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Guide for Further Reading and Bibliography
GENERAL BOOKS
A. S. Tetelman and A. 1. McEvily, Jr., "Fracture of structural materials". John Wiley & Sons, New York (1967).
"Fracture: An Advanced Treatise," edited by H. Liebowitz, Academic Press, New York and London. Volume I: Microscopic and Macroscopic Fundamentals. Volume II: Mathematical Fundamentals (1968). Volume III: Engineering Fundamentals and Environmental Effects (1971). Volume IV: Engineering Fracture Design. Volume V: Fracture Design of Structures. Volume VI: Fracture of Metals. Volume VII: Fracture of Nonmetals and Composites.
« La Rupture des Metaux », edited by Franr,;ois and Joly, Masson & Cie, Paris (1972). 1. F. Knott. "Fundamentals of Fracture Mechanics," Butterworths, London. (1973). H. D. Bui. « Mecanique de la rupture fragile», Masson, Paris (1978). R. Labbens. « Introduction a la mecanique de la rupture », Editions Pluralis, Paris
(\ 980). D. Broek. "The practical use of fracture mechanics," .Kluwer Academic Publishers, Dor
drecht, The Netherlands, (1989). P. F. Thomasson. "Ductile fracture of metals," Pergamon Press, Oxford (1990). T. L. Anderson. "Fracture mechanics - Fundamentals and applications," CRC Press, Boca
Raton, Florida, USA (1995). "Topics in fracture and fatigue," A. S. Argon editor, Springer-Verlag, New-York (1992). W. M. Garrison, Jr and N. R. Moody. "Ductile fracture ," J. Phys. Chern. Solids, 48, II,
pp. \035-1074 (1987). M. F. Kaninen and C. F. Popelar. "Advanced fracture mechanics," Oxford University Press (1985). "Metals Handbook-ninth edition. Volume 8, Mechanical testing," Newby, coord.,
American Society for Metals, Metals Park, Ohio, USA (1985). "Impact Testing of Metals," ASTM STP 466, ASTM, Philadelphia (1970). "Instrumented Impact Testing," ASTM STP 563, ASTM, Philadelphia (1974).
442 GUIDE FOR FURTHER READING AND BIBLIOGRAPHY
H. Kolsky. "Stress waves in solids," Dover Publications, Inc., (1963). L. B. Freund. "Dynamic Fracture Mechanics," Cambridge University Press (1990). Y. Bai and B. Dodd. "Adiabatic Shear Localization," Pergamon Press, Oxford (1992). W. Johnson. "Impact strength of materials," Edward Arnold, London (1972). N. Jones. "Structural impact," Cambridge University Press (1989). 1. Finnie and W. R. Heller. "Creep of Engineering Materials," McGraw-Hill (1959). A. H. Evans. "Mechanisms of Creep Fracture," Elsevier Applied Science Publishers, UK (1984). H. Riedel. "Fracture at High Temperatures," Springer-Verlag (1987). R. K. Penny and D. L. Marriott. "Design for Creep," Kluwer (1995). G. A. Webster and R. A. Ainsworth. "High Temperature Component Life Assessment,"
London (1994). R. P. Skelton. "High temperature fatigue. Properties and predictions," Elsevier, Oxford
(1987). H. Suresh. "Fatigue of Materials," Cambridge University Press (1991). F. Ellyin. "Fatigue damage, crack growth and life prediction," Chapman & Hall, London
(1997). "Fatigue and Fracture, Volume 19," ASM International, Materials Park, Ohio, USA
(1996). 1. Lemaitre and 1. L. Chaboche. « Mecanique des materiaux solides, }) 2nd edition, Du
nod, Paris (1988). 1. Lemaitre and 1. L. Chaboche. "Mechanics of Solid Materials," Cambridge University
Press (1990).
REVIEWS
"Engineering Fracture Mechanics," Dodds and Schwalbe, eds., Elsevier, Oxford. "Fatigue & Fracture of Engineering Materials and Structures," Miller, ed., Blackwell
Science, Oxford. "International Journal of Pressure Vessels and Piping," Ainsworth, ed., Elsevier, Oxford. "International journal of fracture," Williams, ed., Kluwer Academic Publishers, Dor
drecht, The Netherlands. "International Journal of Fatigue," Pook,ed., Elsevier, Oxford. "Journal of Pressure Vessel Technology,"The American Society of Mechanical Engi
neers. "Journal of the Mechanics and Physics of Solids," eds.,Freund and Willis, Eds., Per
gamon, Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK.
"Materials at high temperatures," Nicholls et aI., Eds., Science Technology Letters, PO Box 81, Northwood, Middlesex HA6 3DN, UK.
