tip opening angle 23

11.02.4 FRACTURE BEHAVIOR 2511.02.4.1 Regimes of Behavior of a Specimen in a Fracture Toughness Test 26

11.02.4.1.1 Deformation behavior 2611.02.4.1.2 Crack behavior 27

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Classical Fracture Mechanics Methods

11.02.4.2 Unstable Fracture with Little or No Prior Stable Crack Extension11.02.4.2.1 The Klc standard test method11.02.4.2.2 The CTOD standard rest methods11.02.4.2.3 J testing11.02.4.2.4 Ductile-io-brittle transition of steels

11.02.43 Stable Crack Extension11.02.4.3.1 Introduction11.02.4.3.2 High-constraint testing: J and CTOD R-curves11.02.4.3.3 Low-constraint testing

11.02.4.4 Constraint Effects on Fracture

11.02.5 FRACTURE TOUGHNESS TESTS FOR NONMETALS

11.02.5.1 Ceramics11.02.5.2 Polymers

11.02.6 REPORTING

11.02.7 REFERENCES

27272929293333333537

38

3839

40

40

NOMENCLATURE

a

ae

a(

bBBN

Ej{a\W)

F

JKKlc

KQ

PrReL

V

Wb

Afl

Aflß

AflerrA a m a x

Af/szw

crack lengthoriginal crack lengtheffective crack lengthfinal crack lengthuncracked ligament length, W— aspecimen thicknessnet thickness for a side-groovedspecimenmodulus of elasticitygeometry function in the Ksolutionforceprovisional value of F for a Klc

evaluation./-integralcrack-tip stress intensity factorlinear-elastic, plane strain frac-ture toughnessprovisional value of K\c

failure probabilityyield strength, defined as limit ofelastic deformationtensile strengthyield strength, defined as 0.2%proof stressdisplacementspecimen widthcrack-tip opening displacement(CTOD)CTOD defined for a gauge lengthof 5 mmcrack extensioncrack extension due to crack-tipbluntingeffective crack extensioncrack extension limit for valid/?-curvestretch zone widthcoefficient used for the calcula-tion o f i o r C T O D

Subscripts

0.2

0.2/BL

c

im

u

uc

A nnDiT'vA B I J K J L V

Specimens

A(B)A(T)C(T)C(W)

CMOD

CTOACTODDC(T)LPDM(T)SE(B)

plastic part of rjPoisson's ratiocrack-tip opening angleyield strength, general symbol

definition of initiation at 0.2 mmof total stable crack extensiondefinition of initiation at 0.2 mmof ductile tearing after bluntingcritical, in the test standards:unstable fracture after no or lessthan 0.2 mm of stable crackextensioninitiation of stable crack extensionductile fracture toughness atmaximum forceunstable fracture after more than0.2 mm of stable crack extensionunstable fracture after anunknown amount of stable crackextension

arc-shaped bend specimenarc-shaped tension specimencompact specimencrack line wedge loadedspecimencrack-mouth openingdisplacementcrack-tip opening anglecrack-tip opening displacementdisc-shaped compact specimenload-point displacementmiddle-cracked tensile specimensingle-edge cracked bendspecimen

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Societies

ASTM

BSI

Introduction

ESIS

American Society for Testing andMaterialsBritish Standards Institution

ISOJSA

European Structural IntegritySocietyInternational Standards InstitutionJapanese Standards Association

11.02.1 INTRODUCTION

11.02.1.1 Purpose and Goals of FractureToughness Testing

Fracture toughness is defined as a "generic termfor measures of resistance to extension of a crack"ASTM E 1823-96 (1996). The term 'fracturetoughness' is usually associated with the fracturemechanics method, which deals with the effect ofdefects on the load-bearing capacity of materialsand structures. Fracture toughness is an empiricalmaterial property that is determined by conduct-ing a test following standard fracture toughnesstest procedures. These standard test methods havetraditionally been written by national and interna-tional standards organizations.

The historic developments of fracture tough-ness tests as well as the wide variety of amaterial's response have led to a large number oftest standards. In order to avoid the existence (andthe need of using) too many methods, standards-making bodies have begun to write 'unified' testmethods, comprising the various procedures ofthe individual methods. The evaluation of a testthen follows the actual response of the specimen.

Standard test methods or test procedures forfracture toughness testing are traditionally devel-oped by national societies: in the USA by theAmerican Society for Testing and Materials(ASTM), in the UK by the British StandardsInstitution (BSI), and in Japan by the JapaneseStandards Association (JSA). Many other coun-tries have standards-writing bodies, which are notlisted here. However, the standards-writing activ-ity has become an international effort, in Europethrough the European Structural IntegritySociety (ESIS) and worldwide through theInternational Standards Organization (ISO).Fracture toughness standards from various localsocieties are usually combined to write these inter-national standards. Standards, whether nationalor international, are developed by volunteer com-mittees and are subjected to a balloting procedurein order to gain acceptance. Both the writing andballoting procedures are done carefully and takemany years, typically 5-10 years from the begin-ning of writing to final acceptance.

The standard fracture toughness test methodshave been written primarily for characterizingmetallic materials. Many nonmetal standards aredeveloped based on procedures and analysismethods used for fracture toughness tests of

metallic materials with some modifications toaccount for special needs of the material behavior.Therefore, this review will emphasize those stan-dards written for metallic materials without theintent to make them apply exclusively to metals. Ashort discussion of the fracture toughness testingfor ceramics and polymers is included at the end ofthis chapter.

In a fracture mechanics test, several kinds ofoutput can be achieved:-

• A point value of fracture toughness can bedetermined which is evaluated at unstablefracture either with or without prior stablecrack extension; for details see Section11.02.4.2.

• If stable crack extension occurs during atest, then the development of resistance tocrack extension can be evaluated as a func-tion of the amount of crack extension. Theresult is the crack extension resistance curve(7?-curve); for details see Section 11.02.4.3.

• If stable crack extension occurs during a test,then a point value for fracture toughness canbe evaluated near the onset of stable crackextension; for details see Section 11.02.4.3.

Fracture toughness is determined for variouspurposes:

• characterization of a material;• characterization of a production process, for

example, welding; and• assessment of the severity of a crack in a

structural component.

If the fracture toughness is used to determinethe end point of the useful life of a structure,then it can serve for determining design condi-tions such as allowable stresses, selectingmaterial to provide optimum toughness, deter-mining critical defect sizes to set inspectioncriteria, or, in the case of a failed component,for identifying the conditions that may have ledto failure, by conducting a failure analysis. Nomatter what application is intended for the frac-ture toughness value, the test itself is conductedwith the same set of standard rules.

11.02.1.2 Historic Development

The need for fracture toughness was appar-ent before the term 'fracture toughness' was

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Classical Fracture Mechanics Methods

developed. Failures were reported for railroadcomponents in the 1800s, aerospace compo-nents in the 1900s, as well as bridges, pressurevessels, ships, and many others. Failures ofengineering structures may have severe conse-quences: personal injury, loss of life, costlyproperty damage, and environmental pollution.Often these failures were the result of inade-quate material fracture toughness. Reports offailure cases are given by Hertzberg (1983),Broek (1986), Barsom and Rolfe (1987), andAnderson (1995).

Approaches to the problem of fracture werestudied in the early 1900s (Rossmanith, 2000).These did not provide a complete solution tothe problem of unexpected failure. The work ofGriffith (1921) in the 1920s is acknowledged asthe start of the fracture mechanics approach,but the work of Irwin (1957) in the 1950s led tothe development of the fracture mechanics fieldtheory based on crack-tip stress field analysis.At this point, the ASTM Committee onFracture Testing of High-Strength SheetMaterials has to be mentioned; it developedthe basic fracture mechanics methods beforethe ASTM Committee E24 was established.This led to what is considered the modernapproach to fracture mechanics and fracturetoughness testing which is based on parameterstaken from the crack-tip field, namely thecrack-tip stress intensity factor, K.

During the space race of the 1960s, failures ofrocket motor cases provided the motivation forthe formation of an ASTM committee(Committee E24) to study the fracture tough-ness problem and develop a standard testmethod to be used for the prediction of frac-ture. This committee developed the first widelyrecognized fracture toughness test standard, theplane strain fracture toughness method ASTME 399-70T (1970) for the measurement of Klc.This method was first published in 1970 as atentative standard. It used the crack-tip stressintensity factor, K, as the characterizing para-meter to measure quantitative fracturetoughness values that could be used to predictfinal failure conditions in metallic materials. Itwas limited to the regime of linear-elastic frac-ture mechanics (LEFM), and highlyconstrained geometries, said to be approachingplane strain constraint.

The pioneering ASTM K\c method became aparadigm for other standards. Its format wasused in most of the fracture toughness standardsthat followed. For example, BSI (BS, 1977), JSA(JTSG, 1999), and ISO (ISO, 1996) developed astandard for a plane strain fracture toughness,K]c, testing that had many of the features inthe original ASTM standard. It is still a basicfracture toughness measurement for the

characterization of metallic materials, and ithad been so well written that the actual issue isbasically the same as the original one, with a fewmodifications added. Many other national stan-dards-writing organizations have written Kic

standards. These are too numerous to list here.Although the Klc standards were applicable

to many classes of materials, it was clear earlyin the development of fracture mechanics thatthe limits of linear elasticity that accompanythe use of the K fracture parameter were toorestrictive for many materials including mostmetals used in structural applications.A material with adequate fracture toughnessshould be designed to yield before reachingthe fracture toughness point. Approaches andparameters that could be used to characterizefracture toughness beyond these linear-elasticlimits were already being developed. The lead-ing approach during that time was the crack-tip opening displacement (CTOD) parameter(Wells, 1961). It was first standardized by BSIin 1979 (BS, 1979). In a parallel development,the ./-integral was developed (Rice, 1968) andproposed as a fracture parameter by Begleyand Landes (1972). It was first standardizedto measure J lc in 1981 (ASTM E 813-89,1990). Along with this came methods for R-curve fracture toughness, rapid load tough-ness, crack arrest, weldments, and others.

Fracture toughness standards were devel-oped and added to the literature in a mostlychronological fashion. Individual standardswere not coordinated to see that some of thefeatures common to the various standards weremade compatible, so that more than one stan-dard might be applied in a given test. As aresult, there was some confusion among usersas to which standard fracture toughness testshould be used. Additionally, problems oftendeveloped when the restrictive conditions ofone test method, for example, K\c, could notbe met. When the conditions required by thetest standard were not met, the result of this waslabeled an invalid test result, which was some-times considered to be a useless result.However, if the test standards were properlycoordinated, the test that failed one standard'svalidity requirement could have been evaluatedby another standard that uses a fracture para-meter with another range of validity and maygive a valid result.

In order to try to bring some organizationto the growing list of fracture toughness stan-dards, a large effort to unify and simplify thefracture toughness testing was begun duringthe 1980s. The concept of common or unifiedstandards was proposed and the developmentof unified standards was undertaken by thestandards organizations. As a result, many

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Test Techniques

existing standards were combined into a sin-gle document that used common sets ofequipment and testing procedures. This madethe fracture toughness test more organizedbut not necessarily simpler. Most of the newstandards are a form of a unified standardthat combines various deformation and frac-ture regimes. Examples of these are ASTMC1421-01b (2001), BS (1991, 1997a, 1997b),ESIS P2-92 (1992), and ISO 12135 (2002).

During the 1990s, some special areas of frac-ture toughness testing were considered for thedevelopment of new standards. Examples ofthese include fracture in the ductile-to-brittletransition for ferritic steels (e.g., ASTM E1921-05, 20051; ESIS P2-92, 1992) and the test-ing of thin sections (e.g., ISO DIS 22899, 2005;GKSS EFAM GTP 02, 2002). Some of thestandards developed in the 1990s are stillbeing evaluated and await final acceptance. Itmay be worth mentioning that the procedureGKSS (2002) includes all methods for high-constraint testing, ductile-to-brittle transition,low-constraint testing, and testing of welds.

Test standards are never considered to becompletely finished with a final version in thestandards books. Most standards-writing orga-nizations require that the standards beperiodically evaluated and updated as new tech-nology is developed or as problems withexisting standard practices are discovered.Therefore, the status of new standards is thatthey are under evaluation, either to be morecompletely developed or to be improved. Thisincludes the original K\c standard that was lastre-evaluated in 1996 and editorially changed in1997 (ASTM E 399-90, 2005f). Therefore, adiscussion of fracture toughness testing meth-ods can only relate to current state of testingpractice and standards writing and is not adescription of the final status of any fracturetoughness test method.

11.02.2 TEST TECHNIQUES

11.02.2.1 General Requirements

The main goal of the fracture toughness test isto produce a result that can be used to quantita-tively evaluate a material's or structure'sresistance to the extension of a crack-like defect.This involves the testing of a specimen containinga crack under monotonically increasing force ordisplacement. The test continues until the speci-men reaches or passes a point or region overwhich the fracture toughness is defined. Thiscan be a point that marks the complete separa-tion of the specimen or a point that marks theonset of the process of stable crack extension thatresults from ductile fracture behavior.

As the fracture toughness may depend onthe size and geometry of the specimen, thetest must be conducted with a specimen ofsufficient size of a given geometry to satisfythe conditions given in the standard. Theequipment used must include a test machinehaving the required capacity for applyingforce, a set of loading fixtures, instrumenta-tion with the correct precision, andcalibration to generate the data that areused in the evaluation of fracture toughness.The equipment must meet all requirements ofthe standard. The test procedure and the ana-lysis of the results must follow the rules of thestandard and the results must be subjected tovalidity evaluations specified in the standard.Finally, the test results must be reported in away which includes all of the required infor-mation specified by the standard.

Usually, the standard does not give theserequirements in an order which is easy to fol-low. If these requirements are listed in a step-by-step manner to make the proceduresomewhat like a cookbook recipe, then the non-expert can have a chance of conducting all thenecessary steps and meeting the requirements ofthe standard. The steps to follow are:

1. Choose type of a specimen which is prop-erly machined and introduce a crack into thespecimen.

2. Prepare the test fixtures, choose a testmachine, and obtain the proper instrumentation.

3. Place the specimen in the test fixtures andthe test machine with instrumentation in placeand conduct the test by following the prescribedprocedure.

4. Record the test data.5. Analyze the test data to obtain the frac-

ture toughness parameters.6. Evaluate the qualification or validity

requirements.7. Report the results.

Each step must be performed correctly in orderto conduct a successful test.

