introductory fracture mechanics
TRANSCRIPT
FRACTURE MECHANICS AND INSTRUMENTEDIMPACT TEST
An Informal Introduction with Worked Examples
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CONTENTSPage No.
FRACTURE MECHANICS - An Informal Introduction with Worked Examples 0 CONTENTS i-iii PREFACE iv IMPORTANT NOTE – ACKNOWLEDGEMENT OF SOURCES v
............................................................................................................................................. Chapter 1-Introduction to fracture Mechanics 1
............................................................................................................................................. General 1
1.2 Historic Failures/Developments that Spurred the Emergence of Fracture Mechanics 21.3 Some aspects of fracture in tension/impact/fatigue tests 31.3.1 Ductile vs Brittle - behaviour under the conventional, slow tensile test. 31.3.2. Microscopic Aspects of Ductile and Brittle fractures in Steel 41.3.3. Fatigue of Ductile materials 41.3.4 IMPACT TOUGHNESS and IMPACT TESTS 4Impact tests 5 Pendulm Impact Test Details 6 Charpy Data 6 Ductile- Brittle Transition Temperature (DBTT) 7
1.4 Conventional versus Fracture Mechanics based Design 7 1.5 A Note on Crack Plane Orientation (CPO) in Fracture Test Specimens 8
1.6 Different Regimes of Fracture Mechanics 8
Example 1.1. DBTT Example: TITANIC failure 9
............................................................................................................................................. Chapter 2-Linear Elastic Fracture Mechanics (LEFM) 10
2.1 GRIFFITH THEORY OF BRITTLE FRACTURE 102.1.1 Background 10 2.1.2. Expression for the Critical Fracture Stress of a Brittle Solid 11 Example 2.1.1: Glass Fracture and Griffith Theory 13 2.2 IRWIN’S STRAIN ENERGY RELEASE RATE AND STRESS INTENSITY FACTOR 132.3 COMPLAIANCE AND STRAIN ENERGY RELEASE RATE 16 PREFACE TO GRIFFITH EQUATION PROBLEMS AND PROBLEMS 182.4 MODES OF LOADING, SIF AND LEFM 212.4.1 Three Modes of Loading based on crack surface displacements 212.4.2 The stress intensity factor (SIF) and stress distribution 22 Salient points of the stress distributions 24 Stress Triaxiality, Plane Strain and Plane Stress 252.5 CRACK TIP PLASTICITY 262.6. LEFM FRACTURE TOUGHNESS (KIC) TESTING 28 Relevant features of the ASTM E 399 Standard 30 2.7. TRIANGLE OF INTEGRITY 32
............................................................................................................................................. Specimen SIFs and Typical Fracture Toughness Values 33
................................................................................................................................................................2.8. LEAK BEFORE BREAK (LBB) CONCEPT 34
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WORKED EXAMPLES/PROBLEMS IN LEFM 35
............................................................................................................................................. Chapter 3 - APPLICATIONS OF LEFM 53
................................................................................................................................................................3.1. FATIGUE CRACK GROWTH (FCG) 53
3.1.1. Introduction 533.1.2. Three Stages of Fatigue Crack Growth (FCG) 553.1.3. Stage –II FCG: PARIS LAW 563.1.4. FCG Testing 59Paris Constants for Some Common Steels 62 WORKED EXAMPLES IN FCG 63
3.2. STRESS CORROSION CRACKING (SCC) 733.2.1. General 733.2.2. Treatment of Crack Growth Rate 74
WORKED EXAMPLES IN SCC 77
3.3. ASME CODE APPROACH FOR DESIGN AGAINST BRITTLE FRACTURE:RTNDT-KIR CURVE APPROACH 84
3.3.1. Ductile-Brittle Transition Temperature: Different Approaches 84
3.3.2KIR curve determination for a 9Cr-1Mo steel – Illustration 85
3.3.2.1. Specimen Fabrication 853.3.2.2. Drop-Weight specimen fabrication and Drop-Weight Test (DWT) 863.3.2.3. Charpy specimen 893.3.3. Determination of RTNDT and ASME KIR curve for 9Cr-1Mo Base Material 903.3.3.1. Results from Drop-Weight Test 903.3.3.2. Results From Charpy Test 913.3.3.3. RTNDT and KIR Curve 92
............................................................................................................................................. Chapter 4 - Elastic-Plastic Fracture Mechanics-EPFM 94
............................................................................................................................................. 4.1. Crack Tip Opening Displacement (CTOD) and J-Integral Approach 94
4.2. EPFM in Practical Terms 96
4.3. J-Measurement 974.4. Begley-Landes Multi-Specimen JIC method 100 4.5. Standard Method for crack-tip opening displacement (CTOD) Determination 101
EPFM PROBLEMS 105 Appendix - Chapter 4: A Note on CREEP CRACK GROWTH (CCG) 125
............................................................................................................................................. Chapter 5 - An EPFM Application: ASTM E 1921 Master Curve (MC) 132
5.1. Reference Temperature (T0) and Master Curve Approach 1325.2. Statistical basis of the Master Curve 1335.3. Validity limits 135
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5.4. Multi temperature equation for T0 1365.5. Master Curve equations 1375.6. DETERMINATION OF T0 AND MC FOR A 9Cr-1Mo STEEL – EXAMPLE 1375.6.1. T0
dy from Pre-Cracked Charpy Test (PCVN) 137
Typical Calculation of KJd for Specimen No. 19 Tested at -50 °C (Fig. 5.3) 1435.7. APPLICATION OF THE ASME CODE CASES N-629 AND N-631 1445.8. A NEW UPPER-SHELF FRACTURE TOUGHNESS MASTER CURVE
(USFTMC or USMC) FOR FERRITIC STEELS 145 UPPER-SHELF FRACTURE TOUGHNESS MASTER CURVE (USFTMC or USMC)
CALCULATION PROCEDURE 1465.9. Effect of strain rate or stress intensity factor rate (SIF) rate on T0 146
............................................................................................................................................. Chapter 6 – Instrumented Charpy Impact Test, Charpy-Fracture Toughness Correlationsand Reference Temperature Prediction 148
6.1. Instrumented Charpy Impact Test 1486.1.1. Introduction 148 Additional strength and toughness values from IIT 152 A NOTE ON TD (brittleness transition temperature) 1526.1.2. Calculation Procedures in Instrumented Charpy Impact Test 153Load-time data processing 153 Loading or Strain Rates in Some Fracture Tests 154KId and JId/KJd estimation by conventional methods 154KJd and J1d estimation by the Modified Schindler Procedure 1576.2. Fracture Toughness Correlations with Charpy Energy and other Parameters 1606.2.1. Direct Charpy energy (CV) temperature - T0 Correlations 1606.2.2. Older CVN energy (CV) - KIC Correlations and T0 estimates 160Rolfe, Novak and Barsom (RNB correlation 160Sailors and Corten (SC correlation 161Robert’s lower-bound correlation (RLB correlation 161 Barsom and Rolfe’s Kd-Kc (dynamic to static fracture toughness) temperature shift
(Barsom-Rolfe Shift - BRS) procedure 161Marandez-Sanz Procedure 161Another lower bound correlation for the lower-shelf and lower transition region 1626.2.3. New Reference Temperature Correlations 162
Schindler-Sreenivasan Procedure (SSP)162
RNB, SC and BLB estimates 163 Note on Size Correction 163PRS-Parameter Correlations 163Mean-8 Procedure (M8P) 164
6.2.4. FATT-Master Curve (FATT-MC) approach for lower-bound fracture toughness 1646.2.5. Lower-bound estimate for upper-shelf fracture toughness 1666.3. Dynamic Fracture Toughness from Instrumented Drop-Weight Test 168
Illustrative Example 6.1:Complete IIT data for a service exposed 2.25Cr-1Mo Steel 169 Illustrative Example 6.2: Comparison of actual fracture toughness data for
a modified 403SS 176 Illustrative Example 6.3: Comparison of KId from IIT and Drop-Weight NDT
with RTNDT based KIR Curve for a 403 SS martensitic Stainless Steel – 403SS-IGC 177
Test Standards and Hand Books (Reference – T) 178
Bibliography and Additional References (Reference – B) 179
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PREFACE
When it was suggested that a book introducing Fracture Mechanics in an informal way with emphasis onworked examples should be prepared, the first thought that came to the mind of the authors was that “when somany scholarly treatises are available, why one more?” To our knowledge, the first authoritative bookexclusively on Fracture Mechanics was by Prof. J. F. Knott - "Fundamentals of Fracture Mechanics" (1973). Butit is a scholarly textbook most suited to academic pursuit and, because of the time of its publication, it restrictsitself to mainly Linear Elastic Fracture Mechanics (LEFM). Prof. Knott followed with a sequel, perhaps the firstbook of its kind, Worked Examples in Fracture mechanics; it also restricts itself to mainly LEFM and it containsonly worked examples. Subsequently, as listed in the Bibliography, many treatises have been published, thelatest being the ones by Perez (2004), Ramesh (1999) and Saxena (1998), the latter covering non-linear fracturemechanics and creep crack growth problems. Most of the listed references, which include some freely availableweb-resources, contain numerous solved and unsolved problems and seem to be rather heavy on theory or moredetailed in treatment. So, we felt that a book introducing Fracture Mechanics in an informal way, with emphasison worked examples, will not be out of place. Hence this venture. Accordingly, this volume has been titled asFracture Mechanics - An Informal Introduction with Worked Examples. Its preliminary model is Prof.Knott’s book, but it covers more topics starting from Griffith’s Theory to Non-Linear Fracture Mechanics andsome Applications in ASME Codes. It also introduces the ASTM E 1921 Reference Temperature Approach. Noattempt has been made to derive the relations from first principles, nor a micro-mechanistic nor a metallurgicalview-point adopted. An informal Strength of Materials textbook approach with ample explanation of thenecessary terms involved has been the line adopted during the writing of this monograph. Any engineeringstudent who has completed a Strength of Materials course would benefit in following this book. Of course, itcan be used as a companion or prescribed book, for an introductory course in Engineering Fracture Mechanics.On completion of this book, one will be in a position to appreciate the relative importance of FractureMechanics in safety analysis and assurance of structural integrity. Most of the Fracture testing Techniques havebeen covered with a coverage deep enough to impart an overview of the test. But no attempt has been made tocover the mechanical or instrumentation aspects. Hence, after mastering this book, one is not expected to walkinto a Test Lab equipped with the wherewithals to perform a test; however, with help from the appendedreferences and standards, one would well be on the way to equip oneself for such a task; in short, this book isnot a manual on test techniques.
No originality is claimed for the material presented in this book (see Note and Acknowledgement in the nextpage). About 50 worked examples have been provided. Many problems whose answers are only available, havebeen worked out. The material on KIR and Reference Temperature determination has been provided based on thetest results from our laboratory at Indira Gandhi Centre for Atomic Research. Many glaring omissions may bethere: like, Stress Intensity Factor Measurement or Computational Techniques, many Advanced DefectTolerance Procedures, Creep-Fatigue Crack Growth, Variable Amplitude Fatigue Crack Growth, OverloadEffects on Fatigue, etc. They are intentional as they are perceived to be outside the ambit and aim of the presentbook. Hence this book is neither comprehensive nor exhaustive, but sufficiently introductory and informative.Unusual for a Fracture Mechanics Book, a chapter (Chapter 6 – the final Chapter) on Instrumented Impact
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Testing, Charpy correlations and Prediction of Reference Temperature has been added, mostly based on work atIGCAR, since it has practical implications in many Engineering Applications such as Quality Control, TestTemperature Decisions and Preliminary Design, especially with availability of some new correlations. Thus wehope the material we have provided will serve not only as an appetizer but also as an enhancer and nourisher ofknowledge of Fracture Mechanics, strictly within the Strength of Materials viewpoint (hence no Mathematicsbeyond a knowledge of Preliminary Calculus is assumed). In short, this book will prepare the reader for apleasant first encounter with the subject of Fracture Mechanics and the cited Standards and References will givedirections for further progress. If that is the result, then the aim of this book would have been more than served.
IMPORTANT NOTE – ACKNOWLEDGEMENT OF SOURCES
Two sets of references are appended at the end: (i) “Test Standards and Hand Books” and(ii) “Bibliography and Additional References”. Item (i) pertains to test standards like thoseof American Society for Testing and Materials (ASTM), British Standards (BS), ASMECode, ASM Metals Handbook, IAEA Technical Report etc. Item (ii) pertains to generalbibliographical references, monographs and journal articles etc. relating to FractureMechanics. In the text, references to material under Item (i) are preceded by Letter T: forexample Ref. 1 under Item (i) is referred as T1 in the text. Similarly, references to materialunder Item (ii) are preceded by letter B: for example Ref. 1 under Item (ii) is referred as B1 inthe text. Most of the problems and material have been adapted from References B1, B3,B6, B7, B9, B10, B12, B13, B19, B20, B21, B23, B24, B26, B27, B28, B30, B31, B32, B36,B41, B43, B44, B45, B48 to B55, B56, B59, B60 to B62 and B65 (Item (ii)), where thebold underlined references indicate the source of problems or worked examples.Sources for figures, tables etc. are cited (following the above notation) at theappropriate places in the text.
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Chapter 1-Introduction to fracture Mechanics
1.1. General
Fracture involves breaking up of a body into two or more parts with the creation of newsurfaces. Thus it is destructive and, often, as shown in Fig. 1, catastrophic, resulting in lossof men, materials and money.
Hence “Understanding how things break can avoid having to explain why they brokelater on! [B24]” That is the realm of Fracture Mechanics: understanding fracture ofmaterials and structures and, thereby, predict, prevent and mitigate fracture failures.
(a) Boeing 737-200, Aloha Flight243, 1988T2 Tanker, The Schenectady, 1941
Fig. 1.1. Catastrophic fracture failures [B10]
OUTLINEWhat is fracture? Historical introduction. How fracture wasassessed qualitatively in pre-fracture mechanics days, usingtension and impact tests, fracture appearance, ductile-brittletransition temperature (DBTT) etc. Need and usefulness offracture mechanics. Different Fracture mechanics Regimes.Fracture Specimen Orientation in Steel Plates and an example ofDBTT limited failure.
1.2 Historic Failures/Developments that Spurred the Emergence of Fracture Mechanics
Enormous increase in the use of metals (mainly irons and steels) for structural applications inthe late 19th and early 20th centuries and later led to many accidents, with loss of life andwealth, owing to failure of these structures. In particular, there were numerous accidentsinvolving steam boiler explosions and railway equipment. Some of these accidents were dueto poor design, but it was also discovered that material deficiencies in the form of pre-existingflaws could initiate cracking and fracture.
A new era of accident-prone structures was ushered in by the advent of all-welded designs,notably the Liberty ships and T-2 tankers of World War II. Out of more than 2500 Libertyships built during the war, 145 broke in two and almost 700 experienced serious failures. Thefailures often occurred under very low stresses, for example even when a ship was docked,and this anomaly led to extensive investigations which revealed that the fractures were brittleand that flaws and stress concentrations were responsible. It was also discovered that brittlefracture in the types of steel used was promoted by low temperatures. As clarified in the lastdecade, the failure of the TITANIC (a completely riveted structure unlike the all-weldedstructures of World War-II) in 1912 resulted from the use of poor steel at low temperature.
Indian Connection to Failure and Fracture: On 2 May 1953 exactly one year to the dayafter their introduction, a Comet aircraft (the de Havilland Comet was the first jet-propelledairliner) broke up in flight near Calcutta. In all, 20 Comets crashed between 1952 and 1971taking the lives of almost 500 people. It was eventually discovered that their fuselages hadexploded whilst climbing up to cruising height, weakened by the fatigue of repeatedpressurization and depressurization. The aircraft whose wreckage was discovered had begunto crack at the corner of one of the automatic direction finder (ADF) aerial cut-outs, andanother tested on the ground burst open at the corner of a window. Hatch corners and windowcorners had to be modified to reduce the stress concentration.
In modern times, the emergence of aerospace technology (with use of high-strength structuralmaterials at high stresses for weight reduction) and nuclear power and the attendant concernabout the structural integrity of aero-space structures and nuclear reactors has greatlycontributed to the development of Fracture Mechanics.
To summarise, in the above cases, failures could be attributed to:
the all-welded construction which eliminated crack-arresting plate boundaries present inriveted joints
the presence of crack-like flaws/metallurgical defects in welded joints - like inclusions, lackof fusion, weld cracks etc.
the use of materials whose low resistance to crack advance (toughness) was further reduced bylow temperatures – for example, Titanic.
Poor design - window hatch corners in Comet – resulted in stress concentration and fatigue
cracks.
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1.3 Some aspects of fracture in tension/impact/fatigue tests [B65]
1.3.1 Ductile vs Brittle - behaviour under the conventional, slow tensile test.
ductile
Fig. 1.2
Slow controlled extension; final instabilitydue to gross area reduction. Cup-and-conefailure surface characterised by 45o shearlip.
brittle
Fig. 1.3
Fast catastrophic failure with no warning.Transverse granular cleavage surface inferritic steels - no shear lip.
Fig. 1.4 Fig. 1.5
Relatively high energy absorption capacity(stress-strain area) - "tough". Localisedyielding at high stress concentrationsredistributes stresses advantageously. Shearstresses cause failure, triaxiality effect isrelatively benign.
No significant yielding, so no reduction ofhigh stresses. Low energy absorptioncapacity. Triaxial stresses cause failure.
1.3.2. Microscopic Aspects of Ductile and Brittle fractures in Steel
(a) TransgranularCleavage
(b) IntergranularFracture
Fig. 1.6 [B26]
(c) (d)
1.3.3. Fatigue of Ductile materials [B65]: Fatigue is recognised as a mechanism of crack growthterminated by catastrophic fracture - the S-N diagram, shown in Fig. 1.7, may be used to predictfailure. S-N diagram approach is mostly empirical. Hence a more basic approach involving anunderstanding of the fundamental fracture mechanisms, processes and factors involved can helppredict and assess fracture failure more confidently. Fracture Mechanics provides one suchapproach.
Fig. 1.7a Fig. 1.7b
1.3.4 IMPACT TOUGHNESS and IMPACT TESTS
•Measure of the amount of e
•Low loading ra–Area under the s
•D–
Toughnessnergy a material can absorb before fracture
tes as in normal tension teststress-strain curve up to fracture
ynamic loading
4
Impact energy
5
Impact tests:Charpy (simple three-point bend:TPB) and
Izod (cantilever bend and done only at room temperature) Tests
Fig. 1.8. Pendulum Impact Testing [B26]
Pendulm Impact Test Details•Toughness quantified in terms of the amount of energy required to fracture a notched specimen of thematerial struck by a hammer or pendulum. Notch introduces three factors: (a) Stress concentration;(b) Triaxial stresses which reduce shear stresses favouring brittle fracture; and (c) increased strainrate.
•The hammer has potential energy due to its initial height, h1
–When the hammer is released the potential energy -> kinetic energy
•At the bottom of the stroke, the hammer impacts on the test specimen
•Kinetic energy is used to break or fracture the specimen
–The remaining kinetic energy is not enough to raise the pendulum to its initial height at the end of thestroke : h2 < h1
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•The impact (fracture) energy is the difference in potential energy: U (h1 - h2)
Charpy Data
Fig. 1.9a. Charpy impact energy (toughness) vs. Fig. 1.9b. Charpy impact energy (CV) vsTest temperature curve for a typical ferritic steel. test temperature (T) for various alloys
the almost constant low energy region at low temperatures is called the lower-shelf while thealmost constant high energy region at high temperatures is called the upper-shelf.
•Quantitative results are not directly used in design calculations•These results give a qualitative comparison of the toughness:
–for different materials at the same test temperature–at different test temperatures for the same material
–for the same material subjected to different heat treatments
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Ductile- Brittle Transition Temperature (DBTT) The ductile-brittle transition is exhibited by BCC metals (see Fig. 1.9), such as low carbon
steel; they become brittle at low temperature or at very high strain rates. FCC metals (Alalloys – upper-shelf energy ~ 30 J; austenitic stainless steels – upper-shelf energy ~ 150 to 350J – see, Fig. 1.9b)), however, generally remain ductile at low temperatures without anytransition.
Low temperatures, low energy absorption indicates brittle behaviour. As temperature increases, the energy absorbed increases dramatically, indicating a change of
failure mode from brittle to ductile (for ferritic steels). The effect of a change of loading rate is to move bodily the toughness curve parallel to the
temperature axis. This, together with the shape of the curve, means that a higher rate or lowertemperature will decrease toughness.
Usually, DBTT is defined as the temperature corresponding to a Charpy (CV) value of 28, 30,41 or 68 J, depending on the design codes and types of steels.
1.4 Conventional versus Fracture Mechanics based Design
The traditional design is based on preventing yielding or allowing only local yielding by restrictingdesign stress to a fraction of the yield or ultimate tensile stress (YS or UTS) using a factor of safety(FOS), that is,
Design Stress = (YS or UTS)/FOS
FOS varies from 2 to 4 and 10 for a lift wire rope [B23, B24]. This approach proved inadequate toaddress issues like those described in Section 1.2 or situations involving DBT.
Fracture Mechanics presupposes the existence of cracks/crack-like flaws in the material, which maybe microscopic (for example, grain size, inclusions etc.), or large (for example, casting or welddefects), due to manufacture, to corrosion, fatigue,…
Fracture Mechanics correlates three parameters quantitatively . . .(i) - load – the background stress, σ;(ii) - geometry - the crack size, a (and to a lesser extent, crack shape); and(ii) - material - its resistance to cracking, i.e., its fracture toughness, measured by special tests
and predicts, amongst other things,- degree of safety, or imminence of catastrophic (brittle) fracture- crack growth rate whilst advancing in a controlled manner- remaining component life.
FRACTURE MECHANICS HAS BEEN CALLEDTHE SCIENCE AND ENGINEERING OF ‘LIVING WITH DEFECTS’.
1.5 A Note on Crack Plane Orientation (CPO) in Fracture Test Specimens [B7]
The major CPOs in a rolled plate are indicated in the above figure. L, T and ST or S are thethree principal working directions in the rolled plate, namely, Longitudinal (PrincipalRolling Direction-usually the longest dimension or length), Transverse (Next LongestDimension or Width) and Short Transverse (usually the shortest or thickness direction).
Specimen-1 has its long axis in the T-direction and crack propagates in the T-direction;hence, its CPO is designated as T-L and is referred to as T-L specimen. Similarly for others.
Because of mechanical fibering, inclusion banding etc., ductility and toughness dependsstrongly on the specimen orientation (see Example 1.1 on Titanic steel in the next Box).Usually, the L-T specimen (3 in the above figure) shows the highest and S-T (5) shows the
lowest impact property.1.6 Different Regimes of Fracture Mechanics
1. Linear Elastic Fracture Mechanics (LEFM): Here the body/structure as a whole is in theelastic and plasticity is confined to very small region at the defect/crack tip and failure occursafter very little plastic deformation. The relevant fracture mechanics parameter is linear elasticfracture toughness – critical stress-intensity factor (SIF) - KIC.
2. Elastic-Plastic Fracture Mechanics (EPFM): Here, significant/extensive plasticity occurs atthe crack tip and fracture is preceded by significant plastic deformation. The relevant fracturemechanics parameter is elastic-plastic fracture toughness: critical J-integral – JIC or criticalcrack tip opening displacement (CTOD), δIC.
3. Fatigue Fracture Mechanics: Fatigue crack growth (FCG) is related to (SIF range) ΔK or (J-integral range) ΔJ.
4. Creep Fracture Mechanics: Fracture mechanics applied to creep crack growth (CCG): therelevant fracture mechanics parameter is C* - a time dependent J-integral rate parameter – and
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its variants.
Example 1.1. DBTT Example: TITANIC failure
Metallurgical Cause for the RMS Titanic Failure (on 12 April 1912)
A metallurgical analysis of steel taken from the hull of the wreckage of the 46,000 tons shipTitanic reveals that it had a high DBTT, making it unsuitable for service at low temperatures;at the time of the collision (with an iceberg that was three to six times larger than its ownmass), the temperature of the sea water was -2°C. The analysis also shows, however, that thesteel used was probably the best plain carbon ship plate available at the time of the ship'sconstruction (1912). Presence of massive MnS inclusions and banding resulted in very poor
transverse impact properties. (Adapted from: The Royal Mail Ship Titanic: Did a
Metallurgical Failure Cause a Night to Remember? Katherine Felkins, H. P. Leighly, Jr., andA. Jankovic. J. of Materials, 50(1) (1998) pp.12-18.)
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Fig. 1.10
Chapter 2-Linear Elastic Fracture Mechanics (LEFM)
2.1 GRIFFITH THEORY OF BRITTLE
2.1.1 Background
Fracture strength of a solid material Theoretical cohesive strength of br
modulus of elasticity. Experimentally observed fracture s
E/1000, In the 1920s, A. A.Griffith ascribed
microscopic flaws or cracks at the su Flaws detrimental to the fracture s
concentrated at the tip, with theorientation and geometry.
Fracture stress of brittle solids and GriffiStress Intensity Factor (SIF - K), Irwin’Elastic Fracture Mechanics (LEFM); SigKIC; Compliance and Strain Energy Reletip stress distribution; Crack Tip PlastiLinear Elastic Fracture Toughness (KIC),and its Relation to Structural GeometrExamples in LEFM.
OUTLINEth Theory; Examples of application of Griffith Theory;s Critical Strain Energy Release Rate (GC) and Linearnificance of LEFM Fracture Toughness, critical, K –ase Rate; Different Modes of Fracture; SIF and crackcity; Stress Triaxiality, Plane Strain and Plain Stress;Different Specimens and Testing; Fracture Toughnessy and Quality Factors; Leak Before Break; Worked
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FRACTURE
function of the cohesive forces between atoms.ittle elastic solids ~ E/10, where E is the Young’s
trengths for most engineering materials ~ E/100 to
the above discrepancy to the presence of very small,rface and within the interior of a body of material.trength because an applied stress gets amplified or
magnitude of amplification depending on crack
2.1.2. Expression for the Critical Fracture Stress of a Brittle Solid
(a) (b)Fig. 2.1. (a) Body with central crack, remote stress, σ ; (b) Body with edge crack [B7]
Fig. 2.2 Energy balance during crack growth in a brittle solid [B26]
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Cond. In next BOX
2.1.2. Expression for the Critical Fracture Stress of a Brittle Solid (cond.)Considering an infinite plate with a through-the-thickness elliptical crack of length, 2a (Fig. 2.1a), or an edge-crack of length, a (Fig. 2.1b) and remote stress, σ, Griffith, based on energy balance considerations, obtained anexpression for the fracture stress, σF, of a brittle material-glass. Basic premise of Griffith (1920) was that thecrack will grow if the energy available for growth equals or exceeds that required for growth and therebythe total energy of the system is lowered.
Energy available for growth: An elastically strained (stressed) material contains strain energy. Asthe crack propagates inwards from the surface of a stressed material, the area of material in whichstrain energy is relaxed corresponds to the two shaded triangles shown in Fig. 2.1a. The area of thesetriangles is approximately a2 and, hence, the relaxation of elastic strain energy is proportional tothe square of a. This is confirmed by calculation.
Energy required for growth: As the crack grows, surface energy needed to form the new surfaces,and this equals 2γa (for unit thickness of the plate), where γ is the surface energy per unit area of thematerial. The value of this term increases as the first power of the depth of the crack.
There is therefore an energy balance between the formation of new surfaces and the relaxationof strain energy in the material.
From the above, it can be seen that shallow cracks consume more energy as surface energy thanreleased as relaxed strain energy. Therefore conditions are energetically unfavourable for crackpropagation. As the crack increases in length the conditions are reversed and more strain energy isreleased than is needed to form the surfaces of the growing crack. This occurs beyond the 'criticalGriffith crack length', ag (or ac). Under these conditions the crack is able to grow at an everincreasing rate. This is shown in Fig. 2.2.
Griffith demonstrated that the critical stress c (equivalently represented byf) required for crackpropagation in a brittle material is described by
1
22 sc
E
a
(2.1.1)
where E is the Young’s modulus, s is the specific surface energy and a is one-half the length of aninternal crack.
In practice, the energy required to produce the two new fracture surfaces is significantly greater thans. The surface energy, s, is that required to break all the chemical bonds at the fracture surface.
During fracture the molecular structure of the material around the crack is also disturbed, to a depthwhich is sometimes very considerable. The total energy is known as the ‘work of fracture’, W.
Therefore,
1
22c
EW
a
(2.1.2)
12
2.
Mcrcopl
Fo
Example 2.1.1. Take a glass with E = 69 GPa and W = 0.3 Jm-2. Draw the graph showing thedependence of fracture stress on crack length.
For various assumed ‘a’ values of 10, 20, 30, …..,100 μm, plugging in the above values of Eand W in Eq. (2.1.1), the corresponding critical fracture stress values are obtained and areplotted in the Figure below.
As the crack increases from 10 μm to 100 μm, the fracture stress decreases from 36 MPa
to 11 MPa. The point to be noted is that1
fa
.
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2 IRWIN’S STRAIN ENERGY RELEASE RATE AND STRESS INTENSITY FACTOR
ost metals and many polymers do experience some plastic deformation during fracture; thus,ack extension involves more than producing just an increase in the surface energy. Thismplication may be accommodated by replacing s in Eq. 2.1.1 by s + p, where p represents aastic deformation energy associated with crack extension. Thus,
1
22 ( )s p
c
E
a
(2.2.1)
r highly ductile materials, it may be the case that by p s, such that
1
22 p
c
E
a
(2.2.2)
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In the 1950s, G.R.Irwin chose to incorporate both by s and p into a single term, GC, as
2( )c s pG (2.2.3)
GC is known as the critical strain energy release rate.
After incorporating the new term and rearranging, Griffith cracking criterion becomes
2
c
aG
E
(2.2.4)
Thus crack extension occurs when 2a/E exceeds the value of GC for the particular materialunder consideration. Therefore,
2ca EG (2.2.5)
OR
ca EG (2.2.6)
• This equation says:
–LHS (Some configuration stuff you can calculate) = RHS (Some material property stuff youcan measure)
• LHS = a
•Stuff you can calculate from the cracked structure configuration–Loading,–crack length, a
•This term is referred to as the Stress Intensity Factor (SIF) and given the symbol K
K a (2.2.7)
• K has unusual units: Stress x length = Pam or MPam
• K (quantifies the stress-strain distribution ahead of the crack) is fundamentallydifferent from the concept of Stress Concentration Factor (SCF); SCF describes how muchthe stress has been elevated at a point (crack or notch tip) compared to the remote stress.
15
RHS = Fracture Toughness = cEG
•The right hand side of the critical equation is a combination of material properties–Young’s modulus–Toughness
•It is a measured material property called the critical fracture toughness of the material, Kc.
c cK EG (2.2.8)
Critical Condition•The fracture toughness, Kc, is a constant for a given material and independent of the structuralconfiguration
•The critical condition for fast fracture can be written:
K = Kc (2.2.9)
•When the K reaches the critical value Kc, the crack propagates without limit:–Fracture occurs, as a consequence of: Increase in crack length
Increase in load K a
The tensile strength of a brittle material is determined by the length of the largestcrack existing prior to loading
SIFs for Cracked Bodies
•The SIF defined here is strictly speaking valid only for thin, semi-infinite plates, such that,thickness, t << a << W (and length)
•SIFs have been obtained for a variety of cracked body configurations and have the general form:
K Y a (2.2.10)
•Y is a dimensionless function incorporating finite geometry and crack shape correction factors,that is, a parameter dependent on the–size and shape of both the crack and body–location of the crack–form or mode of loading (described later in Section 2.4.1)
The concepts of strain energy release rate and SIF were developed by G. R. Irwin in the1950s. Thus he can be considered as the father of Modern Fracture Mechanics, while
16
2.3 COMPLAIANCE AND STRAIN ENERGY RELEASE RATE
Fig. 2.3(a) Fig. 2.3(b)
Fig. 2.3. Applied load = P; Crack length = a; Displacement = δ [B7]
In this section, we examine the significance of the strain-energy release rate in a notched/crackedbody subjected to elastic loading and its relation to G in greater detail. Figure 2.3(a) shows how Gcan be measured. A single-edge notch specimen is loaded axially through pins. The sharpestpossible notch is produced by introducing a fatigue crack at the root of the machined notch. Thedisplacement of this crack as a function of the axial force is measured with a strain-gage clipgage attached at the entrance to the notch. Load vs. displacement curves are determined fordifferent length notches, where P = Md (or δ, as indicated in Fig. 2.3(b)). M is the stiffness of aspecimen with a crack of length a and reciprocal of stiffness, i. e., (1/M) is called thecompliance, C; therefore, d = CP. The elastic strain energy is given by the area under the curveto a particular value of P and d.
Griffith can be considered as the father of brittle fracture theory.
