Fracture Mechanics and Fatigue Crack Propagation

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6
FRACTU RE M EC HAN I CS
AND FAT IGUE CRACK
PROPAGATION
JWO PAN AND SHIHHUANG LIN UNIVERSITY OF MICHIGAN
6. 1 INTRODUCTION
Use of crack propagation laws based on stress intensity factor ranges is the most successful engineering application of fracture mechanics. This chapter gives a review of the basic concepts of fracture mechanics. In contrast to the traditional stresslife and strainlife approaches to fatigue, cracks are as sumed to exist in materials and structures within the context of fracture mechanics. Fracture parameters such as K and J can be used to characterize the stresses and strains near the crack tips. A fundamental understanding of fracture mechanics and the limit of using the fracture parameters is needed for appropriate applications of fracture mechanics to model fatigue crack propagation.
In this chapter, the concept of stress concentration of linear elasticity is first introduced. Griffith's fracture theory and energy release rate are discussed. Different fracture modes and asymptotic cracktip fields are then presented. Linear elastic stress intensity factors are introduced. Stress intensity factor solutions for simple cracked geometries and loading conditions are given. Also, stress intensity factor solutions for circular and elliptical cracks are given. Plastic zones and requirements of linear elastic fracture mechanics are discussed. Finally, fatigue crack propagation laws based on linear elastic fracture mechanics are presented.
237

238 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
6.2 STRESS CONCENTRATION BASED ON L INEAR ELAST IC ITY
6.2.1 ELLIPTICAL HOLE
Consider an elliptical hole in a twodimensional infinite solid under remote uniaxial tension. The major and minor axes for the elliptical hole are denoted as c and b, as shown in Figure 6.1. The elliptical hole surface can be described by the equation
x 2 y2 c7 + ~5  1 (6.2.1)
The radius of curvature at x  +c and y  0 is denoted as p, as shown in the figure. The radius of curvature p can be related to c and b as
6 2 p  (6.2.2)
C
The twodimensional infinite solid is subjected to a uniformly normal stress as in the y direction at y  +~. The linear elasticity solution gives the stress solution at x  +c and y  0 as
FIGURE 6.1 tension a~.
O'~c
I I I l x
I I I 1 o'~
An elliptical hole in a twodimensional infinite solid under a remote

STRESS CONCENTRATION BASED ON L INEAR ELASTIC ITY 239
a)..r(xic, y0) a~(1 +b) (6.2.3)
Based on Equation 6.2.2, Equation 6.2.3 can be rewritten as
We can define the stress concentration factor K, for the elliptical hole under the remote uniaxial tension as
K, __ a).).(x  +c , y  0) __ 1 + 2~//~' (6 .2 .5) 0"~ VP
6.2.2 C IRCULAR HOLE
For the special case of a circular hole with a radius of a as shown in Figure 6.2(a), the equations for the elliptical hole will be valid with p = b = c = a. A polar coordinate system for the circular hole problem is shown in Figure 6.2(a). The solutions for the normalized stresses 6rr, 600, and ?fro for this case are
l(.2) ) 6,.,. =  1  cos 20 o~ 5 ~ 2 k r4 1"2 ~ 1 (6.2.6)
6ro = aro _ sin 20 ( 3a4 2a2 ) cr~  2 i~ r2 1 (6.2.7)
~00 =  l(a ) cos20 3.4 )
o~ =2 1 +~ + 2 \ r 4 +1 (6.2.8)
Figure 6.2(b) shows the normalized hoop stress 6o0 distribution as a function of the normalized radius r/a along the x axis (0 = 0). As shown in the figure, the normalized hoop stress quickly reduces from 3 to approach to 1 as r/a increases. The hoop stress along the circumference of the circular hole surface is a function of the angular location 0. Figure 6.2(c) shows the normalized hoop stress 600 as a function of 0 along the circumferential surface of the hole (r = a). As shown in the figure, the normalized stress 60o reduces from 3 to  1 and increases to 3 as 0 increases from 0 ~ to 90 ~ and then to 180". The hoop stresses at the locations of 0 = 0 '~, 90 ~, 180 C~, and 270 ~ are schematically shown in Figure 6.2(a). The stress concentration factor Kt for the circular hole in a twodimensional infinite solid under remote uniaixal tension equals 3. Now consider the case of the circular hole under equal biaxial tension as shown in Figure 6.3. By using the superposition of the elasticity, the stress concentration factor Kt can be obtained as 2. From the example of the circular hole as discussed previously, the stress concentration factor depends not only on geometry but also on loading.

240 FRACTURE MECHANICS AND FAT IGUE CRACK PROPAGATION
(a)
T TOT T
I lol I 3.5
3 ao0
(b)
0= 0 ~
r/a
~Oe 1
r=a
o
(c) FIGU RE 6 .2 A circular hole in a twodimensional infinite solid under a remote tension a~. (a) Cartesian and polar coordinate systems (b) the normalized hoop stress #00 distribution as a function of the normalized radius r/a along the x axis (0 = 0~ (c) the normalized stress #00 as a function of 0 along the circumferential surface of the hole (r = a).

GRIFF ITH 'S FRACTURE THEORY FOR BRITTLE MATERIALS 241
o'~
I I I I v
v
I 1o I I Kt=2
v
FIGU RE 6 .3 A circular hole in a twodimensional infinite solid under an equal biaxial remote tension r~:~.
6.2.3 LIMIT CASE FOR ELLIPTICAL HOLE
As the elliptical hole becomes increasingly flatter or b/c approaches 0, the geometry of the hole approaches that of a crack. In this case, b

T T A
242 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
1 1 1 1 O"
F IGURE 6 .4 A large twodimensional plate of thickness t under a remote uniform tensile stress a.
where V represents the total volume of the plate. The energy release rate per unit crack area can be expressed as
0q~ 1 d4~ o"2r . = = re   (6.3.2)
OA,. 2t dc E
where A,. represents the crack area. In this case, A,.  2ct. Now, the stress at fracture can be denoted as a/. At fracture the energy release per unit crack area growth should equal the surface energy 7 of the newly created crack surface. Therefore,
04, a~.c 0A~. r~ E 27 (6.3.3)
In this case, the fracture stress a /can be obtained as
ar  ~ , ,E (6.3.4) V /re

FRACTURE PARAMETERS BASED ON LINEAR ELASTIC FRACTURE MECHANICS 243
6 .4 FRACTURE PARAMETERS BASED ON L INEAR ELAST IC FRACTURE MECHANICS (LEFM)
6.4.1 THREE PRIMARY LOADING MODES
Consider a segment of crack front as shown in Figures 6.56.7. Because of different loading conditions, the crack front can be subjected to three primary loading modes and their combinations. A Cartesian coordinate system is assigned such that the crack front is in the z direction. We consider idealized planar crack problems, in which the stresses and strains near the crack tip can be expressed in terms of the inplane coordinates x and y only. As shown in Figure 6.5, the crack is subject to Mode I, the opening or tensile mode, where the inplane stresses and strains are symmetric with respect to the x axis. As shown in Figure 6.6, the crack is subject to Mode II, the sliding or inplane
F IGU RE 6 .5 Mode I or opening mode.
FIGURE 6.6 Mode II or inplane shearing mode.
FIGURE 6.7 Mode III or outofplane shearing mode.