11.02.2.2 Specimens and Fixtures

Most test standards allow several choices forthe specimen geometry. For example, in theASTM Klc test (ASTM E 399-90, 2005f), thereare five possible specimen geometries whichcould be chosen. The two most common onesand the ones used in many standards are thecompact specimen, or C(T) specimen, and thesingle-edge cracked bend specimen, or SE(B)specimen. The C(T) specimen shown inFigure 1 makes efficient use of the material,making it a good choice when the amount of

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Classical Fracture Mechanics Methods

H

Root radius 0.1 mmmaximum

Preferred width for highconstraint W= 26

Total width C= 1.25IV+0.01 WHalf height H= 0.6W±0.005W+0.004WHole diameter d= 0.25W ~°Notch width N = 0.06 W max. or

1.5 mm max. if W< 25 mmEffective notch length

M=0AWm\n.Effective crack length

ao = 0A5W-0.65 W

Notes:1. A spark eroded or machined slit

can be used instead of theV-notch profile.

2. Squareness and parallelism tobe within 0.0021V.Holes to be square with facesand parallel.

w

(b)

0.4 S

0.4 • ~ "

Tolerances, surface finish, and dimensions as Figure 1b

Notes:1. Spacing between knife edges depends on type of

clip gauge to be used.2. Side grooves are recommended for tests in the ductile

regime.

Figure 1 C(T) specimens for high-constraint testing (ESIS P2-92, 1992). a, Straight-notched C(T) specimen;b, step-notched C(T) specimen.

material is limited. It is loaded by the pin andclevis fixture shown in Figure 2. The geometryshown in Figure 3 is for low-constraint testing,which requires that the in-plane dimensions aremuch larger than the thickness of the specimen(ISO DIS 22899, 2005). Consequently, for agiven pinhole diameter, which is proportionalto the width of the specimen, the clevis shouldhave a narrower slit than shown by the

proportions in Figure 2. Since a wide and thinspecimen may buckle, the use of antibucklingguides is mandatory (Figure 4). The SE(B) spe-cimen (Figure 5) is easier to machine but usesmore material. It is loaded in three-point bendloading and requires a bend fixture (Figure 6).Some test procedures also allow the four-pointloaded bend specimen (GKSS EFAM GTP 02,2002). These two specimen types are used most

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Test Techniques

0.025 W-0.050 W

I t -0.025 W0.1 W

- Note 3

- Loading rod Thd.

-D-

-R = 0.05±0.01

0.25W ["«-0.5W+0.015W-«-| 0.25WA-Surfaces must be }--o.5iv±o.oo5w-».

f lat- in-line, and perpendicular, asapplicable, to within 0.002 in t.i.r.

Loading flat

Figure 2 Pin and clevis fixture for C(T) specimen (ASTM E 1820-05, 2005k).

Figure 3 C(T) specimen for low-constraint testing(Schwalbe et al, 2004).

often for fracture toughness tests. Other speci-men types are usually specialized geometriesrelated to some product forms.

The fracture toughness test methods havetraditionally been designed for obtaininglower-bound toughness values, which can beachieved by testing the specimens shown inFigures 1, 5, and 9 if their uncracked ligamenthas an approximate square geometry, which isachieved by the thickness-to-width ratio andthe recommended precrack length. The mid-dle-cracked tensile specimen, or M(T)specimen, shown in Figure 7, represents a low-constraint specimen, which is used to charac-terize sheet material. It is typical of the

requirements of the aerospace industry; anothertypical application is the testing of weldedjoints for the pipeline industry, where the speci-men is frequently designated wide plate, oftenwith a part-through crack. If the specimen isvery wide compared to its thickness, then it maybuckle, which affects the test result. In order toavoid buckling, antibuckling guides should beused (Figure 8). This specimen type is addressedin the standards ASTM E 561-98 (2005g), ESISP3-05D (2005), GKSS EFAM GTP 02 (2002),and ISO D1S 22899 (2005). If loading fixturesare not specified in standards, the specimenmay be loaded by hydraulic clamping; pinholeclamps are also in use. As already stated above,a C(T) specimen is also recommended for low-constraint testing if its width is significantlylarger than its thickness.

Other specimen geometries are arc-shapedtension (A(T)) specimen (Figure 9a); disk-shapedcompact (DC(T)) specimen (Figure 9b); and thearc-shaped bend (A(B)) specimen (Figure 9c).The acronyms are standard ASTM nomencla-ture defined in ASTM E 1823-96 (1996), but areused in many other standards.

The choice of the specimen also requires achoice of the specimen size. Since meeting thevalidity criteria of the various standardsdepends on the specimen size, it is importantto select a specimen of a sufficient size beforeconducting the test. However, the validity cri-teria cannot be evaluated before the test iscompleted; therefore, choosing the correct size

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10 Classical Fracture Mechanics Methods

C(T) specimen Crack viewingregion

Biltholes

Antibuckling plates(front and back)

Figure 4 Antibuckling guides for C(T) specimen (Schwalbe el al., 2004).

Thickness = 6Width = W

Preferred width W=2BNotch width N= 0.06 W max. or

1.5 mm max. if W< 25 mmEffective notch length

M = 0.4Wmin.Effective crack length

ao=O.45W-O.65W.

Notes:1. A spark eroded or machined slit

can be used instead of theV-notch profile.

2. Squareness and parallelism tobe within 0.0021V.Notch to be square with specimenfaces and notch faces to be parallel.

3. Side grooves are recommended fortests in the ductile regime.

Figure 5 Single-edge cracked bend (SE(B)) specimen (ESIS P2-92, 1992).

is a guess which may turn out to be wrong.There are guidelines for choosing a correctsize, but no guarantee that the chosen sizewill pass the validity requirements. The testspecimen must also be chosen so that the propermaterial is sampled. This means that the loca-tion in the material source of the orientation ofthe sample must be correct and specified. TheASTM standards use the letter system shown inFigure 10 to specify orientation (ASTM E 1823-96, 1996). ISO uses the system shown inFigure 11. As the specimens are being prepared,requirements for tolerances on locations of

surfaces, size, and location of the notch andpinholes, and surface finishes must be followed.

The preparation of the test specimen has arequired set of rules. The specimen machiningmust be done to a prescribed set of tolerancesand surface finishes. A major consideration inthe preparation of the test specimen is the intro-duction of a crack-like defect into the specimen.This is nearly always done by machining asharp notch that is extended with cyclic load-ing. The defect produced by the cyclic loading iscalled the fatigue precrack. Precracking is alabor-intensive procedure and usually takes

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Test Techniques 11

W/8 min.

Details of roller pins

y//////////////,.

Diameter = W/4 min. 1.25Bmin.

Notes:

1. Roller pins and specimen contact surface of loading rammust be parallel to within 1°.

2. Rollers must be free to move outward.3. Fabricate fixture from a high-strength material sufficient

to resist plastic deformations in general use.

Figure 6 Three-point bend fixture (ESIS P2-92,1992).

f

Panel thickness = B

ooooooooooooooo

2L>ZWUW> 1.5

2L

) Clamping area

Countersunk hole,thin panels

Counterbored hole, thick panelsCentral hole for CMOD gauge mounting. The hole shown in thebottom drawing is used for thicker specimens in order to limit theouter diameter, da.

Figure 7 Middle-cracked tension (M(T)) specimen(Schwalbe et al., 2004).

material fracture toughness. Precracking has aset procedure that must be followed. It is impor-tant in precracking to avoid overloading thespecimen, to get the precrack length within pre-scribed limits, and to get a straight crack front.More details on precracking are given later.

During the test, crack advance can result in acurved crack front. This can be avoided by side-grooving the specimen after fatigue precracking.These side grooves are machined along the sides ofthe specimen in the plane of the crack. Themachining of such side grooves is best done afterthe fatigue precracking is completed to avoid hav-ing the side grooves influence the crack frontshape. Side grooves can be.machined with anotch cutter. These side grooves reduce the netthickness of the specimen. The allowable thicknessreduction may be slightly different fromone standard to another; however, a thicknessreduction of 20%, 10% on each side, is oftenused. This net thickness is usually identified witha subscript N. For example, when thickness usesthe symbol B for the gross section thickness, it uses5 N for the net section thickness that remains afterside-grooving.

11.02.2.3 Test Machine

The next step in the test procedure is the choiceof a loading machine and the preparation of theloading fixtures. All tests must be conducted in amachine that can apply and measure force. Manytypes of machines can be chosen. For the pre-cracking step, it is convenient to have a machinethat can apply cyclic forces at a fairly high fre-quency. Closed-loop servohydraulic or resonancemachines serve this purpose.

For loading the specimen, test fixtures arerequired; they are described in Section11.02.2.2. It is important for the specimen todeform in the loading fixture without extra-neous frictional forces, which may come fromrubbing of a C(T) specimen against the side ofthe pinhole or of an SE(B) specimen against acorner of the bend fixture. These extra forcescould adversely influence the force measuredduring the test. For example, in the clevis, afiat region is required for the pin-bearing area,so that the deforming specimen can rotate with-out a reverse moment being applied by thepinhole edges (Figure 2). The same is true forthe bend fixture (Figure 6). The pins must befree to roll as the specimen deforms.

more time than the actual fracture toughnesstest itself. Schemes have been proposed foreliminating this step but it has not been shownthat fracture toughness results without pre-cracking are truly representative of the

11.02.2.4 Instrumentation and Requirements

11.02.2.4.1 Instrumentation

All tests require force-measuring instrumen-tation. Test machines have strain-gauged load

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12 Classical Fracture Mechanics Methods

Guide plates

Bolt hole

Spacer block

Crack^

I-beam

? Spacer block

(40 in wide M(T) only)1 Specimen

Figure 8 Antibuckling guides for M(T) specimen (ISO D1S 22899, 2005).

Arc-shapedtensionspecimen (A(T))

Disk-shaped compact(b) specimen (DC(T))

(c) Arc-shaped bend specimen (A(B))

Figure 9 Less frequently used specimen for specialpurposes.

cells, which measure force electronically. Allfracture toughness tests also require the mea-surement of displacement on the specimen.Most often a strain-gauged clip gauge is used.The clip gauge is usually an instrument withcantilevered metal arms which have strain gaugesattached to either side of the arms. The gaugesare placed in an electrical bridge, so that they givean electrical signal which varies proportionatelywith the displacement of the arms; usually thereis a linear relationship between displacement andthe voltage output. The precision of these gaugesdepends on type of measurement needed. For abasic K[c test, the precision required is not asstringent as for a test in which elastic slopes areused to measure crack advance. The displace-ments to be measured are as follows.

• The crack-mouth opening displacement(CMOD) measures the opening of the crackstarter notch at the specimen front surfaceon all specimen types but the M(T) specimen(Figure 12a).

• For the M(T) specimen, the arrangementshown in Figure 12(b) is used.

• For the measurement of S5, a special defini-tion of the CTOD, the gauge in Figure 12c isrecommended. The gauge is attached to thespecimen side at the original fatigue crack tipand over a gauge length of 5 mm, ±2.5 mmrelative to the crack line.

• If the fracture toughness test is aimed at thedetermination of the ./-integral, then thedeformation energy of the specimen has tobe measured, which requires the measure-ment of the displacement of the loadingpoints, the load-point displacement, LPD(or load-line displacement). For C(T) speci-mens, this displacement is taken between thesteps in the crack starter notch which definethe origins of the crack length, a, and of thespecimen width, W, in Figure 1. For SE(B)specimens, the measurement of the LPD isrelatively complicated since extraneousdeformations arising from indentations ofthe loading rollers into the specimen haveto be extracted from the measured deflectionof the specimen. Therefore, more recent Jstandards use the CMOD for the J evalua-tion (see Section 11.02.3.3.1). In case of theM(T) specimen, the load-line displacementhas to be measured over a very large gaugelength, which requires the use of a linearvariable differential transducer (LVDT)gauge. During loading of an M(T) specimen,some amount of bending may occur if thespecimen had not been exactly plane priorto testing. This affects the measurement ofthe LPD; also in this case the CMOD maybe advantageous for the determination of J(see Section 11.02.3.3.1)

For crack-length measurement, electricalpotential drop instrumentation may be used(for details see Section 11.02.2.5.3.)

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Test Techniques 13

Figure 10 Orientation code for identifying testing conditions (according to ASTM E 1823-96, 1996).

Grain flow

Figure 11 Orientation code for identifying testing orientations: a, aligned; b, not aligned; c, radial grain flow,axial working direction; d, axial grain flow, radial working direction (according to ISO FDIS 3785, 2005 (E)).

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14 Classical Fracture Mechanics Methods

Foil resistancestrain gauge

Recorder

Optionalintegral machinedknife edge

500 O gauges willprovide greatersensitivity than120Qgauges

(b)

Note: for the gauge R < dJ2and 6< $

(c)

Figure 12 Clip gauges for displacement measurements, a, Schematic clip gauge for measurement of CMODon C(T) and SE(B) specimens (ASTM E 1820-05, 2005k); b, measurement of CMOD on M(T) specimens; c,experimental setup for measuring 65 (Schwalbe, 1995).

11.02.2.4.2 Instrumentation requirements

The selection and calibration of instrumenta-tion is an important part of conducting asuccessful test. The instrumentation type andrequirements depend on the type of test beingconducted. Nearly all tests require a forcetransducer and one or more displacementtransducers.

The calibration required for the various trans-ducers are a function of the type of test beingconducted. For the basic test, the requirementson force and displacement accuracy are not asstrict. For example, in many procedures, an accu-racy of ± 1 % of the full working range and themaximum deviation of a fit to the data of ± 1 %would be required. However, if elastic unloading

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Test Techniques 15

compliance is being used, the maximum devia-tion from the fit could be ±0.2% for both force-and displacement-measuring devices. Also, forthe elastic compliance method, the resolution ofthe displacement signal should be one part in3.2 x 104 and the force resolution should be onepart in 4000. For digital data acquisition, a 16-bitA-to-D converter is required for this. If the work-ing range of the instrument is much greater thanthe range used for the test, separate requirementscould be given for the working range and for thetest range. Each test procedure includes a sectionin which the required accuracies for the instru-mentation are specified.