17
21 1 1= = =
2 2 2U Pd PCP P C (2.3.1)
Consider the case shown in Fig. 2.3 where the specimen is rigidly gripped so that an increment ofcrack growth da results in a drop in load from Px to P2.
1 2 1 1 2 2= =d d C P C P
Since CP is constant,C
+ = 0P
C Pa a
C= -
P P
a C a
(2.3.2)
But the crack extension force, G, is defined as
21= = 2 +
2d
U P CG CP P
a a a
(2.3.3)
Putting Eq. (2.3.2) in Eq. (2.3.3),
21= -
2
CG P
a
(2.3.4)
Thus G can be evaluated by determining C as a function of crack length. In the above, the load Pwas considered for unit specimen thickness (i. e., P/B). For the fixed grip case, no work is doneon the system by the external forces P dd, while for the fixed load case, external work equal toP dd is fed into the system.
PREFACE TO GRIFFITH EQUATION PROBLEMS
Based on more sophisticated analysis, Eq. 2.2.4 can be rewritten as below:
12' C
f
E G
a
where2
'(1 )
EE
.
Therefore, 2
12
(1 )C
f
EG
a
OR
2 2(1 )C
c
f
EGa
Hence, for the common value of ν = 0.3 for metals, the effect of considering or not considering the value of ν, will change the result by about 10%.
Example 2.2.1: MARAGING STEEL FRACTURE
If the fracture stress of a large sheet of maraging steel , which contains a central crack oflength 40 mm, is 480 MPa, calculate the fracture stress of a similar sheet containing a crack oflength 100 mm.
12
640 480 10
0.02C C
f
EG EG
a
;
12
1000.05
C Cf
EG EG
a
Taking the ratios of σf,0.5
100 100
40
0.02
480 0.05
f f
f
; therefore, to σf100 = 303.4 MPa.
Comments: Maraging steel, unlike glass, is both tough and strong; hence, an increase in the
18
flaw-length by 2.5 times resulted in a reduction of the critical fracture stress by about 37%.
19
Example 2.2.2: GLASS SHEET FRACTURE
A sheet of glass measuring 2 m by 200 mm by 2 mm contains a central slit parallel to the 200 mmside. The sheet is restrained at one end and loaded in tension with a mass of 500 kg.What is the maximum allowable length of slit before fracture occurs?Assume the following material property values: E = 60 GPa, surface energy ~ 0.5 J/m2, Poisson's ratio= 0.25 and the fracture stress of sound glass = 170 MPa.
Note on Units: When the critical strain energy release rate is in N/m (=J/m2), E in N/m2, and a in m,the fracture stress is in N/m2 (Pa), which needs to be divided by 106 to convert it to MPa (the standardSI engineering unit). Mass is multiplied by 9.81 m.s-2 to get the force in N. Also note that twice thesurface energy gives GC (see Eq. (2.2.3)).
Then applied nominal stress, σ =500 9.81
0.2 0.002
= 12.26 MPa; the question is, for this σ to be equal
to σf, what is the ac?
Substitution into the Griffith’s equation, i.e.,2 2(1 )
Cc
f
EGa
, gives
94
2 2 12
60 10 0.5 21.355 10
(1 0.25 ) 12.2625 10ca m
This is a central crack in a plate and the full slit length is then, by definition 2a (see Fig. 2.1a). Thusthe maximum length of slit which can be supported is 0.271 mm.
Comments: As the applied stress is only 12.26 MPa, this indicates how critical even relatively smalldefects are in brittle materials, as this value of applied stress is much less than the fracture stress ofsound (uncracked) glass of 170 MPa. This example illustrates that even the apparently superfluousinformation helps in finding the right information!
Example 2.2.3: FAILURE CRITERIA: YIELD vs. FRACTURE
A cylindrical pressure vessel, with a diameter of 6.1 m and a wall thickness of 25.4 mm,underwent catastrophic fracture when the internal pressure reached 17.5 MPa. The steel of thepressure vessel had E = 210 GPa, a yield strength of 2450 MPa a value of GC = 131 kJ/m2.a) Show that failure would not have been expected based on von Mises yield criterion,
2 2 2 21 2 2 3 3 1. ., ( ) ( ) ( ) 2 YSi e
b) Based on Griffith's analysis determine the size of crack that might have caused this failure,stating assumptions that you have made.
Assumptions: (i) Fabrication and the orientation of the fatal defect - vessel is made fromwelded plates with welds running perpendicular to both hoop and longitudinal stresses. As thehoop stress is the maximum principal stress, one should assume that the defect isperpendicular to this stress direction. (ii) 25.4 mm plate is quite thick and hence assume planestrain, Poisson's ratio ~ 0.3.
Solution:
a) Using thin walled pressure vessel theory we get:
1 2 3
1; ; 0
2 4 20
pD pD tif
t t D
1 2
17.5 6100 17.5 61002101MPa; 1050MPa
2 25.4 4 25.4
Substituting these values into the von Mises yield criterion gives:
2 2 2 6 7 2. ., (2101 1050) (1050 0) (0 2101) 6.62 10 1.2 10 ( 2 )YSi e
Thus failure would NOT have been expected on the basis of yield.
b) Substitute into the Griffith formula the relevant values:
3 9
2 2 6 2 2
131 10 210 10
(1 ) (2101 10 ) (1 0.3 )C
c
f
EGa
m
20
Hence: ac = 2.18 mm.
21
2.4 MODES OF LOADING, SIF AND LEFM
2.4.1 Three Modes of Loading based on crack surface displacements
Fig. 2.4. Mode-I (opening or tensile mode),Mode-II (sliding mode), and Mode-III (tearing mode) [B26].
All other loading situations can be expressed as a combination of these three basic modes.
ONLY MODE-I IS CONSIDERED FURTHER
22
2.4.2 The stress intensity factor (SIF) and stress distribution
The SIF or K characterises the elastic stress field for a stressed element near the tip of a sharpcrack – at a distance r and angle θ from the X-axis (Y is in the normal direction to the crack-plane and Z is along the thickness (B) direction). The situation is illustrated below (Fig. 2.5).
Fig. 2.5. Stress components ahead of a (Mode-I) crack in a finite body of thickness B showinglateral contraction in the thickness direction; r is the distance from crack-tip [B59, B48].
Figure 2.5 helps visualise the coordinates for describing the crack-tip stress field for a crack ofdepth or length, a, situated in a body with width, W, and thickness, B. The crack-front (theleading crack-tip, assuming to be straight) lies along the Z-direction (parallel to B) and the crackpropagates in the X-direction (along W); thus, the X-Z plane constitutes the crack plane. σ x, σy
and σz are the three normal stress components, σz being the stress component normal to thecrack-plane.
Figure 2.6 gives the analytical expressions describing the normal stress components in terms ofKI and distance r and angle θ. For an infinite body, KI is given by Eq. (2.2.7), while Eq. (2.2.10)describes the KI for a finite-sized body.
23
Fig. 2.6. Expressions for stress distribution ahead of a crack in terms of SIF, KI,subscript I denotes Mode-I loading (For Modes –II and III, subscripts II
and III will be used respectivey). ν is the Poisson’s ratio [B48].
Fig
The e
Crac(i.e. twher
Alonthe eqto oth
Themeascomboccuris deindeptough
Salient points of the stress distributions (Figs. 2.5 to 2.7) are:
xpressions in Fig. 2.6 are the first terms of a series and hold good for small r.
k-tip stresses are 1/r½ and this causes a stress singularity at the crack tiphe stresses go to infinity as r goes to zero). Thus in the near-tip region, which ise fracture processes occur, the stress field is dominated by the singularity.
g the critical plane for cracking ahead of the crack tip (where the angle is zero),uations for the principal normal stress, that is the Y-direction stress (this applieser stresses also) reduces to the simple form of:
2y
K
r
(2.4.1)
numerator, the stress intensity factor, K, in the above essentially gives aure of the magnitude, or intensity, of the near-tip elastic stress field for anyination of remote stress and crack length. Irwin postulated that fracture wouldat a critical value of K, KC, a material property, called fracture toughness. KC
pendent on thickness and, at sufficiently high thickness, it attains a thicknessendent constant value called plane strain (described in the next box) fracture
24
. 2.7. Distribution of the three normal stresses (see Figs. 2.5&2.6 also).In Fig. 2.7b, z = 0 (the mid-thickness position) [B59, B48].
ness denoted by KIC.
25
Stress Triaxiality, Plane Strain and Plane Stress
In a normal tensile test, a simple bar is pulled along and is subjected to a uniaxialstress (that is, stressing in a single direction) at least up to necking. When a crackedbody is subjected to a remote tensile stress, σy, as in Fig. 2.5, stresses develop in theother two perpendicular directions due to Poisson effect (see Figs. 2.5 to 2.7). Asshown in Fig. 2.7a, the stress in the thickness direction, σz, reduces to zero at thesurfaces (for the simple reason that free surfaces cannot support normal stresses; forthe same reason, as shown in Fig. 2.7b, σx = 0 at the crack tip (r =0)) and is amaximum at the centre (mid-thickness, z = 0). So, as the thickness, B, is reduced, thecentral region of the specimen having the maximum stress will be reduced, andbeyond a point, maximum stress itself will reduce, leading to a situation known asplane stress, where , σz is very small or ~0 and only a biaxial stress field will exist. So,as B increases triaxiality increases, and beyond a certain thickness maximum stresstriaxiality develops.
Plastic deformation depends on shear stresses ( difference of normal stresses; forexample in a simple uniaxial tensile test, shear stress is σy; when triaxiality ispresent, for the same σy, shear stresses are (σy - σz) or (σy – σx) or (σz – σy)). Thushigh triaxiality leads to a reduction in shear stresses and plasticity is reduced orconstrained. So at the highest triaxiality, we have maximum normal stresses leadingto restricted plasticity. Normal stresses promote brittle fracture. This condition ofmaximum triaxiality with restricted plasticity is referred to as plane strain;hence, thin specimens experience a condition known as plane stress with reduced oralmost nil triaxiality.
Hence, plane strain fracture toughness, KIC, described above, is obtained insufficiently thick specimens where maximum stress triaxiality has developed andlarger thicknesses have no further effect on triaxiality. This situation is depicted in Fig.2.8, where KC is plotted as a function of B, the specimen thickness; thinner specimensbelow the plane strain limit show higher and, later, lower KC because of increasedplasticity and because of the interplay of different extents of plane stress (surface) andplane strain (central) regions as in Fig. 2,7a.
26
Fig. 2.8. Dependence of fracture toughness on specimen thickness:plane stress-plane strain transition [B6].
2.5 CRACK TIP PLASTICITY
Fig. 2.9. Development of plastic zone (R or rp) at the crack tip [B6].
27
From the foregoing Section, it is obvious that plasticity and its extent and distribution play animportant role in controlling the plane strain and plane stress regimes. From the definition of thestress intensity, based on the elastic stress field near a crack tip with θ = 0, i.e.:
2y
K
r
(2.5.1)
we can see that, as r tends towards zero, the crack tip stresses become singular. But this isphysically not possible; hence a yielded region will exist in the material ahead of the crack for allreasonable stress values. The shape and size of the plastic zone can be determined, to a firstorder, from the simple model first proposed by Irwin. Consider a material with a simple elastic-perfectly plastic response (i.e. no strain hardening occurs). A first estimate of the plastic zonesize ahead of the crack tip (ry), along the plane of the crack, can be obtained by substituting theyield strength into the above equation (see Fig. 2.9). This truncates the elastic stress field in thenear-tip region, where yielding occurs. The plastic zone size is obtained as:
2
1
2y
YS
Kr
(2.5.2)
Irwin observed that the presence of significant crack tip plasticity caused the specimen to behaveas though it contained a crack of greater length than was actually the case. That is, thecompliance (ratio of displacement to load: /P d , where P and d are load and load pointdisplacements respectively) of the specimen became greater as plasticity developed at the cracktip. This observation led him to propose a 'plastic zone correction' to crack length, based on amore accurate model of crack tip plastic zone size. A more accurate estimate of plastic zone sizecan be obtained by taking the necessary re-distribution of crack tip stresses (which accompaniesyielding) into account. This leads to a larger plastic zone size as indicated in the Fig. 2.9 – R orrp. It was shown that,
R = 2ry with effective crack length, aeff = a + ry (2.5.3)
where ry represents a plasticity correction to crack length which should be applied when crack tipplasticity is relatively extensive, e.g. under plane strain conditions. Under such cases the stressintensity factor is corrected iteratively through taking account of the effective crack length. Theprocedure first calculates K using the actual crack length, then finds ry using this value of K. aeff
is then found and the K value recalculated. This iteration can be continued further if necessary.Plasticity is important in fracture mechanics, as the extent of plasticity, relative to specimendimensions and crack size, determines the state of stress (plane strain or plane stress) andwhether LEFM is applicable or not. In turn, stress state affects the direction of planes ofmaximum shear stress and hence the fracture plane. Thus fracture proceeds perpendicularly tothe maximum principal stress in plane strain, and at 45º to this direction in plane stress.As a general rule, the stress state approaches plane strain when the plastic zone is about 1/15 ofthe crack length and material thickness. Plane stress occurs when the size of plastic zone tendstowards the material thickness. If the plastic zone is of the same order of size as the crack
28
length, LEFM would not be valid and yielding fracture mechanics (YFM or elastic-plasticfracture mechanics-EPFM) parameters must be used. A schematic view of the plane strainand plane stress plastic zones with reference to the cracked body in Fig. 2.5 is given in Fig. 2.10.
Fig. 2.10. Schematic plane strain, plane stress plastic zones (Ref also Fig. 2.5) [B6].
2.6. LEFM FRACTURE TOUGHNESS (KIC) TESTING
Standard fracture toughness tests are designed to allow reproducible determination of therelevant fracture characterising parameter like plane strain fracture toughness, K1C, the J-integralor crack tip opening displacement, CTOD. Hence certain conditions have to be met in the testsregarding specimen and crack geometry, loading parameters and shape of load-displacementcurve, before a valid result can be reported. Bodies like the British Standards Institution (BSI)and the American Society for Testing and Materials (ASTM) have developed standards for
29
testing of metallic and other materials. ASTM, E 399, Standard test method for plane-strainfracture toughness of metallic materials, is discussed in the next BOX.
Fig. 2.11. Bend and C(T) Specimens as per ASTM E 399 Standard [B6, B48].
30
Relevant features of the ASTM E 399 Standard [B21]:
Two of the specimen types: namely, the Three-Point Bend (TPB) Specimen and the Compact TensionC(T) specimens are depicted in Fig. 2.11 along with the K-calibration expressions.
Validity of the K1C result depends on the shape of the force vs displacement record, specimen size andcrack geometry, and the 0.2% proof strength and toughness of the material at the test temperature. Avalid result requires that:
2
, , ( ) 2.5 IC
YS
Ka B W a
(2.6.1)
These conditions restrict the plastic zone size to less than 1/15 of the relevant dimension and therebyensure that plane strain and LEFM conditions prevail.
During the fatigue pre-cracking (to ensure that a sharp enough defect similar to natural cracks is presentto give a lower bound value of toughness), there are restrictions on: (i) the fatigue loading during thefinal 1.3 mm or 50% of pre-crack extension (whichever is lower); (ii) crack length (0.45 < a/W < 0.55 ),difference between crack length measurements on the specimen surfaces (when measured to+/-0.05 mm this shall not exceed 15% of the average of the two measurements). Figure 2.12(i) showsthe appearance of a typical fatigue crack. Crack length is measured at 8 equidistant points (three pointsare shown in Fig. 2.12(ii)c along the crack-front on the fracture surface (plus mean of two surfacecracks counted as one measurement to give 9 values), the difference between any two measurementsmust be less than 10% of the initial crack length (a0 = average of 9 values, called 9-point average); and(iii) the plane of the crack must always be within 10° of the plane of crack extension. A 9-point averagecrack length measurement on a C(T) specimen is illustrated in Fig. 5.1 (Section 5).
During the fracture toughness test, load is applied at a set rate, corresponding to K changing in the range0.5 MPam½s-1 to 3.0 MPam½s-1, while a trace of load versus displacement (Δ) is recorded (see NOTE*).
The appearance of a trace suitable for K1C determination will conform to one of three types shown inFig. 2.12(ii)a. F igu re2 .12 ( i i )b shows the construction used to obtain thefracture load, PQ, which on substitution in the appropriate K expression givesan apparent (tentative) fracture toughness, KQ. The purpose of the 5% secantoffset construction and the read-out of the corresponding secant offset load, P5,
is to ensure that at PQ, maximum crack extension (Δa) is < 2% of the initial cracklength, a0. In Type-I (Fig. 2.12(ii)a), all loads prior to the displacement corresponding to P =P5 (i. e., Δ = ΔP=P5) are less than P5 and hence P5 =PQ. In Type-II curve, thereis a load maximum prior to P5 (this called a pop-in: pop-ins giving load changes of <1% are ignored) which is taken as PQ. In Type-III, maximum in the P- Δ plot, Pmax occurs prior to P5 and Pmax = PQ. Type-III corresponds to brittle fracturewith least macroscopic deformation, and, in such cases, sudden failure oftenresults before reaching P5.
Final validity condition that the ratio of Pmax/PQ < 1.10 ensures too much plasticity is not present priorto the fracture, especially for Type-I and Type-II cases.
*NOTE: Unlike in the J-integral or CTOD tests described later, in the KIC test, the measured Clip-gaugedisplacement serves only to determine the critical fracture load by a suitable geometrical construction asdescribed above; otherwise, it does not enter any calculations.
31
(i)
(ii)Fig. 2.12. (i) Typical Fatigue Crack and Load cycle; (ii) Test Traces
and Data Reduction/Validation as per ASTM E399 [B6, B48].
2.7. TRIANGLE OF INTEGRITY
A very useful way of visualising the equation (2.2.7):
K a (2.2.7)
is via the concept of a 'triangle of integrity' [B26]. It is clear that the equation relates threevariables, applied stress (calculated assuming no crack is present), stress intensity K and crackdepth a. At the critical condition, the value of stress intensity is equal to the fracture toughnessfor the thickness, temperature and strain rate which are relevant to the fracture.
The vertices of the TRIANGLE,
Material Toughness decided by Material Selection Applied Stress decided by Component Design/Sizing Crack Size controlled/assessed by Fabrication/Non-Destructive Testing (NDT)) are
related as indicated below:
The abosafe desgives socommon
TRIANGLE OFINTEGRITY
Material Selection
↓ Material Toughness
Fabrication/NDT
↓ Crack Size
A
ComponentDesign/Sizing
↓
32
ve triangle depicts the inter-dependent relationship between three activities in fracture-ign. The individual parameters are adjusted to achieve an optimum result. Table 2.1me SIF expressions while Table 2.2 gives typical fracture toughness values for somematerials.
pplied Stress
33
Table 2.1.* SIF Calibration Factors for Some Common Specimen Geometries (α = a/W), b here is actually B (the specimen thickness) and P is load [B65]
*NOTE: For actual applications and design, more accurate SIF expressions as given in TestStandards (as in Fig. 2.11) or other References like that of Murakami [B29] should be used.
Table 2.2. Typical Fracture Toughness Values (Representative, not for design) [B13]
Material Young’sModulus,E/GPa
YieldStress,σYS/MPa
FractureToughness,KIC/MPa√m
Thickness2.5(KIC/σYS)
2
in mmSteelsMedium CarbonPressure Vessel (ASTMA533B Q+T)High Strength AlloyMaraging SteelAFC 77 Stainless
Aluminium Alloys2024 T87075 T67178 T6
Titanium AlloysTi-6Al-4VHigh-Yield
ConcreteWC-Co CompositePMMC
210
72
108
401003
260
470146018001530
420540560
10601100
8030030
54
208987683
273023
7338
0.2-1.4131
110
487114.47.9
10.47.94.2
12.63.1
4.72.8
34
2.8. LEAK BEFORE BREAK (LBB) CONCEPT [B26]
It is often advantageous to design pressure containing plant, such as pipework, tubes, vessels, and boilers, onthe basis of leak-before-break (LBB). This means that partial failures which occur by sub-critical mechanisms(fatigue crack growth, stress corrosion cracking etc) are detected by loss of pressure in the plant before finalcatastrophic fracture occurs. This requires a crack to grow in a stable manner through the wall of thecomponent and cause a detectable leak and consequent loss of pressure. This indication of a partial failureallows the plant to be shut down in a controlled manner and repairs/replacement carried out.If it can be demonstrated that a leak-before-break situation exists, other useful benefits may accrue:
supplementing the primary structural integrity safety case alleviating some of the responsibility of non-destructive testing for ensuring safety permitting a reduction in number of restraints engineered into a pipework system to control pipe whip on
failure
The strategy in performing the analysis is as follows, and illustrated by reference to the figures below. Asurface (part-through) crack is assumed to initiate and grow by a sub-critical mechanism. Generally, initiationwill be from the inner surface of the pressurised container, as stresses are usually higher at this point and theremay well be a corrosive environment present (Figure a). However, industrial situations where cracking canoccur from the external surface are relatively common. A typical example might involve intergranular attack ofreactor pipework at elevated temperatures.1. Calculate the length of through-thickness crack which will cause fracture, lcrit.2. Calculate the depth of part-through (or surface) crack which will cause fracture, acrit.3. The value of acrit must be > B, the wall thickness. This allows the part-through crack to penetrate the wall(Figure b).4. Once wall penetration occurs, the part-through crack very quickly grows through the ligaments to become athrough-thickness crack with a length l1 = 2c, where 2c is the surface length of the part-through crack at wallpenetration (Figure c). Hence the aspect ratio of the part-through crack is an important parameter in a leak-before-break analysis. Remember also that the length l of a through-thickness crack is defined as 2a when youare substituting for a in stress intensity equations.5. The value of lcrit must be > l1.
6. Calculate the time for the crack to grow from l1 to lcrit. If the leak rate of fluid is detectable in this time, then aleak-before-break design case is established.
Figure a Figure b Figure c
WORKED EXAMPLES/PROBLEMS IN LEFM
Example 2.1.Rocket motor casings may be fabricated from either of two steels :(a) low alloy steel yield 1.2 GPa toughness 70 MPa√m, (b) maraging steel yield 1.8 GPa toughness 50 MPa√m The relevant Code specifies a design stress of yield/1.5. Calculate the minimum defect sizewhich will lead to brittle fracture in service for each material, and comment on the result (thislast is important).
Solution: Assume KI = σ√πa with centre crack 2a. Fracture occurs at the design stress and thecorresponding critical defect size to be determined.
(a) Low alloy steel: Design stress, σ = (1200 MPa/1.5) = 800 MPa.
Therefore, critical a =2 2
IC2 2
70
800
K
= 2.43.10-3 m = 2.43 mm and 2a = 4.9 mm
(b) Maraging steel: Design stress, σ = (1800 MPa/1.5) = 1200 MPa.
Therefore, critical a =2 2
2 2
50
1200ICK
= 5.53.10-4 m = 0.553 mm and 2a = 1.1 mm
Comments: In case (b), the critical defect size of 1.1 mm is dangerously small, may beeven below the level of detectability; hence, use of a higher toughness alloy or reduction
35
of design stress is suggested.
Example 2.2.
The CTS test piece is from a 1.2 GPa (Yield Stress) steel. If the failure load is 10 kN, what fracturetoughness is indicated? Is the result valid? Specimen dimensions in mm.
Note that width and crack size are reckoned from the load's line of action.
Solution: From Table 2.1, the relevant SIF calibration is that for Finite Plate, Edge Crack, TensileForce (Case(c)-CTS) as given below:
I
PK Y a
bW
with5.23 (5.16 5.88)
1 1.07Y
, where α = crack aspect ratio = a/W.
Given that, W = 20 mm, b = B = W/2 = 10 mm.
a = (W- 9) = 20 – 9 = 11 mm and α = 11/20 = 0.55, which gives Y = 8.644.At failure, P = 10 kN and YS = 1200 MPa.
Then,3
3
3 3
10 108.644 11 10
10 10 20 10IC
PK Y a
BW
= 80.3 MPa√m
Validity condition is given by Eq. (2.5.1) as:2
, , ( ) 2.5 IC
YS
Ka B W a
2
80.32.5
1200
= 0.011 m = 11.1 mm > a (11 mm), (W – a) (9 mm) and < B (20 mm).
Comments: As the evaluated SIF value of the test does not satisfy the ASTM E 399validity condition for a and B, the Plane Stress Fracture Toughness, KC = 80.3MPa√m is not a valid KIC.This is an instructive problem in the sense that, unlike other mechanical tests, like,tension, hardness, etc., in a fracture mechanics test, it is not possible a priori toensure that the test will result in valid/conclusive results. Based on the result, as inthe present case, we may have to repeat the test using larger specimens and alteredtest parameters to obtain a valid result. Hence, a fracture toughness test is costly,time consuming and sophisticated because of the stringent requirements on test
36
conditions/specimen dimensions.
Example 2.7.4.
The long strip may be made from either of the two materials :(a) tough, weak yield 700 MPa plane strain toughness 100 MPa√m (b) brittle, strong yield 1400 MPa plane strain toughness 50 MPa√m A central crack extends through the strip. Plot, as a function of crack length, the failure stressfor each material due to the separate mechanisms of elastic fracture and plastic collapse.Comment on the trends of these graphs.
Assumptions:
(i) Elastic Fracture: The governing equation is: KI = σ√πa, fracture occurring at KI =KIC and critical fracture stress σc = KIC/√πa.
(ii) Plastic Collapse: Plastic collapse occurs when the net section stress equals theyield stress, i. e., for the centre cracked panel considered here,
( 2 ) (1 2 ) (1 2 )net YS
P P
a aB W a BWW W
OR (1 2 )YS
a
W
So we have to plot σc and σnet as a function of 2a or 2a/W for the two cases givenabove.
Solution: The calculated results are plotted for the Cases (a) and (b) in Figs. (a) and (b) belowin the next Box: Example.2.7.4b.
It may be noted that the curve showing the brittle fracture stress is a parabola due itsdependence on √a, whereas the plastic collapse stress is a straight line varying from YS at
Assume an a/W = 0.5 and steel density = 7.9 gm/cc.
Solution: From Table 2.1 (Case (e)), the SIF expression for the SEN Bend geometry is as givenbelow:
6I
PK Y a
bW
with1.12 (3.43 1.89)
1 0.55Y
The relative dimensions of the ASTM E 399 SEN Bend specimen are given in Fig. 2.11; where Bvaries from 0.25 to 1W. Usually B = W or W/2. Here it is assumed that B = 0.5W, with a view toreduce specimen material.
For an a/W = 0.5 = α, Y = 1.424. For a KIC = 140 MPa√m, from the ASTM validity Eq. (2.6.1),
2
, , ( ) 2.5 IC
YS
Ka B W a
,
with σYS = 700 MPa,
2
2.5 IC
YS
K
=
2140
2.5700
= 0.1 m = 100 mm
Hence, for a B = 100 mm, W = 200 mm and a = 100 mm and span, S = 4W = 800 mm and specimentotal length, L > 800 mm.
Specimen Mass:For the above specimen, Volume = 800 x 200 x 100 mm3 = 16000 cc and this gives
a specimen mass = 16000 x 7.9 x 10-3 kg = 126.4 kg, conservatively as actual L > 800 mm.
Load and Test Machine Capacity:For this we must estimate the fracture load. Recasting the SIF expression for the SEN specimen givenabove, load, P is given by:
6ICK bW
Y a
=3 3
3
140 100 10 200 10
6 1.424 100 10
= 584685 N ~ 590 kN
So, the minimum load capacity required for the machine is: 590 kN.For a 100 mm C(T) specimen (with B = (W/2)), following the proportions of Fig. 2.11, the machine
capacity required is 2052 kN, the specimen mass being 48 kg.
Comments: The above is a very illustrative and instructive problem that gives an idea of the specimen dimensions andmasses and machine capacity required to successfully measure a valid fracture toughness. In the present case a medium
2a/W = 0 to zero stress at 2a/W = 1.
Example 2.3.
The toughness of a 700 MPa yield structural steel is estimated to be 140 MPa√m. What size and mass of SEN bend test specimen is necessary, and what capacity of testing machine would be required ?
strength alloy was used, and for low-strength steels specimens with B = 300 to 400 mm have been tested. One disadvantageof the bend specimen is that it requires large amount of material, with comparatively lower machine capacity because of large
37
S/W ratio. C(T) specimen economises on material, hence the name COMPACT.
Example 2.4.
The long strip may be made from either of the two materials :(a) tough, weak yield 700 MPa plane strain toughness 100 MPa√m (b) brittle, strong yield 1400 MPa plane strain toughness 50 MPa√m A central crack extends through the strip. Plot, as a function of crack length, the failure stressfor each material due to the separate mechanisms of elastic fracture and plastic collapse.Comment on the trends of these graphs.
Assumptions:
(iii) Elastic Fracture: The governing equation is: KI = σ√πa, fracture occurring at KI =KIC and critical fracture stress σc = KIC/√πa.
(iv) Plastic Collapse: Plastic collapse occurs when the net section stress equals theyield stress, i. e., for the centre cracked panel considered here,
( 2 ) (1 2 ) (1 2 )net YS
P P
a aB W a BWW W
OR (1 2 )YS
a
W
So we have to plot σc and σnet as a function of 2a or 2a/W for the two cases givenabove.
Solution: The calculated results are plotted for the Cases (a) and (b) in Figs. (a) and (b) belowin the next Box: Example.2.4a.It may be noted that the curve showing the brittle fracture stress is a parabola due itsdependence on √a, whereas the plastic collapse stress is a straight line varying from YS at 2a/W = 0 to zero stress at 2a/W = 1.
38
(cond. To 2.4a)
Example 2.4a (continuation of Example 2.4)
2a/W
0.0 0.2 0.4 0.6 0.8 1.0
Rem
ote
Str
ess
,
0
200
400
600
800
1000
1200
1400
1600
1800
2000
700MPaYS
ICc
K
a
(1 2 )YS
a
W
Fig. a. Plastc Collapse and Elastic Fracture conditions for Case(a)
Tough, Weak Alloy: KIC = 100 MPa.m0.5
Fig. b. Plastc Collapse and Elastic Fracture conditions for Case(b)
0.0 0.2 0.4 0.6 0.8 1.0
Rem
ote
Str
ess
,
0
200
400
600
800
1000
1200
1400
1600
ICc
K
a
(1 2 )YS
a
W
1400MPaYS
2a/W
Brittle, Strong Alloy; KIC = 50 MPa.m0.5
Comments: In the case of the high toughness alloy, Case(a), as shown in Fig-(a), the plasticcollapse stress is below that for brittle fracture. Hence LEFM toughness cannot be measured.
For the case (b), as shown in Fig. (b), critical fracture conditions prevail in the intermediate 2a/Wregions with plastic collapse at low and high stress (high and low crack lengths respectively)
39
regions.
40
Example 2.5. HIGH STRENGTH vs FRACTURE TOUGHNESS
A welded structure is to be fabricated from large sheets of 0.45C-Ni-Cr-Mo steel. The detection limitof available NDT techniques limits the critical defect size to sizes > 3 mm, as cracks smaller than thisare not detectable. A design stress level of half the tensile strength is proposed.To save weight in the structure it has been suggested that the steel could be heat treated to a highertensile strength level. The current grade has a tensile strength of 1520 MPa, and a candidatereplacement grade has a 2070 MPa strength level. Is this change supportable in fracture mechanicsterms?You may assume plane strain conditions in all computations, and the figure below indicates therelationship between fracture toughness and tensile strength for this steel.Compare the allowable stress levels, and hence weights, in both grades of steel for an allowable initialdefect size of approximately 5 mm.
UTS= where =
2K a
Solution:
From the data in the figure, KIC of the 1520 MPa grade is 66 MPa m½, while K1C = 33 MPa m½ for the2070 MPa grade. A through-crack in a large sheet can essentially be treated as one in an infinite plate.
(Cond. To Example 2.5a in the next Box)
Example 2.5a (continued from Ex. 2.5)
For the alloy heat treated to 1520 MPa, we get:
c
66MPa m 760MPa
2.4mm total flaw size = 4.8mm
a
a
This critical flaw size is larger than the minimum NDT detection limit, and this steel is safe touse. For the 2070 MPa grade, however, the equation gives:
c
33MPa m 1035MPa
0.33mm total flaw size=0.66 mm
a
a
Thus it is not possible to detect critical defects in this grade before fast fracture occurs. Toallow a critical defect size of 4.8 mm, i. e.,2a, in both grades, for the 2070 MPa grade:
3
33MPa m380MPa
2.4 10 m
Hence for a similar flaw tolerance level, the allowable stress in the higher strength alloy ishalf that in the 1570 MPa grade - this would imply a two-fold increase in weight of acomponent. Hence, the change is unwise.
Comments: This question illustrates the effect on critical crack length of the loss in fracture
41
toughness that generally accompanies an increase in tensile strength.
