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A two-parameter analysis ofSN fatigue life using Dr and rmax
K. Sadananda a, S. Sarkar a, D. Kujawski b, A.K. Vasudevan c,*
a Technical Data Analysis, VA, United Statesb Department of Mechanical and Aeronautical Engineering, Western Michigan University, Kalamazoo, MI 49008, United Statesc Office of Naval Research, 875 North Randolph Street, Arlington, VA 22203, United States
a r t i c l e i n f o
Article history:
Received 27 October 2008Received in revised form 18 February 2009
Accepted 3 March 2009
Available online 16 March 2009
Keywords:
Stress controlled fatigue
Kitagawa diagram
Crack nucleation and growth
Aluminum and steel alloys
Mean stress effects
a b s t r a c t
The effect of the load ratio, R, or the mean-stress on fatigue life has been recognized for more than a hun-
dred years. In considering the mean-stress effects in the stress-life ( SN) approach, research efforts have
been mostly concentrated in establishing correlating functions in terms of the flow stress or yield stress
or the ultimate tensile stress, etc., by taking, say, R =1 test results as a reference. Very little effort has
been made towards understanding therole of stress rangeDr and the maximum stressrmax, (or rmean) in
the fatigue crack nucleation and propagation and also how to relate this to both the stress-life and the
fracture-mechanics descriptions.
In this paper we first examine crack nucleation based on the stress-life approach using a two-param-
eter requirement in terms ofDr and rmax, and then connect it to crack propagation using the Kitagawa
diagram as the incipient crack grows to become a long crack. Since stress-life data include both nucle-
ation and propagation, the connection of the safe-life approach to the fracture-mechanics analysis is per-
tinent. Comparison of the present analysis with experimental data taken from the literature
demonstrates that a two-parameter approach in terms ofDr andrmax forms a basis for the SNanalysis.
Published by Elsevier Ltd.
1. Introduction
It has been shown previously [15] that an unambiguous
description of fatigue crack growth requires two loading parame-
ters: DK and Kmax. It also has been demonstrated that the fatigue
crack growth phenomena including load ratio effects, underload
and overload effects, environmental effects, acceleration of short
cracks, etc., can be accounted for without invoking any extraneous
factors, such as crack closure. Since both DK and Kmax govern fati-
gue crack growth, the natural consequence of this is the existence
of two limiting thresholds, namelyDKth and K
max;th, which must be
satisfied simultaneously for a crack to grow. For a given material
environment system, these two thresholds measure an intrinsic
material resistance based on the mechanism of cracking. Similarly,
for any nonzero constant da/dN, fatigue crack growth is described
by two-parameter, DK and Kmax which vary with crack growth
rate and correspond, respectively, to the DK value at asymptoti-
cally high Kmax, and the Kmax value at asymptotically high DK. It
has been shown that a map of these parameters in terms of DK
vs. Kmax for the range of da/dN provides a characteristic crack
growth trajectory which is characteristic of the physical crack
growth mechanisms [6]. In this paper the term crack growth tra-
jectory or simply trajectory is understood to refer to this curve,
implicitly parametric in da/dN, in DK* vs. Kmax space. Trajectory-
maps of several engineering materials indicate that mechanisms
operating at the crack tip vary with the material, the environment
and also with transient times during crack increment [6]. In order
to apply this methodology in practice, a two-parameter crack driv-
ing force in terms ofDKand Kmax has been proposed [7,8]. Recently
a UNIGROW model has been developed [9] which takes into con-
sideration this two-parameter requirement. This two-parameter
driving force approach provides an effective predictive methodol-
ogy without any need for adjustable parameters. It has been dem-
onstrated that it predicts the fatigue crack growth behavior under
service loading spectra [9].
If the two-parameter requirement is intrinsic to fatigue, then it
should be applicable not only to fatigue crack growth but to the
crack nucleation as well. Conventionally, two methodologies are
used in fatigue analysis; a safe-life approach based on crack
nucleation using stress-life or strain-life analysis, and a damage
tolerance approach based on crack propagation using fracture-
mechanics analysis. In the past, the integration of the two
approaches has not been very successful. Instead, these two
approaches have been developed independently forcing a designer
to select one or the other for practical fatigue engineering analysis.
At the design stage, a designer relies heavily on the crack nucle-
ation analysis, while at the maintenance stage one is forced to
examine the damage tolerance approach, since cracks do form at
critical locations during service, particularly in aging aircrafts.
0142-1123/$ - see front matter Published by Elsevier Ltd.doi:10.1016/j.ijfatigue.2009.03.007
* Corresponding author.
E-mail address: vasudea@onr.navy.mil (A.K. Vasudevan).
International Journal of Fatigue 31 (2009) 16481659
Contents lists available at ScienceDirect
International Journal of Fatigue
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j f a t i g u e
mailto:vasudea@onr.navy.milhttp://www.sciencedirect.com/science/journal/01421123http://www.elsevier.com/locate/ijfatiguehttp://www.elsevier.com/locate/ijfatiguehttp://www.sciencedirect.com/science/journal/01421123mailto:vasudea@onr.navy.mil7/29/2019 A Two Parameter Analysis of SN Vasudevan IJF 2009
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Additionally, some material factors, for example grain size, ductil-
ity, and etcetera, affect crack nucleation and crack propagation
differently.
It is a fact that it is difficult to differentiate when nucleation
ends and propagation begins. Part of this problem is associated
with the limitations in crack detection methodologies. In fact, the
crack propagation stage has already set in by the time a crack
can be observed by currently available NDE techniques. Additional
problems arise due to the limitations of the conventional fracture-
mechanics methods for short cracks, when only the remote applied
stresses are considered in the analysis [10]. These limitations imply
that short cracks can decelerate even with the increasing applied
stress intensity factor, DK. Hence, short cracks are not easily ame-
nable to analysis using conventional fracture-mechanics consider-
ing only the applied DK. As the Unified Approach [11] to fatigue
ascertains, there are contributions to the crack tip driving force
from internal stresses that are present at the incipient stage of
short cracks, which need to be considered. The short crack problem
is of paramount interest in terms of fatigue life prediction of engi-
neering components, particularly for aircraft structures, since the
time period associated with nucleation and short crack growth
may occupy a significant part of the total life. Hence, evaluation
of crack nucleation and its transition to long crack via short crack
growth are important stages to be considered for reliable fatigue
life prediction.
