International Journal of Fatigue 22 (2000) 743–755

www.elsevier.com/locate/ijfatigue

On the Glinka and Neuber methods for calculating notch tip strains
under cyclic load spectra
M. Knop a, R. Jones a,*
, L. Molent b, C. Wang b

a
DSTO Centre of Expertise in Structural Mechanics, Department of Mechanical, Monash University, Wellington Rd, Clayton,
Victoria 3168, Australia
b
Airframes and Engines Division, Aeronautical Research Laboratory, Defence Science and Technology Organisation, 506 Lorimer St,
Pt Melbourne, Victoria, 3207, Australia

Received 11 January 2000; received in revised form 5 June 2000; accepted 9 June 2000

Abstract

To maintain the continued airworthiness of civil and military aircraft it is essential that the fatigue behaviour of components
subjected to complex multiaxial stress conditions be both understood and predicted. This topic is extremely complex and numerous
criteria ranging from the purely empirical to the theoretical have been proposed. To this end it is necessary to know the localised
stress–strain history. Whilst a detailed finite element analysis can be performed it is often very time consuming. Consequently, the
present paper focuses on the development of a simple method which combines modern constitutive theory with either Neuber’s,
or Glinka’s, approach to calculate the localised notch strains. Here particular attention is paid to the effects of cyclic loading, mean
stress relaxation and its effect of fatigue damage.  2000 Elsevier Science Ltd. All rights reserved.

Keywords: Fatigue initiation; Notch tip stress and strain; Constitutive modelling

1. Introduction lating the localised notch strains. Attention is first

focused on developing a simple method which combines
Structural components are frequently subjected to modern constitutive theory with either the Neuber, or the
complex loading spectra. These alternating loads tend to Glinka, approach to calculate the localised notch strains.
initiate fatigue cracks at notches and at other regions of This method is then illustrated via a range of problems,
high stresses. Historically the field of fatigue has been viz: tension, bending and torsion, and the results pre-
classified into a number of specific areas; viz: high-cycle dicted by the simplified analysis methodology are com-
and low-cycle fatigue; fatigue of notched members; the pared with those obtained from a detailed finite
initiation and propagation of cracks and fatigue life element analysis.
extension techniques. Fatigue initiation and crack growth Attention is also focused on mean stress relaxation
programs require an accurate knowledge of the local and its role in fatigue life prediction. This is considered
notch tip stresses and strains. These quantities can be to be particularly timely since as discussed in [4] it has
determined in several ways, viz: via direct strain gauge recently been found that, for F/A-18 structural compo-
measurements, using finite element analysis or by using nents, traditional approaches predict unrealistically long
approximate methods, such as the Neuber [1] and Glinka fatigue lives, see [4] for more details.
approaches [2,3], that relate local stresses and strains to
their remote values. To this end the present paper briefly
outlines the Neuber and the Glinka approaches for calcu- 2. Glinka and Neuber’s rules for estimating notch
root stresses and strains

* Corresponding author. Tel.: +61-3-9905-4000; fax: +61-3-9905- The strain–life method, which is commonly used to
1825. predict fatigue crack initiation, requires a knowledge of
E-mail address: rhys.jones@eng.monash.edu.au (R. Jones). the notch root stresses and strains. These quantities can

0142-1123/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 6 1 - X
744 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755

