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New approach for the correlation of fatigue crack growth in metals:
Accounting of mean stress effects in the change in net-section strain energy
Article in Acta Materialia · June 2017
DOI: 10.1016/j.actamat.2017.06.013
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Acta Materialia 135 (2017) 201e214
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
Full length article
A new approach to the mechanics of fatigue crack growth in metals:
Correlation of mean stress (stress ratio) effects using the change in
net-section strain energy
K.S. Ravi Chandran
Department of Metallurgical Engineering, The University of Utah, 135 South 1460 East Rm. 412, Salt Lake City, UT 84112, USA
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 11 May 2017
Received in revised form
4 June 2017
Accepted 7 June 2017
Available online 14 June 2017
An unsolved problem in the mechanics of fatigue crack growth is why mean stress or stress ratio in a
fatigue cycle has a profound effect on growth rate when compared in terms of stress intensity factor
range (DK). Specifically, there has not been any fracture-mechanically-consistent theory to explain
the effects of mean stress on fatigue crack growth. In this work, a new and generalized driving force
parameter for fatigue crack growth, which effectively incorporates the mean stress or stress ratio
effect, is developed from energy principles of solid mechanics. A successful explanation of the effect
of mean stress on the growth rate is provided. The driving force is the accumulated change in netsection strain energy, which develops as a function of increasing crack length and decreasing netsection size in fatigue. A generalized mechanics analysis, showing how to account for the stress
amplitude and the maximum stress of a fatigue cycle in the change in net-section strain energy, is
presented. Equivalently, the new crack growth parameter can also be interpreted as the cumulative
work done by cyclic loading at the crack length at which the crack growth data is determined.
Several experimental data, including some historically significant fatigue crack growth data, generated on aluminum, steel and titanium alloys, are used to demonstrate the success of the correlation.
As a further validation of the proposed concept, it is shown that the empirical Walker parameter
(DK0.5Kmax0.5), which had some success in correlating stress ratio effect, is in fact related to the
change in net-section strain energy.
© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords:
Fatigue
Crack growth
Net-section
Strain energy
Stress intensity factor
Mean stress
Stress ratio
1. Introduction
The mean stress level of a fatigue cycle has a profound effect on
the growth rates of fatigue cracks in metals. Maddox [1] made an
excellent review of this subject on developments prior to 1975.
Although a great amount of fatigue crack growth (FCG) data has
been generated till date, a physically sound approach to correlate
the effect of mean stress on fatigue crack growth is not available.
What is meant by “correlation” is that when the mean stress (or
stress ratio, R), which is the asymmetry of the cyclic loading, is
properly accounted for in terms of an effective driving force
parameter for crack growth, then the crack growth data for
varying stress ratios will be practically indistinguishable from
each other.
E-mail address: ravi.chandran@utah.edu.
http://dx.doi.org/10.1016/j.actamat.2017.06.013
1359-6454/© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Fatigue crack growth correlations, even now, rely on empirical
approaches that were proposed several decades ago. Paris, in 1998
[2], stated, “… physical modeling of fatigue crack growth remains
unsuccessful for all these forty years …” This statement is still true,
but with the revision of years to nearly sixty. The fatigue crack
growth data, for the simplest case of constant amplitude loading
cycle, are empirically correlated in the form of da/dN versus DK
graphs (da/dN ¼ CDKm, where a is crack size, N is number of cycles
and C and m are constants). This relationship is known as the Paris
law [3,4]. A change in the mean stress, at a given DK level, usually
causes a change in crack growth rate, either small or large,
depending on the material. A higher level of mean stress usually
increases the FCG rate significantly. The scope of the Paris Law,
however, is limited to the correlation of FCG at a constant stress
ratio. Empirical modifications to the Paris' law have been proposed
to explain the variation of FCG rates with stress ratio. One modification is the Forman's equation [5] in which the crack growth rate is
expressed as a function of R and DK:
202
da
C DK m
¼
dN ð1 RÞKc DK
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
(1)
where Kc is the thickness-dependent fracture toughness. Equation
(1) has been shown to express the stress ratio effects in the fatigue
crack growth data of 7075-T6 alloy [6]. This equation, however, is
also an empirical equation and was constructed by forcing the
limiting behavior (that is, in the limit of K tending to fracture
toughness, da/dN tends to infinity) on the equation, in an attempt
to explain the relative shifts in fatigue crack growth curves with R.
However, Pearson [7], noting that the Forman's equation did not
work well for low toughness materials, proposed a modification,
which placed the denominator in Eq. (1) under the square-root.
These modifications have been made to necessarily fit the fatigue
crack growth data for various stress ratios, but it is evident that the
physical foundations are missing in these approaches. Further,
neither the Paris nor the Forman/Pearson equation correlates well
with the experimental data in the near-threshold regime of crack
growth. This poses a difficulty in predicting fatigue life because
majority of life is spent at low-DK levels, where the mean stress
effects on FCG need to be accurately accounted for. In this regard,
Paris himself has criticized [2] the misuse of his equation for life
prediction, because of extreme sensitivity of predictions to initial
growth rates, that is, the growth behavior in the near-threshold
regime. A principal reason is that fatigue crack growth at low DK
levels is greatly affected by the microstructure [8,9]. There have
been many additional variants since the Forman equation–
Hoeppner and Krupp [10] in 1975 compiled an extensive list of
fatigue crack growth equations, most of which are also empirical
approximations in one way or the other. It is clear that the approaches presented so far to characterize the FCG behavior are not
physically sound and are not adequate to correlate the entire range
of fatigue crack growth data.
In reality, since the Kmax levels at various mean stresses are
different, there should be some combination parameter, which
includes both the mean stress (or maximum stress) and the stress
range, to properly correlate fatigue crack growth rates. Long time
ago, Walker [11] showed that a effective stress intensity parameter
of the form: (DK)m(Kmax)1m successfully correlated (with m~0.5)
the fatigue crack growth data of aluminum alloys obtained at
various stress ratios. Interestingly, an equivalent form:
(Ds)m(smax)1m was shown to work well explaining the mean
stress effect found in the smooth S-N data of the same material.
Walker stated that a distinction between crack nucleation and
propagation is not necessary in applying this empirical approach.