244 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
shearing mode, where the stresses and strains are antisymmetrical with respect to the x axis. As shown in Figure 6.7, the crack is subject to Mode III, the tearing or antiplane shearing mode, where the outofplane stresses and strains are antisymmetrical with respect to the x axis.
6.4.2 ASYMPTOTIC CRACKTIP FIELD FOR LINEAR ELASTIC MATERIALS
Here, the asymptotic cracktip stresses and strains for different modes of loading are explored for linear elastic isotropic materials. As shown in Figure 6.8, we concentrate on a very small area near a crack tip, where the stresses and strains should be naturally expressed in terms of the polar coordinates r and 0. The three stresses art, a00, and ar0 are shown for a material element. The two inplane equilibrium equations for the plane problem are
Oqrr 10arO qrr  r Or + ," ~0 + ~ r = 0 (6.4. l)
OarO 10aO0 2arr 0t 7 +r~ + r  0 (6.4.2)
As schematically shown in Figure 6.8, the traction on the crack surfaces should vanish. This leads to the traction boundary conditions on the two crack faces as
(700 : (YrO : 0 at 0 = +n (6.4.3)
Combining the compatibility equation, the two inplane equilibrium equations and the elastic Hooke's law under both plane stress and plane strain conditions leads to
F I G U R IE 6.8 A crack with the Cartesian and polar coordinate systems centered at the tip.

FRACTURE PARAMETERS BASED ON L INEAR ELASTIC FRACTURE MECHANICS 245
0 1 0 1 0 2 ) JrE~F Jr ~ ~ ( ff rr + a oo )  0 (6.4.4)
Solutions are sought for the three governing equations" Equations 6.4.1, 6.4.2, and 6.4.4, with the homogeneous boundary conditions in Equation 6.4.3. The stresses can be assumed to be separable functions of r and 0 as
ar r  Cr2(Trr(O) (6.4.5)
aoo Cr~6oo(O) (6.4.6)
arO  Cr~?rro(O) (6.4.7)
where 2 represents an eigenvalue. C is an undetermined constant, which represents the singularity amplitude when 2 is negative. 6rr(O), ?fro(O), and ~00(0) represent the normalized stress functions of 0.
When the boundaryvalued problem is solved based on the separable functions in Equations 6.4.56.4.7, the allowable singular solution is with 2 = 1 /2 by consideration of finite strain energy for a small circular region of size r centered at the crack tip. Under plane strain or plane stress condi tions, two primary modes, Modes I and II, are considered. For Mode I, the stresses are symmetrical with respect to the crack line direction (0 = 0). Therefore, the stress intensity factor K~ for Mode I is defined as the singular ity amplitude of the opening stress aoo directly ahead of the tip (0 = 0) as
KI  lim 2x~~raoo(r, 0  0) (6.4.8) r,0
For Mode II, the stresses are antisymmetrical with respect to the crack line direction (0 = 0). Therefore, the stress intensity factor KII is defined as the singularity amplitude of the shear stress at0 directly ahead of the tip (0 = 0) as
KII  lim 2v/2~'~ra,.o(r, 0  0) (6.4.9) r,0
Substituting Equations 6.4.56.4.7 into the governing equations in Equa tions 6.4.1, 6.4.2, and 6.4.4 with use of the boundary conditions in Equation 6.4.3 and the definitions in Equation 6.4.8 for KI, the asymptotic stresses under Mode I loading conditions can be obtained as
KI s3 0 a00 2~co ~ (6.4.10)
o',.r = ~~rcos ~ 1 + sin 2 (6.4.11)
KI sin 0 s20 (6.4.12)

246 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
Similarly, the asymptotic stresses under Mode II loading conditions are
Kn ( 3s in0 ~) o'oo  cos 2 (6.4.13)
KII sin0 ( ~) tTrr : ~ ~ 1  3 sin 2 (6.4.14)
KII O ( ~ ) arO  ~ cos ~ 1  3 sin 2 (6.4.15)
The Mode III stress intensity factor g l I I can be defined as the singularity amplitude of the shear stress tr0 directly ahead of the tip (0 = 0) as
KI I I   lim v/2~nrao:(r, 0 = 0) (6.4.16) r+0 Following the similar procedure as summarized above to solve the antiplane boundaryvalued problem with the homogeneous stress boundary condi tions, the asymptotic stresses can be obtained as
KIII 0 a0= = ~cosx (6.4.18)
v'2nr z
Kill 0 ar=  ~ sin  (6 .4 .17)
2
For the asymptotic analyses, because the boundary conditions are homo genous (aoo = trrO = trOz = 0 at 0 = +n), only the angular dependence of the stresses can be determined. The singularity amplitudes KI, KII, and gll I cannot be determined by the asymptotic analyses. KI, KII, and KIII depend on the crack geometry and loading conditions. In general, the solutions of KI, KII, and KIII for relatively simple geometric configurations and loading conditions can be found in stress intensity factor handbooks such as in Tada et al. (2000).
6.4.3 STRESS INTENSITY FACTOR SOLUTIONS FOR S IMPLE GEOMETRIES AND LOADING CONDITIONS
For small cracks in infinite or semiinfinite solids, dimensional analyses can guide the selection of the form of the stress intensity factor solutions. For example, KI has a dimension of
[F] [L]i/2 [KII  ~ (6.4.19)
where [KI] represents the dimension of the stress intensity factor KI, [F] the force dimension, and [L] the length dimension. The unit of stress intensity factor KI can be MPa m 1/2 or ksi.in 1/2. We now consider three geometric and loading conditions as shown in Figures 6.96.11.

FRACTURE PARAMETERS BASED ON L INEAR ELASTIC FRACTURE MECHANICS 247
i T
I 1ol I
i l
K= t r f~
A center crack in a twodimensional infinite solid under a remote uniform FIGURE 6 .9 tensile stress a.
I 1ol 1 K= 1.12 JE~
F IGURE 6 .1 0 An edge crack in a twodimensional semiinfinite solid under a remote uniform tensile stress a.