be determined in several ways, viz: via direct strain use of a validated constitutive law may be necessary. In
gauge measurements, using finite element analysis or by this case when calculating the ⌬s and ⌬e ranges the use
using approximate methods that relate local stresses and of traditional yield surface plasticity, without consider-
strains to their remote values. The theoretical concen- ing non-linear kinematic hardening, should be avoided
tration factor, Kt, is often used to relate the nominal (or as this tends to produce ‘boxy’ stress–strain loops and
far field) stresses S, or strains, e, to the local values, s gives poor estimates of the ⌬s and ⌬e ranges.
and e. However, upon yielding, Hooke’s law cannot be Stiffener runout Number 2 in the F111 wing pivot fit-
used to relate the local stress, s, to the local strain, e, ting is a good example of this problem. Here when sub-
and the local values are no longer related to the nominal jected to g loads going from ⫺7.3 to 0 g classical
values by Kt. Instead, after yielding occurs, the local incremental plasticity gave a tensile stress range of
stresses and strains are related to the remote values by 苲2000 MPa whilst more exact analysis gave a stress
their respective notch root stress and strain concentration range of only 1200 MPa, see [5]. Unfortunately, the
factors Ks and Ke , viz: s=KsS and e=K⑀e. fatigue relations discussed above do not adequately
The strain life approach frequently makes use of Neu- allow for the load history effects, i.e. load interaction,
ber’s rule [1] to compute the local (notch) stresses and stress relaxation, mean stress relaxation, creep, and
strains. In this approach the theoretical stress concen- overload–underload effects. Here the stiffener runout
tration is taken as the geometric mean of the stress and Number 2 in the F111 wing pivot fitting is (again) a
strain concentration factors. Problems associated with good example of the need to correctly follow load his-
this and other related approximations are: (1) The notch tory. When subjected to g loads going from ⫺7.3 to 0
stress–strain response root must be in phase with the glo- g classical approaches produced an inspection interval
bal load; (2) No account is taken for time dependent of less than 500 h. However, using a state variable based
processes such as creep and mean stress relaxation. constitutive model enabled the inspection interval to be
A number of variants of Neuber’s rule have been used increased to nearly 1500 h, see [5].
to relate the remote stresses and strains to local values. For monotonic loading the work of Shin (pp. 613–52)
The initial Neuber hypothesis [1] can be expressed in [7], who concentrated on plane stress and plane strain
the form: loading, Agnihotri [8] and Knop [10] has enabled the
following conclusions to be drawn:
K 2t ⫽KsKe (1)
or alternatively 1. Neuber’s rule normally overestimates the notch tip
strain while the ESED method normally underesti-
SeK 2t ⫽se (2) mates the notch tip strain.
2. Under plane stress bending or tension loading, Neub-
The work of Glinka [2,3] has shown that there are er’s rule generally makes better predictions of the
instances when the equivalent strain energy density notch tip strain.
(ESED) approach yields more accurate estimates for the 3. Under plane strain bending or tension loading, the
strain. According to this hypothesis the strain energy ESED method generally makes better predictions of
density at the notch in an elastic plastic analysis is the the notch tip strain.
same as the strain energy obtained via a purely elastic 4. Under torsion loading, the ESED method generally
analysis; viz: makes better predictions of the notch tip strain.
5. As the value of the stress concentration factor, Kt

冕 冕
1/2K 2t Se⫽ sij deij ⫽ sde (3)
increases predictions made by the ESED method
improve under all loading regimes.

Once a valid stress–strain relationship is known the use As we have previously remarked accurate predictions of
of the Glinka, or Neuber, approximations can be used to the stresses and the strains in a structure are very
determine the notch stresses and strains. important for fatigue life prediction. Thus one aim of this
For plane stress or plane strain problems it is also paper was to evaluate notch tip behaviour under cyclic
common to use simple power laws to relate s to e, or loading, without utilising the approximations inherent in
for saturated fatigue loops ⌬s to ⌬e. One expression the classical Prandtl Reuss approach to incremental plas-
commonly used to determine the saturated fatigue ticty, and to determine when Neuber’s rule or Glinka’s
loop(s) takes the form method, which we will refer to as the ESED method,
⌬e/2⫽⌬s/2E⫹(⌬s/2K)1/n (4) makes better predictions. To date many authors have
used stress–strain relationships that are only valid for
where K and n are material constants. However, this Massing type materials. In this paper we will evaluate
expression is only strictly valid for Masing type the relative ability of modern state variable based consti-
materials. For more complex 3D structural problems the tutive laws [5,6,9], used in conjunction with Neuber’s
 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755 745

rule and Glinka’s method, to calculate the local stress Table 1

and strain states. In this work the particular constitutive Stress concentration factors used for the various types of loading
model used was as outlined in [6], and represents the Stress concentration factor
stress–strain response of 7050-T7451 aluminium alloy, Loading Kt (plane Kt (plane Loading Kts
a similar model was used in [5]. The accuracy of these stress) strain)
results are evaluated by comparison with the results of
a finite element program, i.e. ABAQUS, which also util- Tension 2.04 2.04 Torsion 1.84
5.2 5.2 4.63
ised the same unified constitutive theory [6]. The formu- Bending 1.6 1.6
lation of this particular constitutive law is summarised 4.07 4.07
in the Appendix A.
In this investigation three basic notch geometries, with
several different Kt’s, were considered, see Fig. 1, viz:
since, on the various stress–strain curves, it is sometimes
앫 A plate with a centrally located hole which was used difficult to visualise the ‘best’ fit curve. In each case the
for tension loading. strain amplitude predicted by Neuber’s rule and the
앫 A plate with notches located at the edge which was ESED method is compared with the value obtained using
used for bending loading. a finite element analysis. The actual strain amplitude
앫 A grooved shaft used for torsion loading. value is given along with the percentage difference
against the associated finite element value. The mean
However, because of space limitations we will only stresses for each cycle are also presented.
report on a representative sub-set of the Kt’s investigated
(see Table 1). 2.2. Discussion