However, this work did not explain why the value of m should be
~0.5, leaving one to wonder whether this value was chosen merely
to fit the data. There are also other works [12,13], which have also
found satisfactory correlation of the mean stress effect on FCG,
empirically, using the Walker approach. Additional empirical approaches such as that by Yuen et al. [14], or Sullivan and Crooker
[15] or Parida and Nicholas [16] are also noted. It is obvious that
there have been several empirical attempts to correlate the meanstress effect on FCG. Interestingly, these works seem not to need the
crack closure concept for explaining the mean stress effect on crack
growth, although several publications on crack closure had
appeared in the same time frame of the said publications. It seems
that there have been two “camps” attempting to explain the mean
stress effect on FCGdone using Walker or alternate parameters and
not needing crack closure, and, the other advocating the concept of
crack closure.
A different approach to explain the mean stress effects on fatigue crack growth was proposed by Elber [17] in 1971 using the
concept of crack closure. This concept suggests that, due to the
residual tensile stretch in the plastically deformed crack-wake region, the crack faces contact, and the crack closes at a positive load
that is higher than the zero load. He proposed an effective cyclic
stress (DSeff, given by Smax-Sop, where Smax and Sop are maximum
and crack opening stresses in a fatigue cycle), or equivalently, an
effective stress intensity range (DKeff) that drives the crack during
the part the crack is open during fatigue cycling. In section 7, a full
discussion of the difficulties with the explanations of mean-stress
effect, on the basis of crack closure, is presented. The advantages
of the present work, which does not rely on crack closure, will be
shown in that context.
The objective of this work is to demonstrate the correlation of
the mean stress effect on fatigue crack growth rates, on the basis of
the change in net-section strain energy in the remaining ligament.
The principal novelty here is the generalization of the concept of
change in net-section strain energy for any mean stress, on the
basis of exact solid mechanics principles. It is then shown that a
successful correlation of means stress effect, on the basis of the
proposed parameter, can be achieved and without resorting to
empirical or crack closure concepts. Specifically, the deduction of
net-section cyclic strain energy (DCR) parameter, for any mean
stress, is presented using strength-of-materials (SOM) analysis of
fracture mechanics specimens. It is demonstrated, using extensive
data from center-cracked-tension (CCT) and single-edge-notchedtension (SENT) specimens, that the mean stress effects can be
successfully correlated on the basis of DCR. The major contribution
of this work is the discovery of the generality of the energy-based
mechanics approach to correlate fatigue crack growth, regardless
of stress cycle asymmetry. Because the proposed driving force
parameter is energetically complete in terms of capturing the
mechanics that drive the crack growth, it is also shown that crack
closure concepts are not needed to explain the mean stress effects
on fatigue crack growth. Further, the change in net-section strain
energy seems to provide a physical basis for the Walker parameter
((DK0.5Kmax0.5) which has had some success to correlate the mean
stress effect on fatigue crack growth.
2. The concept of the change in net-section strain energy
with crack extension
When a fatigue crack extends in a uniformly stressed CCT or
SENT specimen, the stress over the net -section (the uncracked
section or ligament) increases with crack extension until the point
of ligament fracture where the net-section stress level becomes
equal to the tensile strength of the material. It then seems, intuitively, that the change in the net-section cyclic stress (or cyclic
strain energy) that occurs in the remaining section with crack
extension, will have something to do with the rate of growth of the
crack defining that section. A variation in the asymmetry of the
fatigue cycle (due to a change in mean stress), for a given amplitude
of loading, can cause significant changes in the net-section cyclic
strain energy for a given crack length and for a fixed stress amplitude. Hence, for a proper correlation of mean stress effect on FCG,
the change in cyclic strain energy that occurs with crack extension,
as governed by the loading cycle asymmetry, needs to be determined quantitatively.
The principal contribution of this work is to show how the
change in cyclic strain energy can be determined, for any mean
stress, in a fracture-mechanically-consistent manner. In our previous work [18], the change in net-section cyclic strain energy
parameter (DCa) was defined for the simplest case of cycling
loading (R ¼ 0) and was shown to correlate fatigue crack growth
data of several materials very well. That study also provided the
fracture mechanics justification for DCa by showing the equivalence
of the net-section stress determined by SOM calculation and that
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
calculated from fracture mechanics. The accumulated change in
cyclic strain energy in the net-section is the difference in elastic
strain energy between a crack-free specimen and the cracked
specimen.
An illustration of how the change in net section strain energy
can be determined is given in Fig. 1. The energy principles of solid
mechanics [19] state that, when a crack-free specimen is loaded to
smax (R ¼ 0), the complimentary energy (equivalently the work done
at the loading point) and the strain energy of the specimen are given
by area OABO and area OBDO, respectively, as shown in Fig. 1(b). For
linear-elastic deformations, these areas are equal to each other.
After the introduction of crack of length ‘a’ in the specimen, the
complimentary strain energy and the strain energy of the specimen
should change to area OACO and area OCEO, respectively, maintaining the equality between these energies. The change in cyclic
203
strain energy in the specimen, at the crack size ‘a’ and with respect
to the crack-free specimen, is given by the highlighted area OBC in
Fig. 1(b). This represents the accumulated change in cyclic strain
energy (this is cumulative as a result of the fatigue crack growing to
the length, a) in the net-section, DUa, with respect to the initial
cyclic strain energy of a crack-free specimen. This can be written as,
on the basis of Fig. 1(b),
1
2
DUa ¼ Pa dmax;a dmax;0
(2)
where Pa is the load amplitude corresponding to stress amplitude,
sa, and dmax;0 and dmax;a are the elastic deflections of the specimen
at the loading point without and with the presence of a crack of
length, a, respectively. This change in strain energy is accumulated
in the net-section highlighted in Fig. 1(a) and is considered to be
Fig. 1. The accumulated change in net-section cyclic strain energy at a given crack size. The change is the increase the strain energy corresponding to a crack size, with respect to the
strain energy of the crack-free specimen. Figures (aeb) illustrate the increase in net-section strain energy for R ¼ 0 loading. Figures (cee) illustrate the same for R > 0, with (c) for an
infinitesimal extension, a1, and (d) for cumulative crack extension to crack size a.
204
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
responsible for determining the growth rate of the crack with the
length, ‘a’. This strain energy is considered to be accumulated in the
net-section and is elastic and not permanently absorbed in the
material of the net-section, such as that which occurs in plastic
deformation.
