Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 4, pp. 923-931, Warsaw 2012

50th Anniversary of JTAM

MODIFICATION OF THE LSM CRITERIA TO APPROXIMATE TEST DATA

FOR FATIGUE CRACK GROWTH RATE

Sylwester Kłysz

Air Force Institute of Technology, Warsaw, Poland and

University of Warmia and Mazury, Olsztyn, Poland; e-mail: sylwester.klysz@itwl.pl

Janusz Lisiecki, Andrzej Leski

Air Force Institute of Technology, Warsaw, Poland

Tomasz Bąkowski

OLT Express Regional, Warsaw, Poland

Modification of the least squares method criteria is presented. It is intended to enhance
the approximation of test data for fatigue crack growth rate. In particular, changes are
introduced to improve approximation in the cases when the range of values for at least
one of the variables is within the range of several orders of magnitude or differs from the
range of values for other variables.Differences betweenparticular criteria and their influence
on the approximation of test data is illustrated further on. Calculations are conducted for
aluminum alloy 2024 taken fromMi-2 helicopter rotor blades.

Key words: fatigue crack propagation, NASGRO equation, method of least squares (LSM)

1. Introduction

Over the years, a number of relationships have been developed to represent all or parts of a
typical range of fatigue crack growth data, da/dN = f(∆K). Application of the NASGRO
equation derived by Forman and Newman from NASA, de Koning from NLR and Henriksen
from ESA, of the general form ([1]; [8])

da

dN
=C
(1−f

1−R
∆K
)n

(

1− ∆Kth
∆K

)p

(

1− Kmax
Kc

)q (1.1)

has significantly extended possibilities of describing the crack growth rate, tested according to
the standard [3]. TheNASGRO equation represents themost comprehensive formulation of the
crack growth law. It is a full-range model that mathematically represents all three regions of
the da/dN vs. ∆K curve comprises the mean stress effect, threshold, fast fracture and crack
closure (Hudson and Seward, 1983; [9]; Walker, 1970). The coefficients stand for ([2]; Fuchs
and Stephens, 1980; [8]): a is the crack length [mm], N – number of load cycles, R – stress
ratio, ∆K – stress intensity factor (SIF) range depending on the specimen size, applied loads,
crack length, ∆K =Kmax−Kmin [MPa

√
m], ∆Kth – SIF threshold, minimum value of ∆K,

at which the crack starts to propagate: Kc – critical value of SIF, f – Newman’s function
describing closing of the crack (Newman, 1984), C, n, p, q – empirical coefficients.
Determination of the above coefficients for the equation correctly approximating tests data

is difficult and causes some singularities when the Least SquaresMethod (LSM) is used. Curves
fitting to the NASGRO equation are obtained using the NASMAT module contained within
theNASGRO suite of software [8]. TheNASMAT curve fitting algorithms use the least-squares



924 S. Kłysz et al.

error minimization routines in the log-log domain to obtain the corresponding constants. The
constants C and n, i.e. themain fit parameters, are determined through theminimization of the
sumof squares of errors,where the error termcorresponding to the i-th data pair (∆K,da/dN)i
is (Forman et al., 2005)

ei = log
( da

dN

)

i
− logC−n log

(1−f

1−R
∆K
)

i
− log

[

(

1− ∆Kth
∆K

)p

(

1− Kmax
Kc

)q

]

i

(1.2)

Values of da/dN are determined using themethod of differentiating the dependence a−N
with the secant or the polynomial method applied ([1]; [3]; [8]).
The curve fitting of crack growth data is an iterate process that consists in using established

values of various constants (other than C and n), specifying the data sets that typify the
material, applying the least-squares algorithm to compute C and n, and plotting the data for
various Rvalueswith the curve fit of each stress ratio. Theprocess is continued bymaking slight
modifications in the entered values until the best fit to the test data is obtained. In general,
fitting the NASGRO equation is really a multi-step process involving:
• fitting or defining the threshold region

• fitting or defining the critical stress intensity or toughness to be used at the instability
asymptote

• making initial assumptions on key parameters such as p and q

• performing the least squares fit to obtain C and n, and finally

• using engineering judgment to adjust the results for consistency and/or a desired level of
conservatism.