248 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
K= 2 o" J ~ /1;
F IGURE 6 .1 1 A pennyshaped crack in a threedimensional infinite solid under a remote uniform tensile stress ~.
6.4.3.1 Center Crack in a TwoDimensional Infinite Solid
Figure 6.9 shows a center crack with the crack length 2a in a twodimen sional infinite solid under a remote uniform tensile stress e. For the center crack, the stress intensity factor solution is
KI = ev /~ (6.4.20)
6.4.3.2 Edge Crack in a TwoDimensional Semiinfinite Solid
Figure 6.10 shows an edge crack with the crack length a in a two dimensional semiinfinite solid under a remote uniform tensile stress a. For the edge crack, the stress intensity factor solution is
KI = 1.12trx,/~ (6.4.21)
6.4.3.3 PennyShaped Crack in a ThreeDimensional Infinite Solid
Figure 6.11 shows a pennyshaped crack with the diameter 2a in a three dimensional infinite solid under a remote uniform tensile stress e. For the pennyshaped crack, the stress intensity factor solution is
2 KI   ov"~a (6 .4 .22)
717

FRACTURE PARAMETERS BASED ON L INEAR ELAST IC FRACTURE MECHANICS 249
The stress intensity factor solutions for these three crack problems have the same forms except the proportionality constants for different geometries and loading conditions. The stress intensity factor solutions must be propor tional to the magnitude of the remote uniform tensile stress a due to the linearity of linear elasticity. The stress intensity factor solutions must be proportional to the square root of the only significant length dimension, which is the crack length a. For relatively simple geometries and loading conditions, finite element or other numerical methods can be used to find the relevant proportionality constants in the stress intensity factor solutions determined by dimensional analyses.
In general, for a centercracked panel or an edgecracked panel of finite size, the stress intensity factor can be written in the following form:
KI : F (~)Sg~ (6.4.23)
where a represents the crack length; Sg the gross nominal stress, either the nominal stress or the maximum nominal bending stress; and F(~) a geometric function of the normalized crack length ~. Here, ~ = a/b, where b represents the width of the panel.
6.4.3.4 CenterCracked Plate Under Tension
Figure 6.12(a) shows a centercracked panel subject to a tensile load P. For the centercracked panel in tension, Sg and F(~) are
P Sx = 2bt (6.4.24)
1  0.5~z + 0.32672 F(~) = for h/b > 1.5 (6.4.25)
v/1 ~
where ~ represents the normalized crack length, h the height of the panel, and b the width of the panel.
6.4.3.5 EdgeCracked Plate Under Tension
Figure 6.12(b) shows an edgecracked panel subject to a tensile load P. For the edgecracked panel in tension, Sg and F(~) are
P Sg = bt (6.4.26)
0.857 + 0.265~ F(~)  0.265(1  ~)4 + for h/b>_l.O (6.4.27)
(1 00 3/2
where ~ represents the normalized crack length, h the height of the panel, and b the width of the panel.

250 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
J
(a)
h
a
I
J
J
(b) FIGURE 6.1 2 (a) A center cracked panel subject to a remote tensile load P. (b) An edge cracked panel subject to a tensile load P.
Continued

FRACTURE PARAMETERS BASED ON L INEAR ELASTIC FRACTURE MECHANICS 251
M
FIGURE 6. 1 2
\ . _ _ J (c) M
Cont'd (c) An edge cracked panel subject to a bending moment M
6.4.3.6 EdgeCracked Plate Under Bending
Figure 6.12(c) shows an edgecracked panel subject to a bending moment M. For the edgecracked panel in bending, S e and F(~) are
6M S x = b2 t (6.4.28)
I0 41 F(~)  V/~~ tan5rc~ .923 +0.199(lrc~ sin~) for large h/b 's "  cos 2
(6.4.29)
where ~ represents the normalized crack length, h the height of the panel, and b the width of the panel.
Figure 6.13 show the geometric function F(~) for the three cases. As shown in the figure, as ~ becomes smaller, F(~) approaches 1 for the centercracked panel case and 1.12 for the edgecracked panel cases in order to be consistent with the solutions for a center crack in an infinite solid and an edge crack in a semiinfinite solid under remote tensile stress. As shown in the figure, when z~ increases, F( ~) increases.
6.4.3.7 Cracks Emanating from a Hole Under Remote Tension
As shown in Figure 6.14, small cracks with the crack length I are emanat ing from a hole with a radius of c under a remote tensile stress S. When l

252 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
u. 3
0 0
/ K= FSg fE~ ! "
I
(a) % P /! .:" 2bt / 
/ :" (b) %= P / .:.
bt / . /  6M / / . . . . . . . . . . . . (c) sg= ~/
_ / .." / .:
ii .." / ."
 f / ..,.'" l J ..."
2.
1 1 l 1 I I 1 1 02 04 06 08
a = a/b
F IGURE 6 13 Plots of F(~) for the three cracked panels.
j
i ISl l a
I
C
b
p
b .
1111 S
a

FRACTURE PARAMETERS BASED ON L INEAR ELAST IC FRACTURE MECHANICS 253
1s = FSv/ffd (6.4.31)
where a = c + I. In general, the stress concentration factor K can be expressed as (Tada et al., 2000)
K = r(d)Sv/~ (6.4.32)
Here, the geometric function F(d) is in terms of the normalized crack length d as
where
F(d) = 0.5(3  d)(1 + 1.243(1  d) 3) (6.4.33)
l l d =  = (6.4.34)
a c+l
6.4.4 NONSINGULAR ASYMPTOTIC CRACKTIP FIELD FOR LINEAR ELASTIC MATERIALS
As discussed earlier, 2 =  1/2 is adopted for the most singular asymptotic cracktip stress and strain solutions. The next available solution is based on 2 = 0 to satisfy the governing equations and boundary conditions for the asymptotic cracktip fields. The nonsingular stress satisfied the boundary conditions for the inplane modes is the normal stress axx or O'll in the crack line (x or Xl) direction. The nonsingular stress that satisfied the boundary conditions for the antiplane mode is the shear stress a=x or a31. Under generalized plane strain conditions, we can write the asymptotic cracktip fields with the nonsingular stresses T for the inplane modes and S for the antiplane mode (Rice, 1974) as
Kill 1 __KI I ~KII ~!(0) + v/~_ r 6I I (0) ~ ij = 2,/~ ,, ,j ( o ) +
+ 6il 6jl T + 6i36j3 v T + 6il 6j3S +. . .
(6.4.35)
where 6,~(0), 6,~!(0), and &HI(0)are the normalized angular functions for Mode I, II, and llI asymptotic stress solutions, respectively, as specified earlier. The range of the subscripts i and j is from 1 to 3. Here, 6,) represents the components of the Kronecker delta function. For example, ~il equals 1 when i equals 1 and 6;1 equals 0 when i does not equal 1. The T and S stresses, just as the stress intensity factors K~, K~I, and KIII are functions of the geometries and loading conditions [e.g., see Sham (1991)].
Figure 6.15 shows a center crack in a twodimensional infinite solid subject to remote biaxial stressing conditions. Here, a represents the normal stress perpendicular to the crack and Ra represents the normal stress in the crack line direction. The stress intensity factor solution for this case is

254 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
ll ll
RG Ro"
v
v
I I F IGURE 6 .1 5 A center crack in a twodimensional infinite solid subject to remote biaxial stressing conditions.
Kl  ax /~ (6.4.36)
The T stress solution can be expressed as
T R  I = ~ (6.4.37)
Kl x /~
Substituting the stress intensity solution into the T stress solution gives a simple solution for the T stress as
T  a(R  1) (6.4.38)
Therefore, under uniaxial tension, R = 0 and T  0 . Under equal biaxial tension, R=I and T=0. When R> 1, T>0 and when R< 1, T 0, there should be unstable Mode I crack growth (Cotterell and Rice, 1980). The effects of the T stress on the stability of Mode I crack growth is schematically shown in Figure 6.16.
6.4.5 CIRCULAR AND ELLIPTICAL CRACKS
6.4.5.1 Circular Cracks
In practical situations, circular and semicircular cracks are usually found in engineering structures. The stress intensity factor solution for a