For the more general case of stress ratio not being zero, consider
first the introduction of an infinitesimal crack of size a1 in a crack
free-specimen in one fatigue cycle. Fig. 1(d) illustrates the increase
in the cyclic strain energy during the extension of the crack from
zero-size to the size a1 during the load increase from Pmin to Pmax in
one fatigue cycle. The shaded area (BCD) in Fig. 1(d) gives the increase in cyclic strain energy in the net-section per cycle. This is
deduced as follows.
For a crack-free specimen, it is clear that the area OPminDO
represents the complementary strain energy for loading the specimen from zero load to point D (Pmin, dmin,o). Similarly, the area
OPmaxABDO represents the complementary strain energy for
loading the specimen from zero load to point B (Pmax, dmax,o).
Therefore, the complementary strain energy for load cycling the
crack-free specimen, between Pmin and Pmax, is the difference between these two areas, which is the area enclosed by PminPmaxABDPmin. Similarly, the area PminPmaxABCDPmin gives the
complementary strain energy of the specimen with crack size ‘a1’,
as the load is increased from Pmin to Pmax. This is because the line DC
in Fig. 1(d) represents the partition between the increase in the
complimentary strain energy (or work done at the loading point)
and the actual increase in the elastic strain energy of the specimen
(which is concentrated in the net-section), during the crack
extension from Pmin to Pmax in one fatigue cycle. This partitioning
assures the equality between the area DACD and the area DCED,
and is consistent with the complementarity of the two forms of
energy in solid mechanics, which compels the equality between the
increased work done and the increase in the elastic strain energy
upon crack extension.
In Fig. 1(d), the difference between the complimentary strain
energy of a crack free-specimen (area PminPmaxABDPmin) and that of
the specimen with an infinitesimal crack of size ‘a1’ (area PminPmaxABCDPmin) is the area BCD, which is the change in work done or
the change in elastic strain energy for the infinitesimal extension to
size “a1”. On this basis, Eq. (2) can be generalized for the infinitesimal extension, for R > 0, as
1
2
DUR ¼ DP dmax;1 dmax;0
(3)
Applying the same principles as above, the accumulated change
in net-section strain energy for an extended crack of size “a” is
illustrated in Fig. 1(e). This crack is formed by the cumulative increments in crack length in each fatigue cycle, by growing through
a1, a2, a3 etc. and finally making to size “a”. Hence the partition
between the cumulative work done and the cumulative strain energy increase in the net-section of the specimen is also given by line
DC in Fig. 1(e), because we seek the difference between the complimentary strain energy of the crack-free specimen and the
increased level of complimentary strain energy in specimen with the
extended crack size, ‘a’. The accumulated change in strain energy of
the specimen with crack size, a, relative to the crack-free specimen
is thus given by area BCD (Fig. 1(e)) and is written as
1
2
DUR ¼ DP dmax;a dmax;0
(4)
The relative increase in strain energy of the net-section in
Fig. 1(e) will reduce to that in Fig. 1(b) when Pmin ¼ 0, hence, the
accumulation of strain energies in net-section due to asymmetric
cyclic loading is defined here consistently. It is important to note
the distinction between Eq. (2) and Eq. (4). In Eq. (4), Pmin (through
DP) defines the lowest load that the specimen is loaded to in the
fatigue cycle. It is obvious that for a given cyclic load range, a
relatively higher mean stress will necessarily accumulate more
cyclic strain energy in the net-section of the specimen, relative to a
lower mean stress, and hence will determine the crack growth rate
accordingly.
For a specimen of length L, Eq. (4) becomes
1
2
DUR ¼ DP Lεmax;1 Lεmax;0
(5)
where εmax,1 and εmax,0 are the nominal strains with and without
the crack, respectively. Writing Eq. (5) in terms of stresses,
1
2
DUR ¼ Ds ðWLtÞ
s
max;1
E
smax;0 E
(6)
where t is the specimen thickness. It is to be noted that the term
smax;1 =E in Eq. (6) is the maximum strain in a cracked solid and it is
from the deformation of the net-section ligament under the
increased maximum stress in the ligament, which is given by
(Ref. [20])
sl;max ¼ smax þ smax
a
ðW aÞ
(7)
Hence it is clear that smax;1 ¼ sl;max and smax;0 ¼ smax . On this
basis, Eq. (6) becomes,
DUR ¼
1
a
smax
Ds ðWLtÞ smax þ smax
2E
ðW aÞ
(8)
which leads to
DUR ¼
1
a
Ds smax ðWLtÞ
2E
ðW aÞ
(9)
Eq. (9) gives the accumulated change in cyclic strain energy for a
specimen of volume “W.L.t” subject to the cyclic stress amplitude,
Ds and smax for a given R. Hence, the accumulated change in cyclic
strain energy for unit volume in the uncracked net-section, at the
given crack length a, is
DCR ¼
h a i
1
Ds smax
2E
W a
or, on the basis of the fact that smax ¼ Ds =ð1 RÞ,
DCR ¼
1
Ds2 h a i
2E ð1 RÞ W a
(10)
It is to be noted that in Eq. (10) Ds is the stress range, which is
twice the stress amplitude about the mean stress. For R ¼ 0, Eq. (10)
reduces to the following equation, in terms of stress amplitude,
which was determined earlier [18].
DCa ¼
a i
2E W a
s2a h
(11)
where sa is the stress amplitude (zero to smax) for R ¼ 0 loading.
The quantity DCR in Eq. (10) is the average change from the initial
cyclic strain energy density per unit volume that existed in the netsection prior to the arrival of crack. Most of this strain energy will
be concentrated in the plane of the crack. However, in reality, the
strain energy density is expected to be higher near the crack tip and
diminish with distance away from the crack tip. Any extension of
the crack into the net-section region (defined by crack length, a) can
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
thus be expected to be determined by the cyclic strain energy
density of this region, which has been accumulated by the crack
upon growing to the size, a. Even when the crack extension is
interpreted as that caused by the change in crack tip stress/strain
condition, as done in fracture mechanics, this change is embedded
in the global change in cyclic net-section strain energy that has so
far been accumulated in the net-section. Thus, the change in netsection cyclic strain energy is a global concept, as opposed to
fracture mechanics parameter, DK, which is a local or crack tip
concept. A detailed discussion of how stress intensity factor in
fracture mechanics actually evolves from the net-section stress or
strain energy is given in the previous work [18].