"Mechanics of Materials," Nemat-Nasser, ed., Elsevier, The Netherlands. " Journal of Testing and Evaluation," ASTM. "Metallurgical Transactions," Trans. AIME, now replaced by "Metallurgy and Materials Transactions," Trans. AIME. "Acta Materiala," ASM International and The American Institute of Mining, Metallurgi
cal and Petroleum Engineers. "International Journal of Solid and Structures," Steele, ed., Pergamon, Elsevier, Oxford.
.................................................................................................................................................
I Answers to Selected Exercises
1.2. Answer is in Chapter 4. 1.3. T K at NDT increases with decreasing T 105' 1.5.
KIC and KIR MPa m
220 -200
100 t-------r----7''------j
20
·50 ·25
KIC and KIR. MPa m V
% tolerance bound
220 -200
20
+25 +100 T· RTNDT, ·C
+100 T ·RTNDT:C
The two limiting curves are approximately the same. The TO curve underestimates the RTNDT curve at low toughness, and viceversa. The new ASME proposed curve appears very conservative.
1.7. K]C(med) = (Ka - ~in) (In 2)114 + ~in = 0.9124 Ka + 0.0876~in=30.94 + 70.25 exp [0.019 (T - To)]. Inversely, from the variation of the median value given by the standard
Ko=1.095K]Cmed+0.095~in=35.7+76.65 exp [0.019 (T - To)].
1.8. In the figure we read (JSSY / a O"o)max = 0.012. Hence, JSSY(max) = 3 104 J. 1.9. K]C5% = 24.5 + 37 exp [0.019 (T - To)], KJC95% = 34.6 + 102.2 exp [0.019
(T - To)],
444 ANSWERS TO SELECTED EXERCISES
1.10. The maximum strain corresponds to d = D, and therefore is equal to 0.17-0.22 depending on the value of the constant, 10-3 S-1 to 10-1 S-I, 10-4 S-1 to 10-2 S-I.
2.1. The relation between load and displacement is approximated by a power law at small plastic displacements, and by a linear law at large values.
2.2.
2.3.
qa) G(a) PN(a)/W Ve /W q=
Ves/Ws Cs(as) Gs(as)Pn(as)/Ws where C G is constant.
Kr 1.4
1.0r---_
0.2
C G PN IW
Cs G s fPN 1Ws
0'---<'---<'--1--"--'---'---'---'-_ o 0.2 1.0 Lr
2.4. Force and moment equilibria:
F T M
~
~ k bl
F 1 M
F = (21J3)0'0k(W-a)B = kFo;
M = 2 B rW - a)/2 (2 I J3) 0' 0 x dx = (1 .. k 2) Mo . .l(W - a)/2
where PO and MO are the limit loads for pure force and pure moment. Elimination ofk leads to:
:0 + (~r 1,
or in terms of stress,
0' bO ( 0' rnO ) 2
I-lbO'O + I-lrnO'O 1,
ANSWERS TO SELECTED EXERCISES 445
where ~b and ~m are weakening factors taken as equal here. Then solving this equation with respect to the equivalent stress = ~ (Jo as a definition, we get
(Jb 2 ( )2
3 + (Jm ,
which is maximized by (Jm + 0.67 (Jb' 2.5. According to the Illyushin theorem, if there is a relation between the
stress and strain in a body, such as the Ramberg-Osgood law, the same kind of relation stands for the generalized load and displacement, and if P is the load parameter, the stress at every point is proportional to P, while the strain is proportional to pn. Thus, if we consider a horizontal straight pipe of mean radius Rm and assimilate it to a cantilevered beam clamped at one end and to its other one to a given deflection v under a load P, in the case of the elastic-plastic behaviour we have
CM/CMY = MlMy + IX (MIMy)",
where CM is the maximum load in a straight section on the outer fiber. The moment M is given by M= Mop (l-xlL), where Mop is the moment at the embedding. The deflection increment of this section is given by d2<p./dx2 = cM /Rm and the displacement v is obtained by integration of over the length L of the pipe. After calculation, the result is given by
v = (cMy/Rm) (Mop/ My) U/3 + IX (CMy/Rm) (Mop/ My)" U/(n+2).
In the case of the elastic behaviour the displacement is similarly given by
v = (EMy/Rm) (MOE/ My) L2/3.
For the same given displacement, the relation between MOE and Mop IS
given by
(MoE/ My) = (Mop/ My) + IX (Mop/ My)" 3/(n+2).
At the embedding the relations between deformation and moment are given by
EMoPiEMY = Mop/My + IX (Mop/My)n EMOE/EMY = MOE/My
Therefore, the slope joining the point representative of the elastic behaviour to the point representative to the elasto-plastic behaviour in the graph of moment versus deformation is given by
[(Mop/ My)- (MOE/ My)]/( EMOE-EMOE)= - 3/EMvCn-l),
which is the elastic slope multiplied by - 3 / (n-l).