For theLSMapproximation of test data, theanalytical description thereof anddetermination
of coefficients of approximation equations, according to which the criterion used in the analysis,
is the minimum of the square sum

S =
n
∑

i=1

(yi−yi)
2 (1.3)

of deviations between values of the test data yi and those of the approximated function yi. This
method of approximation is characterized with the following properties that in some cases may
be considered as disadvantages (Forman et al., 2005; Huang et al., 2005; Taheri et al., 2003;
White et al., 2005; Zhao and Jiang, 2008):
• value of the sum S increases regarding the order of magnitude as approximated values
increase, e.g. if values of the test data are of the order 10, 103, 106, with the scatter of 10%,
the summed differences are of the order 1, 102, 105 respectively, and hence:

– the same, e.g. the (2-5)-times change in the test value results in different as to the
order of values of the summed differences

– dynamic changes in the total value of the sum S depend on values of the differences
– as a quadratic function it is characterized by a linear function of the derivative,
which alsomeans that for differences close to zero (e.g. 10−5, 10−8, etc.) this dynamic
change is much smaller than for differences of higher magnitudes, which influences
the “flexibility” of the performed approximation

– if the test data differ significantly in magnitude from each other (e.g. from 1 to 105

or from 10−8 to 10−2), the approximated values near the lower threshold contribute
much less to the total sum S than the approximated values near the upper threshold;
this means that, e.g. tens or hundreds of test data with differences in magnitude of
100% from value 1 are less significant in the performing approximation than one or
a few data points which differ by 1% from value 105.



Modification of the LSM criteria... 925

According to the above stated example, the approximation is “asymmetric” since a better
approximation will be achieved for higher values of test data, neglecting differences around
smaller values –anexample of suchanapproximation is shown inFig. 1,whereone can see agood
fit of theoretical description of 9 curves for large values of da/dN (over 10−4mm/cycle), while
there is a visible mismatch-fit for the smallest values (below 10−5mm/cycle). The presented
approximation has been achieved by satisfying the LSMcriterion, i.e. theminimumvalue of the
sum S.

Fig. 1. Example of approximation with the NASGRO equation, LSM according to formula (1.3);
test data with large y value scatter

When the test data are within a wide range of values, e.g. 5 orders of magnitude, i.e. from
10−2 to 10−7mm/cycle, then differences between the highest values and the approximating
function will have the largest influence on the square sum S of deviations while differences for
small values, sometimes of 2-3 orders of magnitude, do not contributemuch to the total sum S.
The LSM can be flexible when the below proposed modification is introduced, and this is

the main aim of this work.

2. Modifications of the LSM method criterion

In order to eliminate the approximationmismatch-fit as shown in Fig. 1 and to improve the qu-
ality of approximation, amodification of formula (1.3) is proposed. It should reach the following
form

S∗ =
n
∑

i=1

(yi−yi
yi

)2

=
n
∑

i=1

(yi
yi
−1
)2

(2.1)

In thisway, the fraction inbrackets, as a relative error, is a stablemeasure of deviationbetwe-
en the approximated and approximating values, i.e. independent of themagnitude of compared
values, since:
• each value among the test data yi has an equal contribution in the sum S

∗, independent
of its magnitude 10−7, 10−2, 1 or 105 (i.e. it fits in any magnitude range) – always a
deviation of e.g. 10-, 50-, 200-percent of the approximating value will give a component of
the sum S∗ equal to 0.01, 0.25, 4, respectively



926 S. Kłysz et al.

• the criterion assures that the achieved approximation is “symmetric”, i.e. the degree of
approximation around lower and higher values is the same

• disadvantages of the criterion described with formula (1.3) are no longer valid.

Figure 2 shows results of approximation achievedwithmodifiedLSMcriterion (2.1).Avisible
improvement in the approximation of the 9 curves can be seenwithin thewhole range of da/dN
values.

Fig. 2. Approximation of da/dN = f(∆K) data with the NASGRO equation, the LSMmodified
according to formula (2.1)

The criterion described with formula (2.1) has also some specific property: if the approxi-
mating value equals zero (i.e. for the approximation smaller by 100%) or it is twice as much
as the approximated value (i.e. for approximation larger by 100%), then independently of the
approximated value, the component of the sum S∗ will be equal to 1.
Both criteria (1.3) and (2.1) have also some disadvantage consisting in that if the appro-

ximating value yi is much smaller than the approximated value yi (i.e. by 3, 5, 7 orders of
magnitude) or simply close to zero, then the component of the sum S and S∗ is close to the
square of value yi (in the case of (1.3)) or to 1 (in the case of (2.1), independently of how these
two values differ from each other.
Obviously, it is important whether the approximation and behavior of the approximating

curve near the value yi at the level of e.g. 10
−6 and lower (i.e. for strongly decreasing values

within the “threshold” range of the graph) take place at the level of 10−8, 10−12 or 10−20