3. Validation by experimental data
In the following, using extensive data generated on CCT and
SENT specimens, the unique correlation of mean stress effect on of
fatigue crack growth behavior, on the basis of DCR, is demonstrated.
The raw crack length and cycle data, presented in literature for
various materials, were used to generate the fatigue crack growth
data used in the present correlation. The crack growth data were
determined using the secant method [21] on two successive crack
lengths and the corresponding fatigue cycles. The DCR values were
determined based on the average of the crack lengths from two
successive measurements using Eq. (10). For comparison, conventional da/dN versus DK plots were generated for the materials. The
DK values for CCT specimens were calculated using the secant
formula:
pffiffiffiffiffiffi
DK ¼ Ds pa
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n pa o
sec
2W
(12)
where a is crack length and 2W is specimen width. The crack
growth rates and the corresponding DK values were determined on
the basis of average crack lengths between the two crack length
intervals.
205
Broek and Schijve [22] investigated the effect of stress ratio on
fatigue crack growth in 2-mm-thick and 160-mm-wide entercracked panels of the 2024-T3 Al alloy. Fatigue testing was performed for various stress ratios ranging from 0.06 to 0.67, and at
various stress amplitudes. The fatigue crack growth data for this
material is shown in Fig. 2(a) and (b) on the basis of DK and DCR,
respectively. The fatigue crack growth data, when plotted in terms
of DK, shows some variation with mean stress level, with crack
growth rates increasing with R, for a given DK. This variation is
approximately a factor of three. It can be seen in Fig. 2(b) that the
use of DCR for crack growth correlation provides a good agreement
between the data. For example, the data for R ¼ 0.06, R ¼ 0.29,
R ¼ 0.57 and for R ¼ 0.67 are on top of each other, indicating the
excellent agreement. Comparing the R ¼ 0.29e49 MPa data with
the R ¼ 0.3e128 MPa data, it appears that the small scatter in the
data in Fig. 2(b) is likely due to material variability.
It is important to note that the X-axis scales of Fig. 2(a) and (b),
and that in similar figures presented in the following, have been
chosen consistently such that the comparisons of crack growth
data in terms of DK, DG and DCR can be properly made. The
parameter DK is a stress-based parameter (with a linear dependence on stress) whereas DG and DCR are energy parameters (with
a dependence on the square of stress). This is also evident from
the fracture mechanics relationship: G ¼ K2/E, where E is the
elastic modulus. Therefore, when crack growth data are plotted in
terms of DK of two orders of magnitude in the X-axis (for example
1e100 MPa√m in Fig. 2(a)), the comparison plots of data in terms
of DG or DCR should be made with four orders of magnitude in the
X-axis scale (for example from 0.0001 to 1 MPa for the DCR plot in
Fig. 2(b)) for DG plots in Fig. 9). This scaling is required by the fact
that when DK varies by a factor of 100, the corresponding variation in energy-based crack growth parameters must vary by a
factor of 10000. Only under such scaling of axes, a valid comparison of the relative merits of crack growth parameters can be
made. Therefore, this scaling principle is applied to all the figures
in this study.
Fig. 2. Fatigue crack growth data for 2024-T3 aluminum alloy (CCT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in-net section strain energy,
DCR. Data calculated from the raw crack length data of Broek and Schijve (Ref. [22]).
206
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
Fig. 3. Fatigue crack growth data for 5052 aluminum alloy (SENT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in net-section strain energy,
DCR. Data calculated from the raw crack length data of Kumar and Gangwar (Ref. [23]).
Kumar and Gangwar [23] generated fatigue crack growth data
for R-values ranging from 0 to 0.5 for 5052 Al alloy, using 3-mmthick, and 50-mm-wide SENT specimens. Fig. 3(a) and (b) illustrate
the correlations of fatigue crack growth data on the basis of DK and
DCR, respectively. The DK values were calculated using the appropriate stress intensity factor solution for uniformly displaced ends,
which is given in Ref. [20].
pffiffiffiffiffiffi
DK ¼ Ds pa 1:11 þ 0:664
a
3:52
W
a 4
a 5 33:26
þ 19:32
W
W
a 2
W
þ 19:92
a 3
W
(13)
Again, excellent agreement of fatigue crack growth data, when
Fig. 4. Fatigue crack growth data for 6063-T6 aluminum alloy (CCT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in net-section strain energy,
DCR. Data calculated from the raw crack length data of Chand and Garg (Ref. [24]).
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
207
Fig. 5. Fatigue crack growth data for 7075 aluminum alloy (CCT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in net-section strain energy, DCR.
Data calculated from the raw crack length data of Hudson (Ref. [25]).
represented in terms of DCR is found in Fig. 3(b) while the DK-based
correlation shows some scatter in data. Fatigue crack growth data
for 6063-T6 Al alloy were generated by Chand and Garg [24] for Rvalues from 0 to 0.6, using 3-mm-thick and 83.5-mm-wide CCT
samples. The fatigue crack growth data, correlated in terms of DK
and DCR, are shown in Fig. 4(a) and (b) respectively. Again, excellent
agreement of fatigue crack growth data, in terms of DCR, can be
seen.
A very extensive fatigue crack growth data for 7075-T6
aluminum alloy was generated by Hudson [25] using relatively
large CCT samples. The specimens were 2.28-mm-thick and 305mm-wide. The data set consisted of tests done at the widest
range of stress ratio, ranging from R ¼ 0 to R ¼ 0.8, with multiple
tests at each stress ratio. The fatigue crack growth correlation for
this material is shown in Fig. 5(a) and (b) on the basis of DK and
DCR, respectively. Again a very good correlation of data on the basis
Fig. 6. Fatigue crack growth data for 7075-T6 (CCT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in net-section strain energy, DCR. Data
calculated from the raw crack length data of Broek and Schijve (Ref. [22]).
208
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
Fig. 7. Fatigue crack growth data for TRIP steel (CCT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in net-section strain energy, DCR. Data
calculated from the raw crack length data of Cheng et al. (Ref. [26]).
of DCR is found. It is important to note that there is still some spread
in the data, which is likely to be due to material variability, rather
than due to error in DCR. For example, in Fig. 5(b) for R ¼ 0, the test
data for two different applied stress levels (68 and 138 MPa) show
some difference in growth rates near low DCR values. This difference at a given R is very likely to be due to material variability. In
Fig. 5(b), if the data for R ¼ 0; 68 MPa and R ¼ 0.33; 68 MPa are set
aside, the rest of the data set still has at least one test data for each
stress ration and all these data form a tight band in the figure. Also,
the multiple data sets for each of R ¼ 0.2, 0.33, 0.5, 0.7 and 0.8, agree
very well each other. Hence the small vertical spread in FCG data in
Fig. 5(b) is likely to be due to material variability.