FRACTURE PARAMETERS BASED ON LINEAR ELASTIC FRACTURE MECHANICS 255
TO
Unstable Mode I Growth
FIGURE 6.1 6 A schematic plot of the effects of the T stress on the stability of mode I crack growth.
pennyshaped crack in a threedimensional infinite solid under a remote tensile stress S is available as discussed earlier"
2 KI   S~/~d (6.4.39)
rt
where a represents the crack length. When the pennyshaped crack is in a finite body, the size of the body can affect the stress intensity factor solution. When the crack length a is small compared with the width or the thickness of the cracked structures, Equation 6.4.39 is a good approximation. Now, we will concentrate on surface cracks. For a semicircular surface crack in a rectangular bar under a tensile force P and a bending moment M as shown in Figure 6.17(a), the stress intensity factor solution varies along the crack front (Newman and Raju, 1986). The maximum values occur at the crack front on the free surface. The gross nominal stresses for the bar due to tension and bending, St and S~, can be expressed as
P St = 2bt (6.4.40)
3M Sb  bt 2 (6.4.41)
where b represents the width and t the thickness. For small surface cracks, the maximum values of the stress intensity factor
solutions due to tension and bending can be expressed, respectively, as
KI  0.728Stff~a (6.4.42)
Ki  0 .728Sb~ (6.4.43)
The stress intensity factor solution decreases by about 10% at the crack deepest front.

256 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
P
(a)
s i s S
e e
l P
L
s~ s Ss S i s ,,,..z, M ,,//
(b) P
F IGURE 6 .17 (a) A semicircular surface crack (b) in a rectangular bar under tension and bending. (b) A semicircular surface crack in a circular bar under tension and bending.
For a semicircular surface crack in a circular bar under a tensile force P and a bending moment M as shown in Figure 6.17(b), the gross nominal stresses for the bar due to tension and bending, St and Sb, can be expressed, respectively, as (Newman and Raju, 1986)

FRACTURE PARAMETERS BASED ON L INEAR ELAST IC FRACTURE MECHANICS 257
4P St = nd 2 (6.4.44)
32M Sh nd 3 (6.4.45)
where d represents the diameter. For small surface cracks, the maximum values of the stress intensity factor
solutions due to tension and bending can be expressed, respectively, as
K~ = 0.728St x /~ (6.4.46)
KI = 0.728Sh x /~ (6.4.47)
6.4.5.2 Elliptical Cracks
The stress intensity factor solution for an elliptical crack in an infinite solid under a remote tensile stress S is also available. Figure 6.18 shows an elliptical crack in an infinite solid under a remote tensile stress S. The major axis for the elliptical crack front is c and the minor axis is a. Figure 6.19 shows a top view of the crack and the coordinate system. The stress intensity factor varies along the crack front. The maximum value of K~ occurs at Point D and the minimum value of KI occurs at Point E. The stress intensity factor solution KI is expressed in terms of the orientation angle ~b as shown in Figure 6.19.
The solution KI is expressed as a function of the orientation angle q5 as
FIGURE 6. 1 8 An elliptical crack in an infinite solid under a remote tensile stress S.

258 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
D
 . . , v i  . . , v I
FIGURE 6 .19 Top view of the elliptical crack and the coordinate system.
K  S ~rt/~fo (6.4.48)
where
[(a) ]88 a f~  2cos2 4~ + sin2 4~ for  1 (6.4.49) c and
f.,. v / V~ = E(k) = 1  k 2 sin 2 ~dB (6.4.50) J0 where Q represents the flaw shape factor and
(a) 2 k 2  1  (6.4.51)
Here, E(k) is the elliptical integral of the second kind. E(k) can be approxi mated as
( (a ) 165) 1/2 v/Q E(k),~ 1+ 1.464 (6.4.52)
The maximum value of KI occurs at Point D where q~ = 90 ~ and f0 = 1. The minimum value of KI occurs at Point E where ~b 0 ~ and f~, x/~/c. Therefore, the maximum value of KI at Point D is
KD  S ~ (6.4.53) E(k)
The minimum value of KI at Point E is
~~ (6.4.54)

PLASTIC ZONE S IZE, REQUIREMENT OF L INEAR ELASTIC FRACTURE MECHANICS 259
For semielliptical cracks, the maximum value of KI still occurs at Point D when the ratio a/c is small. When the ratio a/c is close to unity, the maximum value of/(i occurs at Point E for the semicircular surface cracks. For semielliptical surface cracks, the stress intensity factor solutions should be based on the embedded elliptical crack solutions multiplied by the free surface factor of 1.12.
6.5 PLAST IC ZONE SIZE AND REQUIREMENT OF L INEAR ELAST IC FRACTURE MECHANICS (LEFM)
6.5.1 EST IMATION OF PLAST IC ZONE S IZE AHEAD OF A MODE I CRACK
We consider a crack with the Cartesian and the polar coordinate systems centered at the tip as shown in Figure 6.20. Here, z represents the out ofplane coordinate. For inplane modes (Modes I and II), the outofplane shear stresses are
a:r  a:0  0 (6.5.1)
6.5.1.1 Plane Stress Conditions
Under plane stress conditions, the outofplane normal stress is
a::  0 (6.5.2)
We can estimate the plastic zone size at 0  0 directly ahead of a crack tip. Based on the asymptotic stress solutions in Equations 6.4.106.4.12, the stresses ahead of the crack tip at 0  0 are
KI f f r r : (700   V/~~r (6.5.3)
6rO : 0 (6.5.4)
S FIGURE 6 .20 A crack with the Cartesian and the polar coordinate systems centered at the tip.

260 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
Figure 6.21 shows schematically the distribution of 0.00 as a function of the radial distance r or x ahead of the crack tip. As shown in the figure, 0.oo increases as x decreases. As the stress increases, plastic flow generally occurs for metallic materials. The Mises yield criterion can be expressed in terms of the three principal stresses 0.,,, 0.o0, and 0.: as
1 )2 __)2 0.::)2] 0. g   ~ [ ( 0. rr   0.00 Jr ( 0.r r   0.. J r  (0 .00 (6.5.5)
where 0.o is the tensile yield stress. Under the plane stress conditions with Equation 6.5.2, the yield criterion
for the material element at 0 = 0 becomes
For Mode I cracks, this leads to
0.~  0.~0 (6.5.6)
0.oo = 0.0 (6.5.7)
As shown in Figure 6.21, when the yield criterion is satisfied, 0.o = 0.oo at r = ro~. Therefore, from Equation 6.5.3,
KI 0.0 = 0.oo = (6.5.8)
v/2xroo
Rearranging Equation 6.5.8 gives the plastic zone size ahead of the crack tip as
ro~  ~ ~oo (6.5.9)
Now we consider the elastic perfectly plastic material behavior. The tensile stressstrain curve for an elastic perfectly plastic material is shown in Figure 6.22. For the elastic perfectly plastic material, the plastic zone should be larger than the size of r0o estimated from the linear elastic asymptotic stress solution. An estimated plastic zone size for the elastic perfectly plastic material as schematically shown in Figure 6.21 can be expressed as
aoo
FIGURE 6 .2 1 the crack tip.
a0 !  " , ,
i 
~ X r0o 2r0o
The distribution of aoo as a function of the radial distance r or x ahead of