11.02.2.5 Crack-Length Measurement

For the evaluation of a fracture toughnesstest, some information on crack length isrequired, either the initial crack size, a0, or thecomplete crack extension history during a test.The most basic tests require only a measure-ment of the initial crack size. For tests withsome stable crack extension prior to fracture,the final crack length, ci(, may also need to bemeasured. If a complete Ä-curve characteriza-tion is to be done, a number of crack-lengthvalues between the initial and final cracklengths are required. This can be achieved inone of the two ways, either by using the multi-ple-specimen method or one of several single-specimen methods. The former method needsone specimen for each data point, with visualcrack-length measurement on the fracture sur-face of the broken specimen; the latter methodprovides a complete 7?-curve from a single spe-cimen, using indirect techniques for crack-length measurement. These methods will nowbe described in detail.

11.02.2.5.1 Determination of initial and finalcrack lengths

The determination of initial and final cracklengths requires a visual measurement which isthe oldest method applied. A measurement of thecrack size on the specimen side-surface can easilybe done during a test; however, in many cases, thecrack advance in the interior of the specimen canbe much larger than on the side-surface, in parti-cular in thick specimens. Therefore, side-surfacemeasurements are only recommended for fatiguepropagation tests and /?-curve tests on thin speci-mens. They are usually done using a microscopewith a calibrated traveling length measurementdevice, but remote electronic field measurementsare also available, allowing automated determi-nation of the crack length.

For fracture toughness tests, the usual visualmeasurement of crack length is on the specimenfracture surface. This can only be done as apostmortem measurement, that is, the specimenmust be broken into two pieces. The initialcrack length is taken at the end of the fatigueprecrack. This crack length is identified by thedifference in surface features between the fati-gue crack extension region and the followingfracture region on the fracture surface, whichmay be either slow stable crack extension or fastfinal fracture (Figure 13). The final crack lengthis usually defined at the end of stable crackextension. This can occur at the onset of clea-vage fracture in steels or when the test isterminated by unloading the specimen.

11.02.2.5.2 Visual method

The visual technique described here is used todetermine initial and final crack sizes as well asthe amount of crack extension, Aa, that has

(Not to scale) 1 , 2 3 4 5 6 7 8 9

0.01 S

Measure initial and final crack lengths at positions1-9 from center line of

pinhole 1 2 3 4 5 6 7 8 9

Referencelines

8 I 2 \=2

'A^Center line of the pinhole

Machined notch

Fatigue precrack

Initial crack front

• Stretch zone

Crack extension

Final crack front

Side groove

(b)

Figure 13 Visual crack-length measurement on the fracture surface, compact specimen: a, plane side specimen;b, side-grooved specimen (ES1S P2-92, 1992).

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16 Classical Fracture Mechanics Methods

occurred during a test. It is particularly used inthe multiple specimen method which requires anumber of specimens for the determination ofan 7?-curve. The specimens are loaded to differ-ent amounts of deformation, thus resulting indifferent amounts of crack extension, whichhave to be determined by postmortem investi-gation (see Section 11.02.4.3.1). The visualmeasurement of crack size is on the fracturesurface. For this, the specimen must be brokeninto two pieces. In order to make Aa visible onthe fracture surface, several treatments havebeen developed which mark the crack frontreached at the end of test. After marking thecrack front, the specimen is then broken opento reveal the fracture surface for the crack-length measurements. Some of the crack-frontmarking techniques include

• Specimens made of ferritic steels that can becooled down to low temperatures, for exam-ple, by immersing them in liquid nitrogen. Ifthe specimen is then broken open, final frac-ture occurs in a cleavage mode which can beclearly distinguished from the appearance ofprior stable crack extension.

In materials other than ferritic steels, thefracture surfaces created by slow stable crackextension and fast final rupture may not exhibitdifferent features that could mark the crackfront reached during the test. For these cases,the following two techniques are suitable:

• The fracture surface of some materials, forexample, steels and titanium alloys, createdduring the test can be tinted by heat treat-ment which helps distinguish them from thesubsequent final rupture.

• A frequently used method that can be appliedto virtually every material is post-test fatiguingor refatiguing at a low stress amplitude whichprovides a clearly marked crack front.

The amount of crack extension during the testis then determined on the fracture surface byaveraging several individual local measurements(Figure 13). The crack usually exhibits a curvedfront, and since the fracture mechanics para-meters are based on a two-dimensionaltreatment, the crack must be defined with asingle length, usually an average of the curvedlength. Crack lengths are measured from thefront face for the bend-type specimens (e.g.,SE(B) specimens) and from the center of thepinholes for pin-loaded specimens (e.g., C(T)specimens). In the case of the M(T) specimen,the crack length is taken as one-half of the totalcrack-length measured between both crack tips(Figure 14). Visual crack-length measurements

12 3 4 5 Measure initial and final crackI I I I I lengths at positions 1-5

Referencelines

Figure 14 Measurement of crack length on M(T)specimens. The same procedure is used as outlined inFigure 13. The average of both cracks represents thecrack length of the M(T) specimen.

are usually done with a microscope on a cali-brated traveling stage. The number of individualmeasurements can vary:

• Nine individual values are often required intest methods aimed at determining low-con-straint (plane strain) fracture toughnessvalues under elastic-plastic conditions(ASTM E 1820-05, 2005k; ISO 12135, 2002;ESIS P2-92, 1992): the crack lengths near thetwo surfaces and crack lengths at seven loca-tions through the thickness of the specimen,at 1/8-thickness intervals. The two surfacelengths are averaged to give one point andthe seven middle lengths are averaged withthis surface average to give what is called thenine-point average (Figure 13).

• For the determination of Kic, only threevalues are required: mid-thickness and thetwo quarter-point thickness values areaveraged.

• For specimens that are used to determine thefracture toughness under low-constraintconditions (see Section 11.02.4.3.3), thenumber of individual measurements can bereduced to five; three measurement pointsare sufficient if the specimen thickness issmaller than 5 mm (draft standards: ISODIS 22899, 2005; ASTM E 2472-06, 2006;ESIS P3-O5D, 2005).

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Test Techniques 17

An important aspect of visual crack observa-tion and measurement is concerned with thestretch zone. This zone marks the transitionfrom the fatigue crack to stable crack extensionand is needed for the accurate determination ofthe initiation of stable crack extension. Due tothe microscopically small width of the stretchzone, a scanning electron microscope has to beused for the measurement. The width of thestretch zone varies substantially across the spe-cimen thickness so that a number ofmeasurements have to be taken and then aver-aged (Figure 15). In addition, there is scatterbetween individual specimens; therefore, theresults from at least three specimens have tobe averaged (ISO 12135, 2002; ESIS P2-92,1992; GKSS EFAM GTP 02, 2002).

11.02.2.5.3 Indirect methods

Whereas the multiple-specimen method usingvisual crack-size measurements provides justone data point per specimen on an 7?-curve,continuous or quasi-continuous crack-lengthmeasurement from a single specimen duringthe test allows in principle the generation of acomplete 7?-curve. The choice of method is atthe user's discretion; however, sufficient accu-racy has to be demonstrated. For example, ISO12135 (2002) requires that the final crack exten-sion should be within 15% of the measured

Fatigue crack

Stretch zone

Stable crackextension

/c>5

Beginning ofstretch zones

End ofstretch zonev

Image plane parallel to fatigue surface

/= 1 /=k

Figure 15 Determination of the stretch zone width,AaSZw (ESIS P2-92, 1992).

crack extension or 0.15 mm, whichever isgreater, for Aa < 0.2( W - a0), and within0.03(W-aq) for Aa > 0.2(W- a0). Threemethods will be described in the following,based on the deformation properties of the spe-cimen and electrical potential techniques. Incontrast to measurements on the specimen'sside-surface, which capture only the surfacetrace of a crack, these methods average overthe specimen's cross-section.

For the first crack extension fracture resis-tance curve measured in a series of tests using asingle-specimen method, some standards requirecalibration of the technique used. For example,it is recommended to test at least three specimens(ESIS P2-92, 1992). Two of these are required todemonstrate the accuracy of the test equipmentat small and intermediate amounts of crackextension. One test should be terminatedbetween 0.1 and 0.3 mm of ductile crack exten-sion (Figure 16). The other should be terminatedmidway between the valid crack extension range,Aflmax. Suitable termination points can be esti-mated from data for the specimen covering theAflmilx range. If the difference between the esti-mated and measured crack extension exceeds15% of the measured crack extension or0.15 mm, whichever is greater, then the test isinvalid and the single-specimen technique mayrequire improvement.

(i) Elastic compliance

The elastic compliance of a specimen is afunction of its relative crack length, a\W.Using the appropriate calibration function,which depends on the specimen geometry, thedevelopment of the crack length during a testcould be determined continuously or at certainintervals. The initial part of a force-deforma-tion relationship is linear, thus indicating thatthe crack has not yet increased its size. Withfurther increase of the applied force, the force-

—>o

2-a>octo<n0)

oD

rac

/II

1 tI II I

I/ /1 1i

Blunting line / i

I ^<d^/ /&n / /' / // / // / /' ' /' / /' / // / /

0.1 mm/ Aam

10.3mm/ Crack extension, Aa

Figure 16 Calibration points for indirect methods(ESIS P2-92, 1992).

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18 Classical Fracture Mechanics Methods

deformation relationship becomes nonlinear,and for a material exhibiting little or no plasticdeformation, this indicates crack extension.However, most engineering materials developa plastic zone at the crack tip that contributesto the nonlinear behavior of the test record. Inorder to extract the elastic compliance from thetest record, the unloading compliance techni-que has been developed (Clarke et ai, 1976).The unloading compliance method relies on thefact that a plastically deformed materialbehaves in an elastic manner upon unloading.Small elastic unloadings are taken at intervalsduring the test. The slope of the unloadings ismeasured, and its inverse is the compliancefrom which the crack length can be deducedvia the calibration curve. It should be notedthat in spite of its simple principle, this methodrequires very careful experimentation due to thevery small displacements to be measured anddue to friction effects between specimen andloading device and displacement gauge, respec-tively. This technique is standardized in anumber of national and international standards(ASTM E 1820-05, 2005k; ISO 12135, 2002;ESIS P2-92, 1992). The unloading compliancemethod works best on bend-type and compactspecimens; it is not routinely used on tension-type specimens, such as the M(T) specimen,since they exhibit even smaller deformationsthan the former ones.

(ii) Normalization method

A further way to measure the crack lengthfrom deformation properties is based on thedeformation characteristics of the materialand specimen (Landes et al., 1991; Reese andSchwalbe, 1992; ASTM E 1820-05, 2005k). Aspecimen geometry has a deformation patternwhich relates to the deformation characteristicsdetermined in a tensile test. By knowing theplastic deformation pattern of a specimen witha given crack length, crack-length changes canbe inferred from deviations from this pattern.Usually, the plastic deformation pattern of aspecimen is given by a functional form withfitting constants. These constants can be deter-mined at calibration points on the specimen, forexample, initial and final points of the test. Theprediction of crack extension is based on thesolution of a set of equations that describe theinfluence of crack length and displacement onthe force versus displacement record. In somecases, the method is applied by testing a speci-men without crack extension, for example, ablunt-notched specimen, and a second speci-men which is precracked and hence allowscrack extension. The difference between the

deformation patterns for the stationary crackand the extending crack can be used to infercrack extension during the test. The use of twospecimens assumes that the plastic deformationof the two specimens would be identical for thetwo crack lengths. This technique does notrequire any additional instrumentation and isparticularly well suited for testing at high tem-peratures, in aggressive environments, or athigh loading rates.

(Hi) Electrical potential drop method

The third indirect method is the electricalpotential drop method, using the electricalresistance of the specimen, which depends oncrack length. The method is mainly used to testmetallic materials; however, it can also be usedon nonmetals if a metallic foil is attached to thespecimen's side-surface along the expectedcrack path. Whereas in the former case theresistance of the whole cross-section of the spe-cimen contributes to the electrical resistance,thus providing an average measure of thecrack length, in the latter case the resistance ofonly the foil is determined, whereby only thesurface trace of the crack is captured, restrictingthis method to applications where the surfacetrace is representative of the whole cross-section, for example, when fatigue crackpropagation has to be determined. Whereas thecompliance is a well-defined property of a speci-men, the electrical resistance depends stronglyon the way the current is fed into the specimenand the locations of the voltage pickup. This iswhy numerous techniques have been developed,the main differences being the use of DC or ACand the locations of the current input and vol-tage measurement. Most methods are self-calibrating, that is, the final crack extension isused for calibration. One DC method uses theJohnson equation (Johnson, 1965) to predictcrack extension (Schwalbe and Hellmann,1981). Since this method appears in several stan-dards, its basic items are shown here. Johnson'sequation relates the crack length to the potentialdrop as follows:

2Wa = cos

71

ny^ a |

where y is shown in Figure 17, <E> is the electricalpotential related to the actual crack length, andthe subscript 0 identifies the values of a and <t>before crack extension starts.

This equation is valid if in the neighborhoodof the cracked cross-section, a homogeneous

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Analysis 19

0.5W

Figure 17 Specimens with electrical contacts suitedfor using Johnson's equation (ESIS P2-92, 1992).

distribution of the electrical potential wouldexist without the presence of the crack. Thiscan be achieved if the DC current is fed intothe specimen remote from the crack, which canbe most easily achieved in an M(T) configura-tion for which this equation was derived.However, the method can also be applied toC(T) and SE(B) specimens if the contactarrangement is as shown in Figure 17. Detailsfor the practical application of the electricalpotential method are given in some standards(ASTM E 647-00, 2005h; ESIS P2-92, 1992;GKSS EFAM GTP 02, 2002).

High plastic deformations in a large plasticzone of the specimen may substantially affectthe voltage output measured on the specimen.Therefore, any calibration should considerplasticity effects.

Proper electrical insulation betweenspecimen and test machine is important toavoid effects of the machine on the measuredpotential. Furthermore, when using a DCmethod, due to the high sensitivity needed forthe voltmeter- typically in the nanovolt range-electrical drift may easily occur. Therefore,during the test, the specimen should be pro-tected from temperature fluctuations. Using adummy specimen which is not loaded in the testor reversing the current in certain intervals havealso been proved to be useful (Dietzel andSchwalbe, 1986).

11.02.2.6 Conducting the Test

11.02.2.6.1 Loading the specimen

The specimen can be force or displacementcontrolled. In a servohydraulic machine, mostof the measuring transducers can be used as

control transducers. Usually the displacementof the loading ram is used to control the loadingof the specimen in a fracture toughness test. If aforce control is used, the machine becomesunstable when the maximum force point isreached. Many tests are loaded past this max-imum force point into the unloading region ofthe force versus displacement behavior.Therefore, testing using force control does notallow the test to go to the full extent that dis-placement control could allow.