In this paper, we first examine the crack nucleation based on
the stress-life approach using the two-parameter requirement in
terms of Dr and rmax. The effect of the load ratio, R, or the
mean-stress on fatigue life [12,13] will be considered within the
two-parameter framework. The Kitagawa diagram will be used as
the connecting link between the crack nucleation and the growth
of a crack leading to failure. Since stress-life data include both
nucleation and propagation, the connection of the safe-life ap-
proach to the damage tolerance approach is pertinent. The condi-
tions under which nucleated cracks do not propagate also will be
discussed. Thus the full range of fatigue damage from nucleation
to propagation using the two-parameter framework is addressed.
2. Two-loading parameter requirement for fatigue
Fatigue-crack growth tests are customarily done at constant R or
at constant Kmax. We have shown previously that all these tests are
complimentary to extract the material behavior in a given environ-
ment [15]. In contrast to the crack growth tests, the stress-life
tests using smooth specimens are usually performed with a con-
stant mean-stress, rm. Most of the tests are done at rm = 0 or
R =1, using rotatingbending tests, since such tests are easy to
conduct. To bring out the analogy between the fatigue crack growth
behavior and the stress-life behavior, we will show a parallelism in
the analysis for these two-sets of data. The data reduction proce-dure that has been used in fatigue crack growth (FCG) analysis will
be adopted to analyze stress-life behavior, as is shown in Fig. 1. The
crack growth rate data in a given environment for constant R or
constant Kmax tests form the basic FCG data. The FCG rate data
are normally plotted in terms of da/dN vs. DK. From these data
one can plot DK vs. R and DK vs. Kmax for any given crack growth
rate, da/dN. Fig. 1a illustrates the reduction scheme. Since DK vs.
R generally is approximately bilinear, one can use this plot to
extrapolate or interpolate the data to extract values at intermediate
values ofR, if available experimental data are limited to only few R-
values. Then, a DKvs. Kmax plot provides the fundamental material
curve for any given crack growth rate, da/dN. The DKvs. Kmax plot
represents the interrelation between the applied values ofDKand
Kmax and the resistance of the material in order to sustain the se-lected crack growth rate. At the threshold (operationally, da/
dN$ 1010 m/cycle) we have a fundamental threshold curve
(threshold is not a single value but a curve) below which a fatigue
crack does not propagate. The threshold curve, thus, defines the
non-propagating condition. The curve shows asymptotic limits in
terms ofDK and Kmax and are called the limiting thresholds, DK
th
and Kmax;th. Similar curves and two limiting values of DK and
Kmax can be also obtained at for any given FCG rate. The curve ob-
tained by plotting of these two limiting values DK
vs.
K
max(para-
metrically as a function of FCG rate, da/dN) forms what we have
termed a crack growth trajectory. When the DK vs. Kmax trajec-
tory lies on a 45 line, we have the condition DK Kmax for all
crack growth rates. The FCG rate data falling on this line implies
that the fatigue damage is occurring purely by cyclic strains [6],
which we refer to as pure fatigue. The crack growth process could
be similar to the Lairds plastic blunting process [14]. The crack
growth trajectory maps may deviate from this 45 line, when pro-
cesses other than pure fatigue contribute to the crack growth. The
superimposed process can be an environmentally-assisted crack
growth (corrosion-fatigue) or the stress-corrosion fatigue or any
other monotonic modes of crack growth, where the Kmax compo-
nent contributes additionally via static load. A companion paper
by the authors in this journal issue discusses various types of mate-
rials behavior that occur, based on the trajectory path [15]. We pos-
tulate that similar behavior can be expected under stress-life when
rmax affects fatigue life in addition to Dr, as described below.
Fig. 1b shows a data reduction scheme for the stress-life behav-
ior parallel to that used for the crack growth analysis, Fig. 1a. Fol-
lowing a similar procedure to that described above, the two stress
values in terms ofrmax andDr can be extracted from the data, for
a given fatigue life, NF. The applied stress range, Dr, for a given fa-
tigue life, NF, can be plotted as a function ofR. The constant ampli-
tude fatigue life data, as a function of R, in an inert environment
should form the reference characterizing the material response to
cyclic loads. Any deviations from that reference can be accounted
in terms of additional forces that contribute to fatigue life. The
additional forces could be those due to internal or residual stresses
(for example, due to notch-stresses, shot peening, quenching, etc.),and environmental factors. Thus, for any given fatigue life, NF,
asymptotic or limiting stress values, rmax and Dr can be deter-
mined. When NF is very large, say 107 cycles or more, the limiting
values are taken as the endurance limit of the material. Note that
the selection of 107 cycles to failure as the endurance limit is only
for convenience. As an evolution to the conventional understand-
ing, we now have two critical endurance limits, rmax;e and Dr
e,
analogous to the two thresholds, Kmax;th andDK
th for crack growth.
Both limiting values,rmax;e andDr
e, have to be met simultaneously
for fatigue damage. We cannot have Dr without rmax, while the
converse is not true. Hence, a fatigue process always involves
two independent loading parameters. Because ofrmax, we can have
superimposed monotonic modes of damage on cyclic damage,
sometimes described as ratcheting or cyclic-creep under fatigue.Similarly, a trajectory path for stress-life can also be defined by
plotting the relative changes in these two limiting values, with de-
crease in NF. Pure cyclic damage constitutes the requirement of
Dr rmax for different NF values, which forms the 45 line on a
trajectory path. The deviations from this 45 line represent the
superimposed rmax-dependent processes that include the static
modes of failure and/or environmental damage. In addition, by
defining a two load-parameter requirement for fatigue life, one
should be able to describe the material response under variable
amplitudes and changing R, similar to that for crack growth. Most
importantly, we should be able to connect the safe-life approach
with the damage tolerance approach using a single framework. In
the stress-life approach, since NF includes both crack nucleation
and crack growth, understanding the crack nucleation part isimportant for connecting the two stages of fatigue.