Example 2.6. QUENCHING and RESIDUAL STRESSES
During water quenching of steel components with a section thickness of 30 mm, heat transfercalculations indicate that a peak stress of 130 MPa is generated in the section. Prior to heat treatment,the components were ultrasonically inspected to detect defects. The inspection technique has aminimum detection size of 0.5 mm.a) What type of defect will be most critical?b) Calculate the size of defect which would cause fracture of the component during the quenchingoperation, given that the aspect ratio of the crack is 2c/a = 10.c) Would this inspection procedure guarantee integrity of the component if the quenching stressesapproached the proof stress of the steel?Note: K1C = 30 MPa m½ and the proof stress = 620 MPa. The stress intensity calibration for thiscomponent and crack geometry is given in the figure below (where the subscript y indicates proofstress).
Where, for surface flaws:1
2
1.1a
KQ
and for embedded flaws:1
2aK
Q
42
Solution Cond. in the Next Box-2.6a
43
Example 2.6a (Continuation of 2.6: QUENCHING and RESIDUAL STRESSES)
Solution:
a) From inspection of the stress intensity solutions for surface and embedded flaws it isobvious that, because of the factor of 1.1 for surface defects, they will become critical atsmaller values of a than embedded defects.
b) For substitution into the K equation for surface defects, Q must be determined. From thegraph,
For 0.21 and 0.1, 1.12y
aQ
c
2 2IC
2 2
1.1 1.1 30= = 15.4 mm
1.21 1.21 130c
Ka
This flaw is very much bigger than the NDT detection limit and there should be very little riskof failure during the quenching.
b) If the quenching stresses approached the proof stress of the material, the situation changesdramatically. Surface defects are still critical, but the value of Q has changed:
For 1 and 0.1, 0.882y
aQ
c
2 2IC
2 2
0.88 0.88 30= = 0.54 mm
1.21 1.21 620c
Ka
As the critical size of defect is around the NDT detection limit, the inspection would notguarantee integrity. A change to the quenching procedure would have to be implemented, e.g.using a slower quenchant.
Comments: This problem brings out the importance of heat treatment conditions incontrolling residual stresses and their effect on critical defect size.
44
Example 2.7: FRACTURE TOUGHNESS TESTS
The figure below shows the load line displacement trace recorded from a standard sizedcompact tension fracture toughness specimen. Specimen thickness was 25 mm, the cracklength at fracture was 25 mm and the steel alloy had a yield strength of 650 MPa.
a) Calculate PQ and hence KQ. Apply the required checks on plastic zone size, stress state andplasticity during the test to determine whether KQ is a valid plane strain fracture toughnessvalue.b) What is the maximum K1C value that can be determined for this steel using 25 mm thickspecimens?
Note: The K equation for C(T) specimen is given in Fig. 2.11 and Table 2.1 (the former ismore accurate).
(Continued to 2.7a)
Example 2.7a (Continuation of Example 2.7: FRACTURE TOUGHNESS TESTS)
a) To find PQ, a line is constructed from the origin with a slope 5% less than that of the tangentto the initial straight line part of the load-displacement record. This line is shown in the figureabove. This line intersects the load-displacement trace at 19 kN, and there is no previoushigher value of load on the trace.
Hence PQ = 19 kN.
To find KQ the function f(a/W) (which is the finite geometry correction factor) must becomputed. From the standard specimen size (see Fig. 2.8), W = 2B = 50 mm, hence a/W =25/50 = 0.5:
-3Q
Q 0.5 0.5
From Fig. 2.8, ( ) = 9.66 and
( )19 10 9.66
= = 32.8 MPa0.025 0.05
af
W
aP f
WK mBW
Validity Check on KQ
(i).
2 232.8
, , ( ) 2.5 = 2.5 , i. e., 6.4 mm650
Q
YS
Ka B W a
(ii). Final constraint to ensure absence too much plasticity:
Pmax/PQ = 21 kN/19 kN = 1.105 < 1.1
As the conditions are satisfied, KQ is KIC = 32.8 MPa√m.
Comments: Essentially, the first check ensures that the crack tip plastic zone is a small enoughpercentage of crack length to ensure LEFM, the constraint on B helps to ensure that plane strainconditions prevail, while the condition on (W - a) ensures that a plastic hinge does not develop aheadof the crack.
b) The maximum K1C value that can be determined from this thickness of specimen in steel ofthis grade, called K-capacity of the specimen, is found when B fails to satisfy the ValdityCheck Condition (i) above; i.e., when,
2
Q YS
0.0252.5 OR K > = 650 65 MPa m
2.5 2.5
Q
YS
K BB
Comments: This question illustrates the application of constraints for determining valid K1C values
45
from fracture toughness tests.
Example 2.8: PLASTIC ZONE EFEECT
A thin plate of steel contains a central through-thickness flaw of length 16 mm, which is subjected to a stress of350 MPa applied perpendicularly to the flaw plane. The 0.2% flow stress of the material is 1400 MPa.Calculate the plastic zone size and the effective stress intensity level at the crack tip, making reasonableassumptions about the state of stress.If, after heat treatment, the flow stress of the steel dropped to 385 MPa, what would the plastic zone size beunder the applied stress of 350 MPa, and what conclusions would you draw about the use of LEFM?
Assumptions: (i) The plate is large compared to the size of the crack so that the simple infinite plate formula forstress intensity factor applies and a = 8 mm (half the central crack length). That is,
=
= 350 0.008 = 55.49 MPa
K a
K m
(ii) The steel plate is in a state of plane stress as it is stated to be 'thin'. Plane stress prevails if the ratio of platethickness to plastic zone size tends towards 1, while plane strain prevails if it tends towards 15. Plane stress isalso a conservative assumption, in that K values are higher (through Irwin's plastic zone correction) when plasticzones are bigger.
Solution:
Irwin's plastic zone correction factor to crack length is given by:
2 2
-4p
1 1 55.49= = 2.5 10 m (i.e., 0.25 mm)
2 2 1400YS
Kr
This is small compared with the crack length and its effect on K will be correspondingly small:
3eff p= ( ) = 350 (8.25 10 ) = 56.35 MPaK a r m
This is around a 1.5% change and thus a single iteration of the calculation is sufficient.However, if the flow stress drops to 385 MPa after heat treatment, the plastic zone size now becomes:
2
-3p
1 55.49= = 3.31 10 m = 3.31 mm
2 385r
3eff p= ( ) = 350 (11.31 10 ) = 65.97 MPaK a r m
This represents a correction of around 18.9% and the use of LEFM becomes dubious. This is confirmed by thefact that the applied stress (350 MPa) is now some 91% of the flow stress. A yielding fracture mechanicsparameter should be used to characterise the propensity for fracture.
46
Example 2.9. SPECIMEN THICKNESS EFECT
Catastrophic fracture occurred in a thick steel plate during proof testing, at an applied stress of700 MPa. The initiating defect was an embedded sharp penny-shaped flaw with a radius of2.5 cm. Calculate the fracture toughness of this steel.It is desired to check this value by determining the plane strain fracture toughness fromstandard tests. The yield strength of the steel is 1100 MPa. A sheet of nominally similar steel,7.5 mm thick, is available. Is this sufficiently thick to obtain a valid K1C value? If not, whatthickness of steel should you order?
Given: The stress intensity solution for an embedded circular crack is:
2=K a
Substitution of values into the above formula gives:
2= 700 0.025 = 124.9 MPaK m
If this is assumed to be a valid plane strain fracture toughness value, then the minimumspecimen thickness required is given by:
2 2
IC
YS
124.9> 2.5 = 2.5 = 0.0322m = 32.2 mm
1100
KB
Comments: This is much larger than the plate thickness of 7.5 mm. Hence the thickness of
47
steel in stock is insufficient to provide a valid K1C value. A thickness > 35 mm is required.
Example 2.10. LBB PROBLEM:
a) The stress intensity solution for a semi-elliptic flaw in tension is given below. The K1C
value for a Ti-6Al-4V titanium alloy with a yield strength of 910 MPa, is 115.4 MPam½.Determine the size of the largest stable surface flaw (a/c = 0.4) in a 40 mm thick plate of thisalloy, for a design stress in the plate of 75% of the yield strength. This requires assuming aninitial value of a/B and iterative calculations of stress intensity, if necessary.
For the semi-elliptic surface flaw, =Y a
K
and the values of Y and Φ are given in
the TABLE below.
b) For the same alloy and design stress, calculate the maximum wall thickness of a pressurevessel which could be designed on a leak-before-break criterion. You may assume that theaspect ratio (a/c) of the surface flaw remains constant at 0.4, and that for the through-thickness crack:
=K a
What thickness of plate would you order for the vessel?
48
(continued to 2.10a)
49
Example 2.10a (continuation of 2.10;LBB PROBLEM)
TABLE
Φ a/c Phi Ya/B
0.2 0.4 0.6 0.8
1.051 0.2 0o
45o
90
0.6170.9901.173
0.7241.1221.359
0.8991.3841.642
1.1901.6571.851
1.151 0.4 0o
45o
90
0.7670.9981.138
0.8961.0751.225
1.0801.2471.370
1.3181.3741.447
1.277 0.6 0o
45o
90
0.9161.0241.110
1.0151.0621.145
1.1721.1821.230
1.3531.2431.264
1.571 1.0 0o
45o
90
1.1741.0671.049
1.2291.1041.062
1.3551.1811.107
1.4641.1931.112
Solution:
a) From the stress intensity solution, the highest K values relate to the maximum depth position (Phi =90o) and this position is used to find crack depth a. As we do not know the critical crack depth, we willhave to assume a value of a/B to calculate acrit using K1C, and then check whether this gives a value ofa/B close enough to our initial estimate. If not, then we must iterate through the calculation using amore refined estimate of a/B.Consider a first estimate of a/B of 0.2, this gives:
IC
3c
1.138 0.75 910= = 115.4 MPa
1.151
a 9.3 10 m = or 9.3 mm
caK m
From this, we can find a/B = 9.3/40 = 0.233. This is reasonable close to our estimate of 0.2, but wecould improve the prediction by linearly interpolating for the 'correct' value of a/B. This interpolationgives a/B = 1.138 + [(0.033/0.2) x (1.225 - 1.138)] = 1.152 - a change of only 1.2%. Hence it isacceptable to leave the critical crack size as 9.3 mm. If one redoes the calculation, however, acrit = 9.20mm.
(continued to 2.10b)
50
Example 2.10b (continuation of 2.10a)
Solution (b):
b) The leak-before-break criterion requires the vessel to be stable (i.e. not to suffer fracture) inthe presence of a surface crack which penetrates the wall (a = B) and very quickly grows tobecome a through-thickness crack with athrough-thickness crack = csurface crack. Essentially, this meansthat we have to check for stability in the presence of both a surface crack and a through-thickness crack. As the surface length of the semi-elliptic crack is required in calculating Kfor the through-thickness crack, we start with the semi-elliptic crack.Surface crack: a/c = 0.4 and a/B = 0.8 (highest value given in the table, but the ratio of a/Bshould really be 1):
IC
c
1.447 0.75 910= = 115.4 MPa m
1.151
= 5.76 mm
caK
a
The value of c for this crack is 5.76/0.4 = 14.4 mm
Through-thickness crack:
IC c= 0.75 910 = 115.4 and a 9.1 mmK a
Comments: However, this value of 9.1 mm is less than the value of 14.4 mm found from thesurface crack. Hence the through-thickness defect is critical. The maximum thickness of platein which both types of crack would be stable is 9.1 x 0.4 = 3.64 mm. Thus one might specifythat the vessel would be made using plate < 3.5 mm thick, although the safety margin then israther low.
Note: This analysis assumes that such a plate thickness would be adequate to carry theapplied load, i.e. that design stresses would have a maximum value 0.75x910 = 682.5 MPa.This may not be feasible with such a thin wall, or the proposed thickness of 3.5 mm may notmeet other design criteria (corrosion, deflectional stability etc) and the design may have to bere-assessed. Thus the problem illustrates the possibility of conflict between different designcriteria. Part of the skill of an engineer lies in optimising these types of conflict.
51
Example 2.11. Radial Cracks around Cylinders [B31]
A commonly encountered surface crack configuration under a remote applied tension, torsion ora combined loading system is that shown the Figure next.
The stress intensity factors for the loading system illustrated in the Figure are:
I III= f and = gd d
K a K aD D
For an applied torque T, the torsional shear stress becomes 3
16=
T
D
and the correction factors
are:Cond. In Next Box 2.11(a)
52
Example 2.11a (continuation of Example 2.11)
2 31 1 3 5 11
f =2 2 8 14 15
d D D d d d
D d d D D D
and
2 2 33 1 3 5 35 16
g =8 2 8 16 128 32
d D D D d d d
D d d d D D D
The crack length (size) is estimated as-
=2
D da .
PROBLEM: Two identical high-strength steel rods are prepared: one for a tension test at 106 MPa andone for torsion at 69 MPa. Calculate KI and and KIII. The rod dimensions are d = 4 mm and D = 8 mm. IfKIIIC = √(3/4)*KIC, (a) will the rods fracture? Explain; (b) Calculate the theoretical tensile and torsionfracture stresses, if fracture does not occur in (a). Use KIC = 25 MPa√m.
Solution:
Given: σ = 106 MPa; τ = 69 MPa; KIC = 25 MPa√m; a = (D - d)/2 = (8 – 4)/2 = 2 mm; (d/D) = 0.5 and(D/d) = 2. Then from the above equations, the correction factors are: f(d/D) = 1.9 and g(d/D) = 2.91.
Hence the applied SIFs and Mode-III fracture toughness values are:
Thus answer to question (a) is that the rods will not fracture as the applied SIFs in tension andtorsion are less than the respective fracture toughness values; KI < KIC and KIII < KIIIC.
(b) The fracture stresses are:IC IIIC25 21.65
= = = 167 MPa and = = = 94 MPa1.9* *2 /1000 2.91 *2 /1000f
f f
K K
d da g a
D D
I
III
IIIC IC
= f = 1.9*106* *2/1000 16 MPa m
= g = 2.91*69* *2/1000 = 16 MPa m
3 3= * = *25 = 21.65 MPa m
4 4
dK a
D
dK a
D
K K
Chapter 3 - APPLICATIONS OF LEFM
3.1. FATIGUE C
3.1.1. Introduction
Steady loads only have been considered socrack size on failure tendency. We now cofatigue process. A typical crack historysketched in Fig. 3.1(a). A crack of sizemanner until the critical crack size (af) ishand and disaster strikes. Examples of comsay, automotive transmission rods, steam tNuclear RPVs subjected to heat-up and cocalled thermal fatigue. Only me
For a given material, the instantaneous ratemainly upon the stress intensity range, ΔK,
It is the near tip field (characteriseknown to be greatly dependent updiagram) which is proportional to th
Growth rate da/dN is affected alcharacterised by the load ratio Rapproaching the critical Kc , ie. of im
If and when the load becomes combut such complications are neglecte
So, accepting that intensity range iis found to be as follows - thismaterial. There are three stages as s
Definition of Fatigue a(FCG and SCC); LEWorked Examples; ACode - KIR – curvdetermination for a 9C
OUTLINEnd Stress Corrosion Crack Growths
FM description of FCG and SCC;pplication of LEFM in the ASMEe; Illustration of KIR – curve
53
RACK GROWTH (FCG)
far, for which we have noted the effect of increasingnsider crack growth under alternating loads, ie. the under a cyclic load of constant amplitude, Δσ, isai exists initially, and grows in a stable, controlledapproached - when crack growth rates increase out of
ponents subjected to fatigue are rotating machinery,urbines etc. These are examples of mechanical fatigue.ol-down or temperature fluctuations undergo what ischanical fatigue will be considered here.
of crack growth, the slope da/dN, is found to dependsince,
d by K) which affects crack advance, and fatigue ison the range of stress and of load (the 'S' of the S-Ne range of stress intensity factor(Fig. 3.1(d)).
so by the mean component of intensity - typically= Kmin/Kmax - but this is largely a reflection of Kmax
pending fracture.pressive, a phenomenon called Crack Closure occurs,d here.s the main contributor to growth rate, the relationshipbeing a unique sigmoidal curve for each particularhown in Fig. 3.2.
r-1Mo steel
54
(a)
(b) (c) (d)
Fig. 3.1. (a) Typical crack growth history under constant stress range fatigue; (b) Constant stressrange cyclic load history; (c) Increasing stress range and (d) Stresses and Stress Ranges
expressed in terms of SIF range: ( min max min= , = , = and so onK K K K
) [B65]
55
Fig. 3.2. Fatigue Crack growth Rate depicted by Sigmoidal Curve [B41].
3.1.2. Three Stages of Fatigue Crack Growth (FCG)
Stage-I: Initiation. Relates to cleavage along grain boundaries or favourably orientedcrystallographic planes at rates of the order of one lattice spacing per cycle(4x10-7 mm/cycl); growth requires the stress intensity range to exceed some thresholdvalue; influenced greatly by environment/microstructure. This stage is bypassed if a crackexists prior to loading.
Stage-II: Stable Propagation. This is the most important stage, dealing with identifiablecracks (say > 0.1 mm) growing in a stable manner. The direction of propagation is lessrandom than in stage I and the material behaves more homogeneously. The materialcharacteristic crack growth rate vs intensity range is approximately log-log linearthroughout stage II.
56
Stage-III: Instability. Although important, this stage exists only for a very small fractionof the component's life, since the instability is catastrophic. The onset of stage III isdictated by the critical crack size being approached, that is by Kmax tending to Kc.
3.1.3. Stage –II FCG: PARIS LAW
Life prediction for fatigue cracks was made very much easier and far more quantitative in the1960's when Paris postulated that the range of stress intensity factor might characterise sub-critical crack growth under fatigue loading in the same way that K characterised critical, or fastfracture. He examined a number of alloys and realised that plots of crack growth rate againstrange of stress intensity factor gave straight lines on log-log scales. This implies that:
log = log( ) + logda
m K CdN
(3.1.1)
Taking out the log gives,
m=da
C KdN
(3.1.2)
Equation (3.1.2) is known as the Paris Law. For the first time, it became possible to make aquantitative prediction of residual life for a crack of a certain size. This simply required findinglimits on the integration in terms of crack size, which could be done by finding the final sizewhich caused fast fracture from the relationship between fracture toughness and crack size:
=K Y a (3.1.3)
Separation of the variables a and N and substitution for the range of stress intensity by theequivalent equation in terms of stress and crack size gives
f
0
mm
0 2
= =
=
( )
faN
mm ma
daC K C Y a
dN
dadN
CY a
(3.1.4)
57
It was later realised that this so-called 'law' applied to growth rates in the range of perhaps10-3 mm/cycle to 10-6 mm/cycle, and that the fatigue crack growth rate curve was sigmoidal inshape when growth lower and higher than this range were included. Typical data foraustempered ductile iron in air, as a function of stress ratio (minimum stress in cycle divided bymaximum stress in cycle - a measure of mean stress in the fatigue cycle) is shown in Fig. 3.3.
Fig. 3.3. FCG curves for austempered ductile iron in airas a function of Stress Ratio-R [B26].
58
The lower growth rate region is termed the threshold regime, because growth rates drop offsteeply and the crack becomes essentially non-propagating. This represents a change inmechanism from double shear continuum growth to single shear non-continuum growth. Thehigher growth rate regime is where values of maximum stress intensity in the fatigue cycle aretending towards the fracture toughness and static modes of fracture (cleavage, intergranular) areadding to the fatigue induced growth rates.
Because it covers the range of growth rates most useful to engineering structures, Paris lawremains a very useful relationship and, moreover, its extrapolation into the threshold regime cangive a conservative estimate for the remaining life. This development was crucial to the adoptionof defect-tolerance concepts and the implementation of a retirement-for-cause philosophy.Increasing the mean stress in the fatigue cycle (R = σmin/σmax = Kmin/Kmax) has a tendency toincrease the crack growth rate in all regions of the sigmoidal curve as shown in Fig. 3.3. ThisR-ratio (stress ratio) effect is generally less in Stage II than in Stages I and III. Modifications ofEq. (3.1.2), which caters to both stages II and III, are:
m
max
C
d CΔK= (3.1.5a)
d1
a
N K
K
m
C
d CΔK= (3.1.5b)
d (1 ) -
a
N R K K
With reference to the former, if Kmax << Kc, then the RHS denominator → 1 corresponding to stage II; alternatively if Kmax → Kc then da/dN tends to infinity (stage III ). The latter formulation(i.e., Eq. (3.1.5b), explicitly includes the R-ratio effect and is known as the Forman Equation.
Though fatigue life testing (smooth specimen testing – Section 1.3.3) is usually carried out underfully reversed stress or strain (R = -1), fracture mechanics fatigue crack growth data is usuallydetermined for conditions of pulsating tension (R = 0). Compression loading cycles are not usedas during compression loading the crack is closed and the SIF is zero. Though compressionloading is generally considered to have little effect on crack propagation, under variableamplitude loading (see Fig. 3.1c) compression cycles can have an effect.
59
3.1.4. FCG Testing
For conducting FCG tests and generating data there are various standards, the most importantbeing the ASTM E647-83 and its Revision and BSI (British Standards Institution) DraftStandard. FCG testing generally involves measurement of growth rates above 10-8 m/cycle underconstant amplitude loading. The main requirement as per these standards is that the materialremains predominantly elastic and buckling does not occur and pertain to Mode-I loading. C(T)or Centre Cracked Tension (CCT) or even Single Edge Notched Bend (SENB) specimens areused. Revision of ASTM E647-83 contains different test procedures and conditions, and covers,in addition to the Paris regime, measurement of FCG rates from near-threshold to Kmax controlledinstability. A working definition of ∆Kth is provided. The most important requirement forensuring elastic conditions as per the ASTM standard are:
C(T) specimens:
2
max4- , derived from plastic zone consideration
YS
KW a
and
CCT specimen: max- 2 1.25 , based on general yielding consideration.YS
PW a
B
Anyone embarking on FCG study and testing should consult, in addition to the relevant ASTMor BS standards, the following excellent interpretative and comparative review [B1] by R. J.Allen, G. S. Booth and T. Jutla: A Review of Fatigue Crack Growth Characterisation by LinearElastic Fracture mechanics (LEFM). Part I – Principles and Methods of Data Generation (pp.45-69) and Part II – Adivisory Documents and Applications within National Standards (pp. 71-108)in Fatigue and Fracture of Engineering Materials and Structures, Vol. 11 (No.2), 1988. Becausethinner specimens and smaller loads are used, FCG testing may be accomplished using lowerload-capacity machines as compared to those used for LEFM fracture toughness tests. It will bebeyond the mandate of this introductory book to discuss the various aspects and techniquesinvolved in FCG testing. Figure 3.4 gives a schematic outline of the steps involved in FCGtesting and application of the results to structural life assessment.
60
Fig. 3.4. Steps in obtaining da/dN vs. ΔK data and using it for an engineering application [B13].
Crack growth during testing is monitored either using a low-power microscope of 20-50Xmagnification or using more sophisticated techniques like Potential-Drop, stiffness orcompliance change technique. Sometimes non-destructive test techniques like acoustic emissioncan monitor the position of the crack tip. Figure 3.5 shows the typical crack length (a) vs.number of cycles (N) data obtained for an AISI 4340 low-alloy steel.
61
Fig. 3.5. Crack length vs. cycles data for four different levels of cyclic loadapplied to C(T) specimens of an alloy steel [B13].
These data have been converted to da/dN vs. ΔK data in Fig. 3.6 and the least square fitted line isalso shown giving the Paris Law. Even if data from different specimen geometries are included,the same fit will be obtained so long as the test conditions follow the ASTM E647 guidelines.Typical values for the Paris constants for the worst case for some common classes of steels aregiven in Table 3.1.
62
Fig. 3.6. Derivation of the Paris law from the data in Fig. 3.5 [B13]
Table 3.1. Paris constants for some common classes of steels in the worst case [B13]
Class of Steel Constants for da/dN = C(ΔK)m
C,(mm/cycle)/(MPa√m)m
m
Ferritic-Pearlitic 6.89x10-9 3Martensitic 1.36x10-7 2.25Austenitic 5.61x10-9 3.25
63
WORKED EXAMPLES IN FCG
Example 3.1.1. PRESSURE SURGES in A PIPEProblema) A long pipe has an outer diameter (OD) of 90 mm, an inner diameter (ID) of 70 mm andworks at a pressure (p) of 40 MPa. Valve failure downstream in the pipe caused a pressuresurge which burst the pipe. Examination of the fracture surface revealed a metallurgicaldefect at the inner surface of the pipe which was semi-elliptical in shape with a depth of 1.6mm and a surface length of 4.5 mm. This flaw was orientated perpendicular to the hoopstress in the pipe.What pressure would have caused this failure?
Note: The formula for hoop stress from thin walled theory is p(ID)/2t, while from thickwalled theory it is p[(L2 + 1)/(L2 - 1)] where t is the wall thickness of the pipe and L is theratio OD/ID. Which formula would you use and why?
Given: The plane strain fracture toughness of the pipe alloy is 25 MPa m½ and the geometrycorrection factor can be found from the graph below.
=K Y a
(continued to 3.1.1a)
64
Example 3.1.1a. (continuation of 3.1.1)
b) A new pipe was manufactured from the same alloy and subjected to NDT prior toinstallation. This showed that the pipe contained a similarly orientated flaw, 1.5 mm deep, butwith a semicircular shape.Assuming normal operating conditions, i.e. no pressure surges and a daily evacuation to zeropressure, will the pipe last for its desired lifetime of 30 years?Assume that the geometry correction factor has a constant value of 0.7 in this second part ofthe question and note that a fatigue crack growth rate of 6.25x10-8 mm/cycle corresponds toan applied delta K value of 10 MPa √m. The Paris law exponent m is 4.
Solution:
a) From the OD and ID we can obtain the wall thickness t as 10 mm, i.e. (90 - 70)/2 mm. Thehoop stress from thin walled pressure vessel theory is:
(ID) 40 70= = = 140 MPa
2 20Hoop
p
t
This formula is strictly valid only for situations where ID > 10t. This is clearly not true in thepresent case, and we should really use thick walled pressure vessel theory. In thick walledpressure vessels, the hoop and radial stresses are not constant through the thickness, and thegreatest absolute values occur at the inner wall of the vessel. Here:
2
Hoop,thick 2
1 OD= , where =
- 1 ID
Lp L
L
Thin walled theory gives the average value of the hoop stress through the wall thickness, andit is easy to compare the difference between maximum and average values of hoop stressthrough the parameter:
2
2
1=
- 1
LS
L
The table below shows the difference for values of L from 1.01 to 2.0:
L 1.01 1.05 1.10 1.20 1.50 2.00
S 1.005 1.026 1.052 1.109 1.300 1.667
(continued to 3.1.1b)
Example 3.1.1b (continuation of 3.1.1a)
The ratio of L in the present question is 1.29, hence we should use the thick walled theory,i.e.:
2 2
Hoop,thick 2 2
1 1.29 1= = 40 = 160.4 MPa or (4.01 )
- 1 1.29 - 1
Lp p
L
The internal pressure must be added to this value in calculating the stress intensity factor,because K values will arise from the hoop stress and the internal pressure loading of thecrack faces. Now:
=K Y a
As a/c = 1.6/2.25 = 0.71 and a/t = 1.6/10 = 0.16, Y from the graph given in the problem isapproximately 0.78. Thus at fracture:
IC= 0.78 5.01 0.0016 =
25= = 90.2 MPa
0.277
K p K
p
Thus a pressure surge of 90.2 MPa would cause fracture of the pipe. If thin walled pressurevessel theory had been used the failure pressure would have been given as 100.5 MPa - asignificant difference of 11.4%.
b) To integrate the ‘Paris law' to obtain the fatigue life, the integration limits must be foundin terms of crack size. The initial flaw size is given as 1.5 mm and the final size causingfracture can be found by substituting into the K equation, the fracture toughness and appliedstress arising from the applied load range. The applied load range goes from zero to 40 MPa,i.e. a stress ratio R = 0, hence the peak stress will correspond to the 40 MPa.
The equation above for hoop stress in a thick cylinder indicates that its value at the inner wallis 4.01p, i.e. 160.4 MPa, and to get the total applied stress intensity factor we need to add inthe internal pressure (which loads the crack surfaces). This gives a total stress of 200.4 MPa atpeak load, hence:
IC
f
= = 0.7 200.4 = = 25 MPa
10.2 mm
fK Y a a K m
a
This is, in fact, slightly larger than the wall thickness, so the final length should be limited to10 mm.
65
(continued to 3.1.1c)
Example 3.1.1c (continuation of 3.1.1b)
The last piece of information needed is the constant C in the Paris law. This can be obtainedfrom the growth rate information given:
m
md= ( )
d
aC K C Y a
N
Now separate out the variables in the Paris law and integrate the equation between the crackgrowth limits.
f
1
0.010
2-12 4 40 0.00152
d dad = =
6.25 10 0.7 200.4
fN a
mm ma
aN
aCY a
0.010 0.010
f 2
0.0015 0.0015
f
d 1N = 41.864 = 41.864 = 41.864 -100-(-666.67)
, 23723 cycles
a
a a
Hence N
Comments: As 30 years is equivalent to 30 x 365 = 10 950 days (excluding leap years!), thepipe should last the required lifetime.As many factors can accelerate growth rates, however, it would be prudent to inspect thiscomponent at regular intervals. These inspection intervals can be set with reference to the lifeintegration. A curve of a versus N is generated from integration to various lengths. This is anexponential curve, and converting N to time (which can easily be done seeing the frequency is1 cycle per day) allows inspection intervals to be chosen such that the crack would notbecome critical in the interval between inspections. In the present case, there is also an initial
66
period of time when inspection is not required.
67
Example 3.1.2.
INCREASING FATIGUE LIFEA structure contains a critical component made from A514 steel. After fabrication of thestructure, a welding defect 7.6 mm deep is discovered in this steel plate. The flaw is essentiallyan edge crack under tension loading, and the required cyclic life of the structure is 100 000cycles. The component is subject to a fluctuating load which causes a stress variation from172 MPa to 310 MPa.
Given: Material properties for the A514 steel are: yield stress = 689 MPa, K1C = 165 MPa m½
geometry correction factor Y = 1.12, and the Paris law is:
10 2.25d1.36 10
d
aK
N
where da/dN is in m/cycle, and:
=K Y a
i) Calculate the fatigue life of this component based on attaining a critical defect size for fastfracture.ii) Accurately construct the curve showing crack length against number of applied load cycles.iii) Discuss the various measures that could be adopted to extend the life of the structure to100 000 cycles.iv) What is the effect of reducing the initial defect size to 5 mm (by weld repair with bettercontrol of process parameters)? Explain this result in terms of the shape of the curve of cracklength versus cyclic life.
i) In order to calculate the cyclic life, we require the critical crack size causing fast fracture.This can be obtained by substituting the appropriate data into the K calibration equation:
2 2
-2ICc
max
165= = 7.19 10 m
1.12 1.12 310
Ka
i.e. 71.9 mm. Hence the limits on the Paris law integration are 7.6 mm and 71.9 mm, while thestress range is (310 - 172) MPa = 138 MPa. Separating the variables and rearranging the Parislaw gives:
cond. to next box 3.1.2a
68
3.1.2a (continued from 3.1.2)
f
0
mm
0 2
= =
=
( )
faN
mm ma
daC K C Y a
dN
dadN
CY a
We can perform the integration symbolically and then substitute the actual values into theequation to obtain the required life.
1 12 2
f if
2 2
12
-1 1= =
1 -21
2
f
i
a
m m
m mm m m m
a
m
a aN
mCY CY
am
0.13 0.13
f 10 2.25 2.25 1.13
1 0.0719 - 0.0076=
1.36 10 1.12 138 -0.13N
f = 87,992 cyclesN
ii) To draw the accurate crack length - life curve, all that is required is to calculate N atvarious lengths between 7.6 mm and 71.9 mm, i.e. repeating the above calculation withvarious af values. The table below gives some typical values, and the curve is drawn inthe figure.
a/ m 0.015 0.025 0.04 0.06N/ cycles 29421 49850 67486 81857
iii) Fatigue life can be extended by a number of measures, all of which rely on changingrelevant parameters in the Paris law – stress range, final crack length and initial cracklength. Stress range can be reduced by decreasing the applied loads, which is usually notfeasible particularly for new structures, or increasing component size (which may befeasible only at the design stage). When a crack is detected in service, e. g., in a pressurevessel, it may be possible to reduce applied stresses and continue operation until ascheduled maintenance period.
Cond.
69
3.1.2b (cond. from 3.1.2a)
Final crack length could be increased if it was possible to increase the toughness locally, e.g. byusing additional material at the crack location, or by replacing the cracked component with oneof a higher toughness (and hence a larger critical defect size for the same value of applied peakstress. The increase in life would, however, be quite low because of the fast growth rates as thecrack approaches the critical size.