The rate of the loading is specified in thestandard. The loading rate must be fast enoughso that time-dependent effects do not influencethe test result, and slow enough so that the testresult does not have any of the rapid loadeffects that influence the fracture toughness.Also for rate-sensitive materials such as poly-mers, the loading rate must be continuallycontrolled. The final point of the loadingdepends on the type of test and the fracturebehavior. If there is a brittle fracture response,the test is conducted until there is an unstablecrack advance. For ductile fracture behavior,the point for terminating the loading may bechosen to obtain a desired amount of stablecrack extension. Often, this is just past maxi-mum force, but it could be further into theunloading portion of the test.

11.02.2.6.2 Recording

As the test is being conducted, the measuredparameters such as force, displacements, testtemperature, and stable crack extension mustbe recorded. The recording of the data wastraditionally done autographically. With newcomputerized systems, the data are oftenrecorded digitally. The digital recording of thedata can allow the in situ calculation of thefracture parameters, sometimes called interac-tive testing and data recording. However,additional autographic recording is still beingused frequently in order to have a visual impres-sion and control of the test.

11.02.3 ANALYSIS

11.02.3.1 Introduction

In this section, the tools needed for evaluat-ing the fracture toughness in terms of severalparameters are provided. Usually, a force-dis-placement record (F-v record) serves thispurpose; the displacement is measured eitheracross the crack mouth or between the pointswhere the force is applied (=load-line displace-ment). This is the classical procedure of linear-elastic (see Section 11.02.3.2) and elastic-plasticfracture mechanics (see Section 11.02.3.3). In

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20 Classical Fracture Mechanics Methods

Sections 11.02.3.3.2 and 11.02.3.3.3, two morerecent methods are described which do not needa force-displacement diagram; however, it isrecommended to record this information inany fracture test. In the context of the variousfracture regimes, some examples of specifictests will be given. These tools will then beused in Section 11.02.4 to characterize aplethora of fracture events.

11.02.3.2 Linear-Elastic Fracture Toughness

11.02.3.2.1 Expressions for the stress intensityfactor

Historically, fracture mechanics was intro-duced to study and quantify brittle fractureevents. Therefore, the K1C test was the firstfracture toughness test appearing in thebooks of standards (for details see Section11.02.4.2.1). Brittle fracture means that failureoccurs within a globally linear-elastic beha-vior, which in a test piece is given by alinear F-v record. Plastic deformation nearthe crack tip does not disturb linear elasticityif it is confined to a small plastic zone and ifit leads to only a small deviation from linear-ity of the test record. Fracture toughness isexpressed in terms of the stress intensity fac-tor, K. The stress intensity factor can beexpressed in various ways. A K expressioncan be based on the solution for an infinitelylarge cracked sheet containing a through-crack of length 2a, with the applied stressa acting perpendicularly to the crack andhaving a homogeneous distribution acrossthe sheet:

K = G\fnci [2]

It is obvious that the K solution for atest piece or a structural component, the geo-metry of which deviates from an infinitelylarge sheet with a homogeneously distributedapplied stress, must be different from eqn [2].In order to keep the simplicity of the expres-sion in eqn [2], the deviation of the actualsolution from that equation is given by adimensionless calibration function, Y(ajW),where W designates the width of the testpiece or component:

Since these expressions are not very suitablefor laboratory specimens, the stress intensityexpressions have been given a different format,with the M(T) specimen being an exception.For this specimen, K is given by (ASTM E561-98, 2005g, ASTM E 647-00, 2005h)

is

2BW[4]

or, alternatively

2BW

[5]

where F/(2BW) is equal to a and B is thethickness of the specimen. Whereas the M(T)specimen is appropriate for low-constraint frac-ture toughness determination, the two otherwidely used specimens, SE(B) and C(T) speci-mens, are mainly used for high-constraintfracture problems. Their respective stress inten-sity formulas are for the C(T) specimen: (ASTME 1820-05, 2005k)

K =F

{BBN W)1/2

•f{a/W) [6]

where

j\a/W) =

x (0.886 + 4.64(a/ W)- 13.32(a/ W)2 [7]

+ 14.72(a/HO3-5..6(a/H/)4)

and Z?N is the net thickness of a side-groovedspecimen. If the specimen is plain-sided, then5 N is replaced by B.

For the SE(B) specimen, the followingexpressions have been derived (ASTM E 1820-05, 2005k):

K =FS

f{a/W)

where

j\a/W)

2(1 \3/2

[9]

and S is the loading span of the specimen(Figure 6).

With these equations, the three most widelyused specimens for fracture toughness testingare characterized. For other specimen geome-tries, see the list of standards in the 'References'section. The application of the K formulas forthe determination of linear-elastic fracturetoughness values is described in Section11.02.4. K solutions for specimens not treatedin standard procedures can be either obtainedfrom handbooks (e.g., Murakami el ai, 1987/1992) or from finite element or boundary ele-ment analysis.

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Analysis 21

11.02.3.2.2 Limits of the applicability of thestress intensity factor

For most materials used in structural appli-cations, high toughness is desirable so thatthe structure would not fail before significantyielding occurs. In these cases, the laboratorytest piece exhibits substantial nonlinearity ofthe test record, which is a consequence ofwidespread plastic deformation and fre-quently also of crack extension. Up to acertain degree of plasticity, the specimenbehavior can still be characterized with K ifa plasticity correction is applied (see Chapter7.04.6.4.2). Beyond this stage, however, thecrack-tip field and hence the fracture behaviorcan then no longer be characterized by astress intensity factor. Nonlinear loading isusually characterized by the parameters./-integral and CTOD. At a given amount ofone of these fracture parameters, the degreeof nonlinearity depends on specimen size. Thespecimen with the larger planar dimensionsexhibits less nonlinearity at that value of thefracture parameter than the specimen withsmaller planar dimensions. If a specimen isvery large, then a plastic zone may developwhich is small as compared to the planarspecimen size; in that case, the specimenbehavior may even be characterized with thestress intensity factor. Thus, the characteriza-tion of the behavior of a specimen as brittleor ductile not only depends on the material,but also on the planar specimen dimensions.In such cases, the stress intensity factor canbe evaluated from the ./-integral (eqn [25]).First the ./-integral method is explained.

11.02.3.3 Elastic-Plastic Fracture Toughness

11.02.3.3.1 /-integral

./-based fracture toughness testing is standar-dized by numerous national and internationalorganizations, such as ASTM, BSI, ESIS, andISO. The individual details of each procedureare not described here, but some of the detailsnecessary for /-testing and features commonto all procedures are discussed. The C(T) andSE(B) specimens are most widely used for/-testing; M(T) specimens are sometimes alsoused in the cases where low-constraint beha-vior has to be investigated (see Section11.02.4.3.3). For the evaluation of/, the rela-tive displacement of the loading points (or adisplacement which can be related to theLPD) and the applied force have to be mea-sured, because / is an energy-basedparameter. The formulas for the evaluationof / from the test record partition the amount

of energy consumed in the test up to thepoint to be evaluated in the linear-elasticand the plastic components. For the C(T)and SE(B) specimens, the following equationsare proposed by ESIS P2-92 (1992):

• /<>=•

K2

£(1 - v2) + B{W - a0)'

where v is the Poisson's ratio and

Vp| = 2.0 + 0.522(l - — \

for C(T) specimens

= 2.0 for SE(B) specimens

[10]

[ l l a ]

[ l i b ]

In these equations, the LPD is used for deter-mining the area, A, shown in Figure 18 whichrepresents the plastic energy consumed by thespecimen. It is the area between the loadingcurve and a straight line which is parallel tothe initial linear-elastic slope and goes throughthe point of the loading curve to be evaluated.Here it is assumed that no significant crackextension has taken place during the experi-ment, because otherwise that straight linewould not be parallel to the original linear-elas-tic slope. It would rather have to have a lowerslope according to the amount of crackextension.

The C M O D can also be used, but then

„ = 3.724-2.244-^ + 0 . 4 0 8 ^

for SE(B) specimens[12]

For C(T) specimens, it does not make muchsense to distinguish between LPD and CMOD,because if the slit in the specimen which isneeded for precracking is appropriatelydesigned (see Section 11.02.2) then the CMODis not needed.

For M(T) specimens, the ESIS method (ESISP2-92, 1992) proposes a slightly differentmethod; here, the area A* is determinedbetween the loading curve and the secant

LPD or CMOD

Figure 18 Definition of area, A.

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22 Classical Fracture Mechanics Methods

LPDorCMOD

Figure 19 Determination of A* from force-CMODdiagram (ESIS P2-92, 1992).

connecting the origin of the diagram with thepoint on the diagram to be evaluated(Figure 19):

, * 2 , A'E B{W-a0)

[13]

It has been shown that the plastic parts of theLPD and C M O D of an M(T) specimen areequal. This eases the determination of J onthis specimen type since the C M O D is mucheasier to measure than the LPD.

The preceding J equations are for the case ofa stationary crack. If there is crack extension, asis the case with Ä-curve development, the equa-tions require a crack extension correction. Asimple correction of the J equation for crackextension is provided by ESIS P2-92 (1992):

= J o \ \ -(0.75>/-

(W -a)[14]

where Jo is the J value as determined by theequations above. This correction is suitablefor the determination of an R-curve using themultiple specimen method where each datapoint on the curve is from one test. The crackextension, Aa, is measured on the fracture sur-face as described in Section 11.02.2.5.2.

The ASTM procedure for R-curve develop-ment uses a different method for crackextension correction (ASTM E 1820-05,2005k). The area, A*, needed for a Aa-correctedJ is not directly measured in the test. It is thearea relating to the instantaneous crack lengththat, for the purpose of evaluation, is treated asa stationary crack (Figure 20). The resultingformula is based on an analysis by Ernst et al.(1981):

cu(J

£

LPD

Figure 20 Correct and measured areas in a ./-basedtest.

J =- v2)

[15]

where

K/-D+-

1 - ?('-•)"('-!)

[16]

where / refers to the current value of J and(/ — 1) the previous one. (Af/j — A(i_\^} is a plas-tic area between loading increments and y is afactor defined in the standard. It takes differentvalues for different specimen geometries.

These J evaluation procedures are just exam-ples; they may slightly deviate in other standards(see the standards in the 'References' section).

11.02.3.3.2 Crack-tip opening displacement

CTOD measures a displacement near thecrack tip and hence provides a direct character-ization of the crack-tip stress and deformationfields. It was the first method of fracture tough-ness measurement that was proposed fornonlinear deformation behavior (Wells, 1961).It has also been the basis for the first compre-hensive structural assessment method, thedesign curve; see Chapters 7.01 and 7.04 ofVolume 7. It is interesting to note that the/-integral approach was developed in the USfor application in the nuclear industry where itis still being used, whereas the CTOD approachhas its origin in the UIC, with the major appli-cation in the offshore industry. Originally, therewas some discussion regarding the merits ofusing a CTOD fracture parameter versus aJ parameter. Later it was acknowledged thatthe two are related and represented different

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Analysis 23

ways of measuring the same toughness (Dawes,1979). The two parameters can be simplyrelated by

S = mJ /o\ [17]

where 6 is the symbol for C T O D , try is the yieldstrength, and in is a constant depending onmaterial and constraint.

The CTOD, as originally developed, mea-sured a value of crack opening at the originalcrack tip using a small paddle (Wells, 1961). Asthis was difficult to measure, a new techniquewas developed and standardized by BSI (BS5762, 1979), which infers the CTOD from aremote displacement measurement, theCMOD. This standard method uses SE(B) spe-cimens (Figure 21) and partitions the totalCTOD into an elastic and a plastic componentfor 6:

[18]rpi(W - a0) + a0 + z

where rp](lV-ao) — 0.44 defines a stationaryhinge point around which the specimen rotates,z is the distance of the displacement measure-ment position from the specimen's front face,and v is Poisson's ratio. Later work has shownthat the same method can be applied to C(T)specimens as their uncracked ligament under-goes essentially bending loading. For thisspecimen type, the hinge point is located atrpi(W-a0) — (0.46 to 0.47), and z is the distanceof the displacement measurement point fromthe load line (ASTM E 1290-02, 2005i, ASTME 1820-05, 2005k).

Whereas this method can only be applied tobend-type specimens, a method developed bySchwalbe (1995) measures the C T O D on thespecimen surface at the fatigue precrack tipover an original gauge length of 5 mm(Figure 12c). The resulting quantity is desig-nated <55 and can be measured on anyspecimen or structural component with a sur-face-breaking crack. It is standardized invarious procedures (ESIS P3-05D, 2005; ISODIS 22899, 2005; GKSS E F A M GTP 02,2002). This technique provides a direct

measurement; it does not need any kind of cali-bration functions. The clip gauge shown inFigure 12c is relatively easy to produce; how-ever, its attachment to a specimen requiresattachment parts that have to be designed indi-vidually for the specimen configurations to betested. Alternatively, remote sensing methodsare also recommended. The 65 has been usedfor the determination of/^-curves and of da/dtdiagrams for stress corrosion and creep condi-tions (Schwalbe, 1998).

11.02.3.3.3 Crack-tip opening angle

Both the ./-integral and the C T O D techni-ques described above measure accumulatedquantities as a function of crack extension.This raises the fundamental question aboutthe conditions at the moving crack tip.Physically, the failure conditions during crackextension should be constant. A further para-meter, the CTOA, has been proposed todescribe stable crack extension. Its particularstrength is in describing large amounts ofcrack extension in thin-walled structures. TheCTOA is the angle included by the flanks of anextending crack (Figure 22). After an initialtransition period, the CTOA remains constant,that is, it is independent of the amount of crackextension (Dawicke et eil., 1995; Heerens andSchödel, 2003; see also Section 11.02.4.3.3.2).This constant angle is designated criticalCTOA, ijjc, which can be used for structuralintegrity assessments.

According to numerical analyses, the deter-mination of the CTOA should be facilitated bythe fact that the crack flanks remain linearduring crack extension in these analyses(Figure 23). In this way, the angle can be deter-mined according to the relationship

[19]

where 6 represents the opening displacement atthe distance from the crack tip, rm (Figure 22).Ideally the distance rm is held constant, whichcan be easily done in finite element analyses.Several techniques for determining the CTOA

Force

Figure 21 SE(B) specimen showing rigid rotationdefinition for CTOD measurements.