K. Sadananda et al. / International Journal of Fatigue 31 (2009) 16481659 1649
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3. Current approaches to safe-life design
While it has been recognized since Goodman [12] that mean-
stress has an effect on fatigue life, several stress-based approaches
have been proposed in the literature to quantify mean-stress ef-
fects. All of these approaches are empirically based and have the
following general form:
Dr De1 rm=rLn 1
where Dre is the material endurance in terms of stress range at
specified number of cycles (say 106 or 107 cycles) at zero mean-
stress. It is a constant value for given material and environment,rm the Applied mean-stress, rL the fatigue limiting condition
material yield stress, rYS, or tensile strength rUTS (i.e. design crite-
rion shifts from fatigue to yielding or to fracture when the limiting
or critical condition is reached), and n is the Exponent is 1 or 2, indi-
cating how fast/slow the fatigue limiting condition is reached.
Here, thesmallerthe exponentis, the faster the rate of approach,
since the ratio of (rm/rL) isless than1. Thus,Dr tends to zero when
the mean-stress approaches the specified limiting value. Various
models differin terms of thedefinition of thefatigue limiting condi-
tion specified by rL and the value assigned to the exponent n. For
example, in the Modified Goodman equation (1922) [12], rL is the
ultimatetensilestress andn = 1.For Gerber[16] n = 2, andSoderberg
and Sweden[17] considersrL tobe yieldstresswithn = 1.The Soder-
berg model is further modified incorporating a quadratic term interms of UTS and is called Quadratic Soderberg and Sweden [17].
Finally Bagci [18] further modifies the power equationbut consider-
ing thelimiting condition as yield stress. Fig. 2 shows comparisonof
these various models [19] in relation to 4340 steel data [20]. Note
that when rmean = ra or R = 0, Dr = rmax, the fatigue damage is
defined as pure fatigue sincecorrespondingDr andrmax values fall
on a 45 line. Conversely, as the mean-stress increases, monotonic
modes become increasingly important. Thus, the fundamental con-
sideration in all of the above models is the recognition that with
increasing mean-stress (or increase inrmax or R) monotonic modes
of failure are getting superimposed on fatigue. Subsequently, the
endurance limit (cyclic damage) decreases and becomes zero when
the limiting condition is reached. These models are somewhat sim-
ilar to the crack growth models, wherethe Kmax component is intro-duced to account for the contribution from the limiting load to
fatigue crack growth [21], with a correction to Kmax at high da/dN.
However, this consideration of Kmax at high end of crack growth isdifferent from Kmax as the fundamental parameter with its own
threshold for crack growth, as introduced in the Unified Approach
to fatigue [15].
In the case of the stress-life approach, the consideration of the
superimposition of the monotonic modes of failure on fatigue (as
in the above empirical models) is different from the two-parameter
consideration discussed in this paper. Superimposition of mono-
tonic modes is an extreme case where the imposed rmax induces
a static mode of damage and its contribution increases as the lim-
iting condition is approached. Sometimes this is referred to as rat-
cheting. In contrast, the two-parameter requirement for fatigue, in
terms ofDr andrmax, is operable even when the damage is of pure
cyclic nature. When R < 0, the rmax value may converge to a con-
stant positive value. Thus we ascertain that to understand the fati-gue life, one has to consider two loading parameters, rmax andDr,
Fig. 1. Data reduction schemes for (a) fatigue crack growth and (b) stress-life fatigue.
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80 100 120
a
(ksi)
m (ksi)
Bagci
Clemson
Gerber
Modified Goodman
Quadratic
Yielding
Experimental Data
(Grover el al.1951)
FromWangel al. 2000
SAE4130 Steel
Su = 117 ksiSy = 98.5 ksi
Sn = 50 ksi
Sn
Sy
= 0.51
Fig. 2. The experiment data are for SAE 4130 Steel from Grover, et al 1951 [20].
Endurance limit is 106 cycles. The limiting stress, Drn is 50% of the yield stress
(Drn/ry = 0.51) for this material. Ultimate stress, ru and yield stress, ry are
provided. Various lines correspond to analytical approximations of the mean-stress
effects from Wang et al. [19], used for design.
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even though under a given condition one may be more controlling
than the other.
4. Analysis of SNdata using the two-parameter approach
Using the data reduction scheme outlined in Fig. 1b, we can
now begin to analyze the published SNdata for various materials.
The method will be illustrated by taking two examples, whereextensive constant amplitude SN data are available for several
Rs. Fig. 3a shows the SN data for Ti6Al4 V alloy by Peters
et al. [22], plotted in terms ofrmax vs. NF. The data can be reduced
to Dr vs. rmax for various selected values of NF, shown in Fig. 3b.
Nearly L-shaped curves are seen which define two asymptotic or
limiting stress values ofDr andrmax for a given NF. The deviation
from the perfect L-shape may arise at the corner, from the interac-
tion between the two terms, Dr andrmax, due to plasticity. At high
NF (108 cycles), the limiting values can be taken as the material
endurance limits, Dre and r
max;e Thus, similar to the two crack
growth thresholds in terms ofDKth and K
max;th, we have two endur-
ance limits,Dre andr
max;e, that must be satisfied simultaneously in
order for the material to fail by fatigue damage. The actual values
needed for a given NF
follow the corresponding curve. Note that the
value ofrmax;e is larger (420 MPa) than Dr
e (120 MPa). These are
the limiting (minimum) endurance limits that must be met for
any fatigue failure to occur, assuming the same mechanism is oper-
ating. Examination of Fig. 3b shows that it is the rmax that varies
significantly (420800 MPa) with increase in NF compared to Dr
(160120 MPa). The observed behavior is similar to that noted
for crack growth wherein Kmax isP DK. In addition, the results im-
ply that the limiting conditions based on empirical laws depicted
in Fig. 2 have not been reached under these experimental condi-
tions. Fig. 3c depicts the variation of the two limiting values with
NF indicating that large variation occurs mainly in the r
max-value
than in the Dr-value. Since Dr* corresponds to cyclic strains
and rmax > Dr
max, we assume that the monotonic deformation
helps the fatigue damage by building up the required internal
stresses to set up the condition for crack nucleation and growth.
