Appendix Linear Elastic Fracture Mechanics. Compendium of Stress Intensity Factors Solutions A.1 Introduction Linear elastic fracture mechanics (LEFM) studies the behavior of materials, work pieces and structures in which cracks are present. Actually, the term fracture refers and identifies those failures caused by the presence of a crack. A crack is defined as an extremely sharp structural discontinuity characterized by a root radius no larger than 0.005 mm. Structural discontinuity having root radii larger than 0.005 are classified as notches. In the presence of a notch LEFM is no longer valid. However, LEFM can still be applied supposing that, conservatively, sooner or later at the tip of a notch a crack will initiate by fatigue or corrosion, in particular. The fundamental result of LEFM is that ahead of a crack an elastic stress field exists that is always self-similar (see Fig. 10.4). Its analytical expression for an infinite body containing a through wall crack of length 2a under remote loading, known as Griffith crack, schematized in Fig. A.1 is given by the G. Irwin expression (see Eq. 10.4) r ij ¼ K I ffiffiffiffiffiffiffi 2pr p f ij ðhÞ K I ¼ r ffiffiffiffiffi pa p ðA:1Þ in which f(h) is a non-dimensional factor that depends on the angle h, r is the distance from the crack tip and a the semi-crack length, as schematized in Fig. A.2. The elastic stress field presents a singularity of the type 1/Hr. Its amplitude is given by the Irwin stress intensity factor K I . The subscript I indicates that the stress intensity factor K I refers to the first of the three fundamentals mode of aperture of a crack, schematized in Fig. A.3. Any other mode can be considered as a combination of two or more fundamental modes. For any real case in which the geometry is not infinite, loaded under whatever conditions and the crack is not central, the expression of the stress intensity factor K I is always given by the second of Eq. (A.1) with the addition of a multiplying factor f(a) P. P. Milella, Fatigue and Corrosion in Metals, DOI: 10.1007/978-88-470-2336-9, Ó Springer-Verlag Italia 2013 807
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AppendixLinear Elastic Fracture Mechanics.Compendium of Stress Intensity FactorsSolutions
A.1 Introduction
Linear elastic fracture mechanics (LEFM) studies the behavior of materials, workpieces and structures in which cracks are present. Actually, the term fracture refersand identifies those failures caused by the presence of a crack. A crack is definedas an extremely sharp structural discontinuity characterized by a root radius nolarger than 0.005 mm. Structural discontinuity having root radii larger than 0.005are classified as notches. In the presence of a notch LEFM is no longer valid.However, LEFM can still be applied supposing that, conservatively, sooner or laterat the tip of a notch a crack will initiate by fatigue or corrosion, in particular. Thefundamental result of LEFM is that ahead of a crack an elastic stress field existsthat is always self-similar (see Fig. 10.4). Its analytical expression for an infinitebody containing a through wall crack of length 2a under remote loading, known asGriffith crack, schematized in Fig. A.1 is given by the G. Irwin expression (see Eq.10.4)
rij ¼KIffiffiffiffiffiffiffiffi
2prp fijðhÞ
KI ¼ rffiffiffiffiffiffi
pap ðA:1Þ
in which f(h) is a non-dimensional factor that depends on the angle h, r is thedistance from the crack tip and a the semi-crack length, as schematized inFig. A.2. The elastic stress field presents a singularity of the type 1/Hr. Itsamplitude is given by the Irwin stress intensity factor KI.
The subscript I indicates that the stress intensity factor KI refers to the first ofthe three fundamentals mode of aperture of a crack, schematized in Fig. A.3. Anyother mode can be considered as a combination of two or more fundamentalmodes. For any real case in which the geometry is not infinite, loaded underwhatever conditions and the crack is not central, the expression of the stressintensity factor KI is always given by the second of Eq. (A.1) with the addition of amultiplying factor f(a)
P. P. Milella, Fatigue and Corrosion in Metals,DOI: 10.1007/978-88-470-2336-9, � Springer-Verlag Italia 2013
Fig. A.1 Central through-wall crack of length 2a in aninfinite body remote loaded
x
y
z
r
σy
σx
σz
θcrack front
Fig. A.2 Polar coordinatesahead of a crack tip
x
z
y
x
z
y
MODE I MODE II MODE III
KI KII KIII
x
z
y
Fig. A.3 Schematic of the three modes of aperture of a crack
808 Appendix: Linear Elastic Fracture Mechanics
KI ¼ rffiffiffiffiffiffiffi
pa�p
f að Þ ðA:2Þ
The non-dimensional function f(a) depends uniquely on the geometry of thesystem and length a of the crack. One of the main objective of LEFM is theassessment of the function f(a) relative to the particular geometry under study.Once the f(a) function is known, the relative KI can be inferred via Eq. (A.2). OnKI depends either the FCGR, da/dN, through the Paris-Erdogan power law (10.10),or the occurrence of SCC when the applied KI becomes equal to the thresholdstress intensity factor KIscc. Also brittle fracture occurs when the applied KI
reaches the critical value of the toughness of the material KIc. Several solutionsrelative to simple, yet important geometries will be given in the next sections.