Fatigue crack growth data for 7075-T6 Al alloy were generated
by Broek and Schijve [22] for R values of 0.29, 0.57 and 0.66, using
2-mm-thick and 150-mm-wide CCT samples. The fatigue crack
growth data, correlated in terms of DK and DCR, are shown in
Fig. 8. Fatigue crack growth data for Ti-6Al-4V alloy (CCT specimens) correlated in terms of (a) stress intensity range, DK and (b) the change in net-section strain energy, DCR. Data
calculated from the raw crack length data of Hudson (Ref. [27]).
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
Fig. 6(a) and (b) respectively. Excellent agreement of fatigue crack
growth data, in terms of DCR, can be seen.
Fatigue crack growth data for TRIP steel were generated by
Cheng et al. [26] for R-values ranging from 0.1 to 0.7, using 2-mmthick and 80-mm-wide CCT samples. The fatigue crack growth data,
correlated in terms of DK and DCR, are shown in Fig. 7(a) and (b)
respectively. When plotted in terms of DCR, the crack growth curves
for various R-values are virtually on top of each other. An exception
is the crack growth data for R ¼ 0.7 at da/dN > 104 mm/cycle. Here,
the crack growth rates are unusually accelerated near the fast
fracture regimedthe crack seems to grow with a wide range of
rates at nearly constant DK, which is not real. This appears to be an
anomaly in the data and it is not present in the data for other stress
ratios. Except for this anomaly, the rest of the data are in excellent
agreement with each other.
Fatigue crack growth data for Ti-6Al-4V alloy (R ¼ 0 to 0.85) and
at various stress amplitudes, were generated by Hudson [27] using
1-mm-thick and 203-mm-wide CCT specimens. The fatigue crack
growth correlation for this material is shown in Fig. 8(a) and (b) on
the basis of DK and DCR, respectively. When plotted in terms of DK,
the FCG rates increase with an increase in R. However, this difference nearly vanishes when plotted in terms of DCRdit is worth
noting that the data for R ¼ 0, R ¼ 0.66 and R ¼ 0.85 line up straight
with respect to each other, and a small deviation in data exists for
the R ¼ 0.25 data. It is possible that this deviation is also due to
material variability, because there is no reason that only this data is
deviated, while the other data are in agreement with each other.
It is clear from the extensive evaluations presented above that
the accumulated change in net-section strain energy parameter,
DCR, provides unique and excellent correlations of fatigue crack
growth data, generated under various stress ratios and at various
stress amplitudes (or ranges). All the data clearly indicate that crack
growth rate increases with an increase in cyclic strain energy of
net-section during fatigue crack growth. The success of this correlation emphasizes the fact that the accumulated state of cyclic
strain energy in the net-section is controlling the crack growth. Any
change in crack tip strain energy state that is linked to crack
growth, as often invoked in fracture mechanics approaches, is in
fact derived by the specimen from the change in the global netsection cyclic strain energy as determined exactly in this work.
Thus, it is obvious why the DCR parameter correlates the meanstress-dependence of FCG data quite well.
It is important to note that the scope of application of the proposed change in net section strain energy parameter is for single
cracks growing in fatigue, whether small or large in size, under
uniform stress field. It is certainly possible to apply this approach to
other situations of fatigue, such as crack growth under bending
and/or growth of multiple cracks, using the expressions that
describe the change in net section strain energy in these situations.
These expressions, however, can be developed following the
approach shown in this work.
4. Significance of the change in net-section strain energy for
the correlation of mean stress effect on FCG
The principal motivation for using the DCR parameter for
capturing the mean stress effect on fatigue crack growth is that it is
easy to derive from solid mechanics principles and the meaning of
it is physically understandable. The calculation of DCR does not have
any arbitrary assumptions as clearly illustrated in section 2. It is
more important, however, to emphasize the physical meaning of
DCR. It is obvious from Fig. 1 that the change in net section cyclic
strain energy, corresponding to a crack length, is the difference
between the cyclic strain energy (per unit volume) of the netsection with the crack and the cyclic strain energy (per unit
209
volume) of the specimen in the crack-free state. This difference,
computed at a given crack length, represents the accumulated increase in cyclic strain energy in the net-section from the inception
of fatigue crack growth from a near-zero crack size. This increase is
relatively smaller for a specimen with a smaller crack length and is
larger for that with a relatively larger crack length. In both cases the
point of reference is the crack-free state. This means that the crack
growth rate at any crack size is determined by the change in the
cyclic strain energy per unit volume that has accumulated in the
net-section, defined by the size of the crack, relative to the crackfree specimen.
It is to be noted that the accumulated change in strain energy
belongs to the net-section and hence it is also manifested as the
change in the distribution of normal stress or strain in the crack
plane or near the crack tip, relative to crack-free state. Following
the stress functions of Westergaard [28] or Williams [29], it is
obvious that any change in elastic stress or strain at a point near
the tip of the crack, or in the crack plane, is linearly related to the
change in remote stress and, thereby, to the corresponding change
in strain energy per unit volume. Thus, regardless of whether it is
the strain energy change in the crack plane or that in the crack tip
region that may drive the crack growth, it has to be linearly related
to the global change in the net-section strain energy, on per unit
volume basis. In other words, the change is the same whether it is
interpreted as the global or any local value of the change in strain
energy.
The proposed correlation based on the accumulated change in
net-section strain energy leads to a postulate that is strikingly
different from that in the fracture-mechanics-based DK correlations of fatigue crack growth. In fatigue literature, it is believed that
DK controls the crack growth, which means that the crack tip parameters, especially the stress/strain state in the crack tip region is
solely responsible for crack extension in fatigue. The author has
shown in the previous work [18] that an increase in K with crack
length, at a constant stress, actually corresponds to increase in netsection stress (or strain energy) in the ligament ahead of the crack.