How the internal stresses play the role in fatigue will be discussed
later.
The fatigue life trajectory map can be drawn using these two
limiting values and plotting Dr* vs. Drmax. This is shown in
Fig. 3d. This is similar to the trend shown in a trajectory map for
cracks growing in many of the Ti-alloys [23], as shown in Fig. 4.
In the case of crack growth, with increase in crack growth rate (also
implies increasing stress intensity factors), the curve runs parallel
to the Kmax axis indicating the crack growth is increasingly Kmaxdependent. Fig. 3d shows the behavior is somewhat similar to
the extent that with decrease in life, the fatigue life is increasingly
determined by the maximum stress. That is, we move from the
high-cycle fatigue conditions where cyclic strains are dominant
to the low-cycle fatigue conditions, where the internal stresses
generated by dislocation substructure becomes a dominant factor
in generating the necessary conditions of crack nucleation and
growth. With increasing rmax, the tensile cyclic-creep strains could
increase due to an unrestricted specimen elongation under the
highly non-symmetric cyclic stresses. In addition, Feltner and Laird
[24] have shown that the dislocation substructure formed in low-
cycle fatigue conditions is similar to that under the monotonic
deformation.
In the next example, we examine the fatigue behavior of a low
alloy steel in air and corrosive environments [25]. Fig. 5a shows the
SNcurve of a low alloy steel in ambient air. In reducing the data,
following the steps outlined in Fig. 1b, it was found that there may
be two distinct mechanisms operating, one at low R and one at
Fig. 3. (a) SN data for Ti6Al4V alloy plotted in terms ofrmax vs. number of cycles to failure for various constant R-ratios. Data are from Peters et al. [22]. (b) Typical L-
shapedcurves for each NF defining twolimiting endurancevalues. The L-shape gets distortedinto smoothcurve-behavior due to second order interaction effects between the
two-parameters. Thetwo stresses then have to be along thecurve to enforce thesame NF. (c) Variation of the limitingvaluesas a function of no. of cycles tofailure. (d) Fatiguelife trajectory map showing the variation of the two limiting endurance values.
K. Sadananda et al. / International Journal of Fatigue 31 (2009) 16481659 1651
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high R. Such changes in mechanisms as a function of R have been
noted before, where the crack growth changes from predominately
intergranular at low R to predominately transgranular at high R
[26]. Fig. 5b shows how the two mechanisms can be differentiated
on the basis of the Dr vs. R curves. The differences are small, nev-
ertheless consistent. Using the interpolated and extrapolated data
based on the Dr vs. R curves, it is possible to generate the Dr vs.
rmax curves for each mechanism, as shown in Fig. 5c. Finally,
Fig. 5d depicts the trajectory path for fatigue life for the alloy in
three different environments; air, free-corroding potential and a
superimposed potential. All of the data were obtained from the
same Ref. [25]. In the high-cycle fatigue regime, the data follow
closely the 45 line indicating that the fatigue life, close to endur-
ance, is determined by cyclic damage. The 45 line means that the
Dr
e$r
max;e, within the experimental scatter. For the same
NFva-
lue, there is reduction in the Dre andr
max;e values between air and
the environment. That is, the environment is reducing the cyclic
stresses needed to cause initiation and failure. Since crack initia-
tion is the major part of life under high-cycle fatigue, the environ-
ment must be influencing the crack nucleation by a reduction in
the surface energy. Similarly, as we move towards the low-cycle fa-
tigue regime, the trajectory deviates in the direction of the rmaxaxis, peaks and then drops down towards the rmax axis, indicating
that the monotonic modes are becoming dominant at the low-cy-
cle-fatigue end. Detailed discussion of the various types of mecha-
nisms governing the trajectory paths are outlined for the case of
crack growth in a companion paper [15]. Similar behavior is ex-
pected for the stress-life or the SN behavior.
Finally, we show an example of an Al-7075-T6 alloy where the
trajectory path for the SN data of a notched specimen with Kt = 5
is plotted along with the trajectory for crack growth, Fig. 6. Data
are collected from [27] (MIL-HDBK-5). Because of high Kt, the S
N life is more dominated by the crack growth process. Qualita-
tively, the trajectory paths for both crack growth and the SNfati-
gue life of a notched specimen follow a single line even though the
scales for the two differ and no special effort was made to match
Fig. 4. Trajectory map for crack growth in many Ti-alloys from Ref. [23] and
references therein.
Fig. 5. (a) SNcurves for a lowalloy steel tested in ambient air. The points are not experimental but a digitized representation of the original data curves in Jones and Blackie
[25]. (b) Data plotted for selected NF values as a function of R. Possible change in the mechanism from Mech.1 to Mech.2 with R. Additional data were extracted by
interpolation. (c) L-shaped curves for mechanism I using interpolated data along with experimental data. (d) SNtrajectory paths for the low alloy steel under three differentenvironmental conditions.
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the scales. The striking similarity in the SN life and crack growth
life is obvious from the plot. Implication is that it is possible to re-
late crack growth analysis to the behaviors in the notched and the
smooth specimens. Approaching in reverse, it is possible to move
from crack nucleation to crack growth using a proper consistent
analytical tool for characterizing the damage evolution.