2.1. Cyclic loading In general, the relative predictive capabilities of Neub-

er’s rule and the ESED method of the notch tip stresses
The results obtained for several cyclic spectra are and strains for cyclic loading are very similar to that for
shown in Figs. 2–9. Tabulated data are also presented monotonic loading, viz:

Fig. 1. Schematic diagram of the various geometries used in the numerical investigation.
746 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755

Fig. 2. Stress–strain response using Neuber’s rule and the ESED Fig. 5. Stress–strain response using Neuber’s rule and the ESED
method for cyclic loading. (Plane stress, tension loading, Kt=2.04). method for cyclic loading. (Plane strain, tension loading, Kt=2.04).

Fig. 3. Stress–strain response using Neuber’s rule and the ESED Fig. 6. Stress–strain response using Neuber’s rule and the ESED
method for cyclic loading. (Plane stress, tension loading, Kt=5.2). method for cyclic loading. (Plane strain, tension loading, Kt=5.2).

Fig. 4. Stress–strain response using Neuber’s rule and the ESED Fig. 7. Stress–strain response using Neuber’s rule and the ESED
method for cyclic loading. (Plane stress, bending loading, Kt=1.6). method for cyclic loading.(Plane strain, bending loading, Kt=4.07).
 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755 747

strain, the optimum conditions for the ESED method

appears to occur for plane strain bending loading with
a high Kt. For Neuber’s rule the optimum conditions
appears to occur for plane stress tension loading with a
low value of the stress concentration factor, Kt. In
between these two cases there is a transition area of pre-
diction for Neuber’s rule and the ESED method which
depends on the loading conditions, constraint conditions
and stress concentration factor. The cases shown in Figs.
3 and 8 are in the ‘transition’ zone between Neuber’s
rule and the ESED method. The results in this transition
zone have resulted in controversy regarding the relative
accuracy of Neuber’s rule and the ESED method.

Fig. 8. Stress–strain response using Neuber’s rule and the ESED 3. Implication to fatigue life prediction: strain–life
method for cyclic loading. (Torsion loading, Kts=1.84).
It has recently been found that, for F/A-18 structural
components, traditional approaches can predict unrealist-
ically long fatigue lives, see [4] for more details. This
is believed to be partially due to mean stress relaxation
effects. One aim of this section is to investigate this
possibility. To this end the present section will compare
the strain life predictions made using state variable
approaches and those obtained using a more traditional
approach. The first algorithm to be considered, which is
termed FAMS, Fatigue Analysis of Metallic Structures,
was developed by the USN Naval Warfare Centre
(USNWC) in the 1980s to determine the fatigue life at
a notch tip, see [11]. The second algorithm which we
will term, FLiPP, Fatigue Life Prediction Program, uses
the same approach as FAMS except that the inelastic
response is modelled using the state variable approach
Fig. 9. Stress–strain response using Neuber’s rule and the ESED described in [5,6]. For comparison purposes, the alu-
method for cyclic loading.(Torsion loading, Kts=4.63). minium alloy 7050-T7451 will be used as the common
material in both algorithms.
The stress–strain material response curve used in
1. Neuber’s rule makes better predictions for plane stress
FAMS is based on the stabilized cyclic stress–strain
cyclic loading.
curve and Massing’s hypothesis, [12] is used. This
2. The ESED method makes better predictions for plane
approach produces a stabilized cyclic stress–strain curve,
strain cyclic loading.
for 7050-T7451, which is essentially identical to that
3. As the value of Kt increases, the ESED method
obtained using the state variable formulation outlined in
improves.
[5,6] (see Fig. 10). Whilst FAMS only allows for plane
4. In torsion loading, the ESED method makes better
stress loading, the state variable approach adopted in
predictions than Neuber’s rule for torsion loading.
FLiPP allows for plane stress, plane strain and torsion
loading.
The above generalisations were not true in all the cases.
For example, Fig. 8 revealed that for torsion loading 3.1. Notch tip prediction techniques
Neuber’s rule makes the more accurate prediction. How-
ever, it should be noted that best prediction method also To predict the fatigue life FAMS uses Neuber’s rule
depends on the magnitude of the loading. together with the Neuber notch factor, KN, instead of the
It appears that whilst the above rules are generally theoretical stress concentration factor, Kt. In FAMS the
true, they require certain optimum conditions. For Neuber notch factor is approximated using the formulae
example, consider the results for the plane strain tension
(Kt−1)
loading case with Kt=2.04 as presented in Fig. 2. As
冋 冉 冊册
KN⫽1⫹ (5)
already mentioned for monotonic loading Neuber’s rule a 2
1+
makes the better predictions. In this case, for plane m
748 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755