PLASTIC ZONE SIZE, REQUIREMENT OF LINEAR ELASTIC FRACTURE MECHANICS 261
%
F IGURE 6 .22 The tensile stressstrain curve of an elastic perfectly plastic material.
r0o   (6.5.10t
6.5.1.2 Plane Strain Condit ions
Under plane strain conditions, e:.~  0. This leads to
o::  v(oxx + o . , )  v (o . + aoo)
Directly ahead of the crack tip at 0  0, ar0  0 and
KI ~rr ~ (700 ~ V/~zr
Based on Equations 6.5.11 and 6.5.12, a_ is
a:  2vaoo
The yield criterion for the material element at 0  0 now becomes
%2 _ [(1  2v)ooo] 2
For Mode I cracks, when the yield criterion is satisfied,
oo  (1  2v)aoo
or
(6.5.11)
(6.5.12)
(6.5.13)
(6.5.14)
(6.5.15)
For example, when v = 0.3, o00 has to be equal to 2.5 o0 to satisfy the yield criterion under plane strain conditions. Therefore, the plastic zone size ahead of the crack tip should be smaller under plane strain conditions when compared to that under plane stress conditions. F rom Equation 6.5.12,
1 o00  ~ o0 (6.5.16)
1 2v

262 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
at r  rot. This leads to
~0 K~ = 0"00  (6.5.17)
1  2v v/21tr0~,
(1  2v)2 (K I ) 2 ro~  2rt ~oo (6.5.18)
Similar to the plane stress case, the plastic zone size estimated for elastic plastic materials is
2ro~. (1  2v)2 (~)2 : (6.5.19)
For example, when v  0.3, (1  2v) 2  0.1 6. Therefore, the plastic zone ahead of the crack tip under plane strain conditions is much smaller than that under plane stress conditions. Estimation of the plastic zone from Irwin can be written as
ro~  ~ ~oo (6.5.20)
1 (K I ) 2 2r0~,  ~ ~00 (6.5.21)
6.5.2 ESTIMATION OF PLASTIC ZONE SIZE AND SHAPE UNDER MODE I LOADING CONDITIONS
Plastic zone size and shape for Mises materials can be estimated based on the asymptotic linear elastic cracktip stress fields. The Mises yield criterion in the cylindrical coordinate system for the inplane modes can be expressed as
1 02 __ 2 [(0rr   0"00) 2 if (0"00  0"zz) 2 ~ (0"zz   0"rr) 2] + 30"r20 (6.5.22)
The asymptotic stress fields in Equations 6.4.106.4.12 can be substituted into the Mises yield criterion in Equation 6.5.22.
Under plane stress conditions, the plastic zone size rp(O) as a function of 0 can be obtained as
 (gl)2[~ 3 ] rp(O) = ~ ~00 (1 + cos 0) + ~ sin 2 0 (6.5.23)
Under plane strain conditions, the plastic zone size rp(O) as a function of 0 can be obtained as
 (gi)2[~ 3 ] rp(O) = ~ ~00 (1  2v)2(1 + cos 0) + ~ sin 2 0 (6.5.24)

PLASTIC ZONE S IZE, REQUIREMENT OF L INEAR ELASTIC FRACTURE MECHANICS 263
The size and shape of the plastic zones based on Equations 6.5.23 and 6.5.24 in the normalized coordinate system under plane stress and plane strain conditions for v = 0.3 are shown in Figure 6.23. As shown in the figure, the size of the plastic zone under plane strain conditions is much smaller than that under plane stress conditions.
The nonsingular stress T can also affect the plastic zone size and shape. However, the size and shape based on the linear elastic asymptotic cracktip fields do not in general agree well with the computational results for elastic plastic materials [e.g., see BenAoun and Pan (1996)]. Figures 6.24 and 6.25 show the general trends of plastic zone size and shape for different T stresses for elastic perfectly plastic materials under smallscale yielding plane strain and plane stress conditions.
6.5.3 REQUIREMENT OF LINEAR ELASTIC FRACTURE MECHANICS
The most important requirement of linear elastic fracture mechanics (LEFM) is that the size of the plastic zone near the crack tip should be much smaller than any relevant length dimensions for the crack problem of interest. In this way, the stress intensity factor K controls the plastic deform ation near the crack tip, and, in turn, the fracture process near the crack tip. Figure 6.26 shows a schematic representation of the K field and the plastic zone near the tip of a crack with the length a in a specimen. The requirement of LEFM is
1.5
1
Plane Strain
1
1.5
20.5 o 0.5 1 1.5 2~Oo2X
0.5
o
0.5
L I I
2 2.5 3 3.5
Plane Stress
FIGU RE 6.23 Estimations of the size and shape of the plastic zones in the normalized coordinate system under plane stress and plane strain conditions for v = 0.3.

264 FRACTURE MECHANICS AND FAT IGUE CRACK PROPAGATION
) '~ ' " "  T < 0
<
T=0
T>0
FIGU RE 6 .24 The general trends of plastic zone size and shape for different T stresses under smallscale yielding plane strain conditions.
    . S T>0
T

PLASTIC ZONE S IZE, REQUIREMENT OF L INEAR ELASTIC FRACTURE MECHANICS 265
The definition of plane stress and plane strain conditions depends on the thickness of the specimens or structures and the load amplitude. For example, consider a cracked specimen with the thickness t and the plastic zone 2r0 on the lateral surface as shown in Figure 6.27. A Cartesian coordin ate system is shown in the figure. The crack front is in the z direction. The plastic zone size is larger on the specimen lateral surface, because the surface material element has no constraint and is under plane stress conditions. However, for the crack front in the middle of the specimen, the plastic zone becomes smaller because the material element has the constraint in the z direction. This concept is illustrated in Figure 6.28. Figure 6.28 shows the side views of the specimen with the plastic zones shown as the shaded regions. When the load P is small, most of the plastic zone along the crack front is small compared to the thickness of the specimen. The specimen is subjected to mostly plane strain conditions. When the load P is large, the plastic zone along the crack front becomes large compared to the thickness of the specimen and the specimen is subjected to plane stress conditions. The requirement of plane strain conditions is
t, a, h, b:>2.5 (K~~/2 (6.5.26)
6.5.4 FRACTURE TOUGHNESS TEST ING
Figure 6.29 shows a threepoint bend specimen and Figure 6.30 a compact tension specimen. These two types of specimens are used for fracture toughness testing according to the ASTM standard E399 (ASTM, 1997). The stress intensity factor KQ can be related to the load PQ by
P
z I~t 2to
a
9 X
/
7
FIGURE 6 .27 A cracked specimen with the thickness t and the plastic zone 2r0.