Figure 22 Definition of the CTOA (ISO DIS 22899,2005).

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24 Classical Fracture Mechanics Methods

have been proposed; they are briefly describedin the following.

The probably most obvious method to mea-sure the CTOA is the optical measurement of i//on the specimen side-surface. However, if i// isto be determined experimentally in the neigh-borhood of the moving crack tip, it turns outthat in reality, on a microscopic scale, the crackflanks are quite irregular, making the determi-nation of \p difficult (Figure 24), and leading tolarge scatter (Figure 25). In addition to thatirregularity, the low values of \p contribute toscatter, as well as the difficulty to identify theactual crack tip which is frequently obscured bythe large amounts of plasticity occurring at thecrack tip. Usually, the CTOA is defined by twostraight lines going through the crack tip and apair of points located on opposite sides of thecrack flanks (Figure 26). The difficulty in find-ing the actual position of the crack tip can beavoided if the crack tip is replaced by a secondpair of points (Figure 26). In order to reducescatter, it is recommended to perform thesemeasurements using several pairs of points asshown in Figure 26 and to take the average ofthese individual measurements, i/f,

[20]

where

1 ' I ' I ' I

0.6,\ -

Stable crackextension

(mm 0.4 <|

:0.2 (

r1 u V~O~) 0 O—TL

)—0—O—<v

>-o—o-o_

1 1 1

0 10

oo-oo.

1

20 30

x1 (mm^50

Figure 23 Finite element analysis of stable crackextension in AI 2024-T3 showing linear crack flanks(deKoning, 1978).

[21]

The digital image correlation techniqueemploys a video camera, which is translatedparallel to the specimen surface, thus keepingthe crack tip within the field of view (Suttonet ai, 1994; ISO DIS 22899, 2005). A high-contrast random pattern is applied to the speci-men by spraying white acrylic paint on thespecimen surface and adding diffusely spreadblack toner powder. The pictures taken by thecamera are evaluated for displacements acrossthe crack and then for the CTOA.

Probably the earliest method for obtaining theCTOA uses the infiltration technique which is amultiple specimen method because one specimenis needed for each data point. A number ofnominally identical specimens are loaded up todifferent displacements. A replica material isthen infiltrated into the crack, whereby the speci-men should be under load. The replicarepresents the open crack and can be sectionedparallel to the crack extension direction to revealthe through-thickness variation of the crack pro-file and hence of the CTOA.

A hybrid method uses a finite element analysisalong with the experimental force-displacementand crack-length data (Sutton et ai, 1994). Thisanalysis extends the crack through the materialby keeping a chosen value of the CTOA constant.This procedure is repeated with different valuesof i/f until a characteristic quantity (e.g., the max-imum force attained in the test) is met within acertain accuracy (Newman, 1984; Seshadri et ai,1999). The same procedure can then be used toanalyze cracks in structural components by usingthe known value of the CTOA for the crackextension analysis for the determination of theload-bearing capacity of the component (Hsuel at., 2002; Seshadri et al., 2002).

There have also been attempts for determin-ing the CTOA from <55 R-curve measurements(Heerens and Schödel, 2003; Brocks and Yuan,1991). Here 6 is given by 65 and rm = Aa. The

Aa= 1.32 mm

Distance frominitial prefatiguecrack position

100 um j

-100|im -

1.5

Figure 24 CTOA determination complicated by irregularity of crack flanks (ISO DIS 22899, 2005). RandyLloyd, Idaho National Laboratory, unpublished work.

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Fracture Behavior 25

A 2 C(T), W- 50 mm. a/W=0.5i A C(T). kV= 150 mm, a/W=0.5<> M(T), W= 150 mm. a/kV=0.2O M(T). W=50mm, a/W=0.3• 2 Cruciform, W= 150 mm. a/W=0.2. X=0.5D Cruciform. IV=150 mm, a/W=0.2. X = -0.5

0 10 20 30 40 50 60 70 80 90 100 110

Figure 25 CTOA data determined as per Figure 26on various specimen geometries showing largescatter (Schödel, 2006).

r-i =0.1-0.2 mm

rn < 1-1.5 mm

Figure 26 CTOA determination with the four-pointmethod (Heerens and Schödel, 2003).

Optical determination

Indirect determination from <55

Figure 27 Indirect determination of the CTOA fromthe slope of the S5 R-curve (Schwalbe et ed., 2004).

principle is depicted in Figure 27. From theillustration, it follows that

AI 5083,= 3 mm,

2 C(T), W=50mmC(T). W=150mm

2C(T), W=150mm2 C(T), W=300mm

C(T), W= 1000mmC(T), W= 1000 mm

^ 4-

IO 0

<" 2

i!

• • •

°n cm'

* » A "

AI5083, M(T) e = 3mmO»lV=50mm, a/W=0.3m W=150mm, a/W=0.2n W=150mm, a/lV=0.2,A W= 150mm, a/W=0.2, ABGA W=150mm. a/H/=0.2, ABG

• V• • * •

0 5 10 15 20 25 30

Aa (mm)

"Äö"f22l

(b)

Figure 28 CTOA determined from the slope of theR-curve compared with optically obtained values:a, C(T) specimens; b, M(T) specimens (Schödel, 2006).

Since the 65 R-curve is relatively easy toobtain, exploitation of eqn [22] provides avery simple technique for determining theCTOA. Figure 28a shows good agreementwith direct measurements, whereas the agree-ment in Figure 28b is less good. Theobservations made so far suggest that eqn[22] can serve as an estimate rather than anaccurate method for determining the CTOA.If the CTOA has to be used for structuralassessment, it is recommended to determinei// on C(T) specimens with a minimum widthof 150 mm.

11.02.4 FRACTURE BEHAVIOR

This section deals with the response of aspecimen to the loading applied in a test.Crack extension is characterized using thetools presented in Section 11.02.3, wherebythe choice of the analysis tool depends onthe global specimen behavior in terms of theforce versus displacement record. Specimenbehavior is also discussed by means of themicromechanisms of fracture and in particu-lar as a function of the deformationconstraint present in the specimen, a para-meter of paramount importance in fracturemechanics. Detailed reference to specific teststandards is made as appropriate.

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26 Classical Fracture Mechanics Methods

11.02.4.1 Regimes of Behavior of a Specimenin a Fracture Toughness Test

11.02.4.1.1 Deformation behavior

Unfortunately, the fracture process dependson numerous parameters, given by material,test temperature, loading rate, environment, aswell as constraint conditions in the specimen orcomponent. The latter in turn depends on sizeand geometry of the specimen or structuralcomponent and the proximity of the appliedforce to the yield force, which is a measure ofthe degree of plasticity, to name only the mostimportant factors. For more details, see Section11.02.4.4. In the following sections, the variouskinds of behavior are described in detail;however, the effects of environment, highloading rate, and (high) temperature arediscussed in separate chapters, in Chapters11.03, 11.04, and 11.05, respectively.

An important aspect of fracture behavior isrelated to the amount of deformation a speci-men undergoes during a test:

• The specimen may fail within or shortlybeyond the linear-elastic slope of the force-deformation record (for the deformation,usually the CMOD or the load-line displace-ment are used). Such behavior is calledbrittle since the global behavior of the speci-men is close to linear elastic.

• If the specimen fails after substantial plasticdeformation, which is manifested by asizeable nonlinear section of the test recordbeyond the linear-elastic slope, then thisbehavior is called ductile since the globaldeformation characteristics of the specimenare elastic—plastic.

The terms 'brittle' and 'ductile' are frequentlyconfused with the micromechanisms of frac-ture: the cleavage micromechanism occurringin steels at lower temperatures represents a'brittle' mechanism since experience showsthat when steels fail in a brittle manner, thencleavage is the dominant micromechanism.Figure 29 shows an example of cleavage frac-ture in a typical pressure vessel steel. However,cleavage may also occur after much plasticdeformation and after stable ductile crackextension. The mechanism giving rise to dim-ples on the fracture surface is usually called'ductile', since these dimples are a consequenceof extremely large deformations on a micro-scopic scale. A fracture surface of the samesteel shown in Figure 29, but now at a highertemperature, exhibits the dimple mechanism(Figure 30). However, dimples can also beformed if the high deformations are restricted

Figure 29 Fracture surface of steel pressure vessel22NiMoCr37 covered with cleavage facets.

Figure 30 Fracture surface of steel pressure vessel22NiMoCr37 covered with dimples.

to a very small plastic zone at the crack tip. Inthis case, fracture occurs also in a brittle man-ner since the global failure characteristics arelinear elastic.

A further aspect is given by the fracturetoughness evaluation related to the amount ofcrack extension:

1. Single-valued fracture toughness, orpoint values of fracture toughness, are deter-mined if little or no stable crack extensionoccurs prior to failure. Then the fracturetoughness value is taken at the point of fail-ure of the specimen.

2. However, even if substantial crackextension occurs, frequently point values offracture toughness are determined near theonset of crack extension, since in many casesstructural assessment is based on a singlevalue of toughness.

3. If substantial crack extension occurs dur-ing a test, then the variation of fracturetoughness (or of resistance to crack extension)with the amount of crack extension can bedetermined. The result is the /?-curve.

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Fracture Behavior 27

In all these cases, the resistance of crackextension can be evaluated in terms of K, J, 6,or i/f, as appropriate.

11.02.4.1.2 Crack behavior

Loading a precracked specimen leads at acertain, material-specific, point to the extensionof the existing crack. The following statementsrefer to tests with force control, which occursalso in many structural situations.

Note: This is the more conservative case,since displacement control leads to morestable behavior of a test specimen or struc-tural component. The problem withdisplacement control is that strict displace-ment control is difficult to achieve. Althoughclosed-loop servohydraulic test machines canreact very fast to sudden load drops and keepa specimen stable in this way, the elasticenergy stored in the loading chain maybe sufficient to break a specimen with lowtoughness. An extremely stiff test setup isdepicted in Figure 31. A further stiff arrange-ment is realized by testing DCB specimensas discussed in Chapter 11.03. In both tests,the stress intensity factor decreases with cracklength and even relatively brittle materials canbe tested in a stable manner. These brief com-ments show that the terms stable and unstablecrack extension have to be seen in the contextof the actual test method.

1. Crack extension may occur under increas-ing force; this process is then called stable crackextension. Crack extension can be stopped ifthe applied force is held constant.

2. Crack extension may also occur in anunstable manner, that is, during this processthe applied force drops and the crack can nolonger be prevented from extending. Initiation

of cleavage cracks in ferritic steels is a typicalexample of such behavior. However, unstablecrack extension may also occur if the dimplemechanism is active. This is typical of high-strength, age-hardened metallic alloys.

3. Unstable crack extension may occur with-out any or after only very little prior stablecrack extension.

4. Unstable crack extension can be observedafter substantial stable crack extension.

11.02.4.2 Unstable Fracture with Little or NoPrior Stable Crack Extension

11.02.4.2.1 The KIc standard test method

Unstable fracture with little or no priorstable crack extension is of greatest concernfor the integrity of an engineering structure,because it may lead to a sudden catastrophicfailure of the entire structure. All kinds ofmicromechanisms may be involved here,because, as it was pointed out above, the globalresponse is not related to a specific microme-chanism of fracture. Therefore, and due to theoccurrence of some spectacular failures, stan-dardization of fracture mechanics test methodshad been focused on just this kind of structuralresponse, leading to the K\c test standard.

The plane strain, linear-elastic, K\c test wasthe first standard fracture mechanics test devel-oped which used a fatigue precracked specimen.It was first standardized by ASTM E 399-70T(1970). The original test method was based onwork by Brown and Srawley (1966) and desig-nated a tentative standard in 1970. Itsorganization and procedure became a modelfor other fracture mechanics test standards.Later, other standards organizations developeda Klc test (BS 5447, 1977; ISO 12737, 1996), but

Wedge

Specimen

Split pin

Bose block

Figure 31 Wedge-loaded compact specimen (C(W)) (ASTM E 561-98, 2005g).

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28 Classical Fracture Mechanics Methods

the form of the standard was very similar to theoriginal ASTM version.

The K]c measures fracture toughness which ischaracteristic of predominantly linear-elasticloading with the crack-tip region subjected tonear-plane-strain conditions through the thick-ness. The test was developed for fractureconditions with a microscopic ductile mechan-ism but can also be used for cleavage fracture.For ductile fracture mechanisms, stable crackextension usually occurs, accompanied by anincrease of the fracture resistance. However, asingle point to define the fracture toughnesswas desired. To do this, a point where the duc-tile crack extension equals approximately 2%of the original crack length was identified; thisrepresents a single measurable point on the7?-curve. This criterion for choosing the measure-ment point gives a fracture toughness which issomewhat dependent on the specimen size.Therefore, validity criteria were chosen to mini-mize the size effects as well as to restrict theloading to essentially the linear-elastic regime.

The five specimen geometries shown inFigures 1, 5, and 9 are allowed. Whereas theC(T) and SE(B) specimens are used in mosttests - including tests other than K\c tests - theother three are special geometries representingstructural component forms. Continuous mea-surement and recording of force and CMOD isrequired during the test. The force must beapplied so that the increase in K is in the rangeof 0.55-2.75 MPa m1/2 s"1. The loading is donein displacement control.

The force versus displacement record com-prises the basic data of the test. The data arethen analyzed to determine a provisional A"Ic,labeled A'Q. This provisional value is deter-mined from a force, FQ, and the crack length.The FQ is determined with a secant line ofreduced slope on the force versus displacement

record (Figure 32). The construction for FQinvolves drawing the original slope of theforce versus displacement record; then thesecant slope which is 5% less than the originalone is drawn. For a monotonically increasingforce, the FQ is taken where the 5% secant slopeintersects the force versus displacement curve;this is illustrated as type III in Figure 32. Forother records in which an instability or othermaximum force is reached before the 5% secantslope, the maximum force reached up to andincluding the possible intersection of the secantline is the FQ. Type II illustrated in Figure 32 isan example of one of the other test records. The5% secant corresponds to about 2% ductilecrack extension, that is, 0.02 of the originalcrack length. This raises two fundamental pro-blems (Münz et al, 1976; Munz, 1979):

1. The compliance does not distinguishbetween nonlinearities of the test record whichare caused by crack extension or plasticity.