446 ANSWERS TO SELECTED EXERCISES
2.7.
2.8.
2.9. Under widespread-creep conditions, fc is reduced to
fALr) = [ E&r~ ]-112 = [Eeref ]-112 Lr O'02 aref
2.10. (i) secondary creep i = Can, e = Cant, /1j ~c =e/ l: =t, (ii) primary creep
i = C C rC -l)a" e = C tC2a" /1 / ~ . =e/ i =tIC2) 1 2 2, 1 ,c' c
2.11. By analogy with the case of a Hollomon material the slope is ...
3.1.
3.2. 3.3.
3.4.
3.5.
2 s..-2
cl
ANSWERS TO SELECTED EXERCISES 44 7
1 - 2 v ----, = 0.5 for u = 0, = 0 for v = 0.5. 2 (1 + v)
Carry out the numerical application for verification. Equation of motion for the weight at time t: M (dv 1 dt) = A cr = P A v co. By integration: v = Vo exp (-p A Co tiM), and cr = p Vo Co exp (-p A Co t 1 M). By putting the expression in sinh equal to zero, we then obtain 3cr 0 = (3/2) p(bo 1 t R ), that is the same expression as the one for the
hollow sphere in expansion. The governing differential equation is dt/dy = at 1 a y + (a t 1 a T)( dT/dy). Catastrophic shear occurs at a plastically deforming location within a material when the slope of the true stress-true strain function becomes zero. Then the criterion can be written as
o :s; (at 1 ay)/[- (at 1 aT)(dT/dy)] :s; 1.
4.1. c' is close to Cs for modes II and III, and to cd for mode I.
4.2. Ifp(t) is the load history, the raise in load during the time interval dT at time Tis p'(T) dT. In elasticity theory, the superimposition principle can
be applied. Therefore dK(t) = p'(T) dT ~c'(t - T) for t>T, where c' is a
parameter easily identified according to the loading mode. Then it is sufficient to integrate from time t=O.
4.3. Under mode I and for a stationary crack, the transverse diameter D of the virtual caustic in a plane at a distance Zo of the front surface of the
specimen is related to the stress intensity factor K I by
E D5/2 K = [
10.708 Zo d v
4.4. t*= 1.04 (2a)/cR. t* increases with 2a. For 2a and K]c fixed, t* = ... For a
brittle material for which CR = 1 000 ms·1 and KIC = 1 MPa rm , t*= lOllS
and the minimum pulse amplitude leading to fracture is cr*=6Mpa. For a
ductile material for which CR =3000 ms·1 and KIC = 100 MPa rm , t*= 3
IlS and the minimum pulse amplitude leading to fracture is cr*=600 MPa. 4.5. Integrate the second law P= md2x/de. dxldt = Vo - Pdt. Integrate, etc. 4.6. 10= 2R/co ; cro= (112) pCoVo ; K](t) = n(v)cro co1l2[t1l2 - (t_tO)112 R(t-to)] with
n(v) = [2/(I-v)][(1-2v)/n] 112 ;.= [lIn2(v)cO](K]dcro) ; It Id = K]d 1 2 t; ta =
(t/4)(1 +toltt 5 .. 2 Straightforward identification for the stresses. For the deformation, use
the intermediate expression
lim"~f:i) / 'l ~ _1°0":1",
448 ANSWERS TO SELECTED EXERCISES
5.3.
o O!-::-'--~6~O:-' ---t12:;;O;O':---:--~ Plastic zone develops in three distinct regions.
5.4.
u 2G'fi x K r -----6 0.7
0.6
2 0.4
1 a = (vic ~ 2
0 A· 60' 120' B 180'
/" 10 2
u 2G'fi (vic 5) = .7
Y K r
6
4
0-
a !o!=======:.q.:-:------:l::-=--....L-o· 120' B 180'
The tip curvature increases with crack tip.
5.5. KI< 4a d (l-a s2)
Klcr (K+1)[4U dU s -(l+u s2 )2]
This ratio tends to infmity when v tends to cR.
ANSWERS TO SELECTED EXERCISES 449
1 ad (1- a/ ) ( )2 GJ =-[ ] KIa-
2G 4adas -(1+as2 )2
For v = 0, the static value is found. The correction is small for v/cs lower than O. This correction is limited to a velocity equal to the Rayleigh wave speed.
5.6. Elastic-strain- energy density per unit length in the x direction far ahead of the crack tip= (Euo)%(1_y2). Elastic- strain- energy density per unit length in the x direction far behind the crack tip. Without any energy exchange with the surroundings, the energy balance gives G= (Euo?!h(1_y2).