266 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
t
Plane Strain
y
z_[' Plane Stress F IGURE 6 .28 Side views of the plastic zone size distribution along the crack front. Shaded regions represent the plastic zones.
F IGURE 6 .29 A threepoint bend specimen.
1 /a \ KO  PQ t W ,vv . (6.5.27)
where F(a /W) represents the geometric function for the threepoint bend specimen and the compact tension specimen.
There are several ways to determine the load PQ from different types of the loaddisplacement curves. Figure 6.31 shows a simple rule for a Type I loaddisplacement curve. In the figure, P represents the load and v the displacement. The initial slope of the loaddisplacement curve is denoted as

PLASTIC ZONE SIZE, REQUIREMENT OF LINEAR ELASTIC FRACTURE MECHANICS 267
(3 7
i_ a I
(3
:ip
h
J
5 J
F IGURE 6.30 A compact tension specimen.
milx PO . . . . . . . .
F IGURE 6 .3 1 A simple rule to determine the load PQ for a typical Type I load displacement response.
K. The intersection of the loaddisplacement curve and a line with the slope 0.95 K from the origin defines the load Ps. PQ is the smaller value of P5 or the maximum load Pmax.
Figure 6.32 shows a schematic plot of KQ as a function of the thickness t. In general, as the thickness t increases, the value of KQ decreases. The fracture toughness K1c is the asymptotic value of K 0. Under the plane strain conditions, the thickness and other dimensions of the specimens must satisfy Equation 6.5.26. The values of Kit are in the range of 20200 MPax/~ for metals, 15MPax/~ for polymers, and 15MPav/m for ceramics. Typical values of the fracture toughness KI~ for several metals are listed in Table 6.1 (Dieter, 1988). In general, as the yield stress increases, the value of Kit usually decreases.

268 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
Ko
Ktc
t FIGURE 6 .32 A schematic plot of KQ as a function of the thickness t. The asymptotic value represents the fracture toughness Kt,..
TAB L E 6.1 Typical Values of Kh.
Fracture Toughness Material Yield Strength (MPa) KI,. (MPa v/m)
Maraging steel 1730 90 4340 steel 1470 46 Ti6AI4V 900 57 7075T6 A! alloy 500 24 2024T3 AI alloy 385 26
6.5.5 PLASTIC ZONE UNDER MONOTONIC AND CYCLIC LOADING CONDITIONS
As discussed earlier, the plastic zone sizes ahead of the crack tip under Mode I plane stress and plane strain loading conditions can be expressed as
2(rocr)monotonic I (K I ) 2 = ~ ~o (6.5.28)
1 (K I ) 2 2(r0*:)monotonic = 31t ~0 (6.5.29)
where a0 represents the uniaxial tensile yield stress under monotonic loading conditions.
However, under cyclic loading conditions, if we adopt the elastic perfectly plastic material behavior with an anisotropic kinematic hardening law, the effective Mises yield stress for the fully reversed plastic reloading is 2a0. Therefore, the estimations of the plastic zone sizes under cyclic loading conditions are one quarter of those under monotonic loading conditions.
I (K , ) ~ 1 2(r0cr)cyclic  ~ ~a0  4 2(r0a)monotonic (6.5.30)

PLAST IC ZONE S IZE , REQUIREMENT OF L INEAR ELAST IC FRACTURE MECHANICS 269
..r '  . /5

270 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
v (displacement)
, /
F IGURE 6 .34 A nonlinear elastic cracked solid under displacementcontrolled conditions.
/,
4
U = Strain Energy
v
F IGURE 6 .35 The loaddisplacement curve of a nonlinear elastic cracked solid. The area under the loadingdisplacement curve represents the strain energy U.
The energy release rate q is defined as
10~ IOU Ca =  t Oa = t Oa (6.5.33)
Y
Estimation of the energy release rate q based on the loaddisplacement curve is illustrated in Figure 6.36. For a small amount of crack growth Aa, the change of strain energy, A U, can be used to estimate the energy release rate q as
1AU q ~ (6.5.34)
t Aa
For linear elastic materials, the loaddisplacement curve is linear, as shown in Figure 6.37. The slope of the curve represents the stiffness of the

PLASTIC ZONE SIZE, REQUIREMENT OF L INEAR ELASTIC FRACTURE MECHANICS 271
a
area =  / tO
/ta
V V
F IGURE 6 .36 Estimation of the energy release rate q based on the change of the strain energy, A U, for a small amount of crack growth Aa.
I I
K = Stiffness
V
F IGURE 6.37 The linear loaddisplacement curve of a linear elasticcracked solid.
cracked solid as shown. The stiffness of the cracked solid, K(a), depends on the crack length a. Under displacementcontrolled conditions, the potential energy ~b of the cracked solid can be expressed as
1 1 cb U  Pv  K(a)v 2
The energy release rate G is defined as
q IOU t Oa
(6.5.35)
Estimation of q based on the change of the strain energy, A U, is illustrated in Figure 6.38.
q can be related to stress intensity factors for linear elastic materials. For a crack front under combined Modes I, II, and III loading conditions, the local energy release rate q (Irwin, 1957) is
1  v 2 1 + v K21 (6.5.37) G= E (K2 +K2I )+ E
 1K'(a)v2 (6.5.36) t

272 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
P
a i \ ,
area = .4U
" 8+.48
v
v
FIGURE 6 .38 Estimation of the energy release rate q based on the change of the strain energy, A U, for a linear elasticcracked solid.
Under plane stress conditions,
1 (KI2 + K~) (6.5.38) q2
For nonlinear elastic powerlaw hardening material, q can be related to the pathindependent J integral (Rice, 1968), which represents the singularity amplitude of the HRR cracktip singular stress and strain fields (Hutchinson, 1968; Rice and Rosengren, 1968)
6.6 FAT IGUE CRACK PROPAGATION BASED ON L INEAR
ELAST IC FRACTURE MECHANICS
6.6.1 EXTENDED SERVICE LIFE
Different inspection methods that use visual, Xray, ultrasonic waves, and electric potential drop methods can be employed to detect cracks. Once cracks are detected, fracture mechanics methods can be used to determine the extended life of the structures or components.
Figure 6.39 shows the concept by plotting the crack length a as a function of the number of cycles, N. In the figure, a; represents the initial crack length within a structure or component, aa a detectable crack length, and ac the critical crack length when the stress intensity factor of the crack reaches the critical value or the structure or component reaches fully plastic condi tions. The number of cycles when a crack is detected is denoted as Nd. The number of cycles when the crack length reaches the critical length ac is denoted as No. The extended life Noa equals Nc Na. The damagetolerant approach used in the aerospace industry is based on the worstcase analysis with the assumption that the largest cracks missed by each inspection exist in the structures or components.