2. The intersection point is not a point on the7?-curve at an absolute amount of crack exten-sion, but an amount that increases withspecimen size; therefore, the thus-determinedfracture toughness may somewhat depend onspecimen size. Unstable failure before reachingthe 5% secant also marks a measurement pointfor FQ.

The FQ value is used to determine the cor-responding KQ value from the K expressionfor the specimen type tested. The subscriptQ used with F and K indicates provisionalvalues of these quantities. If KQ passesthe requirements for a valid test, then it isa valid K\c value. The two major validityrequirements are

1-10 [23]

CD•2OJ_

Fc — Fm ax

// /I II/

' / ' 'IIII

ll1'I -s.

// X

Ai /

/ ii i

.95a

F c ^

/

liI /

/ /IIi

1/li

liII

A

//

i

1 S~Fm

^ F 5

/

ax

i/

III // /ll

llll

IIII

1/

Aj

ii

II' ii

/ /^ Fmax

~> C =F 5

) 0 0Displacement

Figure 32 Typical force-displacement records in a A"|c test.

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Fracture Behavior 29

which limits the R-curve behavior to an essen-tially flat character and ensures some physicalcrack extension, and

[24]

This is to guarantee that the specimen is largeenough to ensure linear-elastic loading and planestrain thickness. In this equation, o\ is the mate-rial's yield strength, either the 0.2% offset yieldstrength or, for materials with a Lüders plateau,the lower yield strength, 7?eL. Equation [24]shows clearly that low-strength/high-toughnessmaterials sometimes require very large speci-mens. As a consequence, the limits of the testmachine capacity may be reached; furthermore,the available material may not allow the fabri-cation of specimens with a size meeting therequirement of eqn [24], The standards containmany more requirements which have to be metfor a valid plane strain fracture toughness test.

11.02.4.2.2 The CTOD standard test methods

This method appears in several standards(ASTM E 1290-02, 2005i, ASTM E 1820-05,2005k; BS 7448-1, 1991, 7448-2, 1997a; ESISP2-92, 1992; ISO 12135, 2002). All methods usethe standard C(T) and SE(B) specimens with anear-square ligament which is achieved by thegeometric conditions W=2B and a/i¥~0.5.When testing weldments, which is the primearea of application of this test method, a furtherSE(B) geometry is allowed, where B=W.

The CTOD test method is useful when thethickness of the material to be characterized isnot sufficient to determine a valid Klc value,which is the case for high-toughness structuralsteels. The specimen thickness is usually equalto the thickness of the material. Therefore, novalidity criteria related to the specimen size areapplied to the test result.

For the determination of a 6 value, eqn [18]represents the original evaluation method. Toavoid using this rigid rotation model, a new setof CTOD equations have been formulated inASTM (2005i) using the area under the forceversus displacement record modified by coeffi-cients which relate the area directly to theCTOD. In this way, the calculation is similarto that for J and emphasizes the similarity ofboth parameters.

The basic idea of the test method is to calcu-late a fracture toughness point for brittlefracture or to evaluate a safe point for the caseof ductile fracture. The brittle fracture tough-ness is measured at the point of instability. Theductile fracture toughness is measured at theonset of the maximum force; this is not included

in the standards ESIS P2-92 (1992) and ISO12135 (2002). The fracture points to be evalu-ated are:

• 6C, 65c is a point of unstable fracture after lessthan 0.2 mm of stable crack extension. Insteels, stable and unstable crack extensioncan be distinguished from each other sincethe former exhibits a dimpled fracture sur-face whereas the latter occurs by cleavage.Both mechanisms can be easily recognized.

• <5u> 5u ' s a point of unstable fracture aftermore than 0.2 mm of stable crack extension.Some standards (ESIS P2-92, 1992; ISO12135, 2002) require that the amount ofstable crack extension has to be reported.

• <5uc> suc is the symbol to be used if theamount of stable crack extension cannot bemeasured.

• <5m; 5m 'S the CTOD at the point of max-imum force.

The CTOD test is frequently used to charac-terize the fracture behavior of steels and theirweldments in the ductile-to-brittle transitionregime (see Section 11.02.4.2.4). In addition tothe determination of single-valued fracturetoughness, the CTOD is also used for the deter-mination of/?-curves (see Section 11.02.4.3).

11.02.4.2.3 J testing

Characterization of a material with the/-integral follows the same principles outlined inthe previous section. In particular, the same sub-scripts are used to indicate the kind of crackextension behavior. J is evaluated as describedin Section 11.02.3.3.1 Critical J values are fre-quently converted to critical K values (ASTME 1921-05, 20051) for use in structural assessment:

[25]

11.02.4.2.4 Ductile-to-brittle transition ofsteels

It has been known for about a century thatCharpy tests on specimens made of ferritic steelsexhibit a sharp drop in the toughness value whenthe test temperature is lowered. This is known asthe ductile-to-brittle transition of steels. The sameeffect is observed in fracture toughness tests.Figure 33 shows this behavior in a schematic dia-gram which also shows the regimes of the varioustoughness designations introduced above. Thebroad band shown in this diagram indicates thelarge scatter observed in the transition region.Individual data points are plotted in Figure 34which were obtained on a large number of C(T)

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30 Classical Fracture Mechanics Methods

T5(0

LL

Instability due to cleavage initiationafter ductile crack extension

Instability due tocleavage initiation

Instability givenby fl-curve/

P, = 97% Size independent

Transitiontemperature(P, = 3%)

Temperature

Figure 33 Schematic showing the behavior of ferritic steels in the ductile-to-brittle transition range.

•5 700

E

inCOo>c.cO)o

o2CD

O

-g 600 H

öH 500.sto

5'S 400-1

_5CD

•A

0D

ftBT

C(T)M(T)M(T)C(T)C(T)C(T)C(T)C(T)

C(T)DENT

(W= 100, 6 = 50)(W=45, 8=18, a/W= 0.22)(W=45, 6=18. a/W= 0.61)(W=50, 6=20, 8, = 16)(W= 100, 6=20, 8„=16)(IV=2OO, e = 20, fl, = 16)(W=50,S = 25, 8n = 20)(W= 100, 8=50, 6„ = 40)

(W=200, 8=100, 8n = 80)(W=45, 8=18, a/W=0.B)

oooooo

Q

"cO

initi

<DO lCD

o

300-

200-

v/mss/sss.

o>c 100-

All measurements in mm

20MnMoNi55

-150 -130 -110Temperature

r-90

(°C)

-70 -50

Figure 34 Influence of specimen size, specimen geometry, and test temperature on cleavage fracture toughness(Zerbst et al., 1993). Reproduced from Schwalbe, K.-H. 1998. The engineering flaw assessment method(EFAM). Fatigue Fract. Eng. Mater. Struct. 21, 1203-1213, with permission from Blackwell Publishing.

and some M(T) and DE(T) (double edge crackedtension) specimens (Zerbst et al., 1993). Below aJ value of 150Nmm~', cleavage occurs rightfrom the blunting phase or with ductile crackextension less than 0.2 mm; hence these valuesare labeled as Jc. The data above 150Nmirr 'are Ju values, since here cleavage is preceded bymore than 0.2 mm of ductile crack extension.Among other less obvious items, the diagramdemonstrates that scatter of toughness dataincreases with temperature.

A different way of demonstrating fracturetoughness scatter is depicted in Figure 35, wherefor different temperatures and specimen sizes thecritical 7 values are plotted versus crack extension.This way an 7?-curve is constructed where each

data point was obtained on one specimen, that is,the diagram represents a multiple-specimen7?-curve. It is seen that the R-curve is inde-pendent of specimen size and temperatureand also that the amount of crack extensionprior to cleavage tends to be smaller forlarger specimens. The diagram shows thetwo mechanisms of fracture in ferritic steelswhich are typically observed in the transitionregion, namely, stable crack extension andcleavage. Stable crack extension is character-ized by the trend of the J versus Aa curveand cleavage is characterized by the scatterof the point on the 7?-curve where an indivi-dual specimen fractures. The amount ofstable crack extension prior to cleavage

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Fracture Behavior 31

1400Brittle fracture due tocleavage initiation

a

AÄj^y* O Q

• S Ü ^ D O

(f * *O

22NiMoCr37 A

DO0

1/2TC(T)

1TC(T)

1TC(T)s.g

2TC(T)

4T C(T)

PrecrackedCharpy

ao/W=0.55

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Aa (mm)

Figure 35 Correlation between cleavage fracture toughness (Jc, Ju) and the amount stable crack extension visibleat the fracture surface of C(T) and precracked Charpy specimens (Heerens el al., 2002).

11109876543210

* 1/2TC(T)o 1TC(T)a 2TC(T)o 4TC(T)

ao/W=O.55

Unstably fractured specimens

22NiMoCr37

Side-grooved o £o L

o i

4°JLs_

-40 -20 -10

T(°C)

20

Figure 36 Influence of test temperature and specimen size on the amount stable crack extension visible on thefracture surface of C(T) specimens, which failed due to the initiation of cleavage (Heerens et al., 2002).

increases significantly with increasing testtemperature (Figure 36), as does scatter.The reverse trend is observed with increasingspecimen size.

In order to obtain a meaningful characteriza-tion of cleavage initiation toughness, severalstatistical methods for analyzing small specimentest data have been developed. The statisticalmethods are based on the assumption that clea-vage initiation can be described by a weakest linkmodel. It is assumed that cleavage initiation spots- representing the weakest links - are randomlydistributed in the material where each spot has anindividual critical stress for cleavage initiation.Brittle fracture of the specimen occurs if at oneof the randomly distributed initiation spots thecritical stress is reached. In case of testing pre-cracked laboratory specimens, this idea isqualitatively visualized in Figure 37. The maxi-mum principal stress ahead of a bluntedprecrack tip in relation to randomly distributedcleavage initiation spots ahead of the bluntedcrack tip is shown. With increasing load, the max-imum principal stress is shifted toward theuncracked ligament. Depending on the distance

between the crack tip and the initiation spots andtheir individual cleavage initiation stress, brittlefracture initiation may occur at lower or higherloads. Because of the statistical spatial distribu-tion of the cleavage initiation spots including theirindividual critical stresses, large scatter of thecleavage initiation toughness has to be expected.

According to Figure 37, an increase of thespecimen size, in particular the specimen thick-ness, increases the amount of material sampledby the crack-tip stresses; hence, the probabilityto find a cieavage initiation spot near the cracktip increases, which promotes smaller tough-ness scatter and lower cleavage initiationtoughness. Initiation of stable crack extensionis expected to occur in cases where the distancebetween the active cleavage initiation spot andthe fatigue precrack tip is large. In such cases, itis likely that the displacement applied to thespecimen has to be increased beyond the levelof stable crack initiation, until the stresses reachthe critical value at the initiation spot.

In view of the large toughness scatter, assess-ment of structural components, which may failby cleavage, should be performed on a

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32 Classical Fracture Mechanics Methods

statistical basis for low failure probability. Tothis end, methods for analyzing cleavage initia-tion toughness data have been developed,which are aimed at estimating a lower-boundcleavage initiation toughness of the material.

A standardized statistical procedure is out-lined in ASTM E 1921-05 (20051) as the mastercurve method. It was developed by Wallin(2002). The main items of that method areshown in Figure 38. A minimum of six tough-ness data are required in order to apply themaster curve approach (see Chapter 11.09). Inthis method, the toughness scatter is modeledusing a three-parameter Weibull distribution.Two parameters of the distribution are fixedand one parameter has to be calculated from aset of cleavage initiation toughness data. The

CTyy

'c (min)

SZW

Figure 37 The stress distribution ahead of a crackacts as a probe for weakest links. Increasing appliedload shifts the stress distribution along the x-axis.

Fitting to data scatter at T= const.

Reference temperalure T

Temperature, T

Figure 38 Basic outline of the master curve methodfor statistical treatment of cleavage fracture toughnessdata (ASTM E 1820-05, 2005k; Wallin, 2002).

method also includes the prediction of speci-men size effects. In addition, the master curvecan be used to assess the temperature effect oncleavage initiation toughness. At the end of theanalysis, the method delivers a temperature-dependent lower-bound cleavage initiationtoughness, related to a particular probabilityof cleavage initiation. For more details on themaster curve method, see Chapter 11.08.

Besides the master curve method, othernonstandardized methods have been deve-loped, which are mainly used as in-houseprocedures for benchmarking. The engineeringlower-bound method, developed at GKSS(Zerbst et ai, 1998), allows the determinationof a lower-bound toughness value from single-temperature data sets. A minimum of six validdata points are also required in order to obtaina lower-bound toughness estimate. The methodis summarized in Figure 39. It shows a prob-ability plot Pf—fiJc), which is fitted to anexperimentally derived single-temperaturetoughness data set. In this method, the upperpart of the probability curve is modeled using a

reoi/)

AB 6 T= const.

Aa

Linear fit

J L B = 0.26 Jc ß

ß = 1 + 1.286p

Jc = mean of all valid Jc-data in a data set

p = fraction of nonvalid data in a data set

Validity requirement: Jc < <T7 (W- ao)/3O

Figure 39 Engineering lower-bound method for thestatistical treatment of cleavage fracture toughness data(Zerbst et ed., 1998). Reproduced from Schwalbe,K.-H. Heerens, J. 1993. .R-curve testing and itsrelevance to structural assessment. Fatigue Fract. Eng.Mater. Struct. 21, 1259-1271, with permission fromBlackwell Publishing.

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Fracture Behavior 33

two-parameter Weibull distribution and thelower part, Pt < 50%, is modeled by a linearfit. This linear fit is motivated by the observa-tion that experimentally derived probabilityplots based on very large data sets seem tofollow a straight line rather than a curved one.In this procedure, a lower-bound toughnessvalue is obtained by the intersection of thestraight line with /c-axis.

The exponential curve-fitting method shown inFigure 40 has been developed to derive a lower-bound toughness from very large multiple-temperature cleavage initiation toughnessdata sets (Neale, 2002). In contrast to themaster curve method, no predictions of tem-perature and specimen size effects areinvolved in this data-fitting method. In thismethod, all toughness data, preceded bystable crack extension, have to be replacedby the initiation toughness of stable crackextension. In Figure 40, J02 designates ameasure of ductile crack initiation (seeSection 11.02.4.3). Jc data greater than Jc

are set equal to /0.2- Then, two exponentialcurves are fitted through the toughness dataas illustrated in Figure 40. These exponentialcurves describe the mean of the toughnessscatter. A log-normal distribution is assumedfor describing single-temperature toughness scat-ter. Similar to the master curve method, a lower-bound estimate can then be obtained by choosinga toughness level that corresponds to a low clea-vage initiation probability of about Pf = 5% orlower. In contrast to the master curve method, asuccessful application of the exponential curve-fitting method requires a very large number of

Datacensoring:

Data fitting:

EXDO J ' ' "^ O

0.2 mm Aa

Mean

\ P, = 5%

T<Tt

T>Tt

Figure 40 Exponential curve-fitting method for thedetermination of a lower bound of cleavage fracturetoughness (Neale, 2002).

cleavage initiation toughness data. This isbecause of the three independent parametersAo, A\, and A2, which need to be determinedwith high confidence by the statistical analysis.