(what is not hard to achieve for curves showing strong vertical courses on graphs plotted with
the logarithmic scale applied); it is much better when the possible difference between values yi
and yi is not too large.
Due to dynamic changes around the value yi equal to zero (completely monotonic as for the

second-degree polynomial), functions (1.3) and (2.1) are practically insensitive to the approxi-
mated values 10−2, 10−5, 10−8 or 10−20. Hence, it is most preferable if the LSMapproximating
criterion takes such cases into account.
Therefore, a modification is proposed to transform the criterion into the following form

S∗∗ =
n
∑

i=1

(yi
yi
−1
)(

1−
yi
yi

)

(2.2)



Modification of the LSM criteria... 927

Owingto this, forboth large values yi (muchdifferent fromtheapproximatedvalue yi) and small
values (approaching zero) with respect to yi, the components of the sum take significant values,
i.e. in both cases, they give a significant (although – as it can be seen – diverse/unsymmetrical
for each case) contribution to the total approximation error – as shown in Fig. 3a. In order to
make the Si components of sum (2.2) and the total sum S

∗∗ as an approximation criterion
reach theminimum (not the maximum, as in Fig. 3a, also when reversal of the sign takes place
between the approximated yi and approximating value yi), the following formwould be better

S∗∗ =
n
∑

i=1

(
∣

∣

∣

yi
yi

∣

∣

∣−1
)(

1−
∣

∣

∣

yi
yi

∣

∣

∣

)

(2.3)

Both extremes of the Si function for both positive and negative values of are theminima shown
in Fig. 3b.

Fig. 3. Component of the sum for approximation criterion (2.2) – (a), and approximation
criterion (2.3) – (b)

Obviously, due to the discontinuity of functions (2.2) and (2.3), the reversal of sign of the
approximating value yi in relation to the approximated value yi is not beneficial as the question
of achieving the minimum of the total sum S∗∗ appears. However, it is not a significant disa-
dvantage, since quite often such a situation means that the test data are not physically logical
(e.g. when mass, body height, time or energy are measured), when it should not be expected
that the approximated value becomes negative. Then the approximating values yi with signs
opposite to those of the approximated yi have to be neglected as incorrect, and the discontinuity
effect for the Si components does not occur, since the graph shown in Fig. 3 consists of only
positive or only negative parts.
In other cases (when the approximating value yi and the approximated value yi can have

opposite signs, due to e.g. temperature measurements, income assessment, value gradient me-
asurement, etc.). The criteria (2.2) and (2.3) show an advantageous feature of reversal of the
sign (e.g. the component of the sum Si rapidly increases). Owing to this, they contain a kind of
“mathematical barrier” preventing easy reversal of the sign between the approximating and the
approximated values. However, one should remember that a possible iteration step should be
kept within some reasonable range so that it does not cause the omission of such discontinuity
of the changes in Si, since it may lead to improper estimation of the approximated value or
disturb the convergence of calculations.
Approximation criterion functions for (1.3), (2.1) and (2.2) (and their components) as related

to the approximating values yi, for:

• different approximated values yi equal to 5, 2, 1, 0.25, 0.01, 0.00001

• the same range of variability of yi, i.e. (−3yi,3yi), in order to show the yi → 0 effect, are
shown in Fig. 4.



928 S. Kłysz et al.

Fig. 4. Approximation criterion functions for (1.3), (2.1) and (2.3) for different approximated values yi
for constant approximating values scatter range within (−3yi,3yi) range

All advantages and disadvantages of the above presented LSM approximation criteria can
be seen on the graphs above, in particular:

• significant dependence of values of components of the sum S (formula (1.3)) on the ap-
proximated value yi

• invariability of values of the components of sums S∗ (formula (2.1)) and S∗∗ (formula
(2.2) and (2.3)) on all the graphs, i.e. for any approximated value yi

• no response of the components of sums S and S∗ to the yi → 0 effect, and dynamic
change in the components of the sum S∗∗ near value yi =0.

The only curve that changes in the graphs presented in Fig. 4 is the plot for components of
the sum S graph, i.e. for the standard form of the LSM.
The result of approximation with criterion (2.3) applied is shown in Fig. 5c – for the same

set of data as in Figs. 1 and 2. Exemplary results of approximation for different test data (with
slightly smaller scatter between individual da/dN − f(∆K) curves) is presented in Figs. 5a
and 5b.



Modification of the LSM criteria... 929

Fig. 5. Approximation of da/dN = f(∆K) data in different variants with the NASGRO equation, LSM
modified according to formula (2.3)

Favourable effects of the approximation (in comparison with the results shown in Fig. 1)
after implementation of the modified LSM criterion can be easily seen. They tend to represent
all the test data within the whole range of data variability, independently of their absolute
values, independently of the number of described curves – 3 (Fig. 5a), 6 (Fig. 5b), 9 (Fig. 5c)
and variant of the approximation (Fig. 5d – for each curve being described individually).