Because the definition of K applies only to the crack tip and since
this definition relies only on the singular stress state ignoring the
net-section stress, it seems erroneous to think that DK solely controls fatigue crack growth. This is because the control exerted by DK
on fatigue crack growth arises from the fact that the change in K (or
DK) evolves entirely from the change in net-section stress state. The
change in net section strain energy, which directly relates to the
accumulated work done at the loading ends, is an uncomplicated
and physically understandable parameter. This is also easy to
calculate because it does not require complex finite-width correction factor of fracture mechanics.
5. Evaluation against Griffith's cyclic energy release rate, DG
There is some opinion in literature for metals and polymers [30]
or for delamination fatigue crack growth [31] that the fatigue crack
growth rate should be correlated in terms of DG, because this
parameter, which may be called as Griffith cyclic energy release
rate, can be expressed as a function of stress ratio. In this section, it
is shown why this approach is not correct for the characterization of
fatigue crack growth. Using the relationship between stress intensity factor K and energy release rate G, DG for cyclic loading can
be written as
DG ¼ Gmax Gmin ¼
This reduces to
2
K2
Kmax
min
E
E
(14)
210
DG ¼
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
DK 2 ð1 þ RÞ
E
(15)
ð1 RÞ
where E is the elastic modulus and other parameters are obvious. It
can be shown that the physical meaning of Eq. (15) is that it is the
Griffith's energy release rate for cyclic loading and it corresponds to
the area DBCF as shown in Fig. 1(d). Based on the standard fracture
mechanics principles, this area is often thought of as the release of
energy upon a small crack extension a1 in Fig. 1(d). The question of
whether it is the initial infinitesimal crack of size “a1” that forms in
a crack-free specimen or it is the infinitesimal crack extension
(da ¼ a1) that occurs from a prior crack length is irrelevant here,
because from the energy release rate perspective, both are equivalent. The Griffith cyclic energy released, corresponding to area
DBCF in Fig. 1(d), is written as
1
2
DUG ¼ Pmax dmax;1 dmax;0 1
P
d
dmin;0
2 min min;1
(16)
Writing Eq. (16) in terms of specimen compliances C1 and C0, for
the cracked (a1) and crack-free states, respectively,
1
2
DUG ¼ Pmax ðC1 Pmax Co Pmax Þ 1
P ðC P
Co Pmin Þ
2 min 1 min
(17a)
DUG ¼
1 2
2
Pmax Pmin
ðC1 Co Þ
2
(17b)
It is to be noted that ðC1 Co Þ ¼ vC for an infinitesimal crack
extension. Hence, the cyclic energy release rate for an infinitesimal
extension of the crack can be obtained by rearranging and dividing
both sides by va:
vUG 1 2 vC 1 2 vC
¼ Pmax Pmin
2
va 2
va
va
(18)
According to the definition of Griffith energy release rate by
Irwin and Kies [32,33] in terms of the change in specimen
compliance, the first term in Eq. (18) is the energy release rate
under static loading to Pmax and the second term is the energy
release rate under static loading to Pmin. Hence, it follows from Eq.
(18) that
vUG
¼ DG ¼ Gmax Gmin
va
(19)
which is the same as Eq. (14) or (15). Two examples of FCG correlations based on DG are shown in Fig. 9(a) and (b) for 5052 and
7075 Al alloys, respectively, The raw data are the same as that used
for the DK-based plots, which are shown in Figs. 3(a) and 5(a),
respectively. It is evident that the FCG correlations based on DG are
not good and the data show significant deviations from each other.
The main reason for the poor correlation is the tacit assumption
that area DBCF (in Fig. 1(d)) may represent the energy release
during crack extension under cyclic loading. However, if the area
DBCF is considered as the energy released, it would imply that the
crack extends at Pmin (along the line DF). This is because in this path
the compliance of the specimen changes from C0 to Ci, which can be
possible only if the infinitesimal crack extension takes place at the
constant load Pmin. This extension is in violation of the physics of
fatigue crack growthdcrack growth actually occurs upon loading
from Pmin to Pmax in a fatigue cycle. Thus, this analysis also reinforces the correctness of the area BCD, defined by line DC, for the
calculation of the accumulated change in cyclic strain energy, DCR,
as outlined in section 3. The parameter DCR appears to be the
proper measure of the change in work done (or increase in net
Fig. 9. Poor correlation of fatigue crack growth rates in terms of Griffith DG-approach (a) for 5052 using the data of Kumar and Gangwar in Fig. 3(a) and (b) for 7075 alloy using the
data of Hudson in Fig. 5(a).
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
section strain energy specimen) associated with the crack advance
during fatigue cycling.
6. Evaluation in terms of Dsna parameter by Nisitani and
others
Some fatigue crack growth characterizations, mostly by researchers in Japan, make use of the parameter Dsna, (named here
as Nisitani parameter; where Ds is the cyclic stress and a is the
crack length) to correlate the crack growth data. Nisitani [34] and
others [35] have used this parameter to show that, for small cracks,
the da/dN data at various stress levels can be consolidated by taking
arbitrary n values. The n values varied between 3 and 8 for different
materials [36]. However, conceptually, the Nisitani parameter has
serious problems because (i) its physical basis has not been
demonstrated (whereas DK is at least partially justified on the basis
of fracture mechanics: DG ¼ DK2/E), and (ii) entirely arbitrary
values of n, without any basis, are used to “collapse” the data, in
terms of Nisitani parameter, in several publications in literature.
Further, it was stated [34] that this parameter is suitable only at
high stress amplitudes (>0.6sy), while DK was found to be
adequate for low stress amplitudes (<0.6sy). A more serious difficulty is that the unit of the Nisitani parameter will be in MPanm,
and this unit will change with material (because of the changing
values of n taken to fit the data), which means that it is not a
reliable parameter. Based on the above, it seems that there is no
valid reason to use Nisitani parameter for the characterization of
fatigue crack growth.