Examples of application and validation of allthree statistical methods are provided byHeerens et al. (2002, 2005), Neale (2002), andWallin (2002).

11.02.4.3 Stable Crack Extension

11.02.4.3.1 Introduction

The typical microscopic process of stable crackextension in metallic materials is characterized bythe formation of dimples on the fracture surface.This mechanism is frequently called ductile tear-ing. The deformations near the tip of a crack leadto the formation, growth, and coalescence ofvoids in the ductile matrix of the material (seeChapter 2.06 in Volume 2). As demonstrated inSections 11.02.2.5.2 and 11.02.2.5.3, stable crackextension, Aa, is measured either visually orusing an indirect method. The values thusobtained of Aa are usually plotted as functionsof K, J, 6, or S5. The resulting curve shows thevariation of the crack extension resistance of thematerial tested and is designated R-curve, with Rbeing the symbol for resistance. The data pointsare fitted by an analytical curve; alternatively, theR-curve can be presented as a table of the datapoints. If the visual determination of crack exten-sion is used, then one speaks of the multiple-specimen method, whereas the single-specimenmethod is based on indirect methods as, inprinciple, these allow the determination of theR-curve with just one specimen. From theinitial part of the 7?-curve, point values offracture toughness can be determined whichare based on various definitions of the initia-tion of stable crack extension.

11.02.4.3.2 High-constraint testing: J andCTOD R-curves

These R-cur\e methods are usually applied tohigh-toughness materials in relatively thick-walled configurations, such as pressure vessels.Consequently, elastic-plastic R-curve test stan-dards have been developed for this area.Various J and CTOD R-curve test standardshave been developed by, for example, ASTM,BSI, ESIS, and ISO. Since the various stan-dards deviate from each other in severalplaces, the following discussion is based onISO 12135 (2002) and ESIS P2-92 (1992), ratherthan national standards.

The specimens for elastic-plastic R-curve test-ing are the same as those used for K\c testing,with the C(T) and SE(B) specimens being the

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34 Classical Fracture Mechanics Methods

most popular. As pointed out in Section11.02.3.3.1, the specimens for J testing requirea measurement of displacement on the load line(or a displacement measurement which can berelated to the load-line displacement), because Jis an energy-based parameter. The CTOD iseither inferred from measurements of theCMOD (see Section 11.02.3.3.2). For the deter-mination of /^-curves, eqn [18] has been modifiedfor crack extension (Hellmann and Schwalbe,1986):

< - * 2 ( l - v 2 )2ER p0.2

[26]

Alternatively, the surface measurement in termsof 65 according to Figure 12c can be used.

For the determination of an i?-curve, infor-mation on the crack extension as a function ofthe fracture parameter must be available. Thiscan be achieved using either the single- or multi-ple-specimen techniques.

Validity limits, Aamax , have to be imposed inthe Aa data. These limits are

Aamm = 0.25( W - a) for S and 85 [27a]

or

Aa m a x =0.\{W-a) {or J [27b]

For the multiple-specimen method, at leastsix nominally identical specimens should betested to provide a data distribution satisfyingthe requirements of the standard. The speci-mens are loaded to different amounts ofdisplacements to achieve different amounts ofcrack extension. The specimens are thenunloaded and the amount of crack extension ismade visible, as described in Section11.02.2.5.2. The data pairs of J or 6 versus Aaare plotted as shown in Figure 41. Ideally, thedata points should be evenly distributed. Eachof the four crack extension regions in Figure 41should contain at least one data point. Single-specimen methods provide a large number ofdata points so that the required data point dis-tribution can be easily achieved. The varioustechniques for determining the amount ofcrack extension are demonstrated in Section11.02.2.5.

The data points are fitted using the equation

<5,<55, o r 7 = C{Aa)L [28]

where A and C > 0 and 0 < D < 1.On this curve, validity limits for the fracture

parameters are imposed since beyond certainvalues these parameters are no longer represen-tative of the crack-tip field, and to achieveconditions of plane strain,

OC(0

1

1

" 2~

Exclusion linesi

Crack

A a ™ " "*"

3 Aa — ^4 m«

^ *

A a m a >

extension Aa

Figure 41 Data point distribution for /?-curvedetermination.

" m a x i <?5,max —

and

or

and

30

B

30

•/max = (W -Cl0)

•'max — -° .-),-.

20

[29a]

[29b]

[30a]

[30b]

where the flow stress, Rf, is (Rp02 + Rm)ß-From an 7?-curve, information on single-

valued toughness parameters, which character-ize initiation of stable crack extension, can bederived. As there is a gradual transition fromcrack-tip blunting to crack extension by ductiletearing, the onset of crack extension has to bedefined; this is done in a manner similar to thedefinition of a proof stress in a tensile test.Three different definitions have been devel-oped; these have been proved to be feasible ina comprehensive round robin (Schwalbe et ai,1993):

1. (50.2, <*>5,o2> JQ.2- This engineering definitionof initiation is defined by a vertical cutoff of the7?-curve at a value of stable crack extension of0.2 mm, that is, by the intersection of the/?-curve with a straight line parallel and offsetby 0.2 mm to the vertical axis. This magnitudeof Aa has been chosen since it can be easily

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Fracture Behavior 35

measured using a low-magnification opticalmicroscope.

2. S0.2/BL, 65,0.2//?/., JO.IIBL- This is the secondengineering definition of initiation; it is basedon the assumption that crack-tip blunting(which is included in the parameters listedabove) does not contribute to stable crackextension which is supposed to be due to ductiletearing only. In this case, the data point distri-bution (Figure 42) follows in principle that ofFigure 41, however, with the straight linesseparating the four crack extension regionsdrawn parallel to the blunting line:

[31a]

for the <57?-curve,

ÜB~

for the <55 7?-curve, and

AaB =3J5Rn

[31b]

[31c]

which is an estimate of the blunting process atthe crack tip. Figure 43 depicts the construction

of the fracture parameters. Additional datapoints needed for this construction are shownin Figure 43.

3. <5„ <55i, and J\. These parameters are sup-posed to represent true values of initiation.Their determination requires the use of a scan-ning electron microscope to determine thewidth of the stretch zone which develops atthe fatigue precrack tip before the dimplemechanism of stable crack extension becomesactive. Due to its microscopic nature the stretchzone width, Aa s z w , exhibits large scatter; there-fore, the initiation values of stable crackextension are also subject to large scatter. Thestretch zone width has to be determined at thenine local positions shown in Figure 13. At eachof these positions, at least five individual mea-surements have to be performed (Figure 15) andthe results have to be averaged to obtain the

< stretch zone width representative of the speci-men investigated. Since the stretch zone widthvaries substantially from specimen to specimen,at least three specimens have to be analyzed.The thus-obtained A«Szw values are thenplotted as shown in Figure 44. The intersectionof the average stretch zone width with the7?-curve defines initiation of stable crackextension.

//// .

/ // ;

1//

* ////

1

I

i

>

i

i i

Bluntingt

^ 11I

!

line

•*W

/

//

/i

I*—

Exclusion lines

I 1' 2 / 3

i

1I1I

/

4

* " /

/

/1

I

1 /

1////1/

/A a max

_l_^0A_mm_ C r a c k extension, Aa

Figure 42 Data point distribution for determiningfracture parameters (ESIS P2-92, 1992).

Figure 43 Determination of initiation parametersafter 0.2 mm of ductile tearing (ESIS P2-92, 1992).

11.02.4.3.3 Low-constraint testing

As already mentioned above, the fracturemechanics test standards are designed forlower-bound fracture toughness measurements.The increasing interest in lightweight structureswith high exploitation of their load-carryingcapacity and residual life has initiated the devel-opment of test methods for thin-walledmaterials whose fracture toughness is so highthat the limits of LEFM are exceeded.

<5U, <$5 iorj ,

• &-, 65-, or J-Aa data

o Valid stretch zonewidth data

© Invalid stretch zonewidth data

AaQ 1/ / 0.2 mm

Crack extension, Aa

Figure 44 Determination of initiation parameters(ESISP2-92, 1992).

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36 Classical Fracture Mechanics Methods

Although a test method for this purpose, basedon the 65 parameter, was proposed over 20years ago (Hellmann and Schwalbe, 1984),and later written in procedural form (GKSSEFAM GTP 94, 1994), and although theCTOA has long been known as a crack exten-sion parameter (e.g., de Koning, 1978; Shihet al., 1979), standardization using these para-meters has been started only recently.

. (i) The K R-curve standard test method

This is also one of the earliest fracture tough-ness tests, however, designed for testingspecimens made of thin sheet material whichbehave in a linear-elastic manner. Due to pre-vailing plane stress conditions, thin materialsusually exhibit substantial increase of fracturetoughness with crack extension which is repre-sented as an R-curve. In that case, structuralassessment with a single-point interpretation ofthe fracture behavior would lead to unduly con-servative results. The standard ASTM E 561-92(2005g) determines the R-curve as a plot of Kversus the effective crack extension, Aas. Themethod allows three specimens: the C(T) speci-men, the M(T) specimen, and the crack linewedge-loaded (C(W)) specimen; the latter oneis shown in Figure 31. The C(W) specimen doesnot need a test machine; it is wedge-loaded toprovide a stiff displacement-controlled loadingsystem. This can prevent rapid, unstable failureof the specimen under conditions where theR-curve toughness is low, so this allows theR-curve to be measured for larger values of Aae-

The instrumentation required on the speci-mens is similar to that of the Klc test exceptfor the case of the C(W) specimen. The basictest result is a plot of force versus CMOD.From this, an effective crack length is deter-mined from secant slopes (Figure 45). Thesecant slopes are analyzed using elastic compli-ance for the determination of an effective cracklength, ae. The effective crack extension is thedifference between the original crack length andthe current effective crack length. The effectivecrack extension can be a combination of physi-cal crack extension and crack-tip plasticity.Therefore, the effective crack length is equalto or greater than the actual physical cracklength.

K is determined as a function of the force andcorresponding effective crack length, using theappropriate equation for the C(T) or M(T) spe-cimens in the Section 11.02.3.2.1. In a C(W)specimen test, the force is not measured. Thedata collected are a series of displacementvalues taken at two different points along thecrack line, one near the crack mouth andanother near the crack tip. From the two

Elasticslope

Displacement, v

Figure 45 Construction of K R-curve. (ASTM E561-98, 2005g).

different displacement values, an effectivecrack length can be determined from the ratioof the two displacement values using a calibra-tion given in the standard. From the cracklength and displacement, a force is inferredand a K value can be determined and the K R-curve constructed, which is a function of thematerial thickness. Therefore, there is no valid-ity requirement relating to a thickness level aswith the K\c standard.

The resulting R-curve is subjected to a valid-ity requirement that limits the amount ofplasticity. For the C(T) and C(W) specimens

[32]

where b is the uncracked ligament length andKmax is the maximum level of K reached in thetest. For the M(T) specimen, the net sectionstress evaluated using the physical crack sizemust be less than the yield strength.

(ii) 55, CTOA, and J tests

The K R-curve discussed above has so farbeen the only means for characterizing materi-als in thicknesses so small that plane strainconditions are either not possible due to thethickness available, or plane strain characteri-zation would lead to overconservatism.Furthermore, due to the validity limits for lin-ear-elastic behavior, in many cases, thespecimen sizes have to be very large, thus lead-ing to substantial material consumption fortesting purposes and large test machines.

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Fracture Behavior 37

Therefore, standards are under development atASTM and ISO (ISO DIS 22899, 2005) whichare aimed at providing test methods using elas-tic-plastic concepts. These concepts are basedon the <55 definition of the CTOD and theCTOA.

The standardized specimens are the C(T) andM(T) types, both of which have to be testedusing antibuckling guides (see Section11.02.2.2). The specimen dimensions have tomeet the following requirement:

ao,(W-ao)>4B [33]

so that the 7?-curve is independent of the in-plane dimensions of the specimen. However, itis in general dependent on the thickness of thematerial tested.

Validity requirements for the <55 7?-curve deter-mined on C(T) specimens are the same as inSection 11.02.4.3.2. If M(T) specimens are tested,then the maximum allowable crack extension is

= W - a0 - AB [34]

In GKSS EFAM GTP 02 (2002), guidance isalso given for evaluating tests for the /-integral.

In thin-walled materials, the crack plane ofthe fatigue precrack (which is perpendicular tothe applied force) may deviate during stablecrack extension from its original plane to formshear planes at the specimen surface. Whenthese shear planes have the same slope to theprecrack, then crack extension takes place in asingle shear mode. When they develop differentslopes forming a roof-type fracture (the matingfracture surface then forms a V-groove), crackextension occurs in a double shear mode. Shearfracture slopes are typically 30°^0°. In thatcase, the fracture resistance is higher than thatof single shear mode, and the results are notqualified according to the standards. In addi-tion, the direction of crack propagation may

deviate from that of the fatigue precrack. Ifthe angle included by the two directions exceeds10°, the result is not valid.

The <55 tests are basically identical with thosedescribed in Section 11.02.4.3.2.

11.02.4.4 Constraint Effects on Fracture

Experimental research has shown that theresistance to fracture depends substantially onthe size and geometry of the specimen, on theloading geometry as well as on some otherparameters. Therefore, the fracture properties -either in terms of a single-valued fracturetoughness or an 7?-curve - cannot in generalbe regarded as material parameters. As aresult, a transferability problem arises in thata structural component may exhibit a fractureresistance that is very different from thatdetermined on a specimen fabricated in theform of a standard test piece. A typical caseis given by fracture properties determined on ahigh-constraint specimen, that is, a bend orcompact specimen (see Section 11.02.4.3.2),thus representing lower-bound properties,and a structural component subjected to amembrane stress state which may exhibit rela-tively high fracture values. At first glance, theeffects of numerous parameters on the fracturebehavior is confusing; however, all these para-meters affect the triaxiality of the stress statewhich, in turn, is responsible for the experi-mental observations. A more popular, thoughless precise, designation is constraint effect onfracture. In Figure 46, the most importantparameters affecting constraint are compiledin a schematic manner.