5. Relating crack nucleation to crack propagation
In 1976 Kitagawa and Takahashi [28] provided an important
link connecting the endurance limit Dre of a smooth specimen to
the crack growth thresholds Drth
in a fracture-mechanics speci-
men. No physical explanation was suggested. The first interpreta-
tion of Kitagawa diagram came from El-Haddad et al. [29] who
added an empirical crack length,
a0 1
p
DKth
Dre
22
to the actual crack, a, in order to make a smooth transition from
slopedDrth line to the horizontal Dre line in the original Kitagawa
diagram (not shown here). This smooth line is given by the follow-
ing relationship
DKth FDrthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipa a0
p3
where Drth is the far-field stress range and F is the geometry cor-
rection factor. For F= 1 and assuming that a is very small, the aboveequation reduces to Drth =Dre. On the other hand, when a much
larger than a0, it reduces to
DKth FDrthffiffiffiffiffiffiffiffiffiffipa
p4
In the present work, it is proposed that the smooth transition from
the sloped rmax,th line (designated as the Kmax,th line) to the hori-
zontal rmax,e line in the Kitagawa diagram (Fig. 7) is due to internal
stresses generated by plasticity. These fatigue generated internal
stresses alter the applied threshold stress intensity factor according
to the following relation depending on the tensile or compressive
internal stresses.
Ktotal Kappl Kinternal 5
Throughout the analysis we assume that the small scale yieldingconditions prevail at the crack tip. For a crack emanating from an
elasticplastic notch, the effect of the notch plasticity on the stress
intensity factor needs to be included. That is, in the evaluation of
Kinternal, plasticity corrections have to be incorporated. How this is
done will be described later. Here, it is important to note that the
Kmax;th is an intrinsic material threshold for a given material/envi-
ronment system and is independent of crack size; long or small.
Fig. 7 shows a modified Kitagawa diagram. Normally, log of
nominal stress range, Dr, is plotted by Kitagawa and Takahashi
[28] against the log of crack length. Since we have defined that
there are two endurances, Dre and r
max;e, and for all materials
rmax;e P Dr
e, and K
max;th P DK
th, we propose a modified Kitagawa
diagram, in terms of the log of nominal maximum stress, rmax,nom,
vs. the log of crack length with the trend line for rmax,th = Kmax,th/
F(pa)0.5 (instead ofDrth andDKth) for a fracture-mechanics speci-
men. For convenience we call this line as the Kmax,th-line within the
spirit of the original Kitagawa diagram. We now add two other lim-
iting conditions, the true tensile failure stress rF, and the critical
fracture line rmax,cr = KIC/F(pa)0.5, which will be referred to as the
KIC-line. Under fatigue conditions, rF, represents the limiting stress
(similar to rL in Fig. 2) where the fatigue failure occurs in one half
cycle. The region between the KIC-line and the Kmax,th-line is the fa-
tigue crack growth region. For the case when the applied maxi-
mum stress, rmax, falls below rmax,e and the Kmax,th line, the
growing crack becomes arrested, as often happens under suffi-
ciently high overloads or spike loads, as well as during propagating
of a short cracks at low loads. For example, the compressive inter-
nal stresses that form at the crack tip can bring the total K (due to
applied and residual/internal stresses) at the crack tip below the
Kmax,th. Similarly, tensile residual stresses can augment the applied
stress which will result in an increase in the total K, say, by apply-
ing an underload. Since the crack growth threshold [15] does not
vary with crack length, a growing crack can get arrested if the total
stress intensity factor falls below the thresholds. Hence, the entire
Kmax,th-line represents the threshold crack growth boundary for all
crack lengths. This threshold condition is valid for any given R. In
Fig. 7, the regime bounded by rmax,e (below the endurance stress
value) and to the left of the Kmax,th-line is designated as the non-propagation regime where crack lengths in that regime cannot
grow to failure.
A fundamental question that needs to be clarified in under-
standing the Kitagawa diagram is how a smooth fatigue specimen
that has no noticeable crack at rmax,e will end up with a crack size
of ac, since the crack can only grow at the nominal stress rmax,eafter reaching the ac value; that is, when the crack growth thresh-
old condition is met (where ac = (Kmax,th/Frmax,e)2/p). The assump-
tion that a short crack of length less than ac would have a lower
threshold (as is often assumed in the literature) would not address
this issue, since there is an increasing threshold with increasing
crack length that has be satisfied without increasing the applied
loads. Addition of an arbitrary crack length, ac, as was done by
El-Haddad et al. [29], requires a physical justification.Let us now examine the region bounded by the stresses above
rmax,e and to the left of the Kmax,th-line. We label the region as
the internal stress build up for propagation. In principle, for
any stress above the endurance limit, failure should eventually oc-
cur, at some number of cycles less than the endurance value. We
have shown that for any given number of cycles to failure, NF, there
are two limiting values ofrmax, and Dr (see Fig. 3b). Conversely
for any given rmax and Dr above the endurance limits, even the
smooth specimen will eventually fail at some NF value. For failure
in a smooth specimen, crack formation occurs at an in-situ gener-
ated stress concentration site due to heterogeneity in the deforma-
tion. The physics of the damage process indicates that some grains
at the surface region of the sample are always more favorably ori-
ented than the interior grains to initiate slip and protrusions. Theseheterogeneities lead to localized internal stresses due to strain gra-
Fig. 6. The trajectory paths for crack growth and SNfor notch life data with Kt = 5.
Note the striking resemblance of the two even though the continuity of the line
drawn is somewhat fortuitous since the scales for the two are different. Data are
from MIL-HDBK-5 [27].
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dients. For the purpose of illustration, let us consider rmax, a stress
above the endurance stressrmax,e. Without loss of generality, let us
assume that a smooth specimen has an arbitrary incipient crack
or defect of size
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cient enough to sustain the continuous growth of the initiated
crack. The fatigue endurance limits are therefore the stresses in
terms ofDre andrmax,e that are needed to build up required inter-
nal stresses via localized plasticity to initiate and grow the cracks
under the loading conditions. Below these limits, local plasticity
and thus the internal stresses can be generated and they may be
sufficient to initiate but not to propagate. The stresses in some
cases may not be sufficient even to initiate the cracks. Shot peen-
ing, for example, can suppress the generation of these required
internal stresses [33]. Likewise, periodic electro-polishing the fati-
gue specimen can remove any internal stress build up due to dis-
locations thereby extending or rejuvenating the fatigue life [34].