Fig. 10. Stablized cyclic stress–strain curve for 7050-T7451. (Torsion loading, Kts=1.84).

where a is a material constant, and m is the notch tip 3.2. The e–N curve and mean stress effects
radius.The values used by FAMS for the variable, a are
shown in Table 2 below. The strain–life approach is most generally used for
The aluminium alloy, 7050-T7451 has an ultimate low cycle fatigue where the fatigue life is highly depen-
strength sUTS of approximately 505 MPa (73.2 ksi), so dent on the plastic strain. The total strain amplitude is
from Table 2, the value for a was taken to be: the sum of the elastic and plastic terms such that:
a=0.015 in. ⌬e ⌬ee ⌬epl
The computer program FLiPP has the option of using ⫽ ⫹ (6)
2 2 2
either the ESED method or Neuber’s rule depending on
the loading situation. Here it should be recalled that the
ESED method generally makes more accurate predic- In 1910, Basquin [13] observed that the stress–life (S–
tions under plane strain while Neuber’s rule makes better N) data can plotted linearly on a log–log scale and can
predictions under plane stress. Neuber’s rule has not be expressed as:
been modified in FLiPP. The form used is given Sec- ⌬s
tion 2. ⫽s⬘f(2Nf)b (7)
2
Both FLiPP and FAMS use the rainflow counting
method which will allow both algorithms to consider
sequence loading events. Both programs have the capa- The elastic strain can be written as:
bility to count full cycles and half cycles. In FLiPP, the ⌬ee ⌬s
rainflow counting procedure was based on the standard ⫽ (8)
2 2E
ASTME 1049 while in FAMS the method is based on
the original work by Matsuishi and Endo [11]. from which we obtain:
⌬ee s⬘f
Table 2 ⫽ (2Nf)b (9)
2 E
Values for a used in FAMS

sUTS (ksi) a (in) Coffin [14] and Manson [15] worked independently
in the 1950s and found that the plastic–strain life data
Steel 145.00 0.00300
290.00 0.00008 can also be linearized on a log–log scale and be
Aluminium 43.00 0.02500 expressed as:
87.00 0.01500
⌬epl
Titanium 150.00 0.00008 ⫽e⬘f(2Nf)c (10)
2
 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755 749

The total strain–life curve would then be given by: FLiPP allows the user the choice of Eqs. (12)–(14). Both
FAMS and FLiPP both use the Miner–Palmgren rule to
⌬e s⬘f
⫽ (2Nf)b⫹e⬘f(2Nf)c (11) sum the damage resulting from a complex loading spec-
2 E trum.
where
3.3. Description of the numerical investigation
s⬘f is a fatigue strength coefficient
The problem of a 7050-T7451 aluminium alloy rec-
2Nf are the reversals to failure
tangular plate in plane stress tension with a centrally
b is a fatigue strength coefficient
located hole, Kt=3.04 was used to evaluate differences
e⬘f is a fatigue ductility coefficient
resulting from using the state variable constitutive model
c is a fatigue ductility exponent
(FLiPP). The stress and strain states at the notch were
calculated using Neuber’s rule, ESED and a finite
element method analysis which also used the state vari-
Similar to the S–N diagram, the e–N diagram is gener- able constitutive model. When using FLiPP fatigue life
ated from empirical data for fully reversed cycles. For a predictions were made using finite element data, and pre-
realistic loading history it is important to take account dictions using Neuber’s rule and the ESED method.
of residual stresses or mean stresses at the area of inter- Six different loading spectra were applied to the plate
est since they will have an effect on the fatigue life esti- to observe the material response at the notch tip. The
mation. load spectra consisted of constant amplitude loading,
To take account of mean-stress effects Morrow [16] overloads followed by a constant amplitude spectrum
suggested that the mean stress effect could be taken into and finally an overload followed by an underload and
account by modifying the elastic term in Eq. (11). The then by a constant amplitude load spectrum. These spec-
strain–life equation would then become: trums were selected so that residual stresses were pro-
⌬e s⬘f−sm duced at the notch tip and to observe how the residual
⫽ (2Nf)b⫹e⬘f(2Nf)c (12) stresses were affected by the subsequent loading.
2 E
The loading spectra are summarized below:

Manson and Halford [17] modified both the elastic 3.3.1. Load spectrum 1
and plastic terms of the strain–life equation to maintain Remote constant amplitude loading varying between
the independence of the elastic–plastic strain ratio from 0 and 300 MPa (see Fig. 12). A total of 100.5 cycles
mean stress. In this form the strain–life equation were considered.
becomes:

冉 冊
c 3.3.2. Load spectrum 2
⌬e s⬘f−sm s⬘f−sm b
⫽ (2Nf)b⫹e⬘f (2Nf)c (13) Remote constant amplitude loading varying between
2 E s⬘f 0 and 400 MPa (see Fig. 13). A total of 100.5 cycles
were applied.
Smith, Watson and Topper (SWT) [18] proposed
another equation to also account for mean stress effects. 3.3.3. Load spectrum 3
This equation can be written as: This consisted of an initial remote stress loading to
400 MPa followed by constant amplitude remote stress
⌬e (s⬘f)2 loading varying between 200 and 400 MPa (see Fig. 14).
smax ⫽ (2Nf)2b⫹s⬘fe⬘f(2Nf)b+c (14)
2 E A total of 100.5 cycles were considered.
where 3.3.4. Load spectrum 4
⌬s This consisted of an initial remote stress overload to
smax⫽ ⫹sm (15) 400 MPa followed by constant amplitude remote stress
2
loading varying between 0 and 200 MPa (see Fig. 15).
In strain life approaches the damage for a given cycle A total of 111 cycles were applied. The overload was
is determined from the corresponding e–N curve for the repeated on every 10th cycle.
material, see Fig. 11. However, the data given on the e–
N curve is based on fully reversed cycles, hence with a 3.3.5. Load spectrum 5
mean stress equal to zero. Usually the mean stress for a This consisted of an initial remote stress loading to
given cycle is not equal to zero and a relationship is 200 MPa, followed by a remote stress underload to
required to take into account mean stress effects. ⫺300 MPa followed by constant amplitude remote stress
FAMS uses Eq. (13) to predict the fatigue life whilst loading varying between 0 and 200 MPa (see Fig. 16).
750 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755

Fig. 11. Strain–life curve for 7050-T7451.

Fig. 12. Graphical display of load spectrum 1. Fig. 14. Graphical display of load spectrum 3.

Fig. 15. Graphical display of load spectrum 4.

Fig. 13. Graphical display of load spectrum 2.

between 0 and 200 MPa (see Fig. 17). A total of 110.5

A total of 110.5 cycles were applied. The underload was cycles were applied. The overload followed by the
repeated on every 10th cycle. underload was repeated on every 10th cycle.

3.3.6. Load spectrum 6 3.4. Numerical results

Initial remote stress overload to 400 MPa, followed
by a remote stress overload to ⫺200 MPa followed by In this section the predictions of the fatigue life made
constant amplitude remote stress loading varying by FAMS, and FLiPP will be compared. Figs. 18–25
 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755 751

Fig. 16. Graphical display of load spectrum 5.

Fig. 20. Stress–strain notch tip predictions using Neuber’s rule with
the unified constitutive model for loading spectrum 2.

Fig. 17. Graphical display of load spectrum 6.

Fig. 21. Stress–strain notch tip predictions using FAMS with a yield
surface plasticity model for loading spectrum 2.

Fig. 18. Stress–strain notch tip predictions using the ESED method
with the unified constitutive model for loading spectrum 2.

Fig. 22. Stress–strain notch tip predictions using ABAQUS with a

unified constitutive model for loading spectrum 3.

display the stress–strain predictions at the notch tip.

Only those figures depicting the main trends are given.
Figs. 18–21 display the predicted stress–strain
response at the notch tip for load spectrum 2. For clarity,
Figs. 18–20 display only parts of the resultant load spec-
trum.
In the previous sections we found that for a plane
stress problem in tension Neuber’s rule gives a better
Fig. 19. Stress–strain notch tip predictions using ABAQUS with the prediction of the notch tip stress–strain response than the
unified constitutive model for loading spectrum 2. ESED method. This can again be observed in Figs. 18–
20. Neuber’s rule overestimates the notch tip strain and
the ESED method underestimates the notch strain. This
752 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755

Table 3
Predicted accumulated damage for load spectrum 1 and load spec-
trum 2

Accumulated damage (×10⫺4)

FLiPP FAMS
Strain–life equation Neuber ABAQUS ESED Neuber

Load spectrum 1
FAMS 7.0972 6.4577 5.2987 14.8656
Morrow 7.0107 6.3902 5.2573 –
Manson/Halford 7.2917 6.6128 5.3953 –
SWT 10.1768 9.3624 7.1778 –
Load spectrum 2
Fig. 23. Stress–strain notch tip predictions using ABAQUS with a
FAMS 20.7677 18.6873 17.7867 57.3690
unified constitutive model for loading spectrum 4.
Morrow 20.3491 18.3784 17.5452 –
Manson/Halford 21.7515 19.4064 18.3501 –
SWT 20.8100 19.4996 17.5058 –

the mean stress relaxation effect as shown in Fig. 21.