A.3 Fracture Mechanics Specimens with Increasing KI
Plain Uniaxial Specimens
Same simple geometries will be considered first. These are plane geometriesbelonging to the category of KI-increasing specimens with increasing crack length(see Fig. 10.7). The specimens considered in this section where the first to be usedin fracture mechanics applications and are shown in Fig. A.4. They have side or
2W
2a
4WW
/6
F
8 W
F
0.3-0.4 W
CC(T)
(a)
W/3
DEC(T) SEC(T)
(b) (c)
a a
0.15-0.2 W
F
2W
2a
2W
0.3-0.4 W
F
F
W/3
F
W/3
Fig. A.4 Geometries ofstandard fracture mechanicsplane specimens underremote force F
central notches fatigued to develop a crack at the tip and are subjected to a uniaxialstress state. The first geometry, Fig. A.4a, is the central crack panel CC(T) (the Tstands for traction). The second, Fig. A.4b, is the double edge crack panel orDEC(T). The third one, Fig. A.4c, is the single edge crack panel or SEC(T). Foreach geometry the relative f(a) or h(a) function are given, depending on whetherthe KI expression is given in terms of force F, Fig. A.4, or stress r, Fig. A.5 actingon the extremities of the specimen. Figure A.4 and A.5 also indicates the standarddimensions of the specimens.
For all the geometries considered the general expression of KI is
KI ¼F
sffiffiffiffi
wp � h a
w
� �
KI ¼ rffiffiffiffiffiffi
pap
� f a
w
� �
:
ðA:3Þ
The f(a) and h(a) functions, that in the specific geometries considered are f(a/w)and h(a/w) functions, are:
2W
2a
0.3-0.4 W
CC(T)
(a)
DEC(T) SEC(T)
(b) (c)
a a
0.15-0.2 W
2W
2a
2W
0.3-0.4 W
σ σ σ
σ σ σ
Fig. A.5 Geometries of standard fracture mechanics plane specimens under remote stress r
.Graphically the two functions are shown in Fig. A.8.
Fig. A.6 Diagram of the f(a/w) and h(a/w) functions for a CC(T) panel
Appendix: Linear Elastic Fracture Mechanics 811
For this last geometries the expression are valid only for a \ 0.6w because forlarger a value a non-negligible bending component must be taken intoconsideration.
Biaxial Specimens
Soon after the introduction of CC(T) and DEC(T) panels the attention of fracturemechanics researchers focused on a particular specimen that could introduce acertain degree of biaxial stresses. This was due to the fact that in the early 060 s, inparticular, concerns were arising about the triaxial or plain strain state on thetoughness of materials. A material that under uniaxial stress state exhibited largeductility and, therefore, had an apparent high toughness could become ratherbrittle under a plain strain state condition. Concerns were arising from the high
Fig. A.7 Diagram of the f(a/w) and h(a/w) functions for a DEC(T) panel
Fig. A.8 Diagram of the f(a/w) and h(a/w) functions for a SEC(T) panel
812 Appendix: Linear Elastic Fracture Mechanics
fatigue pre-crack
WOL Type X
WOL Type T
fatigue pre-crack
dialgage
Compact C(T)
fatigue pre-crack
side groove
(c)
(b)
(a)
Fig. A.9 Fracture mechanics specimens type WOL-X, WOL-T and compact C(T)
Appendix: Linear Elastic Fracture Mechanics 813
pressure industries and, in particular, from the nuclear industry that built heavysection steel pressure vessels that were subjected to high pressure and biaxial stressstate. It was an engineer of the nuclear industry, Manjoine [4] that at WestinghouseNuclear introduced the historical WOL (wedge opening load) type X specimenshown in Fig. A.9a.