Hence understanding the role of internal stresses is very important
in the fatigue life prediction. Here we are providing the physical
meaning for the Kitagawa diagram using the Internal Stress Con-
cept. This will eventually help us to develop criteria for crack ini-
tiation and its growth to incorporate in the UNIGROW fatigue life
prediction model [9].
6. Analysis of the stress gradient required for crack growth
Experimentally, it is well known that the acceleration and
deceleration of crack growth can occur by underloads and over-
loads. This has been accounted for by the excess internal stresses
generated by localized plasticity that can cause increase or de-
crease of the stress intensity at the crack tip. The same concept
should be applicable for acceleration and deceleration of short
cracks. Hence, the above analysis shows that total stress intensity
factor from applied and internal stresses will determine if an incip-
ient crack will grow or not to cause failure. For short cracks, which
are nucleated at some stress concentrations, plasticity at the local-
ized stress concentration provides the necessary internal stresses
that augment the applied stresses to meet the crack growth condi-
tion. In addition, the total stresses (applied and internal) must
satisfy the requirement of a minimum stress gradient condition
in order to sustain continuous crack growth. For example, in
Fig. 7, the internal stresses not only have to move the point A to
the point B but their gradient also should be sufficient to move
the crack from point B to the point C. Thus the Kitagawa diagram
provides the minimum requirements for both magnitude and gra-
dient in internal stresses to sustain continuous crack growth. Ob-
served crack arrests and non-propagating cracks during short
crack growth or spike overloads are the result of not meeting
simultaneously the above minima criteria. Since smooth speci-
mens and fracture-mechanics specimens form two extremities that
are connected via the Kitagawa diagram, to understand the com-
plete physical significance of this diagram, we will consider a
notched specimen with different values of stress concentrations,
Kt. In the limit of Kt = 1, we arrive at a smooth specimen and with
increase in Kt we converge to the behavior of a cracked specimen.
Fig. 8 is redrawn from a 1957 classical work [35,36] on the fati-
gue of notched specimens with varying Kt Fig. 8a shows the con-
stant stress range Dr required to cause a crack nucleation and
failure as a function ofKt of a notch. The triangles denote the min-
imum stress required to initiate a crack while the circles denote
the minimum stress required to cause failure. For Kt = 1, that is
for smooth specimen, the endurance limit is around 260 MPa. With
increase in Kt, the endurance drops rapidly. However, with increase
in Kt > 3, the curve bifurcates; the minimum stress for crack nucle-
ation decreases gradually and levels off while the notch endurance
limit based on fracture becomes independent of Kt. It is believed
that the bifurcation point corresponds to the critical gradient of
the stresses ahead of the notch. The lower curve can be estimated
[37] asDr =Dre/Kt which means that the stress range at the notch
root, Drnotch, is equal to the smooth specimen endurance limit,
Dre. Below this curve, i.e.Drnotch = KtDr
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cal stress range. However, the initiated crack may or may not grow.
For it to grow, higher local stresses are needed. In fact Fig. 8a shows
the endurance limit decreases by an amount Kt until Kt% 3 and
then remains constant with further increase in Kt. Thus, the endur-
ance limit for a smooth specimen is around 260 MPa, while the
constant value in Fig. 8a is around 90 MPa, about 1/3 of 260 MPa.
For the initiated crack to propagate, higher local stresses than
Dre
(which must also satisfy the minimum gradient requirement)
are needed, as can be seen in Fig. 8b. The requirements for propa-
gation are governed by the crack growth thresholds in terms of
DKth and Kmax,th, which must be met simultaneously. Note that
the estimated elastic local stress at the notch tip can be very high
of the order of 1400 MPa even when the nominal stress is only
90 MPa. As a result, local yielding will occur, which in turn, will re-
duce the local peak stress and stress gradient but increase the local
peak strain and strain gradient. From Fig. 8a and b, it is clear that
the incipient crack nucleation energetics may be different from
the kinetics of growth. Nucleation is fully governed by the local stress
level alone, whereas propagation is by both the stress level and its gra-
dient. Whether nucleation or propagation controls the fatigue life
of a specimen depends on which of the two is an easier process
for a given condition. If there are already pre-existent stress con-
centrations, as in the case of notches, the local stress can reach
its required value for crack nucleation early in life; hence for those
cases propagation will be the life-limiting factor. Conversely,
where the local stresses have to reach their maximum by localized
plasticity as in smooth specimen, then crack nucleation can be a
large part of the fatigue life.
From the above analysis of crack propagation of an incipient
crack that is nucleated at a stress concentration, it is clear that a
simple elastic analysis of the notch tip stress fields is inadequate,
and we need to resort to elasticplastic analysis. For the purpose
of illustration, we consider below two cases; (a) specimen with
root radius ofq = 3 mm, but increasing depth starting from Kt = 3,
and (b) specimen with Kt = 3, but with changing root radius, q.
We use a simplified elasticplastic analysis to determine K of a
crack growing in the plastic strain field of a notch. The simplifiedexpressions are deduced recently by Kujawski [38] using Neubers
rule and considering the RombergOsgood stressstrain relation
for the material [13] Eq. (6) was used to estimate the stress inten-
sity factor for cracks emanating from an elasticplastic notch
K limq!0
ks
ffiffiffiffiffiffiffipq
4
r6
In the above equation q = qnotch + a, where q is the notch tip radius,
a is the crack length, ke corresponds to an elasticplastic strain con-
centration factor at the distancex = a from the notch tip. The values
of ke can be calculated numerically using FEA software or can be
estimated utilizing the well-known Neubers rule.
The Kplastic for a crack growing in the plastic field of a notch is
calculated using the above expression. Fig. 9a and b show the
two cases. In Fig. 9a, the stress intensity factor, K, for an incipient
crack growing in the plastic strain field of a notch are plotted as
a function of crack length. The K values are normalized with re-
spect to remote stress and plotted as a function of normalized
crack length, for increasing Kt, but with a constant root radius,
q = 3 mm. The initial sharp increases of K in Fig. 9a and b are in
the process zones involving the incipient crack formation, and
the analyses are beyond the scope of the continuum mechanics.