5.9. During growth, G r (v = 0) = TC a 2 a. The energy required for propaga
tion is equal to G r (v) = TC 0'2 ao' Hence, v = c R (1 - al a o).
5.10. A calculation similar to that of the static case, where a virtual infinitesimal extension of the crack is considered, leads to the relation G J =( 1 -
y2) K/(J'KJE:IE.Then,as K/(J.=KJ' hence, KJE:=A(v) K/(j'
5.11. Elastic-strain-energy density per unit length in the x direction far ahead of the crack tip= (Euo)%(1_y2). Elastic- strain-energy density per unit length in the x direction far behind the crack tip. Without any energy exchange with the surroundings, the energy balance gives G= (Euo)2/h(1-y2).
5.12. For a semi-infinite crack; (a) from the approximation G(t, it) = (1 -it IcR) G(t, 0); (b) from the approximation K(t, it )=[(1 - it IcR)/(l -0.5 it IcJJ K(t, 0).
5.13. (a) U= 0'2/2 E == K2/2rE to within a multiplier of order unity. T= p y2/2 == P v2K2/2 E2r, where Y is the particle velocity given by EV. Hence, T/U= v2/(E/p)= v2/co. The ratio is independent of r and not significant as long as the crack speed is less than about one-third of the elastic wave speed. Above this speed, the static-stress field is modified. (b) T/U=lO (v/co? In(~/r) to within a multiplier of order unity. This ratio depends on rand becomes infinite for r approaching O. R must be compared with the process-zone size. However, inertial effect can be higher in plastic field than in elastic field.
6.2. d = 0.5 cm, kT = 10-13 erg at T = 750oK, D = 10-10 cm2s-1 and Q = 10-23
cm3, the viscosity coefficient T]=2.5 1019 poises. If d=0.005 cm , T]=2.5 1015 poises to be compared with the viscosity of a liquid: deformation occurs slowly and the crystal is a solid, on the other hand, the crystal is almost fluid.
6.3. See Cocks and Ashby, 1982.
tcr
\15t,-----b ,c ~ ,~ , , ~, I ____ I
c' b' a'
450 ANSWERS TO SELECTED EXERCISES
With a denoting nonnal stress and't denoting shear stress, the boundaries be, bb', and b' c' slide freely and cannot therefore support any shear stresses. Equilibrium requires that
abc = (1/2) (3a - aT)
a bb, = aT'
6.4. Because it contains contributions from the decelerating primary creep and the accelerating tertiary creep.
6.6. Po = p[e;;}n(~;) r· a e = .J3 p(Rilr) [ 1-(R/Ro)2]
R"p = Ro~2In(Ro IRJ/[I-(R j IRo)2].
6.7. RI: Force equilibrium: a h Ah = a j A j • Definition of strain, s = In (A/A)
and above equation gives: Shn' AOh exp(-sh)= st' AOi exp(-sJ Notation
Ao/Aoh=fo gives: S h exp(-sh/n')= S t folln' exp(-s/n'). Integration gives:
6.9.
6.10.
Sh = -n'ln{(fo)lIn'[exp(-s/n') - 1] + 1}.
10
Eh 1
0.11-....L...-Jt;--...J 0.01
(i) If the jacking apart of the grains by the removal ov is oz: ov= n(l2 - R2) oz = n12( 1 - fh) oz. This produces an additional component of void growth ov'=nR2oz. Hence, the total volume increase of the void is oV=ov+ov'=ov/(I-fh). (ii) Equating the volumes of the masses: (4/3)nR3-nR2u = n(l2 - R2)U leads to u = 4R3/312.
i" = 'o( ::J" {I+ ~I+ :JH:::~ (~ll} 6.11. s c = - S /S where S is the section area of the tensile specimen. tr = 1/n
S c 0, where E C 0 is the initial-creep rate.
6.13. s~ = 2A exp (-H/RT) exp (VaIRT), s~ ~ = CMG leads to T (lntr + C) =
(H - Va)1R = f(a), where C is a constant.
6.14. Cm= [1/(1 +m)] (S 01 ill 0) (a/ao)"-m
Sf = [1/(1 +m-n)]( S 0/0) 0)( a/ao)n-m 'A = [(1 +m)/(1 +m-n)].
6.15. S = Sf [1 - (1 - tltf) 1 - n/(1 +m); tf = [ill o( a/aoYll; Sf = E o( a/ao)"(1 +m)/
[ill o( a/ao)X(1 +m-n)]; Cm = S o( a/ao)"-X ill o.
ANSWERS TO SELECTED EXERCISES 451
6.16. (i) E == Es/(l-p) ~ Es(l+p) for p ~ 0; (ii) dE =Esdp~Edp;
E = Es expp. 6.17.