Numerous papers have been published in thisarea; in fact, the item 'constraint effects' on frac-ture was one of the major research fields infracture mechanics in the 1970s and 1980s, inparticular for elastic-plastic specimen behavior.ASTM dedicated two conferences to this topic

0.5S Crack depth, a/W

Strain hardening

Tension Bending 1aJa. Ligament/Thickness

/I

0 90°Crack front angle, O

Figure 46 Parameters affecting constraint.

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38 Classical Fracture Mechanics Methods

Figure 47 Schematic showing the effect of ligamentslenderness on crack extension resistance forspecimens with predominant bending loading.

(Hackett et ai, 1993; Kirk and Bakker, 1995). Anexample of constraint effects on fracture in theductile-to-brittle transition of ferritic steels isseen in Figure 34: the low-constraint M(T) speci-mens exhibit much higher toughness values thanthe C(T) specimens. Even within a specific speci-men geometry, constraint can exhibit substantialvariations: constraint in a C(T) specimen is highestfor a square ligament, that is if (W — a) ~ B. This iswhy the standards for high-constraint testingrequire W — B and aj W in the neighborhood of0.5. If the ligament becomes greater than B, whichcan be achieved for specimens with W muchgreater than B, then constraint decreases and thefracture resistance increases; this is the area oflow-constraint testing described in Section11.02.4.3.3. The fracture resistance increases withincreased ligament slenderness, S = (W— a)/B,until a saturation value of 3-4 has been reached(Schwalbe and Heerens, 1993; Figure 47).

It may be worth mentioning here that the pro-cedure GKSS EFAM GTP 02 (2002) is a unifiedmethod, containing the methods for low- andhigh-constraint testing, statistical evaluation,testing of weldments, and some other items.

11.02.5 FRACTURE TOUGHNESS TESTSFOR NONMETALS

The standardization of fracture toughness testmethods for nonmetals is relatively new comparedto standards for metallic materials. However, inthe past 10 years, several new standards have beenwritten for ceramics and polymer materials. Theseare usually patterned after similar standards for

metallic materials. The requirements to use thefracture mechanics approach for fracture tough-ness determination is that the materials arehomogeneous, isotropic, and have a macroscopicdefect. Since no material meets this requirement atall levels, it is required that they fit this criterion atsome size scale. Usually, this could be a size scalethat is related to the defect length. If the inhomo-geneities are small compared to the defect size,their effect does not disturb the continuummechanics assumptions on which the fracturemechanics approach is formulated.

To develop the correct test procedure, thedeformation behavior of the material must beconsidered to determine which fracture para-meter will be used to characterize the fracturetoughness results. To determine whether thefracture is characterized by a single point orby an R-curve, the fracture behavior of thematerial must be considered. A brief discussionis given here for fracture toughness testing ofceramics and polymer materials.

11.02.5.1 Ceramics

A discussion of the fracture toughness testingof ceramics should consider two differentgroups, monolithic ceramics and ceramicmatrix composites. The monolithic ceramicsare brittle and fracture in a linear-elastic man-ner. Therefore, the toughness can becharacterized by the K parameter. A fracturetoughness test procedure could be similar to theKlc test procedure given in ASTM E 399-90(20050 or ASTM E 1820-05 (2005k) followingthe methods used for metallic materials. Amajor problem for the fracture toughness test-ing of ceramics is the introduction of the defect.Since the toughness is so low, failure can occurduring a fatigue precracking procedure. Onefracture toughness method that has been usedfor brittle materials including ceramics is thechevron notch fracture toughness test (ASTME 1304-97, 2005j). This is one of the test meth-ods that does not require a fatigue precrack.ASTM E 1304-97 (1997a) was developed formetallic materials but could be used for manybrittle ceramic materials. Since the fracturebehavior of ceramics is brittle, the toughnesscan be measured as a single-point value. Thechevron notch test has a decreasing K drive asthe crack extends through the changing thick-ness portion of the specimen (Figure 48).Therefore, it provides some stability and allowsan easier fracture toughness measurement. Itshould be borne in mind that most ceramicsexhibit a rising 7?-curve, and due to this factdifferent test methods may lead to differentK]c values.

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Fracture Toughness Tests for Nonmetals

11.02.5.2 Polymers

39

Figure 48 Examples of chevron notch specimens.

A new standard, ASTM C1421-01b(2001), has been developed for the fracturetoughness testing of advanced ceramics atambient temperatures. In this standard,three different types of toughness valuescan be generated: fracture toughness froma precracked beam, A"Ipb; fracture toughnessfrom a chevron notch, K]vb; fracture tough-ness from a surface crack, A"Isc. The testspecimen is beam-loaded in three- or four-point loading. For the precracked beam,Klpb, the defect is generally introducedwith a Vickers hardness indentation at oneor more places. The precrack is then poppedin or it forms during loading. The precrackcould be popped in using a compressionloading fixture; however, fatigue loading isnot used for precracking advanced ceramics.The chevron-notched specimen, A^t,, uses aprocedure similar to ASTM E 399-90(20050, so a precrack is not used. The sur-face-cracked specimen, A"Isc, introduces adefect using a Knoop indenter. The threedifferent specimen types can give differenttoughness values with generally K\vb beingthe highest, /flsc being the lowest, and Kipb

being intermediate to these. The validityrequirement centers around the testing of astandard-sized beam specimen. This isusually 3 x 4 m m 2 in cross-section but canhave different lengths for the different testtypes.

Ceramic matrix composites have a more duc-tile-looking toughness character. Thesematerials exhibit an /?-curve type of behavior.In some cases, the deformation has a nonlinearcharacteristic. The nonlinear behavior can berelated to the formation of microcracks or thebreaking of bonds between the particles and thematrix and not to plastic deformation.Although the deformation process may not bethe same as plasticity in metallic materials, thenonlinear fracture parameters may still apply.Fracture toughness testing for these materials isstill largely in the experimental stages, so thetesting procedures have not been standardized.

A comprehensive overview on the fracturebehavior of ceramics can be found in Munzand Fett (1999).

Fracture toughness testing for polymers hasbeen studied extensively (Williams, 1984). Thefracture toughness behavior of polymers usuallyfalls into two classes, below the glass transitiontemperature, T&, and above T&. Below T&, thedeformation is nearly linear elastic and fractureis unstable. Therefore, a single-point toughnessvalue characterized by A" can be used. Above T6,the deformation is nonlinear and the fracturebehavior is stable cracking. A ./-based 7?-curveapproach can be used. Two problems that mustbe addressed in developing test standards forpolymers that make them different from stan-dards for metallic materials are the viscoelasticnature of the polymer deformation behavior andthe problem of introducing a defect by fatigueloading. The viscoelastic deformation charactermakes the fracture toughness result dependenton the loading rate; therefore, the loading ratemust always be specified in the test report.Comparison of toughness results for polymersshould always be made with awareness of theeffect of loading rate. Also, due to the viscoelasticnature of polymers, the introduction of the defectis not easy with fatigue loading and the crack isusually introduced with a razor blade cut.

Fracture toughness testing of polymer materi-als has been standardized in the past few years.ISO has a standard for measuring Glc and K]c inplastics (ISO 13586, 2000). ASTM standardizeda method for determining fracture toughness ofplastic materials that fail under essentially planestrain and linear-elastic conditions (ASTMD5045-99, 1999). It has a basis in the method E399 and follows many of the same methods. Ithas a fracture toughness result given as a Kic orGic. For this standard, only the C(T) (Figure 1)and the SE(B) specimens (Figure 5) can be used.Rate is important for this test. A recommendedloading rate of lOmmmin"' is given in thestandard. A correction for indentation of theloading rollers in the bend specimen can beused. The validity requirement in this standardis the same as in the Kic standard for metals (seeSection 11.02.4.2.1).

For more ductile polymers, a second method(ASTM D6068-96, 1997b) develops the J R-curve fracture toughness for plastic materials.It also uses the C(T) and SE(B) specimens. Theprecrack is introduced with a razor cut. Theanalysis method has provisions for discountingindentations that may occur during the test. Itfollows the original ASTM E 813-89 (1990)method in that a multiple-specimen techniqueis used where each test generates a single pointon the J Ä-curve. It does not have a near-initia-tion J analysis but uses the entire R-curve as thefracture toughness characterization. The

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40 Classical Fracture Mechanics Methods

7?-curve can be fitted with a two-parameterpower law like the one in eqn [27]. To obtain aproper fit, the data must be spaced by a distri-bution law given in the standard.

A single-specimen method for developing theJ R-curve has not been standardized forpolymers; however, the normalization methodfor developing the J R-curve has been shown towork well as a single-specimen method forseveral of the more ductile polymers (Landeset al., 2003).

For more details on the testing of polymers,see Chapter 7.12 in Volume 7.

11.02.6 REPORTING

The results must be reported according to aformat specified in the standard. The list ofitems to be reported can be extensive includingall specimen dimensions, crack length, thedetails of precracking, the test temperature,and information on other mechanical proper-ties of the material tested, for example,hardness, strength, and Charpy value. Theseother properties may come from other tests.With such an extensive list of information thatmust be reported, it is important to rememberto include the fracture toughness result in thetest report. Often a reporting sheet is suppliedwith the standard to ensure that all of therequired data are included in the report.

11.02.7 REFERENCES

Anderson, T. L. 1995. Fracture Mechanics, Fundamentalsand Applications, 2nd edn. CRC Press, Boca Raton.

ASTM E 399-70T. 1970. Standard test method for planestrain fracture toughness of metallic materials, AnnualBook of ASTM Standards, vol. 03.01. American Societyfor Testing and Materials, West Conshohocken.

ASTM E 813-89. 1990. Standard test method for Jlc, ameasure of fracture toughness. Annual Book of ASTMStandards, vol. 03.01. American Society for Testing andMaterials, West Conshohocken.

ASTM E 1152-95. 1995. Standard test method for determin-ing J-R curves, Annual Book of ASTM Standards, vol.03.01. American Society for Testing and Materials, WestConshohocken.

ASTM E 1823-96. 1996. Standard terminology relating tofatigue and fracture testing, Annual Book of ASTMStandards, vol. 03.01. American Society for Testing andMaterials, West Conshohocken.

ASTM E 1304-97. 1997a. Standard test method for plane-strain (chevron-notch) fracture toughness of metallicmaterials, Annual Book of ASTM Standards, vol.03.01. American Society for Testing and Materials,West Conshohocken.

ASTM D6068-96. 1997. Standard test method for determin-ing J-R curves of plastic materials, Annual Book ofASTM Standards, vol. 08.03. American Society forTesting and Materials, West Conshohocken.

ASTM D5045-99. 1999. Standard test methods for plane-strain fracture toughness and strain energy release rateof plastic materials, Annual Book of ASTM Standards,vol. 08.03. American Society for Testing and Materials,West Conshohocken.

ASTM CI421-01b. 2001. Standard test methods for deter-mination of fracture toughness of advanced ceramics atambient temperatures, Annual Book of ASTMStandards, vol. 15.01. American Society for Testingand Materials, West Conshohocken.

ASTM E 8M-04. 2005a. Standard test methods for tensiontesting of metallic materials [metric], Annual Book ofASTM Standards, vol. 03.01. American Society forTesting and Materials, West Conshohocken.

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ASTM E 111-042. 2005c. Standard test method for Young'smodulus, tangent modulus, and chord modulus, AnnualBook of ASTM Standards, vol. 03.01. American Societyfor Testing and Materials, West Conshohocken.

ASTM E 132-04. 2005d. Standard test method forPoisson'sratio at room temperature, Annual Book of ASTMStandards, vol. 03.01. American Society for Testing andMaterials, West Conshohocken.

ASTM E 139-00. 2005e. Standard test methods for conduct-ing creep, creep-rupture, and stress-rupture tests ofmetallic materials, Annual Book of ASTM Standards,vol. 03.01. American Society for Testing and Materials,West Conshohocken.

ASTM E 399-90. 2005r. Standard test method for plane-strain fracture toughness of metallic materials, AnnualBook of ASTM Standards, vol. 03.01. American Societyfor Testing and Materials, West Conshohocken.

ASTM E 561-98. 2005g. Standard practice for «-curvedetermination, Annual Book of ASTM Standards, vol.03.01. American Society for Testing and Materials, WestConshohocken.

ASTM E 647-00. 2005h. Standard test method for measure-ment of fatigue crack growth rates, Annual Book ofASTM Standards, vol. 03.01. American Society forTesting and Materials, West Conshohocken.

ASTM E 1290-02. 2OO5i. Standard test method for crack-tip opening displacement (CTOD) fracture toughnessmeasurement, Annual Book of ASTM Standards,vol. 03.01. American Society for Testing andMaterials, West Conshohocken.

ASTM E 1304-97. 2005J. Standard text method forplane-strain (Chevron-Notch) fracture toughness ofmetallic materials, Annual Book of ASTMStandards, vol. 03.01. American Society of Testingand Materials, West Conshohocken.

ASTM E 1820-05. 2005k. Standard test method for mea-surement of fracture toughness, Annual Book of ASTMStandards, vol. 03.01. American Society for Testing andMaterials, West Conshohocken.

ASTM E 1921-05. 20051. Standard test method for deter-mination of reference temperature, 7~0, for ferritic steelsin the transition range, Annual Book of ASTMStandards, vol. 03.01. American Society for Testingand Materials, West Conshohocken.

ASTM E 2472-06. 2006. Draft standard test method fordetermination of resistance to stable crack extensionunder low-constraint conditions, committee E 08.American Society of Testing and Materials, WestConshohocken.

Barsom, J. M. and Rolfe, S. T. 1987. Fracture and FatigueControl in Structures, 2nd edn. Prentice-Hall, EnglewoodCliffs.

Begley, J. A. and Landes, J. D. 1972. The J integral as afracture criterion. In: Fracture Toughness, Proceedingsof the 1971 National Symposium on Fracture Mechanics,Part II, ASTM STP 514, pp. 1-20. American Society forTesting and Materials, West Conshohocken.

Brocks, W. and Yuan, H. 1991. Numerical studieson stable crack growth. In: Defect Assessmentin Components - Fundamentals and Applications,

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