The best solution is to reduce the initial defect size, either by repairing or replacing thecomponent. If the initial defect size is likely to be an intrinsic problem in the fabrication/repairprocess for this component, the fabrication process must be controlled more closely to reduce theinitial crack length.
iv) Reducing the initial defect to 5 mm gives a cyclic life of 107 423 cycles. The figure aboveindicates that the crack growth rate curve is exponential and hence a relatively small reduction ininitial crack length yields a large life return.
70
Example 3.1.3. Fatigue Crack Growth and Striation Spacing on the Fracture Surface-A [B31]
Because of the cyclic opening and closing at the crack tip during fatigue, at slow and intermediate growthrates in the Paris regime, some alloys develop a characteristic rippled appearance and these are referred toas striations as shown for an Al alloy in the fractograph. Each striation width corresponds to the crackgrowth per cycle (typically of the order of 10-4 mm/cycle or less) and is related to the apparent SIF range(ΔK) and the empirical relation obtained is:
Striations on the fracture surface of an Al alloy-Magnification 12000X
x=
6K E , where x is the average striation spacing in µm, and will be constant during a constant
amplitude FCG test. Striations may not form when large SIF range and large Kmax operate resulting invery large growth rates. This may help in failure analysis.
PROBLEM: Determine the apparent ΔK for the above Al alloy from the striation spacing. Take E =72000 MPa. Based on the magnification given, the average striation spacing is:
-34 mm 4*10 mx = 333.33 μm
12000 12000 ; Hence,
-34*10= 72000 17 MPa m
6*12000K
and the estimated FCG rate is:d
333.33 μmd
a
N
71
Example 3.1.4. Fatigue Crack Growth and Striation Spacing on the Fracture Surface-B [B31]
The fracture surface of a FCG tested 17-4 PH Stainless Steel is shown in the above figure. Asingle-edge cracked plate containing a crack of 6 mm length was tested using a constantamplitude loading with the stress ratio, R = 0. The plate had a thickness of 5 mm and a width of20 mm and was sufficiently long. Calculate the apparent SIF range and maximum load. TheParis equation for the material is given by:
3.512d
(10 m/cycle) Kd
a
N
Cond. to next box 3.1.4a
72
3.1.4a (cond. from 3.1.4)
SOLUTION:
Given: a = 6 mm; W = 20 mm; B = 5 mm; b = (W – a) = 14 mm.
Then, a/W = 0.3; R = Kmin/Kmax = 0; Kmin = 0. The SIF expression from Table 2.1 (Case-b) is:
I
1.12+ (2.19 - 0.64) 1.12 + 0.3(2.19*0.3 - 0.64)= a and = 1.65
(1 - 0.93 ) (1 - 0.93*0.3)K Y Y
From the fractograph, striation spacing x = 7.69*10-8 m. That means crack growth per cycle,da/dN = 7.69*10-8 m
-8 -12 3.5
(1/3.5)-8
-12
d= 7.69*10 m/cycle = 10 (m/cycle)( )
d
7.69*1024.89 MPa m
10=
aK
N
K
Now, based on the striation equation given in the earlier Example,
-8x 7.79*10= = 207000 = 23.43 MPa m
6 6K E
About 6% less than the previous result. This is very much OK.
Since, R= 0, this SIF range gives the value of Kmax. For an average, ΔK = 24.17 MPa√m = Kmax.
3max max max
3
2
max
max max
= 24.17 = a *1.65* *6*10
24.17= = 106.7 MPa
1.65 *6*10
N= * * = 106.7( )*5 (mm)*20 (mm) = 10670 N = 10.67 kN
mm
K Y
B WP
73
3.2. STRESS CORROSION CRACKING (SCC)
3.2.1. General
Stress-corrosion cracking (SCC) is the failure of an alloy from the combined effects of acorrosive environment and a static tensile stress. The stress may result from applied forcesor "locked-in" residual stress associated with welding or cold working. Only specificcombinations of alloys and chemical environment lead to stress-corrosion cracking. Usuallyonly a few chemical species in the environment are effective in causing SCC of aparticular alloy and these species need not be present in large quantities or in highconcentrations. Moreover, the chemical environment which causes SCC does not producegeneral chemical corrosion of the alloy. With some alloy/chemical species combinationstemperatures substantially above room temperature are required to produce SCC.
Mechanisms of SCC are complex and result in a nonductile type of failure.Fractographically they show either intergranular cracking or cleavage and frequentlyundergo extensive branching. Classic examples of SCC are brass in an ammoniacontaining environment (so-called 'season cracking' which was noted in brass cartridgecases stored next to stables during the monsoon season in India), chloride-induced SCCin austenitic stainless steels and aluminium alloys, some ceramics, glasses and polymersin moist air, and steels in caustic, hydrogen containing or hydrogen sulphideenvironments.
The interest in characterisation of crack growth under SCC conditions by the fracturemechanics parameter K arose because it was realised that a number of alloy-environmentcombinations which appeared immune to SCC when tested as smooth specimens, werevery susceptible to this phenomenon in the presence of a crack or crack-like defect.Hence fracture mechanics tests are used to characterise crack velocity in SCC and findthe threshold for stress corrosion crack growth, which is termed K1SCC.
Similar specimens with the same initial crack but loaded at different levels(different initial K-values) show different times to failure as is showndiagrammatically in Fig. 3.7. The specimen initially loaded to KlC fails immediately.Specimens subjected to K values below a certain threshold level never fail. Thisthreshold level is denoted as KISCC, SCC standing for stress corrosion cracking or,more generally, KIEAC (EAC stands for environment assisted cracking). During the SCC,as the load on a specimen is kept constant, as the crack extends, the stress intensitygradually increases and failure occurs at KlC.
As the applied load is constant in SCC, it is more useful to talk about crack velocity(da/dt = v) and plot these against applied stress intensity level to give what are termedv-K curves. A typical v-K curve is shown in Fig. 3.8. As is the case for fatigue crack
74
growth, 3 distinct regions can often be observed which reflect the operation of differentinfluences in the mechanisms of cracking. In region I the crack growth is a strongfunction of K, as the value of K controls the environmental reaction rate at the crack tip.This plot serves to aid in establishing Kt but the magnitude of this threshold value maybe very low. In region II crack growth is essentially independent of K, the ratecontrolling step being the environmental transport to the crack tip, it is still stronglyaffected by temperature and the environment. The plateau growth rate is the maximum thatit is possible to sustain in the alloy environment system by environmental crack growthalone. These rates are typically 10-9 to 10-6 m/s, which are too fast to provide a reasonabledesign life. Region III represents a situation where da/dt varies strongly with K. As Kapproaches KIC the crack growth rate becomes unstable.
Figure 3.7. Schematic time-to-failure curve with KIscc.
3.2.2. Treatment of Crack Growth Rate
The extent of the regions is variable, and region 1 often dominates the life. As the curve inthis region is linear on a log-log plot, it has a simple equation which is easy to integrate toobtain a life estimate. Actual test data mainly of region-I are plotted in Fig. 3.9. The equationof a straight line is:
log = log +n K D
= nDK
75
It is often useful to use the K1SCC data and recast the equation as:
threshold ISCC
=
n
K
K
Separating the variables and integrating this between K limits is straightforward:
== da da dK
dt dK dt
and2
2 2i.e., == K
a aY
K Y
Figure 3.8. Stress corrosion crack growth rate as a function of K.
2 2
2= K dK
Y dt
76
The lower limit on the integration would be the K value corresponding to the combination ofinitial defect size and applied stress, while the upper limit could be either the fracture toughnessKC or an upper limit on the extent of region 1. Typically therefore, separating the variables gives:
f
2 2
0
2= with =
C
i
Kt
n
K
KdKdt DK
Y
and
f
2 2
0
2=
C
i
Kt
n
K
KdKdt
Y DK
Fig. 3.9. A typical v-K curve for the case inorganic glass in a moist air environment(50% relative humidity and a temperature of 25°C) [B26].
2 21
f 2 2 2 2
-2 2= =
2
C
i
K n nn C i
K
K Kt K
Y D Y D n
Note that if the integrand has n = 2, the expression for tf will contain natural log (ln) terms.
77
WORKED EXAMPLES IN SCC
Example 3.2.1. SCC OF GLASS
Note: Stress corrosion cracking (SCC) is another sub-critical crack growth mechanism which isvery suitable for LEFM treatment, because applied stress intensity values are usually low in thiscracking regime. One of the problems with SCC is that crack growth velocities increase verysharply with increase in K, and the lifetime may be correspondingly short. Hence, if thecombination of likely defect size and applied, or residual, tensile stress cause the threshold forstress corrosion cracking (K1SCC) to be exceeded, it is usually necessary to avoid the possibilityof SCC. This is likely to require either a change in alloy, or surface protection. Nonetheless, it isuseful to illustrate the application of fracture mechanics to SCC lifetime determination.
_______________________________________
A glass shelf in a bookcase can be considered as simply supported beam subject to a uniformlydistributed load. A particular manufacturer produces bookcases with shelves that are 1.5 m inlength (L), have a width (b) of 200 mm and a thickness (t) of 10 mm. The design load on theshelves is estimated as w = 100 N/m2.The manufacturer knows that handling of the glass shelves during fabrication may cause surfaceflaws to be present in them, which can be considered as semi-elliptic in shape with a maximumdepth of 0.1 mm. He is also aware of the fact that a moist air environment may cause stresscorrosion cracking to occur in stressed glass. He has therefore approached you, as a fracturemechanics/failure consultant to advise him whether he should offer a one year guarantee on thesebookcases.
In your calculation of the lifetime of a typical shelf under these conditions, you may assume noincubation period is required before the crack extends. You have access to the followinginformation:
(1) The maximum bending stress in a simply supported beam subject to a distributed load isgiven by:
3
2
= , where = second moment of area = and12
maximum bending moment =8
Mt btI
I
wLM
(2) The shelves are to be made from soda glass with a modulus of elasticity E = 70 GPa anda work to fracture R = 0.01 kJ/m2. A crack velocity (v-K) curve for soda glass in a moist
air environment is given below.
78
(3) The stress intensity factor for a semi-elliptic surface defect can be calculated from:
= 1.1K a
where a is maximum depth of defect.
Solution of this problem requires substitution of the value of the fracture stress (which is equalto the applied stress) into the stress intensity equation. This will enable us to find the criticaldefect size and hence we can calculate the initial and final values of K which provide limits onthe integration of the v-K equation, i.e.:
= nDK
We can then express crack velocity as da/dt = (da/dK).(dK/dt) separate variables and henceobtain the time to failure from integrating with respect to time t.
The given information includes work to fracture R and the elastic modulus E, which implies thatthe Griffith equation should be used to calculate the critical crack size causing fracture. Theapplied stress is found by rearranging the bending equation and substituting in for M and I:
79
34200 10
= = 16 667 mm12
I 2 210 0.1 1500
= = 8.43 MPa16 16 16667
tWL
I
Rearranging the Griffith equation gives the critical crack size:
9
f 22 6
70 10 10= = = 0.00314 m, i.e., 3.14 mm
8.43 10
ERa
Hence we can now calculate values for Ki and Kf.
1
-4 2i = 1.1 8.43 10 = 0.164 MPa mK
1
-4 2f = 1.1 8.43 3.14 10 = 0.92 MPa mK
To find the constants in the v-K curve is quite simple, as the plot is linear on a log-log scale, i.e.:
1 2
1 2
log - log= =
log - logn
K K K
Taking two points on the curve, e.g. 10-2 m/s where K = 0.68 MPa m½, and 10-8 m/s whereK = 0.29 MPa m½, gives n = 16.21. The constant D can be obtained by substituting into the v-Kequation.
-2 16.21= , i.e., 10 = 0.68nDK D
= 5.188D
Checking this value with the other chosen point confirms that it is correct, and the v-K equationis therefore:
Checking this value with the other chosen point confirms that it is correct, and the v-K equationis therefore:
It is useful to perform the integration symbolically, as this facilitates finding errors, andsubstituting actual values into the final form.
80
It is useful to perform the integration symbolically, as this facilitates finding errors, andsubstituting actual values into the final form.
== da da dK
dt dK dt
and
2
2 2i.e., == K
a aY
K Y
2 2
2= K dK
Y dt
Integrating this with respect to t gives:
f f
i
2 2
0
2=
t K
K
KdKdt
Y
f f
2 2
0
2=
i
t K
n
K
KdKdt
Y DK
f
i
2 2
1f 2 2 2 2
-2 2= =
2
K n ni fn
K
K Kt K
Y D Y D n
14.21 14.21
f 2 2
2 0.164 - 0.92= s
8.43 1.1 5.188 14.21t
7f = 1.44 10 s or about 167 dayst
As this is less than 6 months, the one year guarantee does not seem like a good option. In reality,there would be an incubation period before crack growth started and the shelves may well lastthe required period of time. Nonetheless, a prudent manufacturer might look to reducing stressesthrough, for example, shorter shelves or reducing the initial defect size by polishing the shelves!It should also be noted that that typical scratches, although around 0.1 mm in depth would not actlike sharp cracks, particularly as high levels of surface compressive residual stress are induced in
81
such shelves through so-called 'tempering' of glass. Hence glass shelves can be routinely foundin bookcases and display cabinets.
Example 3.2.2.
To determine the susceptibility of a material for stress corrosion cracking, 4 specimens withinitial crack lengths of 12.5 mm (specimens with one crack tip) are tested. The times to failureare found as 1, 10, 500 and 5000 hours, while the final crack sizes are 20, 30, 50 and 51 mmrespectively.
Estimate KIscc of the material as accurate as possible. Given: specimen width W = 100 mm planestrain fracture toughness KIc = 40 MPam geometry factor f(a/W) = 1.12 (a/W)2 .
Solution: For each of the above crack lengths, the following Table gives the correspondingf(a/W) (= Y) obtained from the formula given:
a/W 0.125 0.2 0.3 0.5 0.51f(a/W) = Y 1.136 1.16 1.21 1.37 1.3801
σ (MPa) - 137.6 107.7 73.7 72.4initial KI, MPa√m 31 24.2 16.6 16.3Failure time, h 1 10 500 5000
ICI IC
40At failure, = = or = = , where f ( / )
KK K Y a Y a W
Y a Y a
The stress so obtained are given in the third row of the above Table. The stress to which eachspecimen is loaded is the same as the stress at failure for each specimen as the applied load isconstant for a specimen.
Hence the initial KI values for each specimen is given by,
I = = 1.136 0.0125 for a constant initial crack lengthK Y a
where σ is the failure stress calculated above. The initial KI values so obtained are given in thefourth row of the above Table and are plotted against log(time to failure) in the Figure below.
82
Log of Time to failure (h)
0 1 2 3 4
Initi
alK
I,M
Pa.m
0.5
14
16
18
20
22
24
26
28
30
32
It can be seen from the figure that at longer times the KI is approaching a constant value of 16MPam below which failure is unlikely to occur.
Hence KISCC = 16 MPam
Example 3.2.3.
In a very large steel part a crack is found with a length of 2a = 40 mm. The structural part will beapplied under circumstances where stress corrosion plays a role. The incubation time for thecombination of material, thickness and environment can be derived from the plot in figure A.Information about the stress corrosion crack growth rate is given in figure B. (In Figs. A and B,the bold lines are as given in the problem while the light (narrow) lines are those drawnduring solving the problem for estimating the relevant values)
For the geometry of the part the stress intensity as a function of crack length is given by KI =a. The part will be subjected to a constant load resulting in a nominal stress equal to 100MPa. What is the lifetime of the part if possible crack growth is allowed as long as KI KIC/4?
For the constant stress 100 MPa, and initial a = 20 mm (= 0.02 m),
I = = 100 0.02 = 25 MPa mK a
83
From Fig. A, for this KI, the incubation time (log(s)) is approximately read off as 3.7 whichconverts to a time of 5012 s. It may also be noted that in Fig. A the KI corresponding to zeroseconds is 150 MPam (i. e. KIC = 150 MPam).
The problem permits crack growth to occur till this KI reaches a value of KIC/4 = 150 MPam/4= 37.5 MPam.
From Fig. B, it is seen that from the initial KI = 25 MPam to the final KI = 37.5 MPam, thecrack growth da/dt = 0.001 mm/s.
Since the applied nominal stress is constant (100 MPa), the final crack length at the final
KI = 37.5 MPam is:2 2
2 2
37.5= = = 0.045 m = 45 mm
100IK
a
. Hence the time for the crack to
grow from the initial half-size of 20 mm to the final half size of 45 mm at the rate 0.002 mm/s is(45-20)/0.001 = 25000 s.
Hence, Total time = Crack Growth Time + Incubation Time = 25000 s + 5012 s = ~ 30000 s.
84
3.3. ASME CODE APPROACH FOR DESIGN AGAINST BRITTLE FRACTURE:RTNDT-KIR CURVE APPROACH [B25, B27, B28, B49, B51, B53, B55, B56 and B60]
3.3.1. Ductile-Brittle Transition Temperature: Different Approaches
The ductile to brittle fracture transition behaviour of a ferritic steel can be described by a Charpyenergy (CV) vs. temperature plot where the energy absorbed by a Charpy V-notch (CVN)specimen during fracture by impact decreases from a relatively high value at higher temperature(upper-shelf) to low values at lower temperatures (lower-shelf) – see Fig. 1.9. Therefore to avoidcatastrophic failure, operating temperatures of the components are specified to be higher than thetransition region. A reference transition temperature is generally used to locate the ductile-brittletransition temperature (DBTT) above which only the component can be subjected to significantamount of stresses.
The DBTT of ferritic steel depends not only on the material condition but also on the loadingrate and mode of stress distribution ahead of the crack tip. Definition of a meaningful referencetransition temperature for ferritic steels used as structural components therefore requires use of atest method that considers the combined effects of material, loading rate and notch geometry inan effort to match the conditions experienced in actual operations. The reference temperaturecurrently used for reactor pressure vessel steels is RTNDT (ASME Code-Section III andSection XI [T2 and T3]). RTNDT is defined as the higher one of the two temperatures, namely,TNDT and (TCV–33 ˚C), where TNDT is the drop-weight nil-ductility temperature (NDTT) andTCV is the temperature at which both a minimum Charpy V-notch energy of 68 J and a lateralexpansion of 35 mils (~ 0.89 mm; 1 mil = 0.001 inch) are obtained. To prevent catastrophicbrittle fracture in the components, it is recommended that the operating temperature should be33 ˚C or more above the RTNDT, depending on component thickness. It is important to emphasisethat the definition of RTNDT employs both Charpy V-notch and drop-weight test to determine theDBTT instead of only Charpy test results.
ASTM E 208 [T15] gives the standard procedure for drop-weight NDTT testing. In the nuclearpower industry, drop-weight testing and Charpy impact testing constitute the two most widelyused tests for assessing the fracture resistance of materials, particularly their propensity for brittlefracture. The drop-weight test essentially consists of initiating a brittle running crack from a sawcut in a brittle weld deposit and propagating it across the specimen loaded in three point bend(3PB) condition under a gravity driven hammer. Drop-weight NDTT, also termed as TNDT, is thehighest temperature at which the crack initiated in the weld-bead propagates across the entirespecimen within elastic limit. A deflection stopper is used to ensure only elastic deformation ofthe specimen before fracture.
Subsequent development of fracture mechanics methods and test techniques led to incorporationof Linear Elastic Fracture Mechanics (LEFM) concepts in Section III and Section XI of theASME Code in 1972 for design against brittle fracture. This is based on two reference toughnesscurves, namely KIC curve and KIR curve, indexed to RTNDT. While KIC curve guides the operating
85
range of toughness in the static/quasi static loading rates, the KIR curve provides a lower boundtoughness incorporating the (KIC), dynamic (KId) and crack arrest toughness (KIA) of the materialin the operating temperature range. It may be noted that the relationship between the referencetoughness curves and RTNDT is purely empirical and based on the data obtained mainly fromreactor pressure vessel steels and welds. The ASME code prescribes application of KIR curve forsteels with room temperature yield stress level of 345 MPa or less. For steels with higherstrength, the applicability of the KIR curve needs verification by KId measurements over thetemperature range of interest [B25, B49, B51].
In an effort to predict conservative fracture toughness in the transition temperature range, ASMEBoiler and Pressure Vessel Code [ASME Code-Sections III and XI] has merged both thetransition temperature approach and fracture mechanics approach, to predict a lower boundfracture toughness transition curve as:
IR NDT= 29.4 + 13.675exp(0.026( - ))K T RT [3.3.1]
where T is the test temperature and RTNDT is as defined in the previous section [B58, B60]. This iscommonly known as ASME KIR curve, where KIR denotes the lower bound fracture toughness ofstatic, dynamic and crack arrest tests. The ASME KIR curve was developed empirically bydrawing a lower-bound curve by an approximate statistical procedure to a substantial collectionof valid KIC, KId and KIA data (showing considerable scatter) mainly from pressure vessel steels,assembled in the early 1970s. Till date, this curve remains the basis for design against brittlefracture. Similarly, the ASME CODE (Section XI) also gives an expression for the lower-boundstatic fracture toughness curve indexed to RTNDT as follows [B58, B60]:
IC NDT= 36.5 + 3.083exp[0.036( - RT +56)]K T [3.3.2]
For both the curves an upper-shelf cut-off of 220 MPa√m is used for virgin material and a cut-off of 180 MPa√m is used for damaged or irradiated material. The KIR curve is intendedto describe normal operation conditions and the KIC curve emergency and faulted conditions. Inaddition to the different reference curves, safety factors are applied. For normal operatingconditions a safety factor of 10 is applied either on the allowable crack size, or a safety factor of√10 upon KI or KIa (KIR). For emergency or faulted conditions, the corresponding safety factorsare 2 on allowable crack size or √2 on allowable KI or KIC [B58, B60].
3.3.2. KIR curve determination for a 9Cr-1Mo steel – Illustration [B28]
3.3.2.1. Specimen Fabrication
A schematic lay out for the drop weight and Charpy specimens from a weld pad is given inFig. 3.10 (weld metal is of the material to be tested, say, 9Cr-1Mo weld). In the case of the basematerial the specimen configuration is the same except that the specimens are cut entirely fromthe base material plate. Drawing for a weld-pad is shown in Fig. 3.10 to illustrate the method
86
used for fabricating specimens for determining DW-NDT of weld metals. IS 2062 mild steel (oreven the base material plate can be used) side plates are used to reduce the amount of weldmaterial required [B50].
3.3.2.2. Drop-Weight specimen fabrication and Drop-Weight Test (DWT)
Drop-Weight P-2 specimens, as per the dimensions specified in ASTM E 208, were fabricatedfor the base material from a 20 mm plate of 9Cr-1Mo base metal plate (Fig. 3.10). The crackstarter weld bead was deposited on the specimen blanks.
To obtain conservative values of TNDT with reduced scatter, the following procedures werefollowed for weld bead deposition:
1. Use of a short stringer weld bead without oscillation of the electrode especially for P-2 andP-3 specimens as per the latest version of the ASTM E 208 Standard.
2. Application of proper current (ampere) during weld bead deposition and use of properlydried electrode without any pre-heating of the specimen.
3. Use of chiller block during weld bead deposition, i.e., the specimen blank is placed on a largeblock of steel, say 1’ x 1’ x 1’, during weld-bead deposition, to increase cooling rate andconsequently increasing the hardness of the HAZ.
A hard facing weld (RC 52) bead is deposited on drop-weight P-2 specimens of the base material(see Fig. 3.11). The bead is deposited in a single pass, using BOR-C electrodes of 5 mmdiameter, manufactured by M/s D&H Secheron Electrodes Pvt. Ltd, Indore. The chemicalcomposition of the weld is given in Table 3.2. The acceptability of this particular electrode fordepositing hard facing weld bead on drop-weight specimens had earlier been evaluated as perASTM E 208 Standard. For the weld metal drop-weight specimen, the base metal weld forms themid-length of the specimen, and the hard-facing weld bead lies at the centre of the base metalweld with reference to the length and width of the specimen.
Table 3.2.Chemical composition of the crack starter weld bead [B28].
Element Amount(wt %)
C 0.23Mn 1Cr 2Mo 0.4V 0.15
87
Fig. 3.10. Weld detail and specimen lay-out for Drop-weight and Charpy-V notchspecimens of weld material [B51].
The notch on the weld bead is made by a grinding wheel of 1.5 mm width. The notch is placedat the center position of the specimen, perpendicular to the longitudinal direction of thespecimen. The distance from the bottom of the notch in the (hard facing) weld to the specimensurface is kept 1.8-2 mm and the overall height of the weld-bead (in the form of a well-crownedbead) from the surface of the specimen is 4-5 mm. Figure 3.11 shows the schematic view of afinished drop-weight specimen.
The drop weight test (DWT) was developed specifically for the determination of the NDT onfull thickness plates. During the test, the specimen is supported as a simple three-point bend (3PB) specimen as shown in Fig. 3.12, the notch being in tension. Thebrittle weld bead is fractured at near yield-stress levels as a result of dynamic loadingfrom a falling weight. The anvil stop restricts the deflection of the test specimen.Since the specimen is a wide beam loaded in 3PB, this restriction limits the stress on thetension face of the plate to a value that does not exceed the yield stress. If the starter-crackpropagates across the width of the plate on the tension surface to the edges (even one edge issufficient), the test temperature is below the NDT. Complete separation on the compression sideof the specimen is not required. The DW-NDT, TNDT, is the highest temperature at which a nilductility break (B) is produced. Above the NDT temperature the crack does not propagatecompletely even to one edge – called no-break (NB) condition. The test is quite reproducible andthe NDT can be determined to the nearest 10 °F or 5 ˚C. A DWT series for an AISI 403 steel (13Cr-0.1C martensitic stainless steel) is shown in Fig. 3.13 (taken from the literature), with aTNDT = 25 ˚C, illustrating the Break(B)/No-Break(NB) concepts. At the TNDT = 25 ˚C, there are two Bs in the illustration, one showing breaking to both edges of the specimen while the other showing
breaking to one edge only. For confirming this as the TNDT, ASTM E 208 Standard requires thatduplicate tests at + 5 ˚C, i. e., at 30 ˚C in the present case, should demonstrate NB as shown inthe illustration.
Fig. 3.11. Schematic viewhas the same
Fig. 3.12. S
125 5 mm
50 0.5 mmBead
Notch
Base MetalWeld in thecase of weldmetal DWspecimens.
Hard-
facing
weld
88
of a drop-weight P-2 specimen with short weld bead (P-3 specimendimensions as the P-2, but the thickness is 16 mm) [B28]
chematic view of a ASTM E 208 drop-weight test [B12]
20 0.5 mm
25+ 5 mm
89
Fig. 3.13. Illustration of a typical drop-weight test series [B49].
It should also be noted that in all tests the hard weld bead must crack (a test in which the weldbead does not crack is known as a No-Test; in fact, the procedure given in ASTM E 208Standard for certifying the hard facing material as suitable for crack starting purposes in DWTsrequires that the weld bead deposited on P-2 specimens should crack at temperatures at least50 ˚C above the anticipated DW-NDT temperature). As illustrated in Fig. 3.14, the hard-facing weld bead plus its cracked heat affected zone presents a sharp thumbnail shaped elliptical crack(of aspect ratio ‘c/a’ and depth ‘a’; ‘t’, thickness = 20 mm for P-2 specimen) to the base material(base metal or weld as the case may be) on which it has been deposited. In fact, the ASTM E 208DWT determines the highest temperature at which the material can prevent the propagation ofsuch a severe crack formed at yield level stresses and dynamic loading.
3.3.2.3. Charpy specimen
Standard Charpy-V notch specimen (CVN) specimens (Fig. 1.8: 55 mm x 10 mm x 10 mm size;2 mm deep 45˚ V-notch with notch tip radius 0.25mm) were machined as per ASTM E 23 Srandard. Basic feature of an impact machine and test was depicted in Fig. 1.8.
90
→
Fig. 3.14. Crack profile on a DW NDTT specimen [B49].
3.3.3. Determination of RTNDT and ASME KIR curve for 9Cr-1Mo Base Material
3.3.3.1. Results from Drop-Weight Test
The B/NB results obtained in the drop-weight tests, carried out at different temperatures, aregiven in Table 3.3.
Table 3.3.The break/no-break results from drop-weight tests of the 9Cr-1Mo steel basematerial.
Weld-Bead Type
Specimen No. Test Temperature(˚C)
Break (B)/NoBreak (NB)
Result
ShortWeld-Bead
1S -20 NB
TNDT =25 ˚C
7S -20 NB9S -30 B10S -25 NB11S -25 B12S -30 B
91
3.3.3.2. Results From Charpy Test
Figure 3.15 shows the variation of Charpy impact energy with temperature for the 9Cr-1Mosteel, base material. Figure 3.16 shows the variation of lateral expansion with temperature. Boththe Charpy energy and the lateral expansion exhibit scatter at all the test temperatures,particularly in the transition region. As per the ASME specification, the points representing thelowest Charpy energy or lateral expansion over the whole test temperature range, are fitted by asuitable curve (sigmoidal fit in the present case) as shown in Figs. 3.15 and 3.16. The transitiontemperatures corresponding to the Charpy energy of 68 J and a lateral expansion of 35 mils isdetermined as –32 ˚C and –31 ˚C, respectively. Thus, following the ASME criterion, the higher of these two temperatures, i.e. –31 ˚C is taken as the TCV. It may be noted that the ASME Codesuggests an alternative procedure also to determine TCV. This involves testing triplicatespecimens at successively higher temperatures at 5 ˚C intervals, starting the first test at TNDT + 33 ˚C; the lowest temperature at which a minimum Charpy energy of 68 J and a minimum lateral expansion of 35 mils (0.89 mm) are obtained is the temperature TCV. However,the procedure followed here, namely, determining TCV from a smooth curve drawn through thelowest points of all the tests (preferably triplicate tests at a temperature, as in the latter case)conducted over the transition range is likely to give more conservative, i.e., larger, values ofTCV.
Fig. 3.15. Variation of Charpy impact energy with temperature for the 9Cr-1Mo steel,base material [B28].
-120-110-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40
0
20
40
60
80
100
120
140
160
180
200
220
240
260
Lower Bound Fit
68 J Transition Temperature
(-32oC)
9Cr-1Mo Steel
(Base Metal, N&T)
Ch
arp
yIm
pact
En
erg
y,J
Temperature,oC
92
Fig. 3.16. Variation of the lateral expansion with temperature for the 9Cr-1Mo steel,base material [B28].
3.3.3.3. RTNDT and KIR Curve
The nil-ductility transition temperature (TNDT) is –25 ˚C and the Charpy results give a TCV of
–31 ˚C. According to the ASME code, the RTNDT of the material is the higher of the two
temperatures, namely, the TNDT and (TCV 33 ˚C). Following this criterion, the RTNDT of the
9Cr-1Mo steel base material is determined as follows:
RTNDT = 25 ˚C
By plugging in this value in the ASME ‘KIC and KIR curve’ equations, the ‘curves’ for the
9Cr-1Mo base material are obtained and are plotted in Fig. 3.17.
-110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40
0
10
20
30
40
50
60
70
80
90
100
Lower Bound Fit
35 mils Transition Temperature
(-31oC)
9Cr-1Mo Steel
(Base Metal, N&T)C
harp
yL
ate
ralE
xp
an
sio
n,m
ils
(0.0
01
inch
)
Temperature,oC
93
T - RTNDT
/0C
-200 -150 -100 -50 0 50 100
KIC
or
KIR
/MP
a.m
0.5
0
50
100
150
200
250
300
ASME Lower-Bound KIC-curve
ASME KIR-curve
Material: 9Cr-1Mo Base Metal; RTNDT = -25 0C
Fig. 3.17. KIR vs T-RTNDT for 9Cr-1Mo steel, base material - RTNDT = 25 ˚C
Applying the above ‘ASME Fracture Toughness Curves’, to actual structures, say, nuclear
reactor structures, implies that the stresses, rather the SIF values, at the critical points
(say, nozzle corner welds) during operation (say, cool-down or heat-up) are always below
‘the Lower-Bound KIC curve’ or ‘the KIR curve’, depending on static or dynamic loads.
Chapter 4 - Elastic-Plastic Fracture Mechanics-EPFM
4.1. Crack Tip Opening Displacement (C
Linear Elastic Fracture Mechanics can deazone must remain small compared to thebehave in an approximately elastic manntreated elasto-plastically. Due to its coMechanics (EPFM) are not so well underthe crack opening displacement (COD) acrack tip region instead of the stresses, unplasticity a crack tip will blunt when it is lodisplacement at the tip of a blunting crack,as a fracture parameter (see Fig. 4.1).
Figure 4.1. Crack tip op
A simplified descrelastic-plastic concOutline of J-mexamples on EPFM
OUTLINEiption of CTOD and J-integralepts; Relationship of J to K;easurement method; Worked
94
TOD) and J-Integral Approach
l with only limited crack tip plasticity, i.e., the plasticcrack size and the cracked body as a whole must stiller. If this is not the case then the problem has to bemplexity the concepts of Elastic-Plastic Fracture
standable as LEFM theory. In 1961 Wells introducedpproach. This approach focuses on the strains in thelike the stress intensity approach. In the presence ofaded in tension. Wells proposed to use the crack flankthe so-called crack tip opening displacement (CTOD)
ening displacement (CTOD) [B21].