The reduced dimensions of these specimens were due to the limited spaceavailable in a nuclear power reactor in which they were introduced close to corefor neutron embrittlement surveillance programs. Type X specimen is carrying a Vshaped central notch extending at least to 40 % of the total length W fatigued todevelop a sharp crack (q B 0,005 mm). The specimen is loaded by a pin-clevissystem pulling the lower side of the hole while the upper face of the specimen isscrew-fixed to the loading cell. The stress state on the crack plane is equivalent tothe biaxial stress state existing in the wall of a pressure vessel. As shown inFig. A.10, on the A–A section of area S1 is acting a stress whose maximum valuer1 at the crack tip is given by the moment M1 = F�a plus traction F
r1 ¼ FaW1þ F
S1ðA:7Þ
On the B–B section of area S2 the maximum bending stress r2
r2 ¼ FaW2
ðA:8Þ
with W1 and W2 being the strength moduli of sections S1 and S2, respectively.Westinghouse gave a significant impulse to the development of this typeintroducing the WOL type T specimen of Fig. A.10b [5], which was larger and
A
A
A
B
B B
F
F
pD2t
t
σ2
σ1
pressure p
pD4t
S2
S1
a
BA
A
Fig. A.10 Analogy between the biaxial stress state in a pressure vessel wall and that existingahead of the crack tip in a WOL-X or WOL-T or compact C(T)
814 Appendix: Linear Elastic Fracture Mechanics
thicker so to overcome the excessive bending experienced by type X specimen.WOL Type T specimen is wedge bolt-loaded as shown in Fig. 15.19. Type X
specimen is no longer used. The development process continued till the intro-duction of the third type of specimen worldwide known as the Compact C(T)specimen of Fig. A.9c. Today the C(T) type specimen is the most used one infracture mechanics applications. However, WOL Type T specimen is still used inSCC applications for measuring the stress intensity threshold KIscc (see Sect. 15.5).The h(a/w) and V(a/w) functions for the calculation of the corresponding K and Dfor types WOL-T and C(T) specimens are listed in the following [6]. The symbolsrefer to Fig. A.9b and c for types WOL-T and C(T) specimens, respectively.D represents the crack mouth opening displacement (CMOD), i.e., the displace-ment measured at the notch opening on the specimen surface, as shown inFig. A.9.
COMPACT SPECIMEN C(T)
KI ¼F
sffiffiffiffi
wp h
a
w
� �
ha
w
� �
¼2þ a
w
� �
0:886þ 4:64 aw
� �
� 13:32 aw
� �2þ14:72 aw
� �3�5:6 aw
� �4h i
1� aw
� � 3=2
D ¼ F
E0sV
a
w
� �
Va
w
� �
¼ 1þ 0:25aw
� �
" #
1þ aw
� �
1� aw
� �
" #2
1:6137þ 12:678a
w
� �
� 14:231a
w
� �2�16:61
a
w
� �3þ35:05
a
w
� �4�14:494
a
w
� �5� �
:
ðA:9Þ
Functions h(a/w) and V(a/w) are graphically shown in Figs. A.11 and A.12,respectively (Fig. A.13, A.14).
Fig. A.11 Diagram of the function h(a/w) for compact C(T) specimen
Fig. A.12 Diagram of the function V(a/w) for compact C(T) specimen
Fig. A.13 Diagram of the function h(a/w) for WOL type T specimen
Fig. A.14 Diagram of the function V(a/w) for WOL type T specimen
816 Appendix: Linear Elastic Fracture Mechanics
KI ¼F
sffiffiffi
ap h
a
w
� �
ha
w
� �
¼ a
w
� �
30:96� 195:8a
w
� �
þ 730:6a
w
� �2�1186:3
a
w
� �3þ754:6
a
w
� �4� �
D ¼ F
E0sV
a
w
� �
Va
w
� �
¼ exp 4:495� 16:13a
w
� �
þ 63:838a
w
� �2�89:125
a
w
� �3þ46:815
a
w
� �4� �
ðA:10Þ
A.2.3 High Triaxiality Specimens
A particular citation deserve cylindrical specimens containing a circumferentialcrack and those in three point bending since they have a very high triaxiality, evenwith smaller thickness. This are the round notch bar in traction or RNB(T) ofFig. A.15 and the three point bending TP(B) or single edge crack in bending,SE(B) of Fig. A.17, respectively. Their calibration function h(a/w) are given in thefollowing [7] (Fig. A.16, A.18).
KI ¼FL
sw3=2h
a
w
� �
ha
w
� �
¼ 3 � a
w
� �1=21:99� aw
� �
1� aw
� �
2:15� 3:93 � awþ 2:7 � a
w
� �2h i
2 � 1þ 2 � aw
� �
1� aw
� �3=2
ðA:12Þ
V ¼ F � L
E0sw
� �
qa
w
� �
qa
w
� �
¼ 6 � a
w
� �
0:76� 2:28 � a
w
� �
þ 3:87 � a
w
� �2�2:04 � a
w
� �3þ 0:66
1� aw
� �2
" #
ðA:13Þ
A.3 Three-Dimensional Surface and Internal Cracks
Figures A.19, A.20, A.21 and A.22, A.23.