FATIGUE CRACK PROPAGATION BASED ON L INEAR ELASTIC FRACTURE MECHANICS 273
a (Crack Length)
ac
ad
ai
/
v N
Extended Service Life ( N c  Nd)
FIGURE 6 .39 The concept of extended life based on the plot of the crack length a as a function of the number of cycles, N.
6.6.2 CRACK GROWTH BEHAVIOR
Linear elastic fracture mechanics has been successfully used to model the fatigue crack growth behavior. Consider a cracked structure with a crack length a under cyclic stressing conditions, as shown in Figure 6.40(a). The applied stress S as a function of time t is shown in Figure 6.40(b). The maximum and minimum stress intensity factors Kmax and Kmin are linearly related to the maximum and minimum applied stresses Sm~ and Smin, respectively, according to the linear elastic fracture mechanics.
Kmax = FSmaxX~ (6.6.1)
Kmin  rSmin x /~ (6.6.2)
where F represents the function depending on the geometries and loading conditions.
The stress intensity factor range AK is defined as
AK = Kmax  Kmin (6.6.3)
The load ratio R in the stresslife approach is defined as
Smin R = (6.6.4)
Smax
Due to the linearity, R also represents the stress intensity factor ratio
Kmin R = (6.6.5)
Kmax

274 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
i l Sl l
(a)
8
I lsl I ~ax
S~
S~io
(b) = t
F IGURE 6 .40 (a) A cracked structure with a crack length a under cyclic stressing conditions. (b) The applied stress S varies as a function of time t.
Based on experimental results, Figure 6.41 shows schematically the crack growth rate da/dN as a function of the stress intensity factor range AK in a loglog scale. Note that 10 7 mm represents the spacing between atoms. In Region I, when AK decreases, the crack growth rate drops significantly. The asymptote AKth is the threshold of the stress intensity factor range below which no fatigue crack growth should occur. The values of AKth are 516 MPa v~ for steels and 36 MPa x/~ for aluminum alloys, depending on the value of R. In Region III, when AK is large, the crack growth rate accelerates significantly. This can happen when Kma~ approaches Kc, which represents the critical K for crack initiation for a given thickness.

FATIGUE CRACK PROPAGATION BASED ON L INEAR ELASTIC FRACTURE MECHANICS 275
log ( da
(mm/cycle)
103
105 ._
107
I !
I I 
I I
I I I / i i
, I ! i ! ! i ! i i ! i i ! i !
I
zlK, h
FIGURE 6 .4 1 intensity factor range AK in a loglog scale.
da = C(,~K)" EN
ID
Iog(aK) Crack growth rate d~ddN schematically plotted as a function of the stress
In Region II, the crack growth rate da/dN can be approximately linear related to the stress intensity factor range AK in the loglog scale plot. The Paris law for crack growth (Paris et al., 1961; Paris and Erdogan, 1963) is
do = C(AK) m (6.6.6a)
dN
where C and m represent the material constants. To obtain the values of C and m, a leastsquare method can usually be adopted to fit the fatigue data in the loglog scale plot of da/dN as a function of A K as
where
Y A + BX (6.6.6b)
(d~ Y log ~~ ,X  log(AK) ,A  logC, andBm (6.6.6c) The typical value of m is 3.0 for ferritepearlite steel, 2.25 for matensitic steel, and 3.25 for austenitic steel. The load ratio R can affect the fatigue crack growth rate. Typical values of material constants C and m for steels are listed in Table 6.2 (Barsom and Rolfe, 1987). Two approaches with consideration of the load ratio effects are discussed in the following section.
6.6.3 WALKER EQUATION
Load ratio effects can be accounted for by an equivalent zeroto maximum stress intensity factor range AK introduced by Walker (1970)"

276 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
TABLE 6 .2 Typical Values of Material Constants C and m in the Paris Law for R ,~ 0
Material C ~ m
Martensitic steel 1.36 x 10 ...7 2.25 Ferriticpearlitic steel 6.89 x 10 9 3 Austentic steel 5.61 10 9 3.25
"The values of C are obtained with the units of MPa ~ for AK and mm/cycle for da/dN.
AK Kmax(1  R) 7 (6.6.7)
where y is a constant in the range of 0.4 to 0.9. The Walker equation can be rewritten as
AI~ AK(1  R ) ;'~ (6.6.8)
or
AK AK  (6.6.9)
(1  R) 1~'
The Paris law can be rewritten as
da  CI (AK) m' (6 .6 .10)
dN
where Cl and m~ are material constants. For R  0, Equation 6.6.9 becomes
AK  AK (6.6.11)
Therefore,
da = C~(AK) m' (6.6.12)
dN
For R > 0, substituting Equation 6.6.9 into 6.6.10 gives
dN = C~ (1  R) 17 (6.6.13)
Rearranging Equation 6.6.13 gives
da C~ dN  (1  R) '''(l;') (AK)""
Comparing Equation 6.6.13 with 6.6.6(a) gives
C __ Cl
(1  R) ''''~l ;'~
(6.6.14)
(6.6.15)

FATIGUE CRACK PROPAGATION BASED ON LINEAR ELASTIC FRACTURE MECHANICS 277
and
m  ml (6.6.16)
For R < 0, 7  0. Therefore,
AK AK AK = ~ = Kmax (6.6.17)
(1  R) 17 (1  R)
Therefore,
do ml = Cl gmax (6.6.18) dN
I fT  1,
m
AK AK
(1  R) 17 = AK (6.6.19)
Therefore, there is no effect of R on the crack growth rate.
6.6.4 FORMAN EQUATION
The crack growth depends on Kc, which is the critical stress intensity factor for a material and a given thickness. The crack growth rate is written as (Forman et al., 1967)
or
da C2(AK) m2 = (6.6.20)
dN (1  R)K,.  AK
da C2(AK) m2
dN (1  R) (K c  gmax) (6.6.21)
Typical values of material constants for the Walker and Forman equations are listed in Table 6.3 (Dowling, 1999). When the cracktip plastic zone becomes large compared with the relevant physical length of the cracked solid, linear elastic fracture mechanics should not be used. Use of the crack tip opening displacement (CTOD) or the J integral to model the fatigue crack growth is possible.
6.6.5 LIFE EST IMATION
Life estimation under constant load amplitude loading conditions usually needs numerical integration. In general, AK increases as a increases with AS being kept as constant. The crack growth rate da/dN can be expressed as
da = f (AK , R) (6.6.22)
dN

278 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
TABLE 6 .3 Typical Values of Material Constants in the Walker and Forman Equations
Walker Equation Forman Equation
Material C~ ml ~/(R>0) ~(R