. Refs. [B21(Chapter-7), B7]
95
Even for tougher materials exhibiting considerable plasticity critical CTOD values could bedefined corresponding to the onset of fracture. Such critical CTOD value could be used toqualify the materials concerned for a given application.
In 1968, Rice considered the potential energy changes involved in crack growth in non-linearelastic material. Such non-linear elastic behaviour is a realistic approximation for plasticbehaviour provided no unloading occurs in any part of the material (in effect, load does notdecrease; see, Fig. 4.6). Rice derived a fracture parameter called J, a contour integral that can beevaluated along any arbitrary path enclosing the crack tip, as illustrated in Fig. 4.2. He showed Jto be equal to the energy release rate for a crack in non-linear elastic material, analogous to G forlinear elastic material.
Fig. 4.2. J contour integral along arbitrary path Γ enclosing a crack tip in non-linear elastic material. W is strain energy density along Γ, n is outward-directed unit
vector normal to Γ, T is traction acting on Γ and u is the displacement along Γ [B21].
For simple geometries and load cases, the J integral can be evaluated analytically. However, inpractice finite element calculations are often required. In spite of this J has found widespreadapplication as a parameter to predict the onset of crack growth in elastic-plastic problems. Laterit was found that J could also be used to describe a limited amount of stable crack growth. Andnow extension of the J-concept is being applied to fatigue (cyclic plasticity) and even creep crackgrowth problems. Since J-concept itself is a complex subject within the purview of thismonograph, and hence only the parameters will be described to give an idea of what theyaccomplish.
96
4.2. EPFM in Practical Terms [B7]
As described above:J = non-linear elastic energy release rate, whereas
GC = elastic strain energy release rate
In the limit, both are equivalent.
In fact, in the elastic-plastic (el.-pl.) case, like GIC in the elastic case,JIC represents the critical condition for crack initiation/growth in Mode-I loading.
Like, K, in the elastic case, J in the el.-pl. case representsthe elastic-plastic crack tip stress-strain distribution. It was shown in Fig. 2.4 that the normal
stress ahead of the crack-tip (σyy or σy) varies as (K/(√2πr)); similarly, in the case of a Jspecimen, the el.-pl. stress distribution ahead of a crack-tip is as follows:
1
1y
nJ
r
where n is the tensile work-hardening exponent.
Analogous to the KIC-GIC relation, in the el.-pl. case also, J IC is related to KIC, by the followingrelations:
IC IC
2
= '
OR, in the critical case,
= ' ,
where ' = for plane strain and ' = for plane stress,(1- )
with E being theYoung's modulus and being the Poisson's ratio
K E J
K E J
EE E E
(4.1)
Figure 4.3 shows the actual experimental correlation of Eq. (4.1) for many engineering alloys.
97
Fig. 4.3. Relation between experimental KIC-JIC for many engineering alloys [B7].
The CTOD parameter is a crack-tip ductility parameter, represented by δ.
The critical condition in Mode-I crack growth is, δ = δIC (4.2)
Also, δIC is related to JIC, by JIC = m’σYS δIC (4.3)
where m’ has values ranging from 1 in plane stress to 3 in plane strain.
Application of Eq. (4.1) will be described later in describing the Master Curve approach tofracture-safe design/control.
Main advantage of the J or CTOD parameters are that, for elastic-plastic materials, toughnessmeasurement can be accomplished using small specimens, even Charpy-type or even smallerspecimens, unlike KIC which requires very thick (upto 6” to 12” or more) for low to mediumstrength steels.
4.3. J-Measurement
Relevant ASTM (E-813, E-1820 etc.) and BS standards exist. Only J-measurement is brieflydescribed below. Main difference between K and J specimens is that in the former a front faceclip-gauge displacement is sufficient to obtain the critical load whereas in the case of J, load-linedisplacement (LLD), measured in a suitable manner, either by actual physical configuration as
98
shown in Fig. 4.4 or by suitable analytical relations to convert front face displacement to LLD, isrequired to measure energy spent (work done) in deforming and fracturing the specimen.
Fig. 4.4. Front-face displacement for KIC test and LLD for JIC test [B21].
Fig. 4.5. Load (P)-Load Line Displacement (LLD) for a non-linearelastic body [B21].
For a non-linear elastic body, for two specimens differing in crack lengths by ‘da’,1
= , where U is the shaded area between the two curves in Fig. 4.5.U
JB a
99
Fig. 4.6. Load (P)-Load Line Displacement (LLD) for anElastic-plastic body [B21].
If at the particular LLD, crack growth occurs, then the evaluated J will be the critical value. Lateranalytical developments, resulted in a simplified method to evaluate J from a single specimen.The new method depends on the separation of the elastic and plastic contributions to U (areaunder the P-d curve) and LLD (v or d) as shown in Fig. 4.6. Then,
2pl
pl el
2
2= and = ,
'
where ' = for plane strain and ' = for plane stress(1- )
U KJ J
Bb E
EE E E
(4.4)
In the above K is evaluated at the point fracture or unloading relevant to the evaluation.
Based on this, Begley and Landes proposed a multi-specimen method to determine critical J.
100
Fig. 4.7. Begley-Landes Multi-Specimen JIC method [B7].
4.4. Begley-Landes Multi-Specimen JIC method
A series of specimens is provided with the same size of fatigue crack, such that a/W > 0.5. Eachspecimen is loaded to a different point on the load displacement curve and then loaded (Fig.4.7b). After unloading, the crack is marked to enable measurement of stable crack growth. Crackmarking can be accomplished on steel specimens by heat tinting. The specimens are heated to about850 °C for 10 minutes, which will oxidize the fracture surface. Specimens of other materials maybe fatigue cycled at low loads, which will mark the static crack as a result of the differenttopography and light reflection of the fatigue crack.
When the cracks are marked the specimens are broken to reveal the fracture surfaces. Crackadvance that occurred during the initial loading and unloading experiment can now be
101
measured. Usually the crack will have propagated more in the centre of the specimens than atthe specimen surface. Therefore, an average Δa should be determined from a multiple ofmeasurements (Fig. 4.7c).
The value of J for each specimen is determined from the load-displacement curve (Fig. 4.7a).The values of J for all specimens are plotted as a function of Δa as in Fig. 4.7d. A straightline is fit through these data points.
The initially sharp crack tip blunts before stable crack extension occurs. Crack tip bluntingcan be considered to result in a small amount of crack extension, as shown in Fig. 4.7e. It isarbitrarily assumed that crack extension due to blunting Δa = 0.5 (CTOD). Thus, by taking J =σYS(CTOD) = σYS(2Δa), the variation of J due to crack tip blunting is:
= 2 (4.5)ysJ a
This results in another straight line, the crack blunting line, shown in Fig. 4.7d. Theintercept of the two straight lines in Fog. 4.7d is the point where the crack first started toextend by slow stable tear. Therefore, the point of intercept demarcates JIc. Standards likeASTM E-813 and E-1820 recommend use of other sophisticated methods like compliancechanging method (CCM), potential-drop technique or other advanced instrumentation todetect crack inititiation and crack growth measurement from a single specimen. Though noneof these techniques will be discussed here, the compliance method and also the validitycriteria and new regression methods employed will be illustrated in some worked examples.
4.5. Standard Method for crack-tip opening displacement (CTOD) Determination
The standard crack opening displacement (COD) test specimens conform to the three-pointnotched bend (SENB) and the compact tension (CT) configurations following the BritishStandard BS 7448. For CT specimens a JIC type starter notch is allowed also (see Fig. 4.4). Thepreferred W/B ratio is 2, but deviation is allowed within certain limits. In principle the thicknessB must be equal to that of the material as used in service, and the specimens are not sidegrooved. As the tc (critical value of CTOD) resulting from this test method may be affected bythe specimen geometry and size, caution is required when comparing results from differentsources. Requirements for precracking, loading rate, etc. follow the LEFM procedure; only that,instead of Chevron notch, straight crack-fronts are used; measurement of crack length on thebroken specimen follows the a JIC procedure.
As direct measurement of CTOD at the crack tip is impossible, a clip gauge is used to measurethe COD, Vg, at or near the specimen surface (see Fig. 4.8 where clip gage is mounted onattachable knife-edges on the specimen surface). Assuming that the ligament b (= W a) acts asa plastic hinge with a rotation point within the ligament at some distance r.b, from the geometryin Fig. 4.8, one obtains (with rotational factor r as constant) the plastic part of CTOD from thefollowing relation,
102
pl pl
.= (4.6)
.
r bV
r b a z
where z is the thickness of the knife edges and r = 0.4 and 0.46 for deep-cracked SENB and C(T)specimens, respectively (experimentally determined). Vpl is determined from the total Vg as inFig. 4.9.
Fig. 4.8. Relationship between COD, Vg, and CTOD, t [B21].
For reasons of accuracy, Vel from the test Fig. 4.9 is not used, rather the theoretical value fromthe following relation is used:
2 2I
el
(1- )= (4.7)
2ys
K
E
Then, total CTOD, t is obtained from
2 2I
t el pl
(1- ) .= (4.8)
2 .pl
ys
K r bV
E r b a z
The KI in the above cases is obtained from the appropriate ASTM E 399 formula substituting theinitial crack length, a, and the load at which Vpl is measured.
103
Fig. 4.9. Separation of total COD, Vg, into elastic, Vel, and plastic, Vpl, components [B21].
The load-displacement records can assume six different forms as shown in Fig. 4.10. It isnecessary to establish whether stable crack extension occurred during the test and to assess theamount of crack extension associated with possible pop-in behaviour, i.e. a small amount ofunstable crack growth followed by crack arrest.
If post-test examination of the fracture surface reveals that the corresponding crack extensionexceeded 4% of the uncracked ligament, b, then pop-in is considered significant. Otherwise, apop-in is only considered significant if at subsequent crack arrest, the specimen compliance hasdropped by more than 5%. To examine this, the Standard suggests the same procedure forCases 1, 2, and 3 in Fig. 4.10. Cases 1 and 2 are monotonically rising load-displacement curvesshowing no or limited plasticity and no stable crack extension before fracture. Case 3 shows a(significant) pop-in owing to sudden crack extension and arrest. In all these three cases the tc istaken to be c, which is obtained from Eq. (4.8) using PC and VC (see Fig. 4.10).
Cases 4 and 5 may also be treated similarly. Prior to instability, which again is either fracture ora (significant) pop-in, stable crack extension occurs. This should be revealed after the test byexamination of the fracture surface. In these cases tc is calculated as the u at (Pu, Vu). Case 6applies to extremely ductile materials for which stable crack extension proceeds beyondmaximum load Pm: cases tc is calculated as the m at (Pm, Vm).
104
Fig. 4.10. Types of load-COD plots obtained during CTOD testing [B21].
105
EPFM PROBLEMS
Example 4.1.
A JIc test is performed on steel with the following properties: E = 207 GPa; ys = 360 MPa enUTS= 560 MPa ; = 0.28. For this purpose a 3-point bend specimen is used with the followingdimensions:
W = 50 mm; B = 20 mm; a = 30 mm. The load is found to increase linearly with displacement.At the onset of crack extension the load is 25 kN, while the displacement is 4 mm.
a) What value follows for JIC, if this is defined as J at the onset of crack extension? b) Is thisvalue acceptable according to ASTM standard E 813? c) What is the corresponding value for KIc?d) What thickness should the specimen be for a valid KIc determination?
Solution:
Given: W = 50 mm; B = 20 mm; a = 30 mm. Therefore, a/W = 30/50 =0.6 and b = W-a = 20 mm.This a deeply cracked specimen and hence,
J = 2 U/(Bb), where U is the area under the load-LLD curve upto initiation.
As it is given that load varies linearly and that the onset of crack extension occurs at 25 kN withLLD = 4mm,
U = 0.5*25*4 = 50 N.m =50000 N.mm
a) J = 2*50000/(20*20) = 250 N/mm = 0.25 MPa.m
b) ASTM validity criterion, B, b > 25 (J/σYS) = 25*250/360 = 24.3 mm are not satisfied.
c) Corresponding
IC 2 2
207000 0.25 207000 0.25= = = 237 MPa
(1 ) (1 0.28 ) 0.9216ICEJ
K m
d) Following the ASTM plane strain size criterion,
2 2
IC
YS
237> 2.5 = 2.5 1.1 m
360
KB
106
Example 4.2.
For a large welded steel vessel the following data are available: service stress = 200 N/mm2
yield strength weld metalys = 450 N/mm
Young's modulus weld metal E = 205 000 N/mm2 Due to shrinkage after the welding process,residual stresses develop in the weld seam which can be as high as the yield strength of the weldmetal. By means of a heat treatment these stresses can be reduced. It is required that the vesselcan withstand a crack in the weld metal with half crack size a = 60 mm. Furthermore the CTODvalue in this case may not exceed 0.5 mm.
To what level should the residual stresses be reduced to comply with these requirements?
Solution:
Using the plane stress relation relating yield stress to CTOD,
-1 -2C YS CTOD = 450 0.5 = 225 N.mm = 0.225 J.mmJ
C = = 207000 0.225 = 216 MPaCK EJ m
For an a = 60 mm (half crack-length), this gives
C 216= = = 498 MPa
0.06
K
a m
Since 200 MPa is the loading stress,
maximum residual stress permitted is 498-200 = 298 MPa which is 0.66xys.
Hence heat treatment should reduce residual stress to this or lower levels.
107
Example 4.3.
A JIc test is performed on HY130 steel. The results, measured on SENB specimens, are:
U [J] a [mm]45 0.1070 0.4085 0.67105 0.99
117.5 1.22
Specimen dimensions: span L = 4W = 200 mm width W = 50 mm thickness B = 0.5 W= 25 mmcrack length a = 30 mm
Material properties: yield strength ys = 925 MPa; tensile strength uts = 953 MPa; Young'smodulus E = 210 000 MPa Poisson's ratio = 0.28 ; mass density = 7.8×103 kg/m3
a) Approximately (!) determine JIc using a spreadsheet or on paper. Assume that conditionswhich cannot be checked but are necessary for a valid determination are fulfilled.
b) How high is KIc for this steel?
c) At least how many kilograms of weight would be saved by determining KIc through JIc withthis test? Assume that the ratios of the dimensions of the KIc specimen are equal to those listedabove.
Solution:
For each Δa given in the Table above the corresponding J = 2 U/(Bb), where U is the area underthe load-LLD curve. These calculated J values are plotted against Δa in the figure below.
(a) ASTM E 813 procedure requires construction of a crack-tip blunting line, J = 2.σYS.Δathrough the origin and also constructing two exclusion lines at Δa = 0.15 and 1.2 mmrespectively and an offset line at Δa = 0.2 mm, all the three lines being parallel to theoriginal blunting line through the origin (In a spread sheet programme this is easilyaccomplished by setting Δa = (J/(2.σYS)) + Δa(Offset), where Δa(Offset) = 0.15 or 0.2 or1.2 mm as required for each of the above three lines).
108
Then, ASTM E 813 requires that at least 4 valid points lie between the 0.15 mm and 1.2 mmlines. Following the validity criterion, each measured J value should satisfy the condition that Jis less than (b0.σYS/15), the value at the highest Δa will determine the maximum J permitted.
Crack Extension, a/mm
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
J,J.
mm
-2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
J = 0.4465*a0.5196
YS = 925 MPa
1.2
mm
Exc
lusi
on
Lin
e
0.1
5m
mE
xclu
sio
nL
ine
JQ = 0.195 J.mm-2
0.2 mm Offset Line
Blu
nting
Lin
e=
2
YS(
a)
Jmax = 1.16 J.mm-2
open points excluded from power-law regression
For the present case, as indicated in the Figure, the maximum J value is much higher than allthe measured values. This condition ensures absence of ligament yielding.
Then a power-law is fitted through the valid points within the exclusion lines. In the presentcase, the largest value is slightly outside the 1.2 mm exclusion line. However, from the trendof the data it seems OK in making the fit through the 4 points. This is also in accordance withthe guideline “Assume that conditions which cannot be checked but are necessary for a validdetermination are fulfilled”, given in the problem.
All the above are illustrated in the Figure. Then a provisional JQ is determined as the J at theintersection of the power-law regression line and the 0.2 mm Offset line. From the figure, JQ
= 0.195 J.mm-2.
Now this must satisfy the conditions, B and (W-a) > 25 JQ/σYS
-325 0.195= 5.27 10 m = 5.27 mm << = ( - ) = 20 mm
925B W a
b)IC 2 2
210000 0.195= = = 211 MPa
(1 ) (1 0.28 )ICEJ
K m
109
c) For valid KIC test,
2 2IC YS> 2.5( / ) = 2.5 (211/925) 0.13 m = 130 mmB K
Hence for the KIC specimen (dimensions in mm),
3
3
7.8 2 4 2 7.8 16 7.8weight = gm =
1000 1000 1000
16 130 7.8Hence, weight = = 274186 gm = 274 kg
1000
B W L B B B B
Similarly for the JIC specimen,
316 25 7.8weight = = 1950 gm = 1.95 kg
1000
Hence the JIC test will save a weight of ~ 272 kg.
Example 4.4.
On a bend specimen a single-specimen Jc test is performed. Using a non-destructive techniquecrack initiation is observed at a load P = 9315 N. During the test the load is recorded as afunction of the load-point displacement V. A polynominal fit of the P-V curve resulted in:
-5 -14 4= 10 + 10 .P P where V is expressed in mm and P in N.
a) Determine the critical J value at initiation; b) Is the plate thickness sufficient for a validJIc test?
Given: width W = 50 mm; crack length a = 12 mm; thickness B = 28 mm;flow stress o = 1000 MPa.
Solution:
a) Calculation of J requires knowledge of the area, that is the work done, U, under the load-LLD (P-d) curve. This is given by:
-5 -14 3 -5 -14 4= d , where d = (10 + 4 10 )d as = (10 + 10 ) as given above.U P d d P P d P P
110
From the expressions for d and dd given in terms of P, we can set up the integral in terms of Pwith the integration limits 0 to 9315, the load at which crack initiates. Therefore,
9315 9315-5 -14 3 -5 -14 4
0 0
= (10 + 4 10 )d = (10 + 4 10 )dU P P P P P P
9315
-5 2 -14 5
0
-5 2 -14 510 9315 4 10 9315
= + = 561487.4 N.mm2 5
10 4 10+
2 5U
P P
i.e., U = 561.5 J
IC
-22 2 561.5= = = 1.055 J.mm
28 38
UJ
Bb
b) ASTM E 813 validity criterion, Q
YS
25 25 1.06> = = 0.0265 m = 26.5 mm
1000
JB
is
satisfied.
Example 4.5.
For materials with a moderate toughness (e.g. aluminium alloys) KIc can be determined from JIc.Express the minimum required thickness for the JIc test (Bj) in terms of the minimum requiredthickness for the KIc test (Bk).
Given: Young's modulus E = 70 000 MPa; yield strength ys = 345 MPa; tensile strengthuts = 500 MPa ; E' for plane strain = E (for plane stress)
Solution:
C C
YS
jCj C
25 25For , > =
345
34525or =
345 25
J JJ B
BJB J
j
JC C
70000 345= =
25
BK EJ
111
2
jJCJC K j2
YS
2j
K j
70000 345For , > 2.5 = 2.5 21
25 345
For plane strain, will be replaced '=( /(1- )) / 0.91, then 23
25
BKK B B
E E E E B B
B B
Example 4.6.
During a JIc determination using a technique similar to that of Begley and Landes a criticaldisplacement (at the onset of crack growth) is found of 20 mm for an initial crack length of 10mm. In the figure load displacement curves are given for 3 initial crack lengths (5, 10 and 15mm). The tests are performed with bend specimens for which the thickness B and the height Ware both equal to 20 mm. The measured displacement is that of the load point.
a) How high is JIc?
b) Approximately determine the value for the critical displacement at an initial crack length of 15mm.
Solution:
The first thing is to identify which curve in the figure represents which specimen. Normally itcan be expected (given the large differences in initial crack lenghs) that the specimen with thesmallest crack shows the highest loads while the with the largest crack length shows the lowestloads. Based on this argument, the highest curve in the figure corresponds to the specimen withinitial a = 5 mm, the lowest corresponds to the specimen with initial a = 15 mm while theintermediate curve corresponds to the specimen with initial a = 10 mm.
The area under the load-LLD (P vs. v or d) is the work done U. J is determined from,
I
2=
UJ
Bb
Since in the figure the Y-axis is given as (P/B), J expression in the present case becomes simply,
I
2=
UJ
b
It is given that for the 10 mm crack specimen, initiation occurs at a displacement of 20 mm.Hence the area under the a = 10 mm curve upto d = 20 mm will give the required dU forcomputing initiation J.
112
The area can be determined from the figure by counting squares or using a planimeter.
The area determined is 5066.52 N (per unit thickness B). Hence,
a) -1 -22 2 5066.52= = 1013.3 N.mm = 1.013 J.mm
(=( - )) 10i
UJ
b W a
b) Since initiation must occur at almost the same value of J, we must find the d for the 15 mmspecimen at which (2U/5) = 1013.3 (N/mm). Hence, we must find the d for the 15 mm curveat which U = 2533.25 N.
It is found by some trial and error, that for the 15 mm curve, at a d = 18.14 mm, U = 2561.48 N;at a d = 17.93 mm U = 2512.67 N. The above value of U = 2533.25 N falls between these two dvalues. Hence, the required d ~ 18 mm.
Note: By reading off sufficient (P/B) vs d values from the curve for each crack length specimenand fitting them to polynomials, it is possible to find the areas to any displacement moreaccurately by straight integration.
113
Example 4.7.
Using a single edge notched bend specimen of nodular cast iron, a JIc test is performed based onthe unloading compliance technique. In figure A the measured load P is plotted as a function ofthe load displacement V. From these data the slope of the P-V curve during the differentunloading-loading cycles are found as follows:
No. Slope:N/mm
Compliance,C = Inverse
Slope, mm/N(x10
6)
a/W a/mm (W-a) =b/mm
v ord/mm
Area,U,N.mm
Δa/mm J =2U/Bb,N/mm
1 19200* 52.1 0.62 17.36 10.64 0 0
2 18800 53.19 0.622 17.416 10.584 0.21 433.67 0.056 11.713 17600 56.82 0.635 17.78 10.22 0.41 1029.21 0.42 27.784 15800 63.29 0.65 18.2 9.8 0.63 1650.2 0.84 48.115 13300 75.9 0.673 18.844 9.156 0.85 2212.94 1.484 69.066 11000 90.91 0.7 19.6 8.4 ~1.27 3160.22 2.24 107.49
*no crack growth yet; Note: All the bold values in Table were obtained during solution of theproblem.
For specimens with this geometry and made of this material the relative crack length a/W isplotted in figure B as a function of the compliance C (It may be noted that in Fig. B, theparabolic continuous curve represents the given compliance plot. The dotted lines dropped fromthe C-curve to the axes are those made during solution of the problem to determine a/W for eachC determined from Fig. A).
Other data are: Young's modulus E = 200 000 MPa; yield strengthys = 350 MPa; ultimatetensile strengthuts = 450 MPa; Specimen: thickness B = 7 mm ; height W = 4×B
a) Roughly indicate the positions of the J-a points in figure C, as they follow from theexperimental results. Be sure to add numerical values to the vertical axis. J may be calculatedwithout making a distinction between elastic and plastic displacements.
b) Explain how crack tip blunting is quantified for a JIc test. Indicate this in figure C.
The critical J value according to ASTM E 813 is defined as the value after 0.2 mm of stablecrack growth.
c) In figure C, draw the line that represents this stable crack growth.
d) Roughly show in figure C how the critical J-value, JIc, is obtained.
114
e) What is the JIc value (not a calculation but a graphical estimate)?
f) Is this value valid in view of the size requirements?
Solution:
From the given slope values in the Table, the corresponding inverse or C values were computedas given in the Table. For each C, the corresponding a/W was read-off from Fig. B as shown andthe values tabulated in the Table. Also, from Fig. A, at each unloading (2) to (5), the area, U,underload-LLD curve was determined and are given in the Table. The corresponding J values arealso given in the Table.
(a) to (d): As required in (a), Fig. C has been constructed. Answers to (b) to (d) are as detailed inExample 4.3 given previously. The novelty of the problem is that crack length has beendetermined by a Compliance procedeure. The calculations involved show sufficientfamiliarization with the technique.
115
Compliance, C x 106 [mm/N]
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
a/W
0.50
0.55
0.60
0.65
0.70
0.75
0.80
Fig. B
a/mm
0.0 0.5 1.0 1.5 2.0 2.5
J,N
/mm
0
20
40
60
80
100
120
J = 53.5155a0.8353
JQ
15 N/mm
Fig. C
Blu
ntin
gL
i ne
:2
YS
a
1.2
mm
Exc
lusi
on
Li n
e
YS = 350 MPa
Only filled points used forPower-Law Regression
0.1
5m
mE
xclu
s io
nli n
e
0.2 mm Offset Line
(e) JQ = 0.15 N/mm; (f) B (7 mm), (W-a) (= 10.64 mm) > 25JQ /350 = 25*0.15/350 = 0.011 mm.
Hence Valid.
116
Example 4.8 Calculation of CTOD from COD
In a COD test made on a structural steel of yield strength 430 MPa, using a SENB specimen ofdimensions, B = 25 mm, W = 50 mm, and precracked to a depth of 26 mm, to what total crack-tip displacement does a clip-gauge plastic displacement of 0.33 mm correspond, if the load atthis point is 50 kN and the the knife edges are 2mm thick? (Young’s modulus is 200 GPA andPoissons Ratio is 0.3)
For the K value either the expression for the bend specimen in Fig. 2.10 (more accurate) or theexpression (e) in Table 2.8 may be used. Using the latter,
I
6 1.12+ (3.43 -1.89)with = where =
1-0.55
P aK Y a Y
BW W
α = (26/50) = 0.52; therefore, Y = 1.49; taking P = 50 kN,
I
2
6 50 10001.49 0.026 102.2 MPa m
25 50
NMPa and under root sign is in 'm'
mm
K
a
Then from Eq. (4.8),
2 2I
t el pl
2 2
(1- ) .=
2 .
102.2 1000 (1-0.3 ) 0.4 240.33 0.06 0.084 0.14 mm
200 1000 430 2 0.4 24 26 2
pl
ys
K r bV
E r b a z
117
Example 4.9 Initiation CTOD, i, from multi-specimen extrapolation technique
NOTE: As discussed in Section 4.5, the COD standard simply specifies the method for determining a
critical CTOD, tC. For cleavage fracture, where fast fracture coincides with initiation, this may have
extension to structural situations also. In other situations, it may be necessary to measure the initiation
value, i, or a value corresponding to a fixed amount of crack extension; in such cases, and especially in
Case 6 in Fig. 4.10, where a plastic instability value, m, is determined extension to structural cases may
be hazardous. Thus. Though CTOD may be useful in quality control or material development as a
comparative measure, its application to structural integrity analysis is limited.
Using acoustic emission or potential-drop techniques, it may be possible to detect fibrous or ductilefracture initiation in some materials. Another method is the unloading compliance method, where thespecimen is partially unloaded from various load-Vg points and the slopes of the elastic-unloading line arecompared: crack extension would result in change in slope of the elastic unloading line. However, thisrequires sophisticated instrumentation and clip-gauge positioning at the load line (as for the J-integraltest) and may not be useful for very ductile materials. Another more useful technique is to use a doubleclip-gauge (at different heights above the crack-tip); the first deviation from linear proportionality of thegauges is indicative of crack initiation.
Another technique which has been successful in steels, is to generate the full CTOD-a curve by a multi-specimen technique, much like the Begley-Landes procedure for J-integral. Identical test pieces areunloaded from different positions of the Load-displacement line, heat-tinted, if necessary, and brokenopen at liquid nitrogen temperature. Then the CTOD at the original crack-tip can be calculated (includingany change in the centre of rotation due to crack growth) and a plot of these CTOD values against a canbe extrapolated to zero crack extension to yield a i value. Such a procedure is described in the followingproblem.
PROBLEM: Values of COD readings and fibrous thumbnail lengths for an HY-80 steel are givenin the following Table. What is the value of i for this steel?
Crack extension, a, mm CTOD, t, mm0 0.09
0.05 0.150.125 0.180.14 0.220.3 0.250.4 0.300.5 0.340.9 0.5
From the plot of a vs.t given below,
it is seen that, i = 0.13 mm and slope (d/da) = 0.43 mm/mm
118
a/mm
0.0 0.2 0.4 0.6 0.8 1.0
CT
OD
, t
/mm
0.0
0.1
0.2
0.3
0.4
0.5
0.6
t= 0.1254 + 0.4251*a
Example 4.10. Determination of i by double clip-gauge method
A NiCrMo cast steel of proof strength713 MPa is tested by the COD standard using specimenswith dimensions: B = 10 mm and W = 20 mm. Two clip-gauges, gauges I and II, are mounted atheights z1 = 0 mm and z2 = 2 mm above the surface of each specimen. If the first major deviationfrom linearity in a plot of Vg1 vs. Vg2 occurs when Vg1 = 0.295 mm, for a crack length of 9.83mm, what is the value of i for this steel? Young’s modulus = 206 GPa and Poisson’s ratio = 0.3.Test record of load vs.Vg1 is given below.
Solution:
From the Figure, when Vg1 = 0.293 mm, load, P = 11.92 kN and intercept of the line through thispoint and parallel to the initial linear portion of the load-displacement curve (elastic unloadingline) with the X-axis gives a Vpl (or Vp) value of 0.083 mm.
With load P = 11.92 kN and crack length a = 9.83 mm giving an (a/W) or α = (9.83/20) = 0.492,following the same procedure as in Example 4.8 gives Y = 1.4. Then,
3I
6 6 11.92 1000= 1.4 9.83 10 = 87.98 MPa m
10 20
PK Y a
BW
Then from Eq. (4.8)
119
2 2I
t el pl
2 2
(1- ) .=
2 .
87.98 1000 (1-0.3 ) 0.4 10.170.083 0.048 mm
206 1000 713 2 0.4 10.17 9.83 0
pl
ys
K r bV
E r b a z
120
Example 4.11. Calculation of critical defect size using CTOD
A rocket motor case is fabricated from a Cr-Mo low alloy steel of proof stress 1200 MPa, in theform a long cylinder of diameter 0.5 m and wall thickness 2.5 mm. If the design pressure at thehighest thrust is 8 MPa, calculate the size of the largest size that could be tolerated in the rocketmotor case, without the risk of bursting at blast-off. Critical CTOD measured in a small testpiece is 50 µm and Young’s modulus is 200 GPa.
For thin sections plane stress assumption is OK. Longitudinal flaw perpendicular to the hoopstress is the most dangerous.
Hoop stress, σh = pd/2t; Here, Pressure, p = 8 MPa, Diameter, d = 0.5 m and thickness,t = 2.5 mm.
Hence, σh = 8*0.5*1000/(2*2.5) = 800 MPa
In the plane stress case, critical fracture toughness,
KC = √EJC = √E*σYS *δC = √(200000*1200*50/1000000) = 109.54 MPa
Therefore, critical defect size, aC = KC2/(π*σ2) = 12000/(3.14*8002) = 0.006 m = 5.8 mm
This is half-crack length, the total length of the that could be tolerated is 11.6 mm.
Comments: Now a rough estimate of the plastic zone size gives,
ry = (1/2π)*(KC/σYS)2 = 0.0013 m = 1.3 mm and 2 ry = 2.6 mm ~ the sheet thickness, justifyingplane stress assumption.
NOTE: In plane strain, though there is no analytical relation linking δ to KIC, finite elementbased calculations and empirical results give the following expression:
2IC=
YS
K
E
where β takes values between 0.45 to 0.7 (even ‘1’ has been reported). When slow ductile crackgrowth precedes final instability, it is difficult to obtain a critical value, δC; but an ductileinitiation CTOD, δi, will give conservative or lower-bound estimates of K, Ki.
121
Example 4.12. The Toughness of High-Strength Weld Metal
An alloy weld metal has the following properties: proof stress = 1000 MPa; KIC = 95 MPa√m; CTOD tests on thick small pieces gave a δi value of 40 µm; Young’s modulus, E = 200 GPa.Find the value of β in the plane strain relation discussed in the NOTE at the end of the lastExample.
Substitution the above values in2
IC=YS
K
E
gives 40*10-6 = β (952/(1000*200000)),
and, therefore, β = 0.89.