This can have a significant effect on the predicted fatigue
life since, as discussed in previous sections, the predicted
fatigue life depends both on the strain amplitude and the
mean stress.
From Table 3, for load spectrum 2, we see that when
mean stress relaxation is considered, FAMS and FLiPP
predicts significantly different accumulated damage. For
load spectrum 2, FAMS predicts three times the damage
Fig. 24. Stress–strain notch tip predictions using ABAQUS with a compared to FLiPP. However, the amount of difference
unified constitutive model for loading spectrum 5.
is highly dependent on the loading level since the level
of plastic strain affects mean stress relaxation.
For load spectrum 1 (see Table 3), FAMS predicts
twice the damage as compared to the equivalent FLiPP
prediction. Load spectrum 1, consisted of a constant
amplitude loading spectrum similar to load spectrum 2.
However, the remote stress amplitude was 100 MPa less
than for load spectrum 2. This has resulted in a smaller
plastic strain at the notch tip and a smaller magnitude
of the initial mean stress. Consequently, during the mean
stress relaxation process, the mean stress could not relax
to the same extent as it could for spectrum 2.
For load spectrum 3 (see Table 4), the predictions
Fig. 25. Stress–strain notch tip predictions using ABAQUS with a made by FAMS and FLiPP were approximately equal.
unified constitutive model for loading spectrum 6. Fig. 22 displays the stress and strain at the notch tip

Table 4
result is reflected in the accumulated damage for the load Predicted accumulated damage for load spectrum 3
spectrum as shown in Table 3. It can be observed that
the accumulated damage prediction made by Neuber’s Accumulated damage (×10⫺4)
rule corresponds well with that obtained using the finite FLiPP FAMS
Strain–life Neuber ABAQUS ESED Neuber
element analysis. Neuber’s rule tends to overestimate the equation
strain which results in a higher accumulated damage
because the predicted strain amplitude is larger. Load spectrum
It can be observed that by using the unified constitut- 3
ive, that mean stress relaxation of the resultant spectrum FAMS 8.0400 7.9286 7.1490 10.1318
Morrow 7.6175 7.5161 6.8132 –
is predicted, such that the mean stress relaxes down to Manson/Halford 8.5768 8.4531 7.5832 –
zero (see Figs. 18–20). This behaviour is similar to that SWT 11.1949 11.1177 10.5845 –
observed in [4]. The FAMS program does not predict
 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755 753

Table 5 nitude, the relaxation does not have a significant effect

Predicted accumulated damage for load spectrum 4 on the prediction of the accumulated damage (see
Accumulated damage (×10⫺4) Table 7).
FLiPP FAMS In fatigue life calculations a repeated load spectrum
Strain–life equation Neuber ABAQUS ESED Neuber is frequently referred to as a load block. It is often
assumed that for block loading, the damage for one
Load spectrum 4 block can be calculated and that the damage remains the
FAMS 6.0804 5.3498 4.8086 7.7453
Morrow 5.7675 5.1077 4.6178 – same for each subsequent loading block. Each of the
Manson/Halford 6.7714 5.8822 5.2297 – load spectrums considered can be represented as a series
SWT 6.3141 5.6784 5.2034 – of repeating load blocks. However, the response at the
notch tip is usually different in each case because of
mean stress relaxation. This results in a different amount
of damage for each loading block. This result indicates
for load spectrum 3. In this particular example it can be
that for those load spectra it maybe best to consider them
observed (see Fig. 22), that after the initial overload, the
in their entirety and a simplification such as dividing it
subsequent response was elastic and no subsequent mean
into blocks needs to be considered carefully.
stress relaxation occurred. Hence the predicted fatigue
life was similar for both FAMS and FLiPP, since both 3.5. Summary
had the same stress–strain response at the notch tip.
Indeed, it is clear that if mean stress relaxation is to From this numerical study for a notched plate under
occur then the notch tip needs to experience plastic a various loading spectrums the following observations
strains. can be made:
Let us now consider the effects of overloads. Fig. 23
displays the predicted stress–strain response at the notch 1. Taking mean stress relaxation effects into account,
tip for load spectrum 4. From this figure we see that can result in significant differences in the fatigue life
there is a resultant mean stress relaxation after each compared to when it was not taken into account. By
overload which also has an effect on the mean stress on not accounting for mean stress relaxation effects, an
the following ‘elastic’ cycles. In this case neglecting the over conservative fatigue life prediction would occur
effect of mean stress relaxation significantly over pre- for tensile mean stress relaxation and a under con-
dicts the rate of damage accumulation (see Table 5). servative answer for compressive mean stress relax-
The effects of an under load on the mean stress, as ation effects.
given by spectrum 5 are shown in Fig. 24. Here the nega- 2. Plastic strains needs to be present in the spectrum for
tive mean stress relaxes towards zero. A negative mean mean stress relaxation to occur.
stress can be expected to produce a longer fatigue life 3. Load history effects can make a significant difference
prediction than a positive mean stress value. Therefore to the subsequent operating mean stress of the cycles
as the mean stress relaxes a shorter fatigue life is pre- which in turn can also affect the fatigue life predic-
dicted for the same strain amplitude value. Neglecting tion. This illustrates that there will be cases when all
this effect, i.e. using FAMS, significantly underestimated of the load history effects for a given loading spec-
the rate of damage accumulation (see Table 6). trum should be taken into account.
Fig. 25 displays the predicted notch stress–strain
response for a spectrum with an overload followed by
an under load. Similar to previous spectrums the state 4. Conclusion
variable based approach again predicts mean stress
relaxation. However, as the mean stress is small in mag- To the best of the author’s knowledge this work is
one of the first times that modern state variable theories
Table 6 Table 7
Predicted accumulated damage for load spectrum 5 Predicted accumulated damage for load spectrum 6