After the formation of an incipient crack, the K for that crack de-
creases initially and then gradually increases with the crack length.
Thus there is a minimum in the curve at some intermediate crack
lengths. If the material has a long crack threshold (Kmax,th), then the
situation can arise with Kat the minimum falling below the Kmax,ththreshold for crack growth. Then, the growing crack arrests when
K< Kmax,th. At higher Kt, due to increased plasticity, the minima
can be higher than the threshold allowing the crack to grow con-
tinuously. The crack growth rate can decrease as Kdecreases, reach
a minimum, then increase as Kincreases with crack length. Such a
deceleration and acceleration has been observed [10,11] during
both short crack growth as well as crack growth after overloads.
Hence the elasticplastic conditions at the notch tip contribute to
the crack arrest phenomenon, if the K for the incipient crack falls
below the threshold for crack propagation. In Fig. 9b, a case is illus-
trated to account for the experimental data where cracks formed at
smaller holes do not grow while those formed at larger holes can
growto cause failure. In this plot, the Kt is fixed at 3 and root radius
q is increased. For the same Kmax,th used in Fig. 9a, we find that
non-propagating conditions are set for q = 3.5 or less but for larger
hole sizes the minima are above the threshold insuring continuous
crack growth without any arrest. Fig. 10 shows the experimental
data of Murakami and Endo [39] showing that for holes less than
some critical size there are no changes in the endurance limit,
while holes larger than some critical size only contribute to lower
fatigue limit.
Based on Fig. 8a, it appears that the mechanics of crack growth
is not changed significantly by the presence of a notch for all Kt > 3;
the nominal stress for crack propagation remains the same. In allcases the stress gradient is lower than the critical amount needed
for crack growth. That is, the number of cycles required to generate
the necessary internal stresses and their gradients will decrease
with increase in Kt. In addition to nominal stress, local internal/
residual stresses can also affect the stress intensity factor, K. The
crack growth requirement, therefore, is that the total K as a crack
tip driving force has to exceed the threshold Kmax;th. When there
is no crack to start with, an incipient crack has to form by a nucle-
ation process. Once crack has formed the subsequent growth is
determined by the threshold condition for crack growth. Nucle-
ation of incipient crack followed by its growth is the determining
factors for the total fatigue life. Criteria for the two are different,
as shown by Fig. 8a and b. In particular, initiation is independent
of stress gradient whereas propagation is. This is true for a smooth
Fig. 9. (a) Elasticplastic calculations for conditions: (a) at constant q = 3 mm and Kt varying, and (b) Kt = 3 constant with q varying.
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specimen, a notched specimen or a cracked specimen. Only differ-
ence is the nucleation is easier at a notch since the critical local
stress for nucleation can be easily met at lownominal stress ampli-
tudes, while it is difficult for a smooth specimen since conditions of
concentrated local stresses have to be established by local plastic-
ity. That is, the number of cycles required to generate the same
internal stresses for nucleation will be less with increasing Kt.
We have to bear in mind that the Kitagawa diagram connects the
two extreme cases involving nucleation in a smooth specimen
and a crack growth from a pre-existing crack/defect. For all
notches, the local notch root stresses have to be enhanced to meet
the initiation criterion and the stress gradients have to meet the
propagation criterion.
Since the nucleation stress is the same as the endurance stress
of a smooth specimen we can look closely the initial stages of crack
formation and its kinetics of growth. Fig. 11 shows the simplified
Kitagawa diagram (with rmax,e and Dre) and the process that canlead to crack formation. Since Kmax,th has to be met for all crack
lengths, line AB denotes the decrease in stress with increase in
crack length that satisfies the critical gradient requirements along
the path. From the point of stress, it denotes internal stresses that
provide the same Kas the remote nominal stresses. Hence, for any
stress above the endurance limit, crack should form and grow to
failure by generating sufficient internal stresses by the cyclic plas-
ticity to move the crack along the path AB. For engineering mate-
rials which are generally polycrystalline, the deformation is
inhomogeneous due to favorably oriented surface grains, causing
gradients in the internal stresses. As intrusions and extrusions
[40] form due to irreversibility of the slip process (in Fig. 11, see in-
serts), they behave like notches with stress concentrations. Addi-
tional localized slip can lead to further build up of internal
stresses and modify their gradient. A stage is reached when a crack
could form and begin to grow. Thus the localized plasticity sets up
the total equivalent stress and its gradient that meet the minimum
criteria in terms of nominal stresses given by the Kitagawa dia-
gram, Fig. 11. Both nucleation and growth occur by building up
the internal stresses via the formation of a suitable dislocation
structure. The details of the micro mechanics of the process, partic-
ularly the incremental dislocation density in each cycle and how
they build up the necessary and sufficient internal stresses and
their gradients are unknown. Fig. 12 shows the labeled Region of
Internal Stress Build up and the possible variations in the internal
stress gradients that can be set up by an appropriate dislocation
density and its distribution. In Case 1, the internal stresses and
their gradients are more than those needed for the formation of
a crack and its growth. In this case, failure is insured. For Case 2,
the internal stresses are initially higher than the minimum but
drop rapidly below the minimum required leading to crack arrest.
The situations are similar to Fig. 9a and b where a growing crack
can get arrested when Kdue to stresses falls below Kmax,th. For Case
3, the internal stresses are well below the minimum requirement
for crack growth, but above the endurance limit of a smooth spec-
imen. Hence initiation is insured but not propagation. Propagation
ultimately can occur only after building additional internal stresses
by cyclic plasticity, if the nominal stresses are at or above the
endurance limits. Thus internal stresses and their gradient can play
an important role both during initiation and growth stages.Thus, the crack nucleation is governed by the local maximum
stress at the surface of a smooth sample or at the notch tip. The
crack propagation is governed by stress intensity parameter, where
Fig. 10. Relation between fatigue strength and the variation in hole diameter for
0.13% carbon steel [39].
Fig. 11. Intrusions and extrusion formation provide internal stresses and gradients to nucleate cracks (Illustrations from Witmer et al. [40]).