Example 4.13. Toughness of Thick-Section Pressure Vessel Steel (PVS)
Room temperature CTOD tests on a stress-relieved A533B PVS gave a value of 0.19 mm for δi
and almost a linear increase of δ with crack extension giving a slope of (dδ/da) = 0.5 mm/mm. Ifthe YS = 500 MPa, E = 200 GPa and Poisson’s ratio, υ = 0.3, find the values of Ki and possiblevalues of KIC that might be obtained from large LEFM test specimens.
As given in the previous two Examples, the lowest (as also given in the BS standard for CTOD)and highest (from experiments) values of β are 0.45 and 1, respectively. Hence, applying these inthe expression,
2i
i =YS
K
E
, gives
2 2-3 i i0.19*10 = 0.45 or
500*200000 500*200000
K K,
giving Ki = 205.5 or 137.8 MPa√m, respectively.
NOTE: Actual measurement gives a value of 180-200 MPa√m for the KIC of this steel, implyingthat in this steel failure occurs almost after initiation. For a KIC of 200 MPa√m, applying the ASTM E399 validity criterion,
2
IC, ( - ) or 2.5YS
Ka W a B
, will give a value of 0.4 m.
The 5% secant offset construction in the E399 standard (see Fig. 11), implies a crack growth plusplasticity of about 2%, which in the present case would amount to 2% of 400 mm = 8 mm. So forhalf this value, say, 4 mm, CTOD value will be,
122
i
d= + = 0.19 + 0.5*4 = 2.19 mm
da
Hence, the corresponding K values will be 690 and 460 MPa√m, for β = 0.45 and 1, respectively.Such high values may preclude fast fracture above the upper shelf transition temperature.
Example 4.14. Application of CTOD for Quality Control
In such applications, the aim is to specify the minimum crack tip ductility (or CTOD) that willpreclude cleavage or brittle fracture in plate of given thickness, B. Brittle fracture is promoted bytriaxial stresses, and these will be relaxed if the plastic zone size is large to promote throughthickness yielding. The criterion is that the plane stress plastic zone size should be at least equal
to the plate thickness:
2
y
YS
1=
2
Kr B
and
2
YS
(plane stress) =K
E
. Therefore,
YSy 2
YS YS
1 1= =
2 2
E Er B
or YSy2 = 2B B
E
, where (σYS/E) = εy, the yield strain.
Example Application: For a 25 mm plate of structural steel with a yield stress of 400 MPa at265 K, taking E = 200 GPa,
the yield strain, εy = 400/200000 = 0.002, then based on the above criterion,
2*25* *0.002 0.314 mm
Hence if the CTOD is less than 0.314 mm, more detailed fracture mechanics assessmentshould be performed.
Example 4.15. Welding Institute Defect Assessment Procedure
For actual application of fracture mechanics to structures, to relate operating loads to the stressesand CTODs at the stress concentrators in critical regions (say, a crack in a weld around a nozzlein a pressure vessel), detailed finite element based calculations may be necessary. To avoid sucha costly exercise, The Welding Institute (UK) has developed a curve which relates the CTOD tothe local estimated strain. This curve is derived from a large number of precracked wide-plate
123
tests, relating measured CTOD to applied crack-tip strain. Then the strain concentration at acrack tip or defect is equated to the stress concentration.
Welding Institute CTOD Design Curve
Normalised Strain, /y
0 1 2 3 4 5
No
rma
lise
dC
TO
D,
/2 y
a
0
1
2
3
4
5
Fit Line is given by: y = -0.0919 + 0.8585x1.0971
where y = normalised CTOD and x = normalised strain
The CTOD design curve is given above. The points represent data digitized from Knott’sdiagram [ ]. The fit was generated by the authors. For normalized strains greater than 0.2, the fitexpression is a good approximation to the actual curve. ‘a’ refers to the half-length of themaximum allowable through-wall crack. This curve can be used to obtain the minimumallowable CTOD for a given defect or the NDT limit for material with known CTOD. It may benoted that residual stresses (especially in welds and fabricated structures) should be taken intoaccount in calculating the normalized strain.
Example on Use of Welding Institute Design Curve
A pressure vessel of made of low alloy steel of yield stress 500 MPa has a diameter of 25 m anda wall-thickness of 0.25 m. It contains steam at a pressure of 5.5 MPa. Large pipes are weldedinto circular openings in the vessel wall using matching weld metal. If the available NDTtechniques are able to detect defects of size larger than 10 mm, what is the minimum criticalCTOD required in the weld metal to prevent fast fracture? Also, assume that a residual tensilestress equal to half the yield stress is present in the weld metal.
124
Solution:
Because of the large diameter to thickness ratio (d/t), a thin wall tube assumption is made. Then,hoop stress is given by:
5.5*25= = = 275 MPa
2 2*0.25h
pd
t
For circular holes, the stress concentration factor is about 3, i. e., the local stresses exceed theapplied stress by three times. Thus the proof stress is exceeded locally and plastic strainsdevelop. Assuming stress concentration factor is equal to strain concentration factor, it follows,
y y
3*275 + 250 1075= = = 2.15 =
500 500
,
where 3*275 accounts for the external applied load and 250 is the internal residual stress givenas half the yield stress. Then for a strain concentration of 2.15 (i. e., x = ε/εy), from the fitexpression given in the above figure (Welding Institute CTOD Design Curve) one obtains,
1.0971
1.0971
y y
1.0971
= -0.0919 + 0.8585x = -0.0919 + 0.85852
-0.0919 + 0.8585*2.15 1.9
ya
εy = (500/200000) = 0.0025
Assuming an edge crack of length (this will give conservative or safer estimates) 10 mm (NDT
detection limit) gives: 2 *(2.5/1000)10*1.9 = 0.3 mm
Comments: This toughness is achievable provided proper post-weld heat treatment is carried outto reduce residuals stresses.
Note: For other, crack types, like embedded flaws, other curves have been developed. Thisapproach is not feasible when extensive plasticity and plastic collapse occurs. More robustanalytical-empirical methods, like CEGB (former UK Central Electricity Generating Board) R6Procedure or European Structural Integrity Assessment Procedure (SINTAP) are available thatuse a Failure Assessment Diagram (FAD) that uses a failure curve interpolating between pureLEFM fracture to full plastic collapse. As the actual applications of such methods are not sosimple, they are considered beyond the scope of this “elementary strength of materialsapproach” book.
125
Appendix - Chapter 4: A Note on CREEP CRACK GROWTH (CCG) [B32,B41]
Fig. Ch4-A-1: Typical strain-time curve during creep [B32].
Creep occurs when a component is subjected to loading at a high enough temperature relative tomelting point, Tm (usually 0.3 to 0.5 Tm). Creep is a time dependent process that results in non-recoverable or permanent deformation that may lead to failure – creep rupture. In electricalpower generating plants, chemical process industries and aircraft gas turbines – creep is the maindesign failure mode.
Creep is divided into primary, secondary and tertiary modes as shown in Fig. Ch4-A-1. Primarycreep, occurring over a short time, results in decreasing strain rate. Secondary or steady statecreep is characterized by a steady state or constant (rather minimum) creep rate and oftendominates the life of a component. The tertiary creep results in accelerating creep rate and occursat long times.
Since a component is subjected to secondary creep deformation most of its life-time, CreepFracture Mechanics (CFM) based on secondary creep has received wide attention and is thesimplest to deal with. Here only a preliminary account of this will be given for the sake ofcomplete coverage of EPFM. Analogous to power-law plasticity, secondary creep is oftengoverned by a power-law:
126
= (Ch4-A1)ns B
where B is a constant and n the power-law creep exponent.
Analogous to J, a parameter, C*, is defined as a contour integral as follows (see Fig. 4.2):
* *i
*
0
= ( )d - d (Ch4-A2)
with ( ) d , the strain energy rate density.
iuC W y T s
x
W
The terms have the same significance as described with respect to Fig. 4.2., only that strainenergy density, strain and displacement terms have been replaced by their rates with respect totime.
In stress field equations, characterizing steady state creep ahead of a crack, J is replaced by C*.Thus, the stress ahead of a crack in a creeping solid under steady state is:
1* n+1
ij ( ) (Ch4-A3)C
where n is the power-law creep exponent. For example, the stress field and creep zone ahead of acrack is shown schematically in Fig. Ch.4-A-2. Analogous to J, C* can be defined based on theenergy rate interpretation as follows:
* 1 d= - (Ch4-A4)
d
UC
B a
Equation (Ch4-A4) is schematically shown in Fig. Ch.4-A-3. This also provides a basis formeasurement of C* as discussed next.
127
Fig. Ch4-A-2. Schematic of the deformation zones and the associated stressfields ahead of a crack under extensive creep [B41].
Fig. Ch4-A-3. Schematic illustration of the energy rate interpretationof C* integral [B41].
128
Fig. Ch4-A-4. The load-line deflection as a function of time for a crackedbody for a fixed applied load. Determination of steady state
deflection rate (SSV
) is also shown [B41].
Experimental method of measurement of C*: The method based on the energy rate interpretationrequires several sets of identical specimens. The crack size is varied between different sets ofspecimens. Let there be 5 sets of specimens, each consisting of 5 specimens, and thus a total of25 specimens. The crack lengths of specimens in the various sets are a1, a2,…,a5. Five loadlevels, P1, P2,…., P5 sre chosen, and one specimen from each set is subjected one of the loadlevels, measuring the steady state deflection rates. This is repeated for the other four load levels.
The steady state deflection rate (SSV
) is determined as shown in Fig. Ch4-A-4. Next, the (P-SSV
)
relationship for each crack size is plotted and the area under the curve for different values of
(SSV
) is obtained to determine U*, as shown in Fig. Ch4-A-5. Then, as shown in Fig. Ch4-A-6,
U* is plotted as a function of crack size for fixed values of (SSV
) and the slope of these curves can
be related to C*, as shown in the same figure.
Though used in earlier days, this is a very cumbersome method requiring a large amount timeand material. Nowadays, semi-empirical methods are used, which enable determination of crackgrowth rate from a single specimen. ASTM E 1457 Standard governs the method fordetermination of C*.
129
Fig. Ch4-A-5. P-SSV
relationship for a fixed crack size and the stress power, U* [B41].
Fig. Ch4-A-6. U* as a function of crack size for various values ofSSV
[B41].
130
Different specimens like C(T), Centre-Cracked Panels, etc. are recommended. For a C(T)specimen during extensive steady-state creep, the following relation holds:
* SS n= (2 + 0.522(1 - a/W)) (Ch4-A5)
( ) n+1
PVC
B W a
For application, the relation between C* and CCG rate (da/dt), similar to that shown inFig. Ch4-A-7 is made use of. This relation is often expressed in the form:
*da = = ( ) (Ch4-A6)
dqa
A Ct
Fig. Ch4-A-7. CCG rates plotted against C* for two specimen geometries [B32].
where A and q are constants. Based on analytical and experimental work Eq. (Ch4-A7) relatingCCG rate to C* and creep ductility, εf, has been found to work for many materials and predictCCG rate within a factor of 2.
131
*
f
3( )0.85a = (Ch4-A7)
C
The creep ductility is the uniaxial creep ductility (determined from conventional creep tests) forplane stress (thin components) and (1/30) times this value for plane strain (thick components).
Fig. Ch4-A-8. Theoretical CCG rate predictions compared with experimental data [B32].
Figure Ch4-A-8 shows a comparison of experimental CCG rate data shown in Fig. Ch4-A-7 withthose based on Eq.(Ch4-A7) under plane stress and plane stress conditions. Under plane strainCCG rates will be about 50 times higher for the same C* and the experimental data closelycorrespond to the plane stress predictions.
EXAMPLE PROBLEM ON CCG [B51]: A 50 mm wide and 25 mm thick C(T) specimen of304 stainless steel is subjected to constant load at 18 kN at 594 0C. If B and n for this material at594 0C are 2 x 10-18 and 6, respectively, for stress in MPa and strain rate in hr-1, calculate thevalue of C* for a/W = 0.5 and the measured load-line deflection rate is 2.5 x 10-6 m/h.
Solution: Direct substitution into Eq. (Ch4-A5), gives:
-6*
2 2
18 (kN) x 2.5 x 10 (m/hr) 6 kN.m kJ= (2 0.522*0.5) = 0.139 = 0.139
(0.25 m)x(0.25 m) 7 m m hrC
hr
Chapter 5 - An EPFM Application: ASTM E 1921Master Curve (MC)
5.1. Reference Temperature (T0) an
Though the RTNDT based ASME KIR curvfollowing deficiencies: a) the indexing parmeasurement of the true material fracturefollows a lower-bound approach and, hfracture toughness in the transition regionunderutilize the capacity of the material) asteels with room temperature static yield stapproach has been developed to describethe transition temperature regime. This athrough the weakest link model and a thre4. This introduces a new indexing paramdefined as the temperature correspondinginch (25. 4 mm) thick fracture toughnessfracture toughness of ferritic steels thatelastic-plastic KJC instabilities, or both.toughness and has better theoretical andconstraint (thickness) effects. The basic Mthe ASTM E-1921 standard [T8]. A limispecified to ensure high constraint conditio
ASME is also taking steps to incorporate tdirection has been taken by incorporationand N-631. N-629 is proposed for ASMirradiated material as an alternative meanfor ASME Section III, establishes alternatideterminations in RPV [T2]. Though, Aconditions, procedures detailed in it are inalso, using instrumented precracked Charp
Definitions of the New Refere(MC) and their Basis and Bacprocedure for determination oMC for a 9Cr-1Mo steel by InComparison of MC with the KRef.: [B28], [B56], [B60], [T8
OUTLINEnce Temperature (T0) and Master curvekground; Outline of ASTM E-1921f T0; Illustration of determination of T0 andstrumented Precracked Charpy Testing;
IR curve for the 9Cr-1Mo steel.
132
d Master Curve Approach
e has been successful in application, it is having theameter, RTNDT is empirical, and is not based on directresistance, namely fracture toughness, b) the method
ence, does not rigorously evaluate the scatter in theand hence may be overconservative (that is, it may
nd c) restricts itself within its purview pressure vesselress level below 345 MPa. A new Master Curve (MC)the variation of fracture toughness of ferritic steels inpproach interprets the scatter in fracture toughness
e-parameter Weibull distribution with a fixed slope ofeter, known as the reference temperature, T0. T0 isto a median fracture toughness of 100 MPam for 1
specimens. It is determined by actual measurement ofexperience onset of cleavage cracking at elastic, orAs T0 is based on actual measurement of fracturestatistical basis, it can better deal with scatter andaster Curve method has been recently standardised int on KJC values, relative to the specimen size, isns along the crack front at fracture.
his procedure in their code. A preliminary step in thisof T0 concepts in the ASME Code Cases N-629 andE Section XI, recognises measurements of T0 in
s of estimating Reference T0 [T3]. N-631 is proposedve T0 based Reference Temperature for initial property
STM E-1921 is specifically applicable for staticcreasingly being applied to dynamic fracture studies
y impact test.
]
133
5.2. Statistical basis of the Master Curve
The scatter in fracture toughness in the transition region is recognised as being a characteristic ofstructural steels. The present concept acknowledges that data scatter results from randomlydistributed cleavage triggering sources such as carbide cracking, multiple cleavage crackclustering and dislocation pile-ups. Based on weakest-link theory and Weibull statistics, thecumulative probability of failure, pf is given by:
min 0 min
0
1 exp{ ( )[( ) /( )] }bf Jc
Bp K K K K
B (5.1)
where Kmin is the toughness level below which cracks cannot propagate, K0 is the scaleparameter obtained at the point where KJC = K0 and pf = 0.632 and b = 4. By setting pf equalfor two specimen sizes (say, B = B1 and B = B2), the following expression results:
1
41
(2) min (1) min
2
[ ]Jc Jc
BK K K K
B
(5.2)
This is the basis of the size correction equation in ASTM E-1921 method, where B2 is setequal to 1 inch and Kmin = 20 MPam (because ferritic steels of yield strengths ranging from275 to 825 MPa (40 to 120 ksi) will have fracture toughness cumulative probabilitydistributions of nearly the same shape, independent of specimen size and test temperature, whenKmin is set equal to 20 MPam). Thus in the three-parameter Weibull model used to definethe relationship between KJC and the cumulative probability for failure, pf, (pf is theprobability for failure at or below KJC for an arbitrarily chosen specimen taken from a largepopulation of specimens) given above, the only unknown is the scale parameter, K0. Datasamples consisting of six or more valid (defined below) KJC values are used to estimate the truevalue of the parameter K0 from the following relation [ASTM E 1921]:
4 1/ 40 ( ) min min
1
[ ( ) /( 0.3068)]N
Jc ii
K K K N K
(5.3)
where N = number of specimens (valid values) and Kmin = 20 MPam. For a data setconsisting of invalid values (but not less than 6 valid values), N is replaced by the number ofvalid values, r. The standard deviation of the data distribution is a function of the Weibull slope(b = 4) and (median KJC) KJC(med), which is given by [ASTM E 1921]:
1
4( ) 0 min min( )[ln(2)]Jc medK K K K (5.4)
134
where Kmin = 20 MPam and [ln(2)]0.25 = 0.9124. For all valid set data size correction can beapplied either before or after computing K0, while for data set with invalid data (but with not lessthan 6 valid data), the invalid data are replaced by the corresponding limit values (computed usingEq. (5.6)) and then first K0(x) is computed (where x corresponds to the thickness of the specimens-all of the same B) and K0(x) converted to K0 corresponding to 1” B, for computing the KJC(med) for 1”thickness. From the KJC(med), the T0 is determined using the following relation [ASTM E 1921]:
( )
0
301ln[ ]
0.019 70
Jc medKT T
(5.5)
with KJC(med) in MPam and T0 in °C.
For censored data, due to crack length violation (see Fig. 5.1), the invalid data are replaced bythe largest valid data in the data set.
Fig. 5.1. Illustration of the 9-point average crack-length measurement on a 25 mmC(T) Specimen (the slashed area shows side-grooves if present) tested as perASTM E1921 [B52]. (see NOTES for Fig. 5.1 also given next)
135
NOTES for Fig. 5.1:
i 0 max
p 0 0
- = 0.95 mm 0.05 = 0.05*25 mm = 1.25 mm
= - = 0.026 mm 0.05 = 1.216 mm
a a B
a a a b
Here the actual measurements are for a non-side-grooved specimen (BN = B). As shown, 0.01B from thesurface (or the root of the side-grooves, if presented) are omitted. ak, a0 and ap are the average values ofmachine notch depth, fatigue crack length and final crack length, respectively. Please note that thepermitted maximum deviation of any measurement from the average for the precrack length is 5% of B or0.5 mm whichever is larger and valid slow-crack growth extension before cleavage is less than 5% ofinitial ligament or 1 mm, whichever is smaller; otherwise, the tests are invalid and should be used ascensured data.
5.3. Validity limits
The size validity limit for data censor mentioned in the previous paragraphs are applied usingthe relation [ASTM E 1921]:
0.5
0
30
ys
Jc
EbK
(5.6)
where E is the Young’s modulus, b0 is the remaining ligament depth and ys is the yield stress (atthe relevant test temperature and strain rate). This is to ensure sufficient constraint at the cracktip. A lower toughness limit of 50 MPam is imposed, because at the lower shelf and lowtemperatures, there is a change in the cleavage triggering mechanism and size effects areconsidered unimportant below a toughness value of 50 MPam and hence for application ofASTM E-1921 procedure test conditions are chosen such that values much above the lowercut-off limit, preferably close to or above 100 MPam are obtained, a condition difficult to bemet in small specimens, especially Charpy size specimens. It may be noted that Eq. (5.6)pertains to plane stress formulation as was available in the earlier ASTM E1921 versions;in the latest revisions (for example, ASTM E 1921-05) this has been changed to the planestrain formulation, i. e., the denominator in the RHS of Eq. (5.6) is 30*(1 – υ2) instead of 30.
136
Similarly, for conversion of J to KJd, the plane stress formula has been used instead of theplane strain one recommended in the latest revision of ASTM E1921.
5.4. Multi temperature equation for T0
To compute T0 from six or more valid KJC results obtained at different temperatures, thefollowing the multi-temperature equation has been proposed:
40 min 0
51 1min 0 min 0
exp{0.019( )} ( ) exp{0.019( )}0
[31 77 exp{0.019( )}] [31 77 exp{0.019( )}]
i n ni i JCi i
i ii i
T T K K T T
K T T K T T
(5.7)
where the Kronecker i = 1 for valid data and 0 for censored data. Kmin = 20 MPam.
Censoring is done as described for the single temperature case: for size validity Eq. (5.6) is used;for crack growth violation the highest valid value in the data set is used. Then the values are 1”size corrected using Eq. (5.2) before using Eq. (5.7) for computing T0. It may be noted thatASTM E 1921 restricts the test temperatures to within ± 50 0C of the actual T0. Also, for themulti-temperature case, considering increased inaccuracies from data generated between(T0 -50) and (T0 -14), an additional criterion for sufficient number of specimens has beenspecified as follows:
3
i i1
1 (5.8)i
rn
Where ri is the number of valid specimens within the i-th temperature range, (T - T0), ni is thespecimen weighing factor for the same temperature range as given in Table 5.1.
Table 5.1. Weighing-Factors for Multi-Temperature Anaysis
(T - T0) Range, 0C 1T KJC(limit) range, MPa√m Weight Factor, ni.50 to -14 212 to 84 (1/6)-15 to -35 83 to 66 (1/7)-36 to -50 65 to 58 (1/8)
The exact procedure for treatment of censored data is slightly different from that for the singletemperature case, and is detailed in Sections 10.4.1 and 10.4.2 of ASTM E 1921-05. Valuescensored due to crack growth limitation are not size corrected.
137
5.5. Master Curve equations
The following equations [ASTM E 1921] are used for computing the Master Curve and the 5%and 95% tolerance bounds (TB) from T0:
0(median) = 30 + 70exp(0.019( ))JCK T T (5.9)
0(5% TB) = 25.4 + 37.8exp(0.019( ))JCK T T (5.10)
0(95% TB) = 34.6 + 102.2exp(0.019( ))JCK T T (5.11)
The general equation for the tolerance bounds is given by Eq. (5.11a) [T20]:
1
4
JC(0.xx) 0
1= 20 + ln {11+77exp[0.019( )]}
1 0.K T T
xx
(5.11a)
where 0.xx represents the cumulative probability level: 0.5, 0.05, 0.95 and 0.01 being the valuesfor the median (50%), 5%, 95% and 1% TB, respectively. In fact, the 1% TB curve falls closer tothe ASME lower-bound KIC.
5.6. DETERMINATION OF T0 AND MC FOR A 9Cr-1Mo STEEL – EXAMPLE [B28]
This example illustrates, the reference temperature, T0, and the Master Curve (MC) determined forthe 9Cr-1Mo base material following ASTM E 1921 guidelines using instrumented pre-crackedCharpy (PCVN) tests. As the ASTM E 1921 generally determines T0 in the static/quasi-staticregime, the result obtained in this study is termed as T0
dy signifying dynamic conditions.
For developing a dynamic Master Curve for the 9Cr-1Mo steel base material, the Master Curveequations corresponding to static/quasi static cases are used (see previous Sections); only thestatic toughness terms are replaced by the dynamic counterpart, i.e., KJd.
5.6.1. T0dy from Pre-Cracked Charpy Test (PCVN) [B28]
The experimental/computational steps involved in determining T0dy are shown in the flow chart
given in Fig. 5.2.
The dynamic reference temperature, T0dy is determined using the pre-cracked Charpy specimens
following ASTM E 1921 guidelines. To minimise the oscillations in the load signals (common inhigh speed tests – see Chapter 6, Section 6.1 for a more detailed description of instrumentedimpact test), the initial T0
dy was determined by tests conducted at a reduced impact test velocity
of ~1.12 m/s; this was further converted to T0dy at 5.12 m/s, the normal impact velocity, by
estimating the shift in T0dy with change in test velocity from some empirical formulae.
Frvas
Us
T0
T0
KJd from 10x10x55 PCVNspecimens were determined at
138
Fig. 5.2. Flow-chart for T0 d
~1.12 m/s at 50 °C
Validity limit criteria wasapplied following ASTM E
1921
om both the valid and invalid KJd
lues, K0 was determined and thenize corrected to 1 inch following
ASTM E 1921
ing K0, KJd(med) and T0dy were
computed as perASTM E 1921
dy at 1.12 m/s was converted tody at 5.12 m/s applying suitable
shift equation.
Weibull Plot was constructedusing both the valid and
invalid KJd values followingASTM E 1921
Master Curve-DynamicFracture Toughness Test,was constructed using T0
dy at5.12 m/s, following ASTM E
etermination [B28].
1921
139
Fig. Typical load-displacement record from the low-blow testof a PCVN BM specimen
Load-point displacement, d/mm
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Load,
P/k
N
-2
0
2
4
6
8
10
PGY
PF
P-d trace
Linear Fit(initial elastic part of P-d trace)
Sp. No.: 91BM19; Test Temp.: -500C
V0 = 1.103 m.s-1; a0 = 4.601 mm
Fig. 5.3 Typical load – displacement trace [B28].
Though ASTM E-1921 gives approximate test temperatures for quasi-static tests with referenceto 28 J Charpy energy, (T28J), such guidelines are not available for dynamic loading conditions.However, it suggests that the test temperature should be selected such that it yields fracturetoughness values corresponding to 1 inch thickness specimens, close to 100 MPam. As per thisguideline, following some trial and error tests, starting from the T28J temperature, the testtemperature was chosen as –50 °C.
Figure 5.3 shows a typical load-displacement plot for the 9Cr-1Mo base material, obtained froma PCVN test at –50 °C using instrumented Charpy impact test (Instrumented Charpy Impact Testis described in more detail in Chapter 6). This shows clear elastic-plastic fracture with substantialwork-hardening before the fracture. Using these traces, the KJd values for the 10x10x55 sizePCVN specimens are determined following ASTM E 1921 procedures. The test temperature, testvelocity, validity criterion (Eq. 5.6) and 1 inch size corrected KJd values are reported inTable. 5.2. Two tests were conducted at a hammer velocity of 5.12 m/s, at –20 °C and thecorresponding results are also shown in Table 5.2. Both valid and censored data are used forcomputing T0
dy. In the present results only size correction was involved, no violation of crack
growth requirements. Invalid data (only 2 out of 10 in the present case) are replaced by thevalidity limit calculated by Eq. (5.6).
Probability values, pf, are assigned to KJd values following Eq. (5.12) [ASTM E 1921] after theyare ranked in the order of increasing magnitude.
140
pf = (i 0.3)/(N + 0.4) (5.12)
In Eq. (5.12), pf is the cumulative probability of failure, i is the rank of corresponding KJd value(i =1 for the lowest KJd and i =N for the highest KJd), N is the total number of KJd values, 10 inthe present case. The pf, and the Weibull plot points are reported in Table 5.3.
The scale parameter of the Weibull distribution, K0, is estimated following Eqn. (5.3). For all theKJd values (not 1 inch size corrected) obtained at 50 °C, K0 is determined as 149.65 MPam.This was further corrected for 1 inch size using Eqn. (2) as 122.70 MPam. The median KJd wascalculated with Eqn. (4), using the K0 as 122.70 MPam. Thus, the KJd(median) was determinedas 113.71 MPam. With the test temperature as 50 °C and the determined KJd(median) value,the T0
dy was determined using Eqn. (5) as 59.4 °C. The Weibull plot is shown in Fig. 5.4.
The regression line, with a slope 4 was fitted to the data points (shown in Fig. 5.4) and the trendof the data points with respect to the slope 4 line are seen to be in conformity with the ASTM E1921 proposition.
Table 5.2. KJd results from PCVN tests for the 9Cr-1Mo steel, base material,at impact velocity of ~1.12 m/s [B28].
SpecimenNo.
TestTe-mp/
°C
Initialtest
velocityV0/m.s-1
Initialcrack
length,a0/ mm
KJd/
MPa.m
yd/
MPa
Validitylimit/
MPa.m
Validityresults
1 inch sizecorrectedKJd/MPa.m
91BM11 70 1.08 4.408 57.8 - - - 50.0
91BM13 55 1.07 4.427 56.4 - - - 48.8
91BM12 50 1.11 4.505 137.1 717.3 166.1 Valid 112.8
91BM14 50 1.11 4.395 162.3 707.6 166.6 Valid 132.7
91BM15 50 1.15 4.707 187.0 742.6 165.9 Invalid 152.2*
91BM16 50 1.15 4.383 83.5 713.6 167.5 Valid 70.3
91BM17 50 1.13 4.61 118.9 745.6 167.7 Valid 98.3
91BM18 50 1.14 4.541 PF notclear
717 165.6 - -
91BM19 50 1.10 4.601 243.9 721.6 165.2 Invalid 197.3*
91BM10 50 1.10 4.805 60.00 PF PGY
- Valid 51.7
91BM20 50 1.13 4.885 163.00 762.5 165.2 Valid 136.4
91BM8 50 1.11 5.095 70 PF PGY
- Valid 60
91BM9 50 1.13 4.632 134.0 712.1 163.6 Valid 110.3
91BM21 20 5.12 4.870 286.7 779.7 166.7 Invalid 230.6*
91BM22 20 5.12 4.625 296.5 789.2 171.6 Invalid 233.4*
* 1 inch size corrected from invalid KJd
141
Table 5.3. KJd, cumulative probability of failure (pf) and the Weibull plot points [B28]
Rank KJd,MPam
pf Weibull Plot Points
X-axisln (KJd-20)
Y-axisln{ln[1/(1-pf)]}
1 60.0000 0.0637 3.6889 2.66382 70.5000 0.1635 3.922 1.72333 83.5281 0.2596 4.1515 1.20204 118.8664 0.3558 4.5938 0.82175 133.3748 0.4519 4.7307 0.50866 137.1293 0.5481 4.7633 0.23047 160.7989 0.6442 4.9473 0.03298 162.2600 0.7404 4.9577 0.29909 165.1581 0.8367 4.9778 0.5940
10 165.8971 0.9327 4.9829 0.9927
FIg. 5.4. The Weibull plot with all the KJd data, obtained fromPCVN test at 50 °C with hammer velocity of ~1.12 m/s [B28].
As the PCVN tests were conducted at a reduced velocity of ~ 1.12 m/s, to account for the testvelocity effect on the reference temperature, T0, the shift in T0, associated with the change in testvelocity or the stress intensity factor rate is given by Eq. (5.13) [B44].
0 2 4 6 8
-3
-2
-1
0
1
2
fit with slope 4
ln{l
n[1
/(1
-pf)]}
ln(KJd
-20)
142
(22 0.016 ) log (22 0.016 ) logref ref
y y
test test
K VT
K V
(5.13)
where y is the yield stress at the test temperature/strain rate (660 MPa in the present case).
Applying these results in the Eqn. (5.13), for tests at 1.12 and 5.12 m/s, the T0 yielded a shift of7.5 °C based on velocity based estimation. So, the T0
dy corresponding to test velocity of 5.12 m/swas estimated as T0
dy (at ~1.12 m/s) + 7.5 OC = 51.9 °C.
Thus, T0dy (at 5.12 m/s) = 51.9 OC~ 52 °C
The Master Curve at test velocity of 5.12 m/s is plotted using T0dy as 52 °C as shown in
Fig. 5.5.
Fig. 5.5. A Comparison of Master Curves (Dynamic) and KIR
Curve 9Cr-1Mo Steel: T0dy= -52 °C, RTNDT = -25 °C [B28].
Comment: ASME KIR Curve is unrealistically Conservative Compared to the Trend of DynamicFracture Toughness Variation as Shown by the Master Curve in Dynamic Condition.
143
Typical Calculation of KJd for Specimen No. 19 Tested at -50 °C (Fig. 5.3)
The relevant data obtained from the above load-displacement trace are:
Crack Length, a = 4.601 mm (9-point average crack length determination is illustrated for a C(T)specimen in Fig. 5.1)
General Yield Load, PGY = 7.4 kN.m; Brittle Fracture Load, PF = 8.63 kN and Total area up to PF underthe load-displacement trace, U = 8.42 J.
The initial compliance from the linear-elastic portion of the load-displacement trace,
CT = 0.0358 mm/kN (= reciprocal of initial slope)
ASTM E 1921 Standard specifically uses the plane stress version of Eq. (4.3.1) for computing KJC or KJd
(for dynamic tests as in the present case). This requires separation of the total energy into elastic andplastic parts. In the present test, there is no physical crack extension observed prior to brittle cleavagefracture at PF, except crack tip blunting and some miniscule stretch zone formation at the crack tip.
Plastic energy, Upl = U – (1/2)(PF)(del) = U – (1/2)*PF*CT*PF = 8.42 – (1/2)*0.0358*8.63*8.63 = 8.25 –1.47 = 7.1 J
Therefore, plastic part of J, Jpl = 1.9*Upl/(Bb) = 1.9*6.78/(10*(10-4.601)) = 0.250 J.mm-2.
It may be noted that in computing Jpl, the multiplication factor used is 1.9 instead of the usual 2.0 (see Eq.(4.3.1)) and this is as given in the ASTM E 1921 standard.