Nevertheless, it is of interest here to check if the present DCRbased approach will yield a better correlation for small crack
growth data, relative to the Nisitani parameter and DK. Wang and
Fan [37] performed in situ small crack growth experiments, using
AZ50 magnesium alloy in SEM, gathering growth data for cracks
with sizes from ~10 mm to ~1 mm at various stress levels. A constant
stress ratio (R ¼ 0.1) was used. Their crack length data for tests at
150C are used here in making the crack growth plots shown in
Fig. 10(aed) in terms of various parameters. Fig. 10(a) is the plot
based on DK. Fig. 10(b) and (c) are plots made on the basis of
Nisitani parameter with n ¼ 1 and n ¼ 3, respectively. At higher
values of n > 3, however, the correlation in terms of Dsna was found
to be increasingly worse. The quality of correlations of growth data
in Fig. 10(aec) is not great, although it appears to be similar between the plots made using DK and Dsna. On the other hand, it is
interesting to note that the plot based on DCR (Fig. 10(d)) shows a
visible improvement in the correlation of data, relative to the other
parameters, suggesting that the change in net-section strain energy
concept can be a better approach to correlate small crack growth
data. This warrants the used of DCR for a more extensive evaluation
of small crack data, especially the data generated at various mean
stress levels.
7. Comparison against mechanism-based interpretations of
the mean stress effect
7.1. Approach based on crack closure
Several early researches [38e40] showed the role of increased
mean stress in producing increased static modes of crack extension
(void coalescence or cleavage) during FCG. These studies suggested,
based on fractographic evidence, that the increased proportion of
static fractures during cyclic crack growth caused the increased
crack growth rates at high mean stress levels. A vast majority of
interpretations of mean stress effect on FCG, however, are based on
crack closure concepts such as plasticity induced crack closure
(PICC) [17,41,42] and/or roughness induced crack closure (RICC)
211
[43,44]. The premature closure of the near-tip crack-wake region,
during the unloading part of the fatigue cycle at low stress ratios,
has been suggested to shield the crack tip from the applied stress
leading to an apparent reduction in crack growth rates. In
plasticity-induced crack closure, the residual stretch of plasticallydeformed crack-wake region has been suggested to close the crack
at some positive stress. In roughness induced crack closure, the
increased surface roughness near threshold has been suggested to
cause interference of crack surfaces, producing a shielding effect,
similar to that in PICC. However, the reliability of these concepts
appear to be weak because it has been shown that different measurement methods provide different values of crack opening stress
levelsdthe near-tip (optical, SEM) and remote (crack mouth and
back-face) measurements provide different closure results [45,46].
A study attempting to standardize automated closure measurements concluded that global closure measurements (for example,
using a clip-gage) can give misleading information in the nearthreshold regime [47]. Also, there has been no universally agreed
upon method of crack closure measurement that produced
consistent results, which is an issue along with several discrepancies in the concept itself, both of which have been critically
assessed in a 1984 review [48].
A problem with the application of crack closure concept is that
at high stress ratios, the crack opening displacement at the smin can
be larger than the extent of plastic stretch and no crack-wake
interference occurs, but still an effect of R on FCG, due to factors
other than crack closure (such as the increased incidence of static
modes of fracture), can be observed. Also, the conceptual basis can
be questionable when arbitrary functions, such as that proposed by
Elber [17], are used “correct” for the effect of R, in an attempt to
consolidate the data at various R values in to one curve. This may be
acceptable and indeed practiced for engineering purposes, but
there are some conceptual difficulties as discussed below. This
correction implicitly assumes a high Sop value for the high-R FCG
data, which cannot be justified because here the minimum crack
opening displacement is a large fraction of maximum crack opening displacement and hence crack-wake contact is not expected to
occur, unless visual evidence is provided to support it. Although
Elber showed that the mean stress effects on the fatigue crack
growth behavior of 2024-T3 alloy, generated by Hudson [25], could
be explained on the basis of a crack closure function (U ¼ DSeff/
DS ¼ 0.5 þ 0.4R), there was no presentation of the raw loaddisplacement data or visual evidence, especially at high R-values,
to justify the use of such functions. Further, Elber's illustrative data
(Figs. 5 and 6 of that paper [17]) show similar Sop values in the
crack-wake and the crack-tip load-displacement curves, which
raise a fundamental conflict about the definition of crack closure
itself, at least in the context of Elber's paper. If the crack-wake
contact, as inferred from the non-linearity of load-displacement
curve (Fig. 5 in Elber's paper) is accepted as crack closure, then
the comparable nonlinearity in the crack-tip measurement (Fig. 6 in
Elber's paper) is difficult to explain. It should also be noted that
several studies [49e52] have also shown that crack closure could
not explain completely the R-effect on the fatigue crack growth
behavior of materials used in those studies.
Another weakness of crack closure approach is that after the
subtraction of opening stress, the stress ranges (DSeff) used to
compute DKeff values for various R-values are still subject to some
mean stress variations among them. The DKeff values computed
using Elber's crack closure function will place the DKeff data of a
high-R test at a relatively higher mean stress position, relative to
that of a low-R data. In reality, the effective stress ratio that is
driving the crack, in the absence of crack closure, will have to be
defined as Reff ¼ Sop/Smax for tests with crack closure, unless the Sop
levels are the same for all stress ratios, which contradicts Elber's
212
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
Fig. 10. Demonstration of fatigue crack growth correlation by various approaches for small cracks in terms of (a) DK and Nisitani's Dsna approach in (b) with n ¼ 1, and in (c) with
n ¼ 3. The change in net-section energy parameter, DCR used in (d) shows a relatively better correlation, compared to other parameters. The crack length data of small cracks
(10e1000 mm) in AM50 Mg alloy, generated by Wang and Fan by in situ SEM testing, are used.
closure function. In summary, the removal of crack closure stress
does not automatically normalize the variations in the mean stress
levels in the effective stress ranges of different R test conditions.
This discrepancy is not addressed in the closure-based approaches.
Finally, there is a major theoretical difficulty in accepting the
concept of crack closure to explain the mean stress effect on FCG. If
the effective driving force, DKeff ¼ Kmax-Kcl, is accepted, then this
implies that as far as the crack tip mechanics is concerned, K ¼ 0 for
the period of loading: 0 < K < Kop because crack is technically
“closed” in this range. This forces the idea that the crack-tip singular
stress field, or the Westergaard stress function from which the
single-stress field evolves, is correspondingly zero during
0 < K < Kop. This is physically inconsistent because the crack tip
stress distribution (as defined by the singular stress field) or the
net-section stress distribution (as defined by the Westergaard
stress function) is not zero for 0 < K < Kop, because the specimen is
still under non-zero load in this range. It is therefore clear that the
definition of DKeff on the basis of crack closure (that is ignoring the
loading portion up to Kop to define a crack driving force, DKeff), is
seriously in conflict with the applied stress on the sample. This
inconsistency, within fracture mechanics based definition of crack
closure, a serious obstacle to the arguments invoking crack closure
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
213
mechanism to rationalize the mean stress or stress ratio effects on
FCG.
strain energy, as determined for any stress ratio. This is an interesting connection that needs to be explored further.
