a two parameter analysis of sn vasudevan ijf 2009

7/29/2019 A Two Parameter Analysis of SN Vasudevan IJF 2009

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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: [email protected] (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:[email protected]://www.sciencedirect.com/science/journal/01421123http://www.elsevier.com/locate/ijfatiguehttp://www.elsevier.com/locate/ijfatiguehttp://www.sciencedirect.com/science/journal/01421123mailto:[email protected]


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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.

1650 K. Sadananda et al. / International Journal of Fatigue 31 (2009) 16481659


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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.



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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].



7/12

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


8/12

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.



10/12

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]).



11/12

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.



12/12

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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