The elastic part, Jel is obtained from Jel = K2/E, where K is evaluated using the 3PB specimen formula (seeFig. 2.1) at load PF = 8.62 kN and a = 4.601 mm.
K at PF = 81.3 MPa√m and E = 210.1 GPa and Jel = 0.0314 J.mm-2. Therefore, total JC = 0.25 + 0.0314 =0.281 J.mm-2. Then, KJd = √EJC = 243 MPa√m, subscript ‘d’ signifies dynamic test.
Dynamic yield stress, σYd = 2.85PGY*W/(Bb2) for a precracked specimen (for unprecracked specimen, thefactor 2.85 is replaced by 2.99). Substituting the values of W = B = 10 mm and others as given above, σYd
= 724 MPa, the subscript ‘d’ signifies the dynamic test.
Now the ASTM E1921 Standard validity condition is,
KJd ≤ √(EbσYd) = [(210000*(10-4.601)*723/(30*1000))]0.5 = 165 MPa√m.Hence the present result is invalid.
144
5.7. APPLICATION OF THE ASME CODE CASES N-629 AND N-631 [B28]
ASME Code Case N-631 (Section III) defines RTT0 for unirradiated reactor vessel material,while ASME Code Case N-629 (Section XI) defines RTT0 for unirradiated and irradiated reactorvessel material. This new reference temperature is defined as:
RTT0 = T0 + 19.4 0C (5.14)
RTT0 is the equivalent of RTNDT for utilizing the ASME lower-bound (LB) KIC and KIR curves(Eqs. (3.3.2) and (3.3.1), respectively). Since, presently, the design is based on ASME curvesbased on RTNDT, and as the familiarization with and transition to the new ASTM E 1921 basedMC approach will take time (including further validation), this procedure has been proposed.
5.8. A NEW UPPER-SHELF FRACTURE TOUGHNESS MASTER CURVE (USFTMC orUSMC) FOR FERRITIC STEELS [B14 to B16]
The current Appendix G (ASME) method does not consider upper shelf toughness behavior incalculating a resistance toughness. The implementation of a model for fracture toughness on theupper-shelf will be necessary to ensure that appropriate toughness values are used at alltemperatures of interest. For ferritic steels, the master curve (MC) proposed by Wallincharacterizes the median fracture toughness and its scatter in the transition regime; the ASTM E1921 reference temperature, T0, locates the MC on the temperature axis. However, this MCneither does define the upper-shelf nor it quantifies the US fracture toughness. Recently, basedon some theoretical models, a new MC has been proposed that quantifies US fracture toughness;moreover, an empirically defined upper-shelf start temperature (TUS) has been correlated to theASTM E 1921 reference temperature (T0). These correlations are given below:
2
US 0
with the temperatures in C and R = 0.99
50.1 + 0.79 (5.15)T T
The USFTMC is given by Eq. (5.16):
( ) - (288 C) 1.75 exp ( 273.15) + ( 273.15) lnIC IC 1 2 3 refJ T J C C T C T
(5.16)
where, Tref = 288 °C (or 561 K), C1 = 1033 MPa, C2 = 0.00698/K, C3 = 0.000415/K,ε• = 0.0004/sec and σref = 3.33 MPa.
145
Thus, Eqs. (5.15) and (5.16) together with the ASTM E 1921 MC provide a complete descriptionof fracture toughness from lower-shelf through all the way to the upper-shelf based only ondetermination of the single index of reference temperature (T0). Figure 5.6 illustrates theseconcepts for an A533B reactor pressure vessel steel (RPV).
Fig. 5.6. Comparison of Model Predictions of Transition and Upper-Shelf FractureToughness (Based Only on T0) with Experimental Results for a Low-TransitionTemperature A533B Steel [B14].
Using these equations it is possible to estimate the complete variation of initiation fracturetoughness with temperature in both the transition regime and on the upper shelf based only on ameasurement of To. The USFTMC method is summarized in the next Box for easyreference.
146
UPPER-SHELF FRACTURE TOUGHNESS MASTER CURVE (USFTMC orUSMC) CALCULATION PROCEDURE
Step 1. Estimate T0 using ASTM E 1921.
Step 2. Convert the mean toughness curve from ASTM E 1921 in Step 1 to J units by:
2 2
0C
30 70 exp(0.019( )) (1 )T TJ
E
,
where E = (207200 - 0.057T), T is in °C and ν = 0.3.
Step 3. Calculate TUS using the T0 from Step 1 in Eq. (5.15).
Step 4. Calculate JC at TUS (from Step 3) using the Eq. from Step 2. Call this JC(US).
Step 5. Calculate ΔJIC at TUS using Eq. (5.16) and call this ΔJIC(US).
Step 6. Let JADJUST = {JC(US) - ΔJIC(US)} using JC(US) from Step 4 and ΔJIC(US) from Step 5.
Step 7. The variation of JIC with temperature can be estimated as follows:
IC 1 2 3 ADJUST= 2.09 exp ( 273.15) ( 273.15) lnTrefJ C C T C T J
The complete variation of initiation fracture toughness with temperature is now defined in boththe transition regime where failure occurs by cleavage (using the equation for Jc in Step 2) andon the upper shelf where failure occurs by microvoid coalescence (using the equation for JIc inStep 7). Conversion of these equations to K-units can be made by using the conventionalrelationship between J and K in plane strain (Eq. (4.1)).
5.9. Effect of strain rate or stress intensity factor rate (SIF) rate on T0 [B55]
If T0 is known, estimates of T0dy (reference temperature under dynamic loading conditions) can
be made using Wallin’s strain rate shift equation [K. Wallin and T. Planman. Effect of strain rate onthe fracture toughness of ferritic steels. IAEA Specialist Meeting on Master Curve Testing, Results andApplications. 17-19 Sept., 2001, Prague, Czech Republic, 12 pages and E. Lucon and R. Chaouadi.(2002) Radiation damage assessment by the use of dynamic fracture toughness measurements onprecracked Charpy V notch specimens. Effects of Radiation on Materials: 20th Int. Symposium. ASTMSTP 1405, (Eds.) S. T. Rosinski, M. L. Grossbeck, T. R. Allen and A. S. Kumar, ASTM, WestConshohocken, PA, 2002 as in [61], pp. 68-78.], Eqs. (5.17).
147
1.091.66
0
dy 00
= 9.9exp (5.17a)190 722
and
.=
( ln )
ysT
TT
K
(5.17b)
where T0 is the quasi-static (stress intensity factor – SIF- rate ~ 1 MPa√m.s-1) reference
temperature and the SIF rate in Eq. (5.17b),
K , is the dynamic rate, approximately equal to 104
to 105 MPa√m.s-1 for low velocity tests at 1 to 2 m.s-1 and 106 MPa√m.s-1 for the normalvelocity impact tests at ~5 m.s-1. The yield stress in Eq. (5.17a) is the room temperature YS,taking dynamic SIF rate as 106 MPa√m.s-1. Expressions for variation of yield stress withtemperature and strain rate are given in Section 6.1.2 (under Modified Schindler Procedure –MSP).
Chapter 6 – Instrumented Charpy Impact Test, Charpy-Fracture Toughness Correlations and Reference Temperature
Prediction
6.1. Instrumented Charpy Impact Tes
6.1.1. Introduction
Results of instrumented precracked Charpyfracture toughness and reference temperatutest. The fracture toughness obtained frotoughness, is denoted by KId or KJd dependinor Elastic-Plastic mode; in Fig. 5.2 the test spJd value and KJd were obtained. The use of atup) allows a standard Charpy impact macCharpy V-notch specimen during deforminstrumenting the Charpy test is the additionsmall specimens, and simple operation. Thstrain gages to the striker to sense the load-gages are placed on the specimen as well, suc
In instrumented impact test (IIT), often semsignal to noise ratio. A typical load (P) – diFig. 6.2 with the various energy and load feand fast fracture occurs and this gets arrestedof load-time traces from IIT of CVN specimis purely linear-elastic with sudden brittle fato 100% cleavage, while as the tests progrprecedes cleavage fracture and, ultimately,fracture. Such traces can be analysed for obshown schematically in Fig. 6.4. The fracsummarized.
Instrumented Impact Test (IITCharpy-Fracture Toughness CReference Temperature; IllustComparison of Fracture TougMaster Curve. Refs. [B48], [B[B62], [B38], [B40], [B19], [B
OUTLINE) - Brief Outline; Data Reduction from IIT;orrelations – Old and New; Prediction of
rated Examples on IIT Data Reduction,hness Predictions from IIT, Correlations and49], [B54], [B55] [B56], [B56], [B60] to
148
t
tests were analysed in Chapter 5 for evaluating there under dynamic loading conditions of an impactm these dynamic tests, called dynamic fractureg on whether the result is obtained in LEFM modeecimen was showing elastic-plastic mode and hencedditional instrumentation (typically an instrumentedhine to monitor the analog load-time response ofation and fracturing. The primary advantage ofal information obtained while maintaining low cost,e most commonly used approach is application oftime behavior of the test specimen. In some cases,h as for the example shown in Fig. 6.1.
i-conductor strain gauges are used to obtain largesplacement (d) trace for a ferritic steel is shown inatures marked. At the brittle fracture load, cleavage
at the arrest load. Figure 6.3 shows a typical seriesens of a carbon steel: at the lower-shelf, the fractureilure occurring at the maximum load correspondingesses through to the upper shelf, general yieldingat the upper shelf, the traces do not show any fasttaining what is called the load-temperature diagramture mechanisms in the various regions are also
47], [B36], [B66], and [T18]
149
Fig. 6.1. Charpy specimen with additional instrumentation at the supports [T18]
Fig. 6.2. Typical load (P) –displacement (d) trace from an instrumented impact test (IIT) [T18]
150
With reference to Fig. 6.4., above the temperature TD, called the brittleness transitiontemperature – corresponding to the intersection of the brittle fracture load (PF = Pm without anygeneral yield at temperatures lower than TD) curve with the general yield load (PGY) curve.General yield represents a situation where plastic deformation starting at the V-notch spreadsacross the remaining ligament below the V-notch. Dynamic yield stress (σyd) corresponding tothe loading rate in an impact tested CVN specimen (at the usual impact velocity of ~ 5 m/s thestrain rate ~ 1000 s-1) can be obtained from the PGY values. More importantly, the PF = PGY
values at TD can be used to obtain (as shown later) the microcleavage stress, σf, a constant for aparticular steel (especially at low temperatures) and is related to fracture toughness, and hence animportant fundamental parameter controlling micro-fracture processes.
Fig. 6.3. A typical series of IIT load-time traces using CVN specimens for a carbonsteel over its DBTT range ( 1 ft-lb = 1.356 J; 0C = (0F – 32)*5/9)[B48, DynaTup Product Data Sheets (IGCAR-1979)].
151
For dynamic loading, the dynamic yield stress (DYS), σyd, and flow stress, σfl, of standardCharpy V-notch (CVN – blunt-notched, not precracked) specimens can be estimated for postgeneral yield behavior as:
GYyd 2
GY m
fl 2
2.99 (6.1.1a)
and
22.99
P W
Bb
P PW
Bb
(6.1.1b)
Fig. 6.4. Typical load-temperature diagram for a ferritic steel obtained from IITs of CVNspecimens over the transition region (see R. A. Wullaert (1970) in Impact Testing of Metals,ASTM STP 466)
152
where general yield load, PGY, and maximum load, Pm, are as indicated in Fig. 6.2; W is thespecimen width and b is the remaining ligament depth, (W – a). For a standard CVN specimenW = 10 mm and a = 2 mm. For precracked specimen the constant 2.99 is replaced by 2.85.
Additional strength and toughness values from IITApart from DYS (yd), microcleavage fracture stress (σf), the load-temperaturediagram and the DBTT based on it, TD, described above, the following parameters canalso be obtained from an IIT.
Dynamic fracture toughness: KId (linear elastic fracture mechanics (LEFM) parameter) Elastic-Plastic Fracture toughness: JId/Jid or critical crack tip opening displacement
(CTOD) – dynamic J calculation for precracked CVN (PCVN) specimens wasdemonstrated in Chapter 5.
Fracture appearance – from the analysis of P-d traces as in Fig. 6.3, empirical formulaehave been obtained to calculate the Fracture Appearance and therefrom the DBTT, called,Fracture Appearance Transition Temerature (FATT) – conventionally, FATT is obtainedby analysis of fracture surfaces of broken CVN specimens using low magnificationmacro-photographs (see, ASTM E23 standard)
Reference Temperature from T4kN, where T4kN is the temperature at which IIT load-time(or displacement) traces show an arrest load Pa = 4 kN
Reference Temperature, T0, through advanced correlations Testing PCVN specimens gives lowerbound fracture toughness values for ferritic steels Computerised acquisition and processing of data with new analysis procedures can
help determine initiation J values (Ji) from both CVN and PCVN specimens and generatethe entire J-R curves (see Examples 4.3 and 4.7 in Chapter 4 for explanation of J-Rcurves and Ji)
A NOTE ON TD (brittleness transition temperature)Physically, TD represents 100% cleavage fracture (end of the gross elastic region in the load-temperature diagram) and whereas many of the Charpy energy indices like T28J, T41J etc.mentioned later represent different amounts of cleavage or ductile fracture that varies dependingon the type of steel. Hence, TD is said to be capable of truly reflecting the change in DBTT ofsteels resulting from neutron irradiation or ageing embrittlement. Recently, Chaouadi [B11] hascorrelated TD to T0 and a fit to his data gives: T0 = 1.53TD + 39.95, where T0 is the ASTM E1921 reference temperature. It is also claimed that shift in TD (∆TD) correlates with irradiationinduced shift in T0.
153
Figure 6.5 gives a schematic variation of fracture toughness (LEFM) of ferritic steels withloading rate – at higher loading rates the fracture toughness is called dynamic fracture toughness.For ferritic steels increased strain rates result in higher strength, reduced ductility and reducedtoughness. Table 6.1.1 gives an idea of the strain rates or loading rates involved in somecommon fracture toughness tests.
6.1.2. Calculation Procedures in Instrumented Charpy Impact Test ([B45], [B53], [B50], [T18])
Load-time data processingIn computerized systems, the load (P)-time (t) data are acquired by a high speed transientrecorder or digital storage oscilloscope (which is capable of capturing the signals at a fast or slowrate according to the speed of the test) and converted to digital form for storage and furtherprocessing by a computer (older systems record the P-t trace in an analogue storage oscilloscopeand a Polaroid photo of the signal is taken and the trace analysed manually – not so efficient and
Fig. 6.5. Effect of loading rate on fracture toughness of ferritic steels(Schematic – 1 ksiin = 1.099 MPam) [B6].
accurate). As is well known, the signal from the instrumented impact test is superimposed withoscillations – especially so in the case of precracked CVN (PCVN) tests at high velocity; thisresults from inertial load and wave propagation effects related to the mechanics of the impacttest. To reduce the effects of these on the accuracy of the results, either advanced analysistechniques called dynamic analysis are employed or tests are done at a reduced velocity.
154
Table 6.1.1. Strain rates or loading rates in some fracture tests [B50]
So a mathematical curve smoothing is done to obtain a sort of average load values. For example,Fig. 6.2 is a record for a high speed test (~5.5 m/s) of a CVN specimen while Fig. 5.2 is therecord for a PCVN specimen tested at a reduced velocity (~ 1.1 m/s) with the smoothened curvesuperimposed). General yield load, PGY, is determined as the load at the intersection of the initialstraight line portion of the P-d curve (representing elastic deformation) and the average curveafter the general yield as shown in both the Figs. (5.2) and (6.2). Dynamic yield stress (DYS),σyd, and flow stress, σfl, values are obtained from Eq. (6.1.1).
KId and JId/KJd estimation by conventional methods [B45, B53, B50 and T18]Based on observations like those in Fig. 6.3, P-t or P-d traces can be classified into three types asshown in Fig. 6.6. Type I failure is brittle failure exhibiting no gross ligament plasticity orgeneral yielding. Type II failure shows plasticity up to maximum load and sudden failure onreaching the peak load. Type III failure shows post maximum load plasticity before brittle failureor complete ductile failure.
155
For Type I brittle failure, KId (LEFM dynamic fracture toughness) is estimated using the ASTME 399 expression for the three point bend (TPB) specimens using an a/W=0.2 and the finalexpression for K1d applicable to the standard CVN specimen is :
4.67 (6.1.2)Id CK P
where the inertial corrected (see next paragraph for explanation) critical fracture load, PC, is inkilonewtons (kN) and K1d is in MPa√m.
Critical fracture load PC is obtained from the brittle fracture load, PF, by applying corrections foroscillations; correction by using so called Varga’s procedure is illustrated in Fig.6.6 against theType-I P-t trace. Lower bound values of KId are obtained by applying a multiplication factor of0.8 to the values obtained from Eq. (6.1.2) using PF directly (instead of PC) – the 0.8 factor mayaccount for notch acuity effects also at the lowest temperatures for a CVN specimen, becausestandard CVN specimen is a blunt notched specimen.
For Type II P-d traces (Fig. 6.6), JId is estimated by a J-integral approach using the relation
Id
0
η(6.1.3)
AJ
B b
where = 1.384 for a CVN specimen and A is the area under the P-d curve up to crack initiation(for deep cracked TPB specimen with a/W = ~0.5, = 1.9 (ASTM E1921) or 2 (ASTM E1820 orE813)). The point corresponding to (Pm+Pgy)/2 is taken as the crack initiation point, where Pm isthe maximum load on the P-d trace. JId values are converted to KJd values using
Jd Id
2
(6.1.4)
where ' = for plane stress and ' = for plane strain(1 )
K E' J
EE E E
The plane strain and plane stress values differ by 5% for a Poisson’s ratio of 0.3, thus the lattergives conservative (that is, lower) values. Empirically, for blunt CVN specimens, lower boundvalues of KId are obtained by applying a multiplication factor of 0.65 to the values fromEq. (6.1.4).
KJd can also be calculated from CTOD. The total CTOD at initiation, δt , is the sum of elastic(δel) and plastic (δpl) parts given by:
156
TYPE- I
Fig. 6.6. Typical load-time (P-t) traces from instrumented Charpy impact tests (PGY,Pmax, PF above are sometimes referred to as Pgy, Pm and Pf, respectively) [B53].
157
t el pl
2 2
pl
ys
(1 )(6.1.5)
2
K v
E
where K is calculated from Eq. (6.1.1) (or the ASTM E399 expression for TPB specimen withthe appropriate a/W) using the load at the crack initiation point and for dynamic tests σys isreplaced by σyd.
The plastic part of the CTOD for a CVN specimen is calculated using the following relatio
pl pl0.16 (6.1.6)d
Then fracture toughness based on CTOD is given by:
yd tId ( 1.7 (6.1.7)K E
The CVN specimen lower bound value is given by applying a multiplication factor of 0.65 to theresults from Eq. (6.1.7). For deeply cracked PCVN specimens, CTOD can be obtained asdiscussed in Section 4.5. For small specimens like Charpy specimens, using only the plasticcontribution in Eqs. (6.1.3) and (6.1.7) results in only negligible error.
For Type III P-d traces (Fig. 6.6) showing significant ductility and crack extension aftermaximum load, taking crack initiation as for Type II traces will be ultra-conservative (i.e,significantly underestimate the true JId obtainable from precracked CVN specimens). So,methods used in the literature like compliance changing rate or key-curve method can be appliedto identify crack initiation and extension. This aspect is not within the purview of this book.
KJd and J1d estimation by the Modified Schindler Procedure (MSP) [B53 to B55]For the power law J-R curve (see Examples 4.3 and 4.7 in Chapter 4 for explanation of theJ-R curves and Ji) given by
( ) (6.1.8)pJ H a
Schindler [15], by an analytical-empirical procedure, suggested a method for obtaining theconstants H and p from the following equations:
158
0V mp1
0
1-( )2( ) (6.1.9)
{ ( ) }p
p
ppa
H C Ep B b
mp 1
V
3[1 ] (6.1.10)
4
Ep
C
where Emp is the plastic energy up to maximum load (see Fig. 6.2) and CV is the total CVNenergy, i.e, the impact energy and where (a0) = 1.384 for a CVN specimen with a0 = 2 mm. TheJ by the Schindler procedure is referred to as JSch; J from Schindler curve for Δa = 0.1 mm isreferred as JSch0.1 and the corresponding K is KJSch0.1.
Later, Schindler, based on further analysis, suggested that minimum JId (JMSP – MSP as definedin the title to this Section)) can be obtained obtained from the following relation (this JMSP can beconverted to KJd-MSP using Eq. (6.1.4)):
-3V
MSPV
fld
7.33 10(6.1.11)
1 1.47 ( )
m CJ
C
where J is in J∙mm-2, σfld is the dynamic flow-stress (Eq. (6.1.1b)) and m is the power-lawexponent given by
( )ydlog log6010 (6.1.12)n
Knowing n alone from σyd, the ratio of maxd/yd (or equivalently a ratio of Pm/Pgy) can beestimated using Eq. (6.1.13) given below [B55]:
2 3 (6.1.13)= 0.9867 + 1.6647 + 19.137 - 6.6306p n n n
where the load ratio p = Pm/Pgy (or maxd/yd) usually is in the range of 0.8 to 2.0. For variationwith temperature of the Young’s modulus, E, the following relation has been used (with E inGPa and temperature T is in C) [B55]:
= 207.2-0.0571 (6.1.14)E T
When yield load data is not available, the variation of static yield stress can be estimated fromthe following equation [B55]:
-4 2 -7 3ys = 6.895 (73.62-0.0603 +1.32 10 -1.16 10 ) (6.1.15)T T T
159
where T is in F (F = C*(9/5) + 32) and ys is in MPa. The results from Eq. (6.1.15) must bescaled using known YS values for the particular steel at one or two temperatures. For example, atroom temperature (RT) of 25 °C, YS value computed from Eq. (6.1.15) ~ 480 MPa; then for asteel with actual RT YS value of 580 MPa, the scaling factor computed (based on RT YS) is(580/480) approximately 1.21 and all the YS values computed over a range of temperature (saythe whole transition region) using Eq. (6.1.15) must be multiplied by 1.21 to yield the actual YSvalues for the particular steel. If the YS values for a steel are known at two or three temperatures,then the above procedure can be applied at each of these temperatures (i. e., calculating ratio ofknown YS over the YS computed from Eq. (6.1.15)) and then determining an average scalingfactor to scale all the YS values from Eq. (6.1.15) over the temperature range of interest –usually, the transition temperature range for ferritic steels.
In the case of dynamic fracture toughness, corresponding dynamic yield stress values evaluatedfrom instrumented impact tests or estimated from Eq. (15) are employed. Dynamic yield stresscan be estimated from the following equation [B55].
yd ys-RT 10
666500= + -190 (6.1.16)
( +273)*log(2*10 * )T t
where t is the fracture time in ms (usually taken as 0.1 ms for impact tests) and T is in C. Thevalues from Eq. (6.1.16) should be scaled using known values of dynamic yield stress at RT (seethe next paragraph) or at other temperatures as was suggested for the estimation of static yieldstress.
A good way to estimate the RT dynamic yield stress (σyd-RT) is to apply the empirical relation –Eq. (6.1.17) - to obtain the RT dynamic general yield load (Pgy-RT
dy – as will be obtained frominstrumented Charpy V-notch tests) from the easily available RT static yield stress (σys-RT). Theempirical equation for estimating Pgy-RT is as follows [H. Mathy and T. Greday. Interpretation of theimpact strength transition of structural steels by means of instrumented Charpy tests. C.R.M. Report No.52, C.R.M. (Centre De Recherches Metallurgiques), Liege, Belgium (Reprinted by Effects Tech. Inc.)(May 1978) [B55]]:
dygy-RT ys-RT(N) 6300 14.8 (6.1.17)P
where σys-RT is in MPa and σyd-RT is estimated from the Pgy-RTdy using Eq. (6.1.1a).
The general procedure that can be adopted for evaluation of results from IIT of CVN specimensis to obtain the lower-bound estimate for each specimen taking the lowest value from theestimates obtained using the equations given in the various sub-sections of Section 6.1.2. Thiswill be illustrated later.
160
6.2. Fracture Toughness Correlations with Charpy Energy and other Parameters
6.2.1. Direct Charpy energy (CV) temperature - T0 Correlations
One of the simplest and direct correlations is that between the temperature for 28 J CV energy,
T28J, and T0 [B56] as given by Eq. (6.2.1) (all temperatures in this section are °C):
00 28J 18 ( = 15 C) (6.2.1)estT T
Subsequently, the above has been revised by many authors.
Similarly, based on T41J, the following correlations have been given [B56]:
00 41J
0 1 41J
26 ( = 25 C) (6.2.2)
with conservative estimate given by
- 1
estT T
T T
(6.2.2a)
Equation (6.2.2a) is recommended as a very conservative estimate.
6.2.2. Older CVN energy (CV) - KIC Correlations and T0 estimates
The CV-KIC correlations examined in this paper are the following pertaining to the Charpytransition region:
Rolfe, Novak and Barsom (RNB correlation [B38, B36, B6]) given by Eq. (6.2.5),
4 1.5 0.5IC V
V
= ( .1000.(2.28).10 . ) (6.2.5)
(YS range = 270-1700 MPa; range 4 82 J)
K E C
C
where KIC is in MPa√m, E is the Young’s modulus in GPa and CV is the Charpy energy in J.
Sailors and Corten (SC correlation [B40, B36]) given by Eq. (6.2.6)
0.5IC V
V
= 14.63 (6.2.6)
(YS range = 410-480 MPa; range 7 68 J)
K C
C
Robert’s lower-bound correlation (RLB correlation [B36]) given by Eq. (6.2.5),
0.63IC V= 8.47 (6.2.7)K C
161
Barsom and Rolfe’s Kd-Kc (dynamic to static fracture toughness) temperature shift (Barsom-Rolfe Shift - BRS) procedure given by Eqs. (6.2.8a) and (6.2.8b) [B36, B6]:
0.5d V
V
= (0.64 ) (6.2.8a)
(YS range = 250-345 MPa; range 2.7 61 J; in MPa; in MPa m)
K EC
C E K
shifted CVN ys-RT
shifted CVN d
= - (119 - 0.12 ) (6.2.8b)
( = temperature after shift; CVN test temperature at which is
T T
T T K
d CVN IC shifted ys-RTgiven by Eq. (6.2.8a); at = at ; = room temperature yield stress)K T K T
Here a dynamic fracture toughness is computed and shifted to lower temperatures to give thestatic values.
Marandez-Sanz Procedure [B36]
This procedure has been verified for CV ≤ 50 J and involves the following steps:
Determination of the impact CVN transition curve Calculation of KIC-T curve from CV using the following
correlations:0.5
VIC
IC v
= 19 (6.2.9)
where is in MPa m, is in MPa and is in J.
K C
K E C
Calculation of TK100 (temperature corresponding to KIC = 100 MPa√m) using the correlation:
K100 28J
28J
= 9 +1.37 (6.2.10)
where is the 28 J Charpy temperature as described before.
T T
T
Shift the KIC-T curve so that it passes through the coordinates: (TK100 (0C), 100 MPa√m)
Another lower bound correlation for the lower-shelf and lower transition region is given byEq. (6.2.11) [B4]:
25 V
25
= 12 (6.2.11)
where is the estimated material fracture toughness in MPa m for a thickness
mat
mat
K C
K
V
of 25 mm
and Charpy energy ( ) is in J.C
162
Most of the above correlations and a large number of additional correlations compiled andtabulated by Nevasmaa and Wallin [B69] are reproduced in Appendix Tables A1 to A3.
6.2.3. New Reference Temperature Correlations [B67, B68, B54, B55, B53]
(a) IGCAR Procedure
Methodology
2.1. Inverse Wallin Strain Rate Equation – IWSRE
Recently, the author had prepared a paper [1] which provides an inverse relation to the WallinStrain Rate Equation (WSRE) [2,3] for estimating the reference temperature [4] at larger loadingrates (expressed in terms of stress-intensity factor - SIF - rates) from room temperature yieldstress and (quasi-)static reference temperature, T0 [4]; the inverse relation is termed IWSRE.IWSRE enables estimation of the quasi-static reference temperature from a knowledge of roomtemperature yield stress and dynamic reference temperature, i.e., reference temperaturecorresponding to faster loading or SIF rates. The TQSch
dy obtained from the application of themodified Schindler procedure (MSP) to Charpy impact data was taken as the referencetemperature at a SIF rate of 106 MPa√m.s-1 [1,5]. Then applying the TQSch
dy in the IWSRE alongwith the RT-YS and the SIF rate of 106 MPa√m.s-1, results in an estimate of quasi-static T0,namely, TQMSP-IW, the subscript indicating the use of both the MSP based TQSch
dy and the IWSRE
The IWSRE derived in [1] is given below:
125.98 0.409dyys-RT0
dy0
0
' = -41.54exp (1a)72.86 0.142
and
. '= + 10
( ' ln )
T
TT
K
0
(1b)
Correlation Coefficient, R =0.8801; Standard Error of Estimate, SEE = 20 C
wher
e the temperatures are in K, YS is in MPa and SIF rate is in MPa√m.s-1. For our purpose, T0dy is
taken as TQSchdy (which is obtained from Eq. (2)) and the corresponding SIF rate is taken as 106
MPa√m.s-1. The estimate of reference temperature from Eq. (1) is referred as TQMSP-IW, indicatingthe application of both TQSch
dy (by the MSP) and the IWSRE.
2.2. TQSchdy by the modified Schindler procedure (MSP)[1,5,7]
For ease of reference, it is appropriate to repeat the procedure for calculating TQSchdy though the
same has been detailed in [1]. This simply involves use of the Schindler relation (Eq. (2)) [14-15]for computing a dynamic initiation J, Jd as modified in [1,5,7]:
163
3V
V
fd
7.33 10(2)
1 1.47 ( )d
n CJ
C
where CV is the total Charpy V-notch (CVN) energy, i. e, the impact energy in J, Jd is in J∙mm-2,n is the power-law exponent and σfd is the dynamic flow stress; Jd is converted to KJd using theusual relation (KJd = √EJd: plane stress equation is used following the older ASTM E-1921procedure as detailed in [1]). The dynamic work-hardening exponent is computed using thefollowing equation (Eq. (3)):
y(log log60)10 (3)n
where σy = σyd is in MPa [1,5,7]. In the absence of actual instrumented impact test load-temperature data, the procedures for computing σyd (dynamic yield stress) and σfd are detailed in[1,5,7].
Since, usually CV-T (temperature) data from impact tests are fitted to a continuous curve(usually a Tanh or exponential curve for data excluding the upper-shelf values or even a bestfitting polynomial), the above procedure would yield a continuous series of KId values extendingfrom the lower-shelf to the upper-shelf or upper cut-off value. The resulting KJd data areexamined for validity by using the ASTM E 1921 equation [4]:
0.5
0
30
ys
JC
EbK
(4)
where, in the case of dynamic tests, the corresponding dynamic quantities are used (for anunprecracked CVN specimen, b0 = 8 mm). Usually for the KJd estimated from the CV values,validity extends up to the upper-shelf or to much larger KJd values than 100 MPm. Then the KJd
values from the lower-shelf (only those above 50 MPm) to the upper cut-off limit are sizecorrected (to 1T equivalence) using the following ASTM E 1921 equation [4]:
0.25
(1T) ( )
1T
20 [ 20] xJC JC x
BK K
B
(5)
Dynamic quantities are used for dynamic tests, and for a CVN specimen Bx = 10 mm and B1T =25.4 mm (1’’).
Then the T-KJd pairs between 80 to 120 MPm are selected and the reference temperature,TQSch
dy, is computed using Wallin’s multi-temperature equation given below [4,1]:
40 min 0
51 1min 0 min 0
exp{0.019( ) ( ) exp{0.019( )}0
[31 77 exp{0.019( )}] [31 77 exp{0.019( )}]
i n ni i Jd i
i ii i
T T K K T T
K T T K T T
(6)
164
where the Kronecker i = 1 for valid data and 0 for non-cleavage or censored data and Kmin = 20MPm (usually for the range of dynamic fracture toughness values considered, validity is alwayssatisfied for the usual structural steels because of elevation in yield stress at faster loading rates).For some low upper-shelf steels, there was saturation at the upper-shelf at or below the 100MPm. The way in which such cases are dealt with is discussed in detail in [1] as also later whiledealing with TSE5A Charpy data. In addition to applying the computed TQSch
dy to Eq. (1), adirect correlation between T0 and TQSch
dy was also derived in [1] as given in the next sub-section(2.2a)
2.2a. TQMSP - T0 – estimate from the direct correlation between T0 and TQSchdy[1]
The final equation for evaluation of TQMSP (i.e., reference temperature estimate based on directcorrelation of TQSch
dy – obtained by MSP – with the measured reference temperature) is as givenbelow:
dyQMSP QSch
0
-795 + 735.8exp(0.0011 ) (7)
Correlation Coefficient, R = 0.9521; Standard Error of Estimate, SEE = 19.9 C
T T
2.2b. TQSchW - T0 – estimate based on both TQMSP and TQMSP-IW [1]
For reasons discussed in [1], it was suggested to take the larger of the two estimates, namely,TQMSP and TQMSP-IW, as the final estimate based on the MSP and IWSRE procedures, anddesignated TQSchW, signifying the use of MSP procedure originating from modifying anexpression due to Schindler and the IWSRE originating from modifying an expression due toWallin.