7.2. Two-parameter (DK, Kmax) models
8. Conclusions
Recent approaches [53,54] to explain the mean stress effects on
FCG, relay on a two-parameter approach where it is suggested that
both DK and Kmax of the fatigue cycle control the fatigue crack
growth rate. Especially at threshold, it has been suggested [49] that
some minimum or threshold values of both parameters, DKth and
Kmax,th are required to see any measurable crack growth in a material. These proposals have been made as alternatives to the use of
the concept of crack closure to explain the stress ratio effects, as
clearly summarized elsewhere [55]. However, these approaches
have problems of their own. The major problem is that the two
parameters, DK and Kmax, are directly related to each other through
the relationship: DK¼(1-R)Kmax. It is not possible to argue that DK
and Kmax are two separate or independent parameters, because a
change in one also changes the other. Hence, the two-parameterbased diagrams depicting threshold conditions for fatigue crack
growth, although may be appealing from the way the data is presented, can not be justified as scientifically sound. Even if the twoparameter approach is accepted at its face value, the fundamental
mechanism by which the two parameters act simultaneously to
determine the threshold condition, or the fatigue crack growth rate,
is not clarified. More recently, however, the two-parameter
approach has been advocated [12,13] to lead to an effective crack
growth rate parameter (DK0.5 Kmax0.5) which seems to successfully
explain the mean stress effect on fatigue crack growth. This
parameter, however, is the same as that of Walker, proposed long
time ago.
It is clear from the above discussion that crack closure concept
has conceptual problems with respect to its theoretical definition
and experimental problems due to its inability to consistently
explain the mean stress effect on fatigue crack growth. The
alternate two-parameter approach, mainly developed to “fix” the
problems that arise from the crack-closure-based analysis of fatigue crack growth, is quite similar to Walker's empirical
approach, which also does not rely on crack closure. In this light,
the remarkable success of the present approach, which is a
physically-based net-section strain energy parameter to account
for the mean stress effect, demonstrates that this may have indeed
captured, quantitatively, the real crack driving force of fatigue
crack growth.
It should be noted that the change in net-section cyclic strain
energy as proposed in this work also has two stress parameters (Ds,
smax) defining it. It might be tempting to conclude that, since this
seems similar to the two-parameter (DK0.5 Kmax0.5) model, the
present approach is similar to the Walker formula or the twoparameter model. The appearance of a similarity is merely coincidental and the approaches are not related to each other. This is
because the changes in DK and Kmax are that which occur with a
change in crack length, at constant loading, where as the change in
net-section cyclic strain energy, as defined by the two parameters
(Ds, smax) is a cumulative change, with respect to the crack-free
specimen. Nevertheless, the present net-section strain energy
concept seems to validate the Walker's approachdEq. (10) in
essence, contains the product “Ds.smax” to represent the maximum
stress and the stress amplitude as crack driving force. The empirical
Walker parameter (when written in the most-successful form:
DK0.5 Kmax0.5) is also proportional to this product, even though the
difference is the square-root and the finite-width correction factor
of fracture mechanics. Thus, it appears that the reason for the
success of Walker approach actually stems from the fact that the
Walker parameter is related to the accumulated change in cyclic
1. The general solid mechanics approach to determine the change
in net-section strain energy for the characterization of fatigue
crack growth, for any mean stress, is presented. This has led to a
generalized mechanics parameter to correlate the mean stress
effect on fatigue crack growth behavior expressed in terms of
the stress ratio, stress amplitude, crack length and specimen
width:
DCR ¼
1
Ds2 h a i
2E ð1 RÞ W a
2. It is shown that the rate of growth of a fatigue crack, at a given
length, and at a given stress ratio, is uniquely determined by the
accumulated change in the strain energy of the net-section,
relative to the crack-free state. This change is directly related to
the change in cyclic work done at the loading ends, which is
responsible for driving the fatigue crack growth.
3. Extensive experimental data on several materials, spanning a
wide range of applied test stress levels and stress ratios in CCT or
SENT specimen configurations, have been used to demonstrate
the validity of the FCG correlation based on the net change in
strain energy.
4. It is also shown that the common contender for the correlation
of mean stress effect on fatigue crack growth, that is the Griffith's cyclic strain energy rate, DG, is neither correct nor useful
for the description of changes in energy states due to the crack
advancement, because its definition violates the physics of crack
extension in a fatigue cycle.
5. The success of the present approach negates the necessity of
crack-closure-based interpretations of the mean stress effect,
which are plagued with conceptual weakness as discussed in
the paper. The empirical two-parameter approach is also not
needed. In contrast to these, the present approach is physically
sound and contains the necessary parameters to characterize
the fatigue crack growth behavior well.
6. The change in net section strain energy parameter can provide
very good correlation of small crack growth data obtained at
different stress levels. The correlation was shown to be better
than that provided by empirical Nishitani parameter (Dsna) for
small cracks.
7. The present approach, in fact, provides a reasoning as to why the
empirical Walker parameter, (DK0.5 Kmax0.5), is able to correlate
the fatigue crack growth rates well as a function of stress ratio.
Since the Walker parameter is proportional to the product
“Ds.smax” which is contained in the net change in cyclic strain
energy parameter, it appears that the success of Walker
approach stems from the fact that fatigue crack growth behavior
is fundamentally related to the accumulated change in netsection cyclic strain energy.
Acknowledgement
The author gratefully acknowledges the efforts of the researchers, whose publications have been fully referenced here, for
generating the experimental fatigue crack growth data that have
been used here for evaluation. The author gratefully acknowledges
the discussions with Dr. J. C. Newman, Jr., Mississippi State University, on finite-width correction factors. The author thanks Dr.
Emeil Amsterdam of Netherlands Aerospace Center (NLR), The
214
K.S. Ravi Chandran / Acta Materialia 135 (2017) 201e214
Netherlands, for providing a copy of Ref. [22]. Thanks also to Dr. Xu
Cheng, SKF, The Netherlands, for generously providing the raw
crack growth data for the TRIP steel.
[28]
[29]
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