6.18.
6.19.
7.1.
o E
N~=O.014 3 r-------.-.---,
N~=O.063 10 r---::-r--r----"
2 2
o 2 3 o 2 3 S/cr S/cr e e
Enhancement of uni-axia1 creep- strain rate owing to micro cracked facets as a function of stress tri-axiality ratio for two facet-crack density NR3,
for three creep-stress exponents in the case of non grain-boundary sliding (after Rodin and Parks, 1993; and Sester et aI., 1997).
Ell = i;o(O"~)n{l+ P(1+ l )[1+(n-1)(l)]}. 0"0 O"e (n+1) O"~
[ 112 ](n+1)/2
1 + ~1tNR3 8(l+3/~- (Ho + HI + H2 ) E sliding 3 1t 2 vn fn~ ____________________ ~~ __ ~ __ _
[ 3 1/2] (n+1)/2
E non-sliding 1 + 8NR (1 + 3/ n)-
From Ylliushin's theorem, it is possible to write that the stress field is also proportional to the loading, that is
O"ij(r, e, t, pet)) = pet) Lij(r, e), where Lij(r, e) is dependent on polar coordinates rand e, but not on
load and time. Integration of this creep law with respect to time
+. 3 [ ( ) ]nl(I+PI)-1 Eeq PI Eprij = "2Bi Le r, e pet) Sij ,
leads to
Cprij =~[BI (PI + 1) ( p(t,)nl(l+PI) dt,]I/(PI+1) [-I-]O"eq nl-i Sij
2 t POO~ The stress field near the crack tip is thus described by
1 p(t)n, J(t)j1/(n'+1)
cr ij = 1/(p, +1) I r (n1 + I{ B1 (PI + I) i p(t,)n,(l+p,) dt] n
crij(e, n) .
Then differentiating E prij with respect to time leads to
452 ANSWERS TO SELECTED EXERCISES
which is the equation of a viscous material with a power law. Therefore, the stress field near the crack tip is described by the equation
(J .. = j[ (PI +1) !p(t,)nl(I+PI) dtT/(PI+I) CCt)jll(nl+l)
I) I( ) cr iJ (8, n) . BII PI +1 p(t)nIPI In r
Riedel proposed writing this expression under the form
1 )1I(nl+l) 1 C~ ~
(Jij= ~ I) (Jij~'~ , [BI(PI +1)] PI+ In r
which means that the term C~, with the suffix h for hardening, is intro
duce such that
[ ]PI/(PI+I)
• (PI +1) !p(t,)nl(I+PI) dt' ,
Ch = C(t)· p(t)n1P1
This integral is very interesting because it does not depend on time. Moreover, we have the relation
[ (p +1) rp(t,)n1(l+Pl) dt'] J (t) I -lJ _ --en +1) . C(t) - I p(t)nl(Pl+l)
By making PI=O, we find again the results of the viscous material with a power law.
7.3. Ct is proportional to K\ and thus does not coincide with C(t) in the limit of small-scale creep, which is K2.
7.4. Insert the stress field into the material law. The strain-rate field has a singularity of the form (r-n)/(n-l). Because the time derivative at a material point is given by - a (a I ax), the result is obtained.
7.5. Inserting the critical strain and the critical distance into the expression of the creep strain obtained by x-integration, we obtain, etc.
7.6. a 88 (8=11) la 0
P jSPC LSPC SSSC LSSC
~ log time. h
P : plasticity regime ssep: small-scale primary·creep regime LSCP: large-scale primary-creep regime SSSC: small-scale secondary-creep regime LSSC: large-scale secondary-creep regime
" 88 (8=0) I~O
2
~~ 0.5'--_--1:-_--1 __ --1.
o 10 100 log r/J "0
7.7.
7.8. l22v ; ,, ,
ANSWERS TO SELECTED EXERCISES 453
_Id"(' _ d(, ~~ __ ~, I I
ti tf ti tf dV/dt C~xp
For a secondary-creep regime with E = B2a n2, the steady value of C* is used:
C*= ~ llP dV n 2 + 1 B( w - a) dt
For a primary- creep regime with E =PIBlanlt(pl-I)= B\an\ the instantaneous value is used:
C*= n l llP dV
n l +1 B(w-a) dt
Since nl and n2 are relatively high, the two expressions are of the same order of magnitude and very close of each other.
7.10. From the total displacement V = Ve + Vp +Vc, the total instantaneous
elasto-plastic J =Je + Jp = d f VdP Ida with V= Cn pn and n= 1 for elas-
ticity and n=n for plasticity. Hence, J= [pn+l/(n+ 1 )]dC/da. Therefore, at P constant J=[P/(n+ 1 )]dV Ida= [P/(n+ 1)] a dV/dt.