2.3. Other or older correlations/expressions used
2.3a. Charpy energy (CV) temperature - T0 correlations [7,1]
A conservative estimate (at least for the older steels) is given by [7,1]:
Q41b 41J (8)T T
2.3b. TQSLF from Schindler like fit (SLF) procedure [7,1]
The KIC values (KIC = 100 MPa√m is likely to occur in the range CV = 30-70 J) are estimatedfrom the following relation:
165
VIC
V
ys
'(9)
1 ' ( )
A n CK
CB
where
the work hardening exponent n is determined from Eq. (3) using the static yield stress (σys) at theparticular temperature and A’ and B’ are fit constants. For the sake completeness, the fitconstants given in [7,1] as a function of room temperature yield stress (RT-YS: σys-RT) aretabulated in Table A1 in Appendix-A. Thus, knowing RT-YS, A’ and B’ values for a particularsteel can be estimated from to enable computation of KIC values in the transition region. Then,the temperature corresponding to a KIC = 100 MPa√m is determined as TQSLF, the SLF in thesubscript indicating Schindler like fit (similar to Eq. (2), but here directly correlated to KIC).TQSLF is the equivalent of T0 as discussed in [7,1].
2.3c. TQM2 from the Mean 2 Procedure (M2P) [7,1]
The conclusion arrived in [7] was to take the mean of the two estimates, namely, TQ41b and TQSLF,and designated, TQM2, as a consistent and assuredly, but not excessively, conservative, estimateof reference temperature. The most conservative of the two, namely, TQ41b and TQSLF, wasrecommended for the most conservative estimate.
2.3d. T0 from the correlation based on the brittleness transition temperature, TD [6]
The brittleness transition temperature, TD, represents 100% cleavage fracture (end of the grosselastic region in the load-temperature diagram [7,6]), whereas many of the conventional Charpyenergy indices like T28J, T41J etc. represent different amounts of cleavage or ductile fracture thatvaries depending on the type of steel. Hence, TD is said to be capable of truly reflecting thechange in ductile brittle transition temperature (DBTT) of steels resulting from neutronirradiation or ageing embrittlement. Recently, Chaouadi [6] had correlated TD to T0 and a fit tothe data obtained in [1] gives Eq. (10).
TQBT = 1.5TD + 40 (10)
where TQBT is the T0 equivalent estimated from TD, the subscript indicating ‘brittleness-transition’. It was also shown, in the limited cases examined in [1], that the shift in TD (∆TD)correlates well with irradiation induced shift in T0 and T41J. Chaouadi states that TD-correlation isbetter, especially for the newer and advanced steels [6]. However, the general trend of the resultsin [1] was, because of the problem of scatter and lack of robustness of the TD data for welds andinhomogeneous materials, it was unsuitable for making an assuredly conservative estimate ofreference temperature.
2.4. Final comments on the methodology in Sections 2.1 to 2.3It was shown in [1] that the estimate, TQSchW, is a reliable and conservative one, but not undulyconservative as the RTNDT (except for some high reference temperature steels). Moreover, it wasdemonstrated in [1] that even for a worst steel (that is, a steel having a very high transitiontemperature), even with a less conservative TQSchW estimate (compared to TQM2 or TQBT or known
166
T0), the adoption of the ASTM E-1921 1% MC produced a nearly lower-bounding curve. TQM2
was suggested as a more assuredly conservative estimate. For steels with TQSchdy greater than 60
°C, applicability of TQSchW require further exploration and validation. The TQBT estimate ispromising, provided robust estimates of TD are available, which is not usually the case as wasshown in [1]. The final conclusion in [1] was to recommend TQSchW as the reference temperatureestimate (TQ-est) for steels with TQSch
dy ≤ 60 °C; for steels with TQSchdy > 60 °C, the larger of the
two estimates, namely, TQSchW and TQM2, was recommended as the reference temperatureestimate (TQ-est). This whole procedure, based on [1], is termed IGCAR-procedure (after theinitials of the author’s organization – Indira Gandhi Centre for Atomic Research).
6.2.4. FATT-Master Curve (FATT-MC) approach for lower-bound fracture toughness [B53]
Because of the sharp transition and high scatter in fracture toughness in the transition region, forengineering applications, a master curve (MC) or reference fracture toughness curve is used toobtain a lower bound estimate of fracture toughness. One approach uses the 50% shear fractureappearance temperature (FATT) for locating the master curve. Since, nowadays, MC refers to thereference toughness curve indexed to the reference temperature T0 determined by the recentASTM E 1921 standard, in this paper, the MC based on FATT is referred to as FATT-MC, toavoid any confusion. The FATT-MC for low alloy Cr-Mo steels (especially for 2.25Cr-1Mo typesteel) is given by Eq. (6.2.21):
IC
IC-us
(99 % Conf.Curve) 0.0807 1.962 exp[0.0287 ( FATT)]
(for 233 K ( -FATT)) (6.2.21)
KT
K
T
0.623 0.406 exp[ 0.00286 ( FATT)]
( for 623 K (T-FATT) 233 K)
T
where T is temperature in K. An expression for the FATT-MC for 2.25Cr-1Mo steel is notavailable for dynamic test results. However, based on dynamic fracture toughness results ofA508/A533 steels, a 99% confidence FATT-MC given is given by the following expression –Eq. (6.2.22):
Id
Id-us
0.0545 0.1443 exp(0.0187 ( FATT))
(for 368 K ( -FATT)) (6.2.22)
1.1195-0.1674 exp(0.0022 ( -FATT))
KT
K
T
T
(for ( -FATT) 368 K)T
This can be taken as the FATT master curve for predicting 99% confidence lower bound KId.The KIC-us, the upper-shelf fracture toughness (i.e., the fracture toughness at the upper-shelftemperature, which is defined as the temperature at which CVN impact specimens first
167
experience zero-percent brittle fracture: the corresponding CVN energy is called the CVN-us), isgiven by the modified Rolfe-Novak-Barsom upper-shelf (RNB-US) correlation (Eq. (6.2.23)):
2
IC-US V-US
0.2 0.2
0.6478 ( 0.0098) (6.2.23)K C
where the fracture toughness is in MPam, CV-US is in J and yield strength, σ0.2, is in MPa. Fordynamic fracture toughness values, the corresponding dynamic yield strength should be used.
For obtaining the FATT, apart from direct measurement of PSF (percent shear fracture) onfracture surfaces of the broken Charpy specimens as suggested in ASTM E23 method, severalempirical relations have also been proposed that relate PSF to the various load (P) values thatcan be determined on the P-d traces of the IIT. The most widely used are the following:
f a
m m gyPSF2 {1 } 100% (6.2.24a)
( )
P P
P P P
f a
m m gyPSF3 {1 } 100% (6.2.24b)
0.5 ( )
P P
P P P
where, Pa is the crack arrest load.
RNB-US correlation, Eq. (6.2.23), is not a lower-bound correlation.
6.2.5. Lower-bound estimate for upper-shelf fracture toughness
For Charpy upper-shelf (US) region (i. e., for this purpose defined as the temperature at abovewhich 100% shear fracture appearance is exhibited), a lower-bound estimate of US fracturetoughness is given by Eq. (6.2.25) [B4].
mat V(MPa m) = 0.54 (J) + 55 (6.2.25)K C
The expression is recommended for CV ≥ 60 J.
Recently Wallin [B61, B62] has derived a near-lower-bound correlation for predominantly ductilefracture (especially applicable to Charpy US region) applicable in the temeperature region -100to 300 °C. In fact, this new correlation gives not only the initiation J-value, but also the J-Δatearing resistance curve (J-R curve) as a function of standard in the CV values in the US as afunction of temperature. The correlations are as given below:
168
m -21mm [kJ.m , mm] (6.2.26)J J a
where
1.28 -21mm V-US
20= 0.53 exp( ) [kJ.m , J, C] (6.2.27)
400
TJ C
and
ys0.256V-US
20= 0.133 exp( ) + 0.03 [J, C, MPa] (6.2.28)
2000 4664
Tm C
For estimating JIC, Wallin [B61] provides the following relation:
1
IC IC m
f 1mm
2
f ys
ys
+ 0.2 mm - ( ) (6.2.29)2
150 MPawith flow stress approximated by: . 1 +
J J
J
Flow stress can also be obtained by the procedure described in Section 6.1.2 under ModifiedSchindler Procedure (MSP).
169
6.3. Dynamic Fracture Toughness from Instrumented Drop-Weight Test [B49]
Drop-weight (DW) test for determining TNDT temperature was described in Section3.3.2. Some labs have instrumented drop-weight testers with the tup or strikerinstrumented with strain gauges as shown in Fig. 6.1 for an IIT machine. Figure 3.14showed broken halves of a P-3 specimen tested at below the TNDT (Nil-DuctilityTransition Temperature-NDTT) temperature. Half of the same Fig. is reproduced inFig. 6.8, illustrating the semi-elliptical thumb-nail crack formed by the crack-starterweld-bead. Based on available SIF solutions for such a crack, the equation forcalculation of KId is given on the left column of Fig. 6.8; right column illustrates thedistinctitive load-time traces obtained above and below NDTT. For brittle fractures,the measured fracture load, PF, and crack depth, a, KId can be estimated.
Another useful empirical expression is that (KId/σyd) at TNDT = 0.074√m.
170
Fig. 6.8. Dynamic fracture toughness measurement from an Instrumented DW Test [B49].
Illustrative Example 6.1:
Complete IIT data for a service exposed 2.25Cr-1Mo Steel [B53]
Table Ex.6.1.1. Test temperature, impactenergy, primary loads and PSF data frominstrumented impact tests on service-exposedCVN specimens of 2.25Cr-1Mo steel
T(K)
Cv
(J)Pgy
(kN)Pm
(kN)Pf
(kN)Pa
(kN)Type of
P-dtrace
MeasuredPSF
*
213 3 11.6 11.6 0 Type-I 0
213 5 11.6 11.6 0 Type-I 0
243 10 13.6 13.6 0 Type-I 0
243 11 13.7 13.7 0 Type-I 9.8
243 31 12.7 15.7 15.5 0 Type-I/II 0
258 45.5 11.5 16.0 16.0 0 Type-II 9.5
258 47 11.7 16.0 16.0 0 Type-II 6.2
258 78 11.7 16.2 15.0 0 Type-III 21.9
171
273 110 11.5 16.2 13.8 0 Type-III 26.6
273 78.5 11.5 16.0 15.5 0 Type-III 17.7
273 108 11.5 16.0 14.0 0 Type-III 27.1
283 105 11.0 16.0 14.2 0 Type-III 27.4
283 125 11.0 16.0 12.5 3 Type-III 41.0
283 125 11.3 15.8 13.2 0 Type-III 37.4
295.5 160 10.0 15.1 11.0 5.7 Type-III 56.7
295.5 160 10.2 15.3 10.8 2 Type-III 54.4
323 217 9.0 14.9 N.A N.A Type-III 68.2
323 214 9.0 14.8 9.0 2.8 Type-III 76.6
* PSF-Percent Shear Fracure Appearance
Table Ex.6.1.2 Various transitiontemperatures and microcleavage fracturestress obtained from instrumented Charpytest for service-exposed 2.25Cr-1Mo steel
TD
(0C)T28J
(0C)T68J
(0C)T41J
(0C)T0-Sch
dy
(0C)T for 50%
PSF(FATT)(0C)
f
(MPa)
-35 -28 -9 -21 4 22 1561
NOTE: The various Charpy energy indices are from the mean curve fitted to the data.Only for RTNDT smooth curve through the lowest values is taken.
Table Ex. 6.1.3 Displacements and energy values correspondingto different crack initiation criteria determined from load-displacementtraces of instrumented impact test for service-exposed 2.25Cr-1Mo steel
T
(K)
d'(mm)
d'pl
(mm)di
(mm)di-pl
(mm)E'
(J)
E'pl
(J)Ei
(J)Ei-pl
(J)
213
PPRS1
(0C)PPRS2
(0C)T0-PRS1
(0C)T0-PRS2
(0C)T0-SSP
(0C)T0-28
(0C)T0-41
(0C)T0-M5
(0C)-114.3 -18.9 -69.9 -55.9 -66 -47.5 -54.6 -58.8
172
213
243
243
243 1.34 0.71 2.19 1.50 13.1 9.0 25.4 20.3
258 1.64 1.00 3.30 2.57 16.6 12.5 41.3 35.8
258 1.42 0.79 2.50 2.37 13.8 9.6 36.8 31.2
258 1.68 1.00 3.15 2.38 17.6 12.8 39.8 33.5
273 1.78 1.07 3.15 2.34 18.5 13.5 39.0 32.5
273 1.50 0.91 3.19 2.50 14.5 10.7 38.9 33.7
273 3.81 2.88 3.94 3.01 48.3 40.9 50.4 42.9
283 1.60 1.00 3.87 3.15 15.5 11.6 48.3 42.8
283 1.72 1.03 2.80 2.03 17.3 12.7 33.2 26.9
283 1.84 1.06 3.18 2.30 18.4 13.0 37.8 31.0
295.5 1.86 0.86 3.29 2.13 15.7 9.5 34.8 26.5
295.5 1.99 0.81 3.48 2.12 16.7 9.45 37.6 27.7323 1.64 1.00 4.64 3.83 14.0 10.3 55.6 49.7
323 1.95 0.82 3.41 2.08 15.1 8.65 34.4 25.2
Notes: (i) “`” denotes values at (Pm + Pgy)/2 ; (ii) “pl” denotes plasticcomponent; (iii) “i” denotes crack initiation as determined by modifiedcompliance change rate/key curve (KC) procedure (see [B53]).
Table Ex.6.1.4. Various dynamic fracture toughness estimates for service-exposed 2.25Cr-1Moferritic steel
T(K)
KId
(Pf)KJi KJ
(CTODi)
KJ
(CTOD')
KJ' KJSch-
0.1a
KJSch-min Lower-boundKId (IIT-LB)
yd
(MPa)fd
(MPa)Sch-H Sch-p KC
exponent=m
213 43.6 43.6
213 43.6 43.6
243 51.2 51.2
243 51.4 51.4
243 58.2 168 181.9 96.2 112 147.5 62.6 58.2 593.3 663.41 0.325 0.453 0.0844
258 223 226.3 108.6 131.9 190.4 79.8 79.8 537.3 642.85 0.518 0.433 0.09
258 208.2 219.3 97.1 115.9 190.1 82.0 82.0 546.6 647.76 0.523 0.439 0.0918
258 216 220.3 109.5 133.5 201.9 115.1 115.1 548.95 654.06 0.764 0.551 0.1005
273 212.5 216.4 112 136.9 242.9 139.7 139.7 537.3 647.06 0.971 0.495 0.0956
273 216.5 223.3 103.3 122.4 218.6 125.5 125.5 537.3 642.38 0.751 0.475 0.1181
273 244.3 245.3 184.1 238.5 239.8 138.8 138.8 537.3 643.08 0.948 0.496 0.0965
283 243.8 245.1 105.9 127.2 240.1 152.9 152.9 513.9 630.70 0.940 0.491 0.1208
283 193.5 196.9 108.0 132.8 240.4 170.5 170.5 513.9 630.70 1.004 0.518 0.1184
283 207.6 212.3 110.7 134.8 238.5 159.6 159.6 527.9 633.04 0.994 0.521 0.1039
295.5 191.9 192.2 94.1 115.4 225.9 229.4 192 467.2 587.96 1.018 0.579 0.1417
295.5 196.4 193.9 92.2 114.5 227.1 208.0 193.9 476.9 596.59 1.025 0.577 0.1176
173
323 262.7 244.6 95.8 119.7 241.4 294.7 244.6 420.5 558.99 1.241 0.607 0.1561
323 187.1 180.8 87.2 109.6 269.4 291.6 180 (*) 422.5 557.12 1.573 0.614 0.1528
348 234 (RNB)
373 234 (RNB)
Note 1: All KId estimates reported are in MPam; Note 2: “i’ signifies crack initiation point detected by modifiedcompliance change rate/KC (key-curve) methods; Note 3: “`” (prime) signifies crack initiation taken at (Pm + Pgy)/2(used only for Type-II traces which are indicated in bold); Note 4: fd – flow-stress estimated from the dynamicyield stress formula using (Pgy + Pm)/2 for load; Note 5: RNB-Rolfe-Novak-Barsom correlation; Note 6: Theunderlined value in the lower-bound toughness column out of trend and neglected.
Fig. Ex.6.1.1
174
Fig. EX.6.1.2 (PF appears to be PI)
Fig. Ex.6.1.3
Fig. 2 Load-temperature diagram for service exposed2.25Cr-1Mo steel from instrumented impact test
Test Temperature, T/K
200 220 240 260 280 300 320 340
Lo
ad
,P
/kN
8
10
12
14
16
18
Pm
P f
Pgy
T-D
/238
K
13 kN
Fit to Pm data above Q
Fit to P f data above Q
Fit to Pgy data
Fit to Pm=P f
data pointsupto point Q
Q
Fig. 4 Measured PSF (ASTM E23) and calculated PSF3plotted against test temperature
Test Temperature, T/K
200 220 240 260 280 300 320 340
Estim
ate
dP
SF
0
20
40
60
80
100
Measured PSF (ASTM E23)
Fit to measured data
PSF3
Fit to PSF3 data
Mateial: Service-exposed2.25Cr-1Mo Steel
50% PSF ~ 293-294K
175
COMMENTS: For the present material the PSF3 formula seems to predict the percentage fracture
appearance correctly.
Fig. Ex.6.1.4. Fracture toughness estimations from older Charpycorrelations compared with MC prediction
Temperature/ 0C
-200 -150 -100 -50 0 50 100 150
KIC
/MP
a.m
0.5
0
50
100
150
200
250
300
RNB-Correlation
SC-Correlation
RLB-Correlation
BRS-Procedure
1in MC data: T0-M5= -58.8 C
Lower-Bound-IIT-KId-data
Marandez-Sanz
FATT-MC-KIC
FATT-MC-KId
E 399 K-IC from 1in MC data
USMC KJC data from T0-M5
US-LB-K-IC using CV
Wallin-LB-US-KJC from CV
KIC=12*Cv^0.5-LS lower-bound
Material: Service Exposed2.25Cr-1Mo Steel-21IGC
COMMENTS:
Lower-Bound (LB) IIT (Lower bound KId values given in Table Ex. 6.1.4) – diamonds withcentre star - should be compared with the FATT-KId-MC (right extreme curve), while the FATT-KIC-MC bears comparison with all the fracture toughness estimates (KIC) using older correlations.
Older correlations, compared to the MC based on T0-M5, seem to be extremely conservative,especially in the lower-shelf and lower transition regions.
BRS (Barsom-Rolfe dynamic-static shift) does not seem to applicable to this steel as its valuesvery larger (extreme left).
176
Wallin’s US lower-bound correlation seems to applicable to lower-shelf also and gives values inagreement with older Charpy correlations.
Fig. Ex. 6.1.5. Various lower-bound (LB) and MC curve estimatesof fracture toughness compared with ASME KIC-KIR curves
T/ 0C
-100 -50 0 50 100 150
KIC
/M
Pa.m
-0.5
0
100
200
300
400
E 399-KIC from 1in MC data
FATT-MC-KIC
FATT-=MC-KId
USMC from T0-M5
US-LB-KIC from CV
Wallin-LB-US correlations
KIC = 12*CV^0.5, LB-Lower-Shelf
ASME LB KIC: RTNDT = -350
C
ASME KIR: RTNDT = -350
C
ASME LB KIC: RTT0 = -39.30C
ASME KIR: RTT0 = -39.30C
1in MC: T0 = - 58.8 C
IIT-KId-LB
COMMENTS:
Material: Service Exposed 2.25Cr-!Mo Steel as shown above ASME KIR curves based on RTNDT or RTT0 are close together, validating the concept of
the RTT0. Lower-Bound IIT KId values higher than ASME KIR curves New USFTMC (USMC) seems to be reasonable. FATT -KId-MC is very conservative, even compared to the KIR curve.
177
Illustrative Example 6.2:
Comparison of actual fracture toughness data for a modified 403SS (12Cr Martensitic stainlesssteel with RT YS = 677 MPa: W. A. Logsdon and J. A. Begley. Upper shelf temperature dependence offracture toughness for four low to intermediate strength ferritic steels. Engng. Fracture Mechanics, 9(1997) pp. 461-
470) steel with those predicted by Charpy Correlations and MC based on T0-M5 (Measured T0 = -28 C; T0M5 = -4.2 C)
Fig. Ex.6.2.1. Comparison of actual KIC with those estimated using the
older Charpy Correlations, the MC based on T0-M5 and ASME KIR-KIC curves
Temperature/ 0C
-200 0 200 400
KIC
/MP
a.m
0.5
0
50
100
150
200
250
300
0
50
100
150
200
250
300
RNB-Correlation
SC-Correlation
RLB-Correlation
BRS-Procedure-KIC
MC: T0-M5 = -4.20
C
Test KIC Data:1T Size Corrected
Measured ASTM E 399 Valid KIC
USMC-KIC
LB-US-KIC from CV
Wallins-LB-US KIC
ASME RTNDT - KIC: RTNDT = 200
C
ASME RTNDT - KIR: RTNDT = 200
C
ASME RTT0 - KIC: RTT0 = 150
C
ASME RTT0 - KIR: RTT0 = 150
C
COMMENTS: KIC from older Charpy correlations is extremely conservative comared to actual or T0M5-MC data. Conservatism of the T0M5 is obvious. ASME curves follow older Charpy correlation data. US-MC concept seems to be attractive.
178
Illustrative Example 6.3:
Comparison of KId from IIT and Drop-Weight NDT with RTNDT based KIR Curve for a 403 SSmartensitic Stainless Steel – 403SS-IGC [B49]
Fig. Ex.6.3.1
COMMENTS:
RTNDT = 305 K and DW NDTT (TNDT) = 298 K TD = 234 K and σf = 2143 MPa KId from Instrumented DW tests (see Section 6.3 and Fig. 6.7) agree with the KId from IITs. ASME KIR is conservative for the present steel in the lower-shelf and lower transition regions.
179
APPENDIX
Table A1. Summary of lower shelf and lower transition correlationsReference Eq. No. Equation YS
range-MPa
Charpyenergyrange-J
Equation-UNITS
Chaudhuri, S.K. etal.Int. J. of Pressure Vesselsand Piping, 22 (1986),pp.23-30.
(A1-1) logδc = 1.14logCV – 2.33(HAZ, ductile fracture)logδc = 1.3logCV – 2.58(HAZ, brittle fracturelogδc = 1.06logCV – 2.24(Weld metal)
520 40-63
24-35
18-40
mm,J
Girenko, V.S. andLyndin, V.P. AutomaticWelding, 13-19September, 1985.
(A1-2) KIC = 2.5CV0.5 200-
17002-150 MPa√m, J
Imai, J., et al. Proc."Transport and Storage ofLPG and LNG", Brugge,7-10 May 1984, pp. 91-100
(A1-3)0.5IC
V
Y
2.5K
C
NotKnown
5-215 MPa√m, J, MPa
Logan, J.G. andCrossland, BPractical Applications ofFracture Mechanics toPressure VesselTechnology, Institutionof Mechanical Engineers(London), 1971, pp 148-155.
(A1-4) KIC = 20.3CV0.5 820-
14207-37 MPa√m, J
Sailors, R.H. and Corten,H.T ASTM-STP-514,1973, pp 164-191.
(A1-5) KIC = 14.6CV0.5 410-
8157-70 MPa√m, J
Barsom, J.M. and Rolfe,S.T. ASTM STP 466,1970, pp. 281-302 andRolfe, S.T. and Novak,S.T ASTM-STP-463,1970, pp. 124-159.
(A1-6)
(A1-7)
KIC2 = 45.1CV
1.5
2
IC V
Y Y
0.64 0.01K C
270-1700
270-1700
4-82
31-121
MPa√m, J, MPa
MPa√m, J, MPa
EXXON Chemical:'Fitness for ServiceGuide', Part B, Section 5,Material Property Data,Rev. 4, February1995.
(A1-7)V
IC
650
1000
C EK
(claimed to be alwaysconservative)
Notknown
Notknown
MPa√m, J, MPa
180
Table A2. Summary of upper transition and upper shelf correlationsReference Eq. No. Equation YS
range-MPa
Charpyenergyrange-J
Equation-UNITS
Ito, T. et al. Study ofBrittle Fracture Initiationfrom Surface Notch inWelded FusionLine, IIW Doc. No. X-707-73, 1973.
(A2-1)2 5 V
IC
Y
2.9 eC
K
570-840
5-160 MPa√m, J, MPa
Norris, D.M. et al.Fracture Mechanics 13th
Conference, ASTM-STP-743, R. Roberts (Ed.),ASTM, 1981, pp. 207-217.
(A2-2)Y
IC V
[ 1600]
1300J C
447-1696
22-192 N/mm, J,MPa√m
Priest, A.H. et al.Effects of Radiation onMaterials, 11thConference, ASTM-STP-782, H.R. Brager and J.S.Perrin (Eds.),ASTM, 1982, pp. 475-491.
(A2-3) 2
IC IC
Y Y
1662 0.011K K
2
IC V
Y Y
0.65 0.0064K C
190-320(ParentPlate)
290-450(Weldmetal)
5-120 MPa√m, J, MPa
Thorby, P.N. andFerguson, W.G. Mat. Sci.and Engng,22, 1976, pp.177-184.
(A2-4) KIC = 16.9CV0.534 400-
60013-85 MPa√m, J
Witt, F.J.Int. J. Pres. Ves.and Piping, 11, 1983, pp.47- 63.
(A2-5) KIC = 2.78CV – 3.93
KIC = (0.068CV + 9.9)2
414-648
47-203 MPa√m, J
British Standard BSIPD6493:1991: Guidanceon methods for assessingthe acceptability of flawsin fusion weldedstructures.
(A2-6) KIC2 = 0.54CV + 55 <480 0-250 MPa√m, J
181
Table A3. Summary of correlations involving temperature shiftsReference Eq.
No.
Equation YSrange-MPa
Charpyenergyrange-J
Equation-UNITS
PVRCGroup onToughnessRequirements, WRCBulletin175, August1972.
(A3-1)
IC NDT= 1.333 exp[0.0261( - 89)] 29K T T <621 NA MPa√m, ºC
BritishStandard BSIPD6493:1991:Guidance onmethods forassessing theacceptabilityof flaws infusion weldedstructures.
(A3-2)
As above, but with TNDT = T40J <480 NA MPa√m, ºC
Barsom, J.M. Engng.Fract.Mech., 7,1975, pp.605-618.
(A3-3)
A3-4)
KIC2 = 105CV
ΔT = 119 – 0.12σY (σY < 965 MPa)
250-1700
3-61 MPa√m, J,
ºC,MPa
Matsumoto,K. et al.Proc.3rd Int.Symp. ofJapanWeldingSociety.Criteria forPreventingServiceFailures inWeldedStructures',26-28September,
(A3-5)
KIC = 1.615exp[0.038(T – ATT + 140)] + 31
ATT = arrest transition temperature
Notspecified
NA MPa√m, ºC
182
1978,Tokyo,JWS, pp.167-172.Oda, I. Engng.Fract. Mech.,29 6, 1988,pp. 663-671.
(A3-6)
(A3-7)
(A3-8)
VC
Y
= 0.0344ln 256C
(brittle)
VC
Y
= 1.8l 0.1C
(ductile)
ΔT = FATT - Tf
276-834
3-50
50-200
mm, J,MPa
Sanz, G.Revue deMetallurgie,CIT, Juillet,1980.
(A3-9)
TK100MPa√m = 1.37 TCV28J + 9
KIC = 19 CV0.5
274-820
5-50 MPa√m, J
Wallin,K.JernkontoretsForskning, ReportNo. T040-29,December1994
(A3-10)
TK100MPa√m = TCV28J - 18300-1500
NA ºC
Ibid (A3-11)
K0 = 31 + {77 exp(0.019[T0])} 300-1500
T +20 to-109ºC
MPa√m, ºC
Ibid (A3-12)
0.25
IC 28J
25 1= 20 + {11 + 77 exp(0.019[ - + 18]} lnK T T
B P
300-1500
NA MPa√m,ºCmmm
183
Test Standards and Hand Books (Reference – T)
NOTE: In the TEXT References to material under this heading are preceded by Letter T: for exampleRef. 1 is referred as T1 in the TEXT.
1. “ESIS Instrumented Charpy V-Notch Standard,” Proposed Standard Method for the Instrumented Charpy-V Impact Tests on Metallic Materials, Draft 10, European Structural Integrity Society, ESIS (Jan 14, 1994).
2. AMERICAN SOCIETY OF MECHANICAL ENGINEERS, “Use of FractureToughness Test Data toEstablish Reference Temperature for Pressure Retaining Materials Other than Bolting for Class 1 Vessels”,Section III, Division 1, ASME Boiler and Pressure Vessel Code: An American National Standard, CodeCase N-631, ASME, New York (1999).
3. AMERICAN SOCIETY OF MECHANICAL ENGINEERS, “Use of FractureToughness Test Data toEstablish Reference Temperature for Pressure Retaining Materials”, Section XI, Division 1, ASME Boilerand Pressure Vessel Code Case N-629, ASME, New York (1999).
4. ASTM E 1152. “Standard Method for Determining J-R Curves.” Annual Book of ASTM Standards, Vol3.01, ASTM (1999).
5. ASTM E 1290-93: “Standard Test Method for Crack Tip Opening Displacement (CTOD) FractureToughness Measurement.” Annual Book of ASTM Standards, Vol 03.01, ASTM (1999) pp. 831–840.
6. ASTM E 1457: “Standard Test Method for Measurement of Creep Crack Growth Rates in Metals.” AnnualBook of ASTM Standards, Vol 03. 01(1998), American Society for Testing and Materials.
7. ASTM E 1820-99: “Standard Test Method for Measurement of Fracture Toughness,” Annual Book ofASTM Standards, Vol 03.01, ASTM (1999) pp. 972–1005.
8. ASTM E 1921-02: “Standard Test Method for Determination of Reference Temperature, To, for FerriticSteels in the Transition Range.” Annual Book of ASTM Standards, Vol 3.01, ASTM (2002).
184
9. ASTM E 23-98: “Standard Test Methods for Notched Bar Impact Testing of Metallic Materials.” AnnualBook of ASTM Standards, Vol 03.01, ASTM (1999) pp. 138–162.
10. ASTM E 399-90: “Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials,”Annual Book of ASTM Standards, Vol 03.01, ASTM (1999) pp. 422–452.
11. ASTM E 561: “Standard Practice for R-Curve Determination.” Annual Book of ASTM Standards, Vol 3.01,ASTM (1999).
12. ASTM E 647-95a: “Standard Test Method for Measurement of Fatigue Crack Growth Rates.” AnnualBook of ASTM Standards, Vol 03.01, ASTM (1999) pp. 577–613.
13. ASTM E 813. “Standard Method for JIC, a Measure of Fracture Toughness.” Annual Book of ASTMStandards, Vol 3.01, ASTM (1999).
14. ASTM E1221. “Standard Method for Determining Plane-Strain Crack Arrest Toughness, KIa, of FerriticSteels.” Annual Book of ASTM Standards, Vol 03.01 (1999) ASTM.
15. ASTM E208–87a: “Test method for conducting drop-weight test to determine nil-ductility transitiontemperature of ferritic steels.” Annual Book of ASTM Standards, Vol 03. 01(1990), p. 360, ASTM,Philadelphia, PA.
16. BS 7448: “Fracture mechanics toughness tests — Part 1: Method for determination of KIc, critical CTODand critical J values of metallic materials “, The British Standard Institution (BSI) (1991) London.
17. BS5762 - 1979: “Methods for Crack Opening Displacement (COD) Testing.” The British StandardsInstitution (1979).
18. Mechanical Testing and Evaluation (2000). Volume 8 of the ASM Handbook, ASM International,Materials Park, OH.
19. Design and Construction Rules for Mechanical Components of PWR Nuclear Islands, RCC-M Code (1988)– French Design Code.
20. TECHNICAL REPORTS SERIES No. 429: GUIDELINES FOR APPLICATION OF THE MASTERCURVE APPROACH TO REACTOR PRESSURE VESSEL INTEGRITY IN NUCLEAR POWERPLANTS. INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 2005.
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