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we switch from the local maximum stress to the stress intensity
factor K of the nucleated crack. The value of K depends not only
on the local internal stress but also on its distribution along the
length of the crack. Thus there appears to be change in the
mechanics of damage from a local peak stress during crack nucle-
ation to stress intensity factor that is somewhat global during crack
propagation. Fig. 8a and b depict this change. However, this has to
be understood in terms of the local plasticity affecting the local
strains and strain gradients which in turn affecting the Kof a crack
through strain-energy function. Thus we do have local strain
strain behavior at the notch or crack tip affecting the Kplastic, and
thus the kinetics of crack growth. This switch from the elastic to
the elasticplastic constitutive behavior is unavoidable in fatigue,
since fatigue is a plasticity-induced damage, while we are charac-terizing it on the basis of an extended linear elastic fracture-
mechanics, assuming that all the plasticity is localized or it corre-
sponds to small scale yielding conditions.
7. Crack nucleation in a smooth fatigue specimen: key issues
It is well known that for a polycrystalline annealed material, the
crack nucleation occurs at a surface of a grain that is favorably ori-
ented to the deformation slip. Localized deformation leads to a for-
mation of intrusions and extrusions (or protrusions), which help to
build up the local internal stresses. Incremental deformation in
each cycle along the slip planes sets up the dislocation dipole ar-
rays inside the grain forming protrusions at the surface, therebyaugmenting the local stresses or more specifically the internal en-
ergy of the system. Crack formation is ensured if there is sufficient
energy to nucleate a crack; similar to Griffiths fracture condition.
Since the dislocations arrays are formed on the slip planes, crack
can form along the slip plane if there is a reduction in the total en-
ergy when the dislocated material is replaced by a crack. The
nucleation kinetics can be somewhat similar to that proposed by
Mura [41]. The cleavage planes could become favorable if environ-
ment can lower the surface energy and thus reduce the energy of
the crack formation. Excluding those special cases, the crack nucle-
ation occurs generally along the slip plane forming Stage I. As the
crack grows, it changes to the Stage II crack growth when the
growth condition is met, i.e. the K at the crack tip exceeds the
Kmax,th. The number of cycles required to nucleate a crack dependson the number of cycles needed to build up the necessary local
stresses via the dislocation processes. Both slip and its degree of
reversibility determines the rate of accumulation. Grain size and
local microstructure will have a strong bearing on the process of
internal stress build up via dislocations. All these aspects are
embedded in the determining the endurance limit as well as the
criteria for the growth of the incipient crack formed, as described
in the modified Kitagawa diagram (Fig. 7). Fatigue life prediction
in terms of crack nucleation and growth therefore depends closely
in the rate of accumulation of internal stresses and their gradients.
These are related at micro level to dislocation density and their
gradients and at continuum level to localized strains and their
gradients.
8. Summary
In the above analysis, we have analyzed several aspects that are
involved in the crack nucleation and growth and thus the total fa-
tigue life. First, we have shown that there are several common fac-
tors between crack nucleation and crack propagation as well as
some divergent factors. Common factors include the two-parame-
ter requirement of fatigue damage which manifests as Kmax andDK
for crack growth andrmax andDr for SNlife. Just as there are two
thresholds, Kmax;th and DK
th, for crack growth, we have two endur-
ance limits for the SN fatigue, rmax;e and Dr
e. In all materials,
Kmax;th P DK
th, and similarly r
max;e P Dr
e. Most life prediction
methodologies, including the simple Miners rule, ignore the two-
parameter nature of fatigue, hence are empirical. We believe that
consideration of these two-parameters in each cycle would help
in better prediction of fatigue life, as has been done in the UNI-
GROW model [9] where the role of both Kmax andDKare included.
Finally, the trajectory maps for both crack growth as well as for
fatigue life can be developed that give details of the changing
mechanisms as a function of crack growth or the SNlife. The pure
fatigue behavior where the damage is governed by only cyclic
strains can be seen under the high-cycle fatigue conditions where
rmax Dr, somewhat similar to the crack growth condition where
under pure fatigue crack growth occurs when Kmax DK
. The
deviations from pure cyclic strain controlled process occur under
both crack propagation and the SNlife. In the SNlife, monotonic
modes become dominant at the low-cycle fatigue regime, while
the cyclic strain controlled process dominates in the high-cycle fa-
tigue regime. The SNlife includes both crack nucleation and crack
growth. In the high-cycle fatigue region, the nucleation life may be
major part of the life, while in the low-cycle fatigue region the
crack propagation is dominant.
The analysis also shows that in some respects the mechanics of
crack nucleation is different from crack propagation. The uncer-
tainties and ambiguities in defining when the nucleation ends
and propagation begins are not related to the mechanics of the pro-
cess but to the limitations in the detection of a nucleated crack. Thedifferences between the two get magnified when we have condi-
tions where the crack nucleation is possible without their growth,
resulting in non-propagating cracks. Nucleation is governed by the
maximum local stress and not maximum nominal stress. This max-
imum local stress is the same as the stress required to nucleate a
crack in a smooth specimen. On the other hand, the propagation
is governed by stress intensity factor which has to meet the prop-
agation threshold, Kmax;th.
In the UNIGROW model [9] the total crack tip driving force Dj
was expressed in terms of the two-parameters DK and Kmax in a
form that could collapse all the crack growth data into a single
curve. Similarly it would be convenient to express the SN life in
terms of a single parameter that incorporates both rmax and Dr.
Then, one can develop a consistent fatigue model that can be usedfor spectrum loads. Efforts in that direction are currently being
Fig. 12. Schematic illustration showing three possible internal stress profiles
indicating the required minimum for steady growth of an incipient fatigue crack.
This internal stress triangle is above the endurance line in Fig. 7.
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pursued. By separating the nucleation life from the propagation life
in the SN fatigue, one can go from the nucleation model to the
propagation model incorporating rmax and Dr for nucleation and
Kmax and DK for propagation. These aspects are brought together
using a modified Kitagawa diagram for which a physical interpre-
tation has been provided.
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