FATIGUE CRACK PROPAGATION BASED ON L INEAR ELASTIC FRACTURE MECHANICS 279
1 ~f 1 N~f  (6.6.29)
C(FAS~)m m _ 1 a~ i
Equation 6.6.29 indicates that N,y is very sensitive to the selection of a~. Because F is a function of geometry, loading, and crack length, a numerical integration is usually needed. The numerical integration is usually carried out by taking F as a constant for a small range of a.
From the fracture mechanics viewpoint, a structure of interest contains initial defects or cracks. Therefore, a dominant crack with an initial crack length can be found to correlate the fatigue life of the structure under cyclic loading conditions. From the strainlife or stresslife viewpoint, the fatigue initiation life for the material near the notches or stress concentrators in the structure can be estimated. Then, the fatigue life due to crack propa gation can be estimated by the fracture mechanics methodology based on the Paris law. The total fatigue life of the structure is the combination of the initiation life and propagation life [e.g., see Dowling (1979) and Socie et al. (1984)].
Example 6.6.1. Consider a very large plate with a small center crack. The initial total crack length 2a is 20mm. The plate is subjected to repeated remote stress cycles from O'min  0 MPa to O'max  75 MPa. The initial crack grows as the stress cycles are applied. The critical stress intensity factor Kc for fracture initiation for the plate is 25 MPa v/m. The material is assumed to have a very high yield stress such that the plastic zone size is small compared to the crack size under the applied stresses. Assume that the Paris law will be applicable to the plate material to characterize the fatigue crack growth
da = C(AK) m (6.6.25)
dN
where C  1 x 10 9 and m  3. The value of C is obtained with the units of MPa x/~ for AK and mm/cycle for da/dN.
1. Determine the final half crack length af at fracture initiation. 2. Determine the number of cycles before the fracture initiation of the plate.
Solution. For a center crack in a large plate,
KI  ov"~ (6.4.20)
The critical Kc for fracture initiation is 25 MPa v/m. Fracture initiation takes place when Kl Kc at the final half crack length af under the maximum applied stress O'ma x . Equation 6.4.20 becomes
gc  O'maxx/~ (6.6.30)

280 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
The final half crack length af can be obtained by substituting Kc = 25 MPa and O'ma x  75 MPa into Equation 6.6.30 as
af = 35.37 mm
The initial half crack length ai is
ag = 10mm
The stress range Aa is determined as
Ao" = O'max   O'min " 75 MPa
For a center crack in a large plate, the geometric parameter F in Equation 6.6.29 is 1. Equation 6.6.29 becomes
( (ai) "71 ) 1 ~ 1
Ur C(Ao'~/~)m(2  1)ai~I (6.6.29a)
Based on Equation 6.6.29 and the given values of C and m, the number of cycles Nr for crack propagation from the initial half crack length ai to the final half crack length af under the stress range Aa is obtained as 3.99 106.
Example 6.6.2. Consider a spotweld cup specimen of dualphase steel as shown in Figure 6.42(a) under cyclic loads of Pmax  200 N and Pmin = 0. Figure 6.42(a) only shows a half specimen to represent the weld nugget clearly. As shown in the figure, the sheet thickness is represented by t, the nugget diameter by d, and the specimen diameter by D. Here, t is 1 mm, d is 5 mm, and D is 40 mm. Figure 6.42(b) shows a symmetric cross section of the cup specimen containing the weld nugget before testing. The notch near the weld nugget can be considered as an original crack. When the cup speci men is under pure opening loading conditions, the stress intensity factor KI for the original crack can be obtained as (Pook, 1975; Wang et al., 2004)
KI  ~ )5 2  1 (6.6.31)
1
The stress intensity factor KI I is 0 for the specimen under pure opening loading conditions. During the testing, a kinked crack is initiated near the weld nugget and then propagates through the thickness with a kink angle c~ as shown in Figure 6.42(c).
Figure 6.43 shows a schematic plot of a main crack and a kinked crack with the kink length a and the kink angle ~. Here, KI and KII represent the global stress intensity factors for the main crack, and kl and kll represent

FATIGUE CRACK PROPAGATION BASED ON LINEAR ELASTIC FRACTURE MECHANICS 28 1
Weld Nugget
t
I (b)
(a)
Typical Crack Path
(c) FIGURE 6 .42 (a) A schematic plot of a half specimen with the sheet thickness t, nugget diameter d, and the specimen diameter D under the applied force (shown as the bold arrows). (b) A schematic plot of the symmetry cross section near the spot weld with the notch considered as an original crack. (c) The typical fatigue crack propagation path shown with a kink angle 2.
K,
Main Crack
Kinked Crack
1 FIGURE 6 .43 A schematic plot of a main crack and a kinked crack with the kink length a and the kink angle 2.

282 FRACTURE MECHANICS AND FATIGUE CRACK PROPAGATION
the local stress intensity factors for the kinked crack. The arrows in the figure represent the positive values of the global and local stress intensity factors KI, KII, ki, and kii. In general, the local stress intensity factors of the kinked crack, ki and k~i, are different from the global stress intensity factors of the original crack, KI and KII. When the kink length a approaches 0, the local stress intensity factors kl and kll can be obtained from the global stress intensity factors KI and Kii as (Bibly et al., 1978; Cotterell and Rice, 1980) l(
kl  ~, 3 cos ~ + cos KI + ~ sin ~ + sin KII (6.6.32)
kII   ~ sin ~ + sin KI + ~, cos ~ + 3 cos KiI (6.6.33)
For the cup specimen subjected to pure opening loading conditions, the stress intensity factor Kn is 0. Therefore, the local stress intensity factors kl and kn are
k i ~ 3cos~+cos KI
kll =  ~ sin ~ + sin KI
The effective local stress intensity factor is assumed to be
(6.6.34)
(6.6.35)
keq = v/k21 + k21 (6.6.36)
Assume that the local stress intensity factors remain constants during the crack propagation and the Paris law is applicable to characterize the fatigue crack growth. In this case, the material constants C and m are chosen for martensitic steels as C = 1.36 x 10 7 and m = 2.25. The value of C is obtained with the units of MPa x/~ for AK and mm/cycle for da/dN. 1. Determine the range of local stress intensity factors for the kinked
crack. 2. Determine the number of cycles when the kinked crack grows through the
thickness and the cup specimen is separated.
Solution. Based on Equation 6.6.31,
AKI  Zlg l , max  z Ig I , min :
(d ) 2In (d ) "
2 1 1
PmaxPmin(3) 1 /22n ~
= 5.62 MPav/m
(6.6.37)

REFERENCES 283
For the cup specimen under pure opening loading conditions, the Mode II stress intensity factor of the original crack is 0. The range of local stress intensity factors of the kinked crack can be calculated by Equations 6.6.34 and 6.6.35 with the kink angle ~ = 90 ~ as
Akl  1.99 MPax/m
Akii   1.99 MPa
The range of effective local stress intensity factors for the kinked crack can be calculated by using Equation 6.6.36 as
Akeq  ~/Ak 2 + Ak 2,  2.81 MPav/m
Based on the Paris law in Equation 6.6.25, the number of cycles Nf for the cup specimen as the kinked crack propagates through the thickness can be obtained as
It j i / sin ~ da N  C(Akeq)m (6.6.38) Because the local stress intensity factors are assumed to be constant during crack propagation, Equation 6.6.35 can be integrated as
t Nf   C(Akeq)m sin ~t (6.6.39)
Therefore, the number of cycles Nf to failure is calculated as 719,236.
REFERENCES
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