Appendix: Linear Elastic Fracture Mechanics 817
2R
b a
F
F
(A.11)
RNB(T) SPECIMEN
Fig. A.15 Cylindrical specimen with circumferential crack RNB(T)
Fig. A.16 Diagram of the function h(a/w) for RNB(T) specimen
F
TP(B) OR SE(B) SPECIMEN
t
L (=4W)
Wa
δ
Fig. A.17 Schematic ofSE(B) specimen
818 Appendix: Linear Elastic Fracture Mechanics
Fig. A.18 Diagram of the function h(a/w) for SE(B) specimen
2 W
σm
σb
t
a
2c
tφ
2 c
a
A
Fig. A.19 Semi-elliptical surface crack with a c [8]
Appendix: Linear Elastic Fracture Mechanics 819
2 W
σm
σb
t
a
2c
t
2c
a
φ
A
Fig. A.20 Semi-elliptical surface crack with a [ c [8]
820 Appendix: Linear Elastic Fracture Mechanics
2 W
σm
t
2a
2c
t
2c
2a
φA
σm
d
Fig. A.21 Elliptical central crack [8]
Appendix: Linear Elastic Fracture Mechanics 821
Fig. A.22 � Ellipse corner crack with a c [8]
822 Appendix: Linear Elastic Fracture Mechanics
A.4 Cylindrical Geometries Under Pressure
Figures A.24, A.25 and A.26.
W
σm
σb
t
a
c
t
a
c
φ
Aa
Fig. A.23 � Ellipse corner crack with a [ c [8]
R t
2a
Fig. A.24 Through-wall crack [9]
Appendix: Linear Elastic Fracture Mechanics 823
References
1. Tada, H., Paris, P.C., Irwin, G.R.: The Stress Analysis of Cracks Handbook. Del Research Co.,St Louis (1985)
2. Tada, H., Paris, P.C., Irwin, G.R.: The Stress Analysis of Cracks Handbook. Del Research Co.,Hellerton (1973)
3. Brown, W.F., Srawley, J.E.: Plain Strain Fracture Toughness Testing of High StrengthMetallic Materials. American Society for Testing and Materials, ASTM STP–410 (1967)
4. Manjoine, M.J.: Biaxial brittle fracture tests. J. Basic Eng. Trans. ASME 293–298 (1965)5. Wilson, W.K.: Optimization of WOL Brittle Fracture Test Specimen. Westinghouse Research
Report 66–B40–BTLFR–R1, January 4 (1966)6. E 399–90, Annual Book of ASTM Standards, Section 3, Metal Test Methods and Analytical
8. Newman, J.C., Raju I.S.: Stress Intensity Factors Equations for Cracks in Three-DimensionalFinite Bodies Subjected to Tension and Bending Loads. NASA Technical Memorandum85793. NASA Langley Research Center, Hampton (1984)
9. Zahoor, A.: Closed form expression for fracture mechanics analysis of cracked pipes. J. Press.Vessel Technol. 107, 203–205 (1985)
R (cont.)Robinson, S.L., 765Roe, C., 581Rolfe, S.T., 580, 622, 764, 774Rone, J.W., 188Rosemberg, H.M., 191Rozendaal, H.C.F., 190Ruiz, A., 581, 650Rungta, R., 107Ruther, W.E., 727Rybicki, E.F., 650Ryder, D.A., 101
SSaanouni, K., 363Sakamoto, I., 475Sakane, M., 519Saraceni, M., 474Sarrazin-Baudoux, C., 804Sato, K., 579Savaidis, G., 491, 518Schaffer, J., 107Schijve, J., 107(2), 307, 403, 441,
multiaxial fatigue, 480maximum shear stress theory orTresca
theory, 152, 482von Mises theory, 152, 483
Error function, 200Euler’s function. see Gamma function
extrusion, 77, 116Evans diagram
corrosion potential, 669Exchange current
at electrode potential, 669
FFailure theories
maximum normal stress, 481Coulomb-Mohr, 481modified Mohr theory, 481Tresca or maximum shear stress, 482distortion strain energy, 483triaxiality factor, 492
effect on fatigue, 178Polarization diagram, 668Potentio-dynamic polarization method, 690,
695Polarization
in electrochemical reactions, 541Porosity, 626Potential drop test method, 535Power spectral density, 450Purbaix diagram, 653Postweld heat treatment, 636Prestressing
effect on fatigue, 155, 636Probability density function, 196Probability of failure, 200Probability of survival, 200Probability paper, 203Process volume
modified, 110Spectral power density, 442Spring back effect. See shacke-down effectSpring index, 302SSY. see small scale yieldingStacking-fault energy, 34, 506Stainless steel, 549