8.1. Lla/2=(a'F- arn)(2NF)'b. However, a combination of mean stress and stress amplitude may result in a cyclic-dependent creep (ratcheting) strain that can lead to premature creep-fatigue fracture.
8.2. 1110 /2 = (E'F-EOrn) (2NF)-c. This equation can be rewritten as LlE /2 + Earn ( 2N F) -c = (10' F- Earn) ( 2N F) -c.
Assuming that Earn is of the order of the plastic strain amplitude, and setting c=-0.6, the second term on the left-hand side is small in comparison with LlE /2, and can be neglected for 2NF>4000((2Nf),,=(4000)-06=0.007). When the mean strain is of the order of the fatigue ductility coefficient, 10' F' it is likely that a crack initiates at the start, and thus the fatigue life would be reduced. Therefore, for moderate mean stains, the low-cycle fatigue life will not be appreciably affected by the introduction of a mean strain.
8.3. 1110 12 = LlEE/2 +LlEp/2= [(a'F - am) IE)] (2NF),b+ E'F( 2NF)-C 8.4. The equation of the Masing curve is given by
LlE= Llal E + 2 (Lla12 K,)lIn', etc.
LlWP = [(l-n')/(l+n')] Lla LlEP ; LlWE=(1/2E)( Lla)2; LlWT = LlWE + LlWP.
90% of the energy is dissipated into heat and vibratory energy, and 10% is stored and converted to damage associated with dislocations.
454 ANSWERS TO SELECTED EXERCISES
8.6. ~WP=Is, (2 Nf),dp" where Is, and ~ are constants to be detennined from the best fit to experimental data. For a Masing material, dp = b+c and kp = 4[(1-n')/(1+n')] cr' FE' F'
8.7. da/dN= C(~K)m = C(F ~cr & r. Hence,
8.8.
8.9.
8.11.
8.12.
Nf= 1 m-2 m-2 [ 1 1] (m_2)Cpm 1t m/2L\cr m ~-~ ,
for m*- 2,
Nf- 1 In(~) , for m=2. Cp2 1t L\cr 2 ao
R( C) = (~Ku,/cro)2/(24 1t)=d. NR = fo+R(O)-R(C) da
o q,R[R(o),R(C),L\a] (dadN)(L\K,R)
0'
KtS
cyclic curve
€
The stress at point 1 is given in the first-coordinate system by crE = cr2/E + cr( crlK ,)l/n'= (Kf S)2/E = constant.
The stress at point 2 is given in the second-coordinate system with the origin at point 1 by ~cr ~E = ~cr/E + 2(~cr/2K')lIn'== (Kf S)2/E=constant.
These equations are solved relatively easily using the Newton-Raphson iteration technique and standard numerical methods.
L\cr/2
Index
adiabatic shear bands, 173-176
Ainsworth simplified expression, 85
Annex A 16 approach, 114 application mode, 247 ASME flaw elastic analysis, 98 available-energy method, 37
bending stress, 97, 98, 108, 110, Ill,
135 bi-axiality ratio, 36, 66, 67
blunting process, 122 brittle metals with high strain rate and
low ductility, 283
brittleness transition temperature, 6
calibration function method, 200 cavities, 288, 291-297, 299, 302, 306,
307,311,312,315,323,329,331, 334,338,368,377,379,383-385, 387
characteristic safety factor, \3 7
Charpy impact energy, 5, 55 cinematic stress-intensity factor, 227,
236 cleavage, 2, 6, 8, 12,22,35,37,40,42,
52,53,58,59,63,66,68,69,71,
72,74,75,9~ 177, 198,208,217,
249,262,264,266,291,438
C-Mn mild steels, 10 CODAP, 27, 76 compressive plastic zone, 74, 410
constraint correction, 30, 35, 36, 52,
58, 103
constraint parameter, 64, 66, 67, 95,
96,103 contour integral, 110, 118, 121,227,
377,388,391,392,426 Corten-Sailors correlation, 25 crack arrest temperature, 16 crack arrest toughness, 251, 269
crack closure, 71,418,419,420,426,
436 crack extension force, 242, 272
coupled-pressure-bars technique, 210
crack propagation resistance curve, 215 crack propagation velocity, 252
crack resistance toughness, 22 creep-constrained cavity growth, 374
creep damage, 319, 334, 336, 338, 372, 378,387,388,397,428,430,431
creep-damage tolerance, 321 creep mixity or mode parameter or
factor, 392 creep zone, 356, 357, 358, 364, 365,
366,370,378,392,393,432,434 critical normalized (maximum)
principal stress, 35 critical volume, 35
cruciform-beam specimen, 65 cyclic hardening, 402
cyclic plastic zone, 411, 412, 416, 432,