The above described test data come from the examination of aluminum alloy 2024 taken
fromMi-2 helicopter’s rotor blades [2].

This effect has been achieved only bymodifying the LSMcriterion, since the idea underlying
the approximation method for all the presented graphs is identical – the minimum of the sum
of squared deviations between the approximated test data and the approximating values.



930 S. Kłysz et al.

3. Conclusion

Theapplication of theLeast SquareMethod in its classical form for determination of coefficients
of theNASGRO equation that describes the fatigue crack propagation curve is ineffective, since
data of the approximated function da/dN = f(∆K) take values from the range of a few orders
of magnitude.

The paper offers some technique tomodify the LSMcriterion for significant improvement of
the approximation results.

The proposed modification of the LSM criterion brings favourable effects for results of the
test data approximation (unachievable with the classical LSM method). These effects are as
follows:

• the provision of equal “weights” of each test data point in the total sum that determines
this criterion (i.e. the sum of deviations between the approximated and approximating
values) – independently of the magnitude of difference between the values of data subject
to approximation and that of the difference between the approximated and approximating
values

• flexibility of the process of approximating in response to the reversal of sign between the
approximated value and the approximating one, or to the approximation with values close
to zero.

References

1. AFGROW Users guide and technical manual, AFRL-VA-WP-TR-2002-XXXX, Version
4.0005.12.10, James A. Harter, Air Vehicles Directorate, Air Force Research Laboratory,WPAFB
OH 45433-7542

2. AFIT Report No 8c/07, Fatigue crack growth rate testing report,Warsaw, 2007 [in Polish]

3. ASTME647 Standard test method for measurement of fatigue crack growth rates

4. Forman R.G., Shivakumar V., Cardinal J.W., Williams L.C., McKeighan P.C., 2005,
Fatigue Crack Growth Database for Damage Tolerance Analysis, DOT/FAA/AR-05/15,U.S. Dep.
of Transportation, FAAOffice of Aviation Research,Washington, DC 20591

5. Fuchs H.O., Stephens R.I., 1980, Metal Fatigue in Engeneering, A Willey-Interscience Publi-
cation

6. Huang X.P., Cui W.C., Leng J.X., 2005, A model of fatigue crack growth under various load
spectra, [In:]Proc. of Sih G.C., 7th Int. Conf. of MESO, Montreal, Canada, 303-308

7. Hudson M.C., Seward S.K., 1982, Compendium of sources of fracture toughness and fatigue
crack growth data for metallic alloys, International Journal of Fatigue, 20

8. NASGRO RO Fracturemechanics and fatigue crack growth analysis software, v5.0, NASA-JSC and
Southwest Research Institute, July 2006

9. nCode, User guide manual, ICE-FLOWcracks growth, 2003

10. Newman Jr. J.C., 1984, A crack opening stress equation for fatigue crack growth, International
Journal of Fracture, 24, 3, R131-R135

11. Taheri F., Trask D., Pegg N., 2003, Experimental and analytical investigation of fatigue
characteristics of 350WT steel under constant and variable amplitude loadings,Marine Structures,
16, 69-91

12. Walker K., 1970, The effect of stress ratio during crack propagation and fatigue for 2024-T3 and
7075-T6 aluminum,ASTM STP, 462



Modification of the LSM criteria... 931

13. White P., Molent L., Barter S., 2005, Interpreting fatigue test results using a probabilistic
fracture approach. International Journal of Fatigue, 27, 752-767

14. ZhaoT., JiangY., 2008, Fatigue of 7075-T651aluminumalloy, International Journal of Fatigue,
30, 834-849

Modyfikacja kryterium Metody Najmniejszych Kwadratów do aproksymacji wyników

badań prędkości propagacji pęknięć zmęczeniowych

Streszczenie

W pracy przedstawiono modyfikację kryterium Metody Najmniejszych Kwadratów do poprawienia
dokładności aproksymacji wyników badań prędkości propagacji pęknięć zmęczeniowych. W szczególno-
ści wprowadzono zmiany do poprawy aproksymacji w przypadku, gdy przedział wartości przynajmniej
jednej ze zmiennych jest w zakresie kilku rzędów wielkości lub różni się znacznie od zakresu zmienności
drugiej zmiennej. Przedstawiono również różnice pomiędzy poszczególnymi kryteriami i ich wpływ na
aproksymację wyników z badań. Obliczenia przeprowadzono dla danych uzyskanych z badań próbek ze
stopu aluminium 2024 pobranego z łopat wirnika helikopteraMi-2.

Manuscript received September 23, 2011; accepted for print November 23, 2011