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.