434 cycling with hold time, 429 cycling without hold time, 428
damage function, 75, 318, 409
456 INDEX
deformation-mechanism map, 277, 288
deformation plasticity failure
assessment diagram, 132
design transition temperature, 19
diffusional-flow creep, 276
diffusive cavity-growth, 295
dimple, 217
dislocation-flow creep, 276
drop-weight test, 12, 13, 15, 19,38,56
ductile, 1,2,8,10,14,28,33,40,47,
49,50-53,58,59,70,71,81,87,89,
93,95,96,98,118-121,125,134,
162,168,179,181,198,209,211,
213,217,218,220,224,225,226,
249,256,259,265,266,270,283,
290,312,331,334,370,371,373,
374,379,395,396,406,414,437,
447
ductile fracture method, 93
ductile-tearing initiation, 118, 218
ductile-to-brittle transition, 1, 2, 22, 30,
47,70,217
dynamic elastic-strain energy-release
rate, 236 dynamic fracture toughness, 38, 181,
195,246,248,253,256,259 dynamic recovery, 288
dynamic-strain energy-release rate, 241
dynamic stress-intensity factor, 187-
189,192,202,214,229,234,236,
239,245-247,251,252,261,271,
272
dynamic stress-intensity factor at the
instant of arrest, 252
dynamic yield stress, 6
effective mode-I values of C*, 393
elastic-strain energy, 228,438
elasto-plastic fracture mechanics, 29,
121 EPRI-GE J-integral estimation scheme,
83 equivalent-energy principle, 36
Esso test, 16
exponential visco-elastic law, 150
facet stress, 280, 282, 294, 311,329,
393,394
facet-crack-opening rate, 310
failure assessment diagram, 86, 87, 88,
89,95,96, 132, 133, 134, 135
fatigue-crack growth rate, 418
fatigue-ductility coefficient, 405
fatigue-ductility exponent, 405
fatigue-notch factor, 425
fibrous (shear) fracture, 2, 6
first order reliability method, 139
fracture analysis diagram, 1, 12, 17, 18,
85, 87, 107
fracture appearance transition
temperature, 2, 6
fracture process zone, 214, 411
Fracture Transition Elastic (FTE), 12
French approach, 100, 318
generation mode, 40, 247
grain-boundary cavities, 280, 290, 383
grain-boundary sliding, 280, 291, 292,
294, 306, 308, 310-312, 320, 330,
332,339,342,385,388,451
harmonic loading, 189 high cycle fatigue, 404
HR asymptotic-stress field, 368
importance factor, 138
incubation time, 122, 124
inertia effect, 141, 146, 164-166, 169,
171, 180,200,201,203,206-208,
217,218,219,227,251,256,262
initiation toughness, 196, 211, 212,
250,254
in-plane constraint effect, 36
intergranular creep fracture, 288, 299
inverse-geometry-impact hammer, 206,
207
Johnson-Cook equation, 158
key curve method, 199,200
kinetic-energy density, 240, 273
Larson-Miller parameter, 314
lateral expansion, 4, 5, 7, 28, 29, 47,
49
law RR, 344, 346
limiting crack speed, 229, 264
line integral, 122,240,329,344-346,
351,353,356,366,389
low-cycle fatigue, 81,405,409,435,
453
lower shelf, 2, 5, 18,26,50, 70
maximum principal stress material,
315
mean level of loading, 405
mean safety factor, 137
membrane stress, 98, 108
microstructurally small crack, 424
miniaturized disk bend test, 50
model of Bodner and Partom, 155
modified R-6 procedure, 125
Monkman-Grant relation, 314, 338
monotonic plastic zone, 411, 412
moving crack, 141, 189, 191, 192,229,
245,249,257,268,273
multi axial stress rupture criterion, 317,
318
mUltispecimen method, 209
nucleation strain, 168, 294
nucleation stress, 293, 385
nucleation time, 294
one-point-bend method, 207
opening stress, 64, 72, 164, 218, 232,
236,257,258,272,358,399,424,
425
optical method of caustics, 190,203,
205,221,222,251
overload, 73, 75, 76,421,422,439
partial safety factors, 119, 121, 134,
136,140
peak stress, 97, 98, 111,286
Pellini, 10, II, 12, 17-19,56
INDEX 457
penny-shaped crack, 297, 306-308,
324,325-328,334,336
plane-strain fracture toughness, 200,
222
plastic-collapse, 89, 99, 337
plastic-strain energy, 175,245,438
power-law creep, 122, 128,275,285,
298,299,301,302,304,307,320,
329,333,359,362,363,377,396,
398,431
Prakash and Clifton model, 150
pre-cracked Charpy specimen, I, 36,
177
pre-cracked Charpy V specimen,
54
pressure loading, 187
pressurized thermal shock, 39, 42, 74
primary creep, 123, 136,283,350,
360-363,366,396,398,399,446,
450
primary-creep zone, 361
primary stress, 96, 98, 99,104,105,
108, 109, 121, 129
principal facet stress, 280, 332, 335,
393
process zone, 71,131,175,179,221,
264,265,378,415
R-5 procedure, 124, 125
R-6 procedure, 83,90, 116, 125
rate of energy flow, 241
rate of recovery, 277 RCC-MR, 112, 114, 125, 129, 130,
131, 133
reconstituted CVN specimens, 55
recrystallization, 173,283,288
reference stress approach, 124
reference temperature, 23, 27, 28, 33,
34, 38, 39, 58, 59, 65, 160
reference toughness, 27
reliability index, 119,136-140
reversed cyclic plasticity on creep,
432 reversed plastic zone, TJ, 411,416,439
Robertson test, 15
458 INDEX
Rolfe and Novak correlation, 26
RSEM code, 117
rupture, 20, 21, 57, 79, 81,125,128,
130,134,163,164,180,198,201,
204,207,217,265,266,275,288,
290,300,307,313-319,323,332,
334,335,386,393,396,397,404,
408,414,436,438,441
rupture time, 288, 313, 315, 316, 323
safety margin, 83
secondary creep, 123, 124, 127, 128,
136,314,320,322,338,344,349,
361,363,366,386,399,446
secondary stress, 91, 96, 97, 98, 99,
100-105,108,114,121,133
secondary-creep zone, 360, 361
shallow flaw, 36, 66
shear-band toughness, 175, 176
Sherby-Dom parameter, 314
short crack, 422-424, 437
sigma-d approach, 131, 135
SINTAP,100
sintering stress, 293, 297, 306 skeletal point, 286, 287, 317, 318, 337,
352
small punch (SP) test, 50 static yield stress, 152
static-fracture toughness, 28, 246 stationary crack, 141, 191, 197,212,
227,233,234,239,245,261,262,
271,352,367,380,390,396,447
steady-state crack growth, 366, 382
strain energy-release rate, 245, 262,
271 strain-aging phenomena, 147
strain-energy density, 51, 240, 272,
438,449
strain-hardening law, 284
strain-hardening rate, 154, 277 stress-enhancement factor, 294, 387 stress-intensity factor some time after
arrest, 252 striation, 414, 415, 420
structural ledges, 291
Taylor-Quinney coefficient, 156
tension Double Cantilever Beam (DCB) crack arrest test specimen,
250
tertiary creep, 283, 314, 321, 323, 327,
333,344,350,387,388,450
theoretical cohesive-fracture stress, 290
thermal loading, 108, 134
thermal shock, 39,42, 60, 74, 76, 110,
111,112,250
thermal stress, 20, 75, 97, 100, 108
three-parameter Weibull distribution,
30
threshold stress-intensity factor, 412,
420,429
time-hardening law, 283, 284, 408
transgranular creep fracture, 288
transient-growth-rate behavior, 372
transition time, 124,203,204,205,
354,356,361,365,373,389,393,
398,432
transverse Compact Wedge Loaded (CWL) crack arrest specimen, 250
upper shelf, 2, 5, 10, 18,26,28
void coalescence, 163, 169,266,331,
397 void expansion, 165
void growth, 163, 171.172, 179,211,
296,299,304,305,306,319,333,
382,450
void nucleation, 168, 170, 179,225,
291,382
void-growth map, 305
warm pre-stressing, 70, 71, 73, 75
wavefront, 144, 146, 182, 183, 186, 234
wave-propagation equation, 143 wide-plate crack arrest, 40
widespread creep, 126, 128, 136
Mechanical Engineering Series (continued from page jj)
R.A. Layton, Principles of Analytical System Dynamics
C.V. Madhusudana, Thermal Contact Conductance
D.P. Miannay, Fracture Mechanics
D.P. Miannay, Time-Dependent Fracture Mechanics
D.K. Miu, Mechatronics: Electromechanics and Contromechanics
D. Post, B. Han, and P. Ifju, High Sensitivity Moire: Experimental Analysis for Mechanics and Materials
F.P. Rimrott, Introductory Attitude Dynamics
S.S. Sadhal, P.S. Ayyaswamy, and IN. Chung, Transport Phenomena with Drops and Bubbles
A.A. Shabana, Theory of Vibration: An Introduction, 2nd ed.
A.A. Shabana, Theory of Vibration: Discrete and Continuous Systems, 2nd ed.