Accumulated damage (×10⫺4) Accumulated damage (×10⫺4)

FLiPP FAMS FLiPP FAMS
Strain–life equation Neuber ABAQUS ESED Neuber Strain–life equation Neuber ABAQUS ESED Neuber

Load spectrum 5 Load spectrum 6

FAMS 9.7631 9.3163 9.3860 8.1117 FAMS 16.0777 14.6287 13.0535 16.5260
Morrow 9.8730 9.4102 9.4590 – Morrow 15.6600 14.2841 12.8130 –
Manson/Halford 9.3835 8.9899 9.1278 – Manson/Halford 17.2439 15.5846 13.7177 –
SWT 10.1936 9.9002 10.2426 – SWT 15.1350 13.8897 12.4007 –
754 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755

have been used to evaluate the Neuber and Glinka, mathematical law. In contrast the unified constitutive
approach for cyclic loading. Indeed, this paper has theory expresses the strain tensor eij as
presented a brief summary of several of the criteria com-
monly used to compute the notch tip stresses and strains. eij ⫽eeij ⫹eIij ⫹eTij (A2)
A simple fortran program which uses a modern constitut- where eIij are the inelastic components of the strain.
ive law for the stress–strain response together with either The present formulation consists of an inelastic flow
the Neuber or the Glinka approaches has been developed equation and ‘evolution equations’ for the state vari-
and the notch strains compared with those obtained from ables; back stress ⍀ij, and drag stress Z. In this formu-
a finite element analysis. From this numerical investi- lation the inelastic strain rate tensor is related to the
gation, the following general conclusions can be made: stress field by a flow equation, viz:

앫 Neuber’s rule tends to overestimate the notch tip

stress and strain while the ESED method underesti-
ėIij ⫽Dexp ⫺冋 冉 冊册
A Z2
2 3K2
n
(Sij −⍀ij )
冑K 2
(A3)
mates the notch tip stress and strain.
앫 Neuber’s rule makes better predictions for plane stress where ėIij is the deviatoric inelastic strain rate tensor, Sij
loading conditions. is the deviatoric stress tensor, ⍀ij is the inelastic back
앫 ESED method makes better predictions for plane stress and D, A and n are temperature dependent material
strain loading conditions. parameters. For the aluminium alloy considered in this
앫 As the value of the stress concentration factor investigation D=10,000, A=1 and n=3.0. Here the second
increases the predictions made by the ESED invariant of the effective deviatoric stress tensor (K2) is
method improve. defined as:
앫 For tension loading the predictions made by Neuber’s
1
rule are better than for bending loading. The converse K2⫽ (Sij ⫺⍀ij )(Sij ⫺⍀ij ) (A4)
is true for the ESED method. 2
앫 For torsion loading predictions made by the ESED and the evolution equation for the inelastic back stress
method are better than Neuber’s rule. ⍀̇Iij is:

This work supports the findings presented in [4] that for 3 ⍀ij I
⍀̇Iij ⫽f1ėIij ⫺ f1 ė (A5)
structures subjected to complex load the mean stress can 2 ⍀max eff
relax during the loading process. This has important
where
implications for fatigue life prediction since the majority
of techniques used to predict the stress–strain response ⍀ij ⫽f2Sij ⫹⍀Iij (A6)
due to complex load spectra do not allow for this relax-
ation process and hence have the potential to result in For this particular aluminium alloy the parameter f1 satu-
poor predictions of the fatigue life. rates to a limiting value f1f in accordance with the formu-
lae:
I
f1⫽f1f⫹(f10⫺f1f)e−0.6W (A7)
Appendix A. The constitutive model
where WI is the inelastic work and m is a material con-
stant. For D6ac steel the typical material constants are
The particular unified constitutive model used in this E=72,000 MPa, f10=132,000 MPa, f1f=205,000 MPa,
investigation is as outlined in [6]. A metallurgical f2=0.861, and ⍀max, the maximum value of back stress,
interpretation for this law, in terms of the movement of is 414 MPa. Here the effective inelastic strain rate, ėIeff
dislocations as a result of the inelastic response to load, is defined by:
is presented in [9].

冪冉3ė ė 冊
This approach differs quite markedly from conven- 2
tional formulations where the strain tensor is expressed ėIeff⫽ I I
ij ij (A8)
in the form:
Here the drag stress Z is defined as:
eij ⫽eeij ⫹eve
ij ⫹eij ⫹eij ⫹eij ⫹eij
p vp c T
(A1)
Z⫽Z1⫹(Z0⫺Z1)exp(⫺m0syeIeff) (A9)
Here the total strain eij is divided into ⑀Tij the compo- where Z0 , the initial value of drag stress, is 40 MPa
nents of the strain induced by thermal effects, a viscoel- respectively Z1, the saturated value of drag stress, is 182
astic component eve p
ij a plastic component eij , a viscoplas- MPa respectively, the rate of cyclic softening m0 is
tic component eij and a creep component ecij . In classical
vp
0.00012 respectively and sy=500 MPa is the yield stress.
formulations each component is governed by a different Here the effective inelastic strain is:
 M. Knop et al. / International Journal of Fatigue 22 (2000) 743–755 755

冪冉 冊 fatigue crack propagation in metallic structures. Amsterdam:

2 I I
eIeff⫽ e e (A10) Elsevier Science BV, 1994.
3 ij ij [8] Agnihotri GG. Calculation of elastic–plastic strains and stresses
in notches under torsion load. Engng Fract Mech
1995;51(5):823–35.
Details of the methodology for evaluating the material [9] Stouffer DC, Dame LT. Inelastic deformations of metals, models,
constants n, f10, f1f, f2, Wmax, Z0, Z1, m0 and sy are given mechanical properties and mettalurgy. New York: John Wiley,
1996.
in [6]. The terms D and A are not true material constants [10] Knop M, Jones R. On the Glinka and Neuber methods for calcu-
and are generally set to 10,000 and 1 respectively. lating notch tip strains. In: Wang CH, editor. Structural integrity
Variants of this constitutive model have been success- and fracture. Australia: Australian Fracture Group Inc Publishers,
fully used to study the behaviour of D6ac steel F111 1998:93–101.
components [5], the post-buckling failure of rib stiffened [11] Ayling J, Molent L. An investigation into the program FAMS
(Fatigue Analysis of Metallic Structures). DSTO Tech Report
composite panels [20] and adhesively bonded joints [21]. 0681, June 1998, Melbourne, Australia.
[12] Massing G. Stretching and hardening for brass (in German). In:
Proceedings 2nd International Congress on Applied Mechanics,
Zurich, 1926:332–5.
References [13] Matsuishi M, Endo T. Fatigue of metals subjected to varying
stress. Paper presented to Japan Society of Mechanical Engineers,
Fukuoka, Japan, 1968.
[1] Neuber H. Theory of stress concentration for shear strained pris- [14] Basquin OH. The exponential law of endurance tests. Am Soc
matic bodies with arbitrary nonlinear stress strain law. J Appl Test Mater Proc 1910;10:625–30.
Mech 1969;28:544–51. [15] Coffin LF Jr.. A study of the effects of cyclic thermal stresses
[2] Glinka G. Energy density approaches to calculation of inelastic on a ductile metal. Trans ASME 1954;76:931–50.
stress–strain near notches and cracks. Engng Fract Mech [16] Manson SS. Behavior of materials under conditions of thermal
1985;22(3):485–508. stress. Heat Transfer Symposium, University of Michigan Engin-
[3] Glinka G. Calculation of inelastic notch-tip strain–stress histories eering Research Institute, 1953:9–75.
under cyclic loading. Engng Fract Mech 1985;22(5):839–54. [17] Morrow J. Fatigue design handbook. In: Advances in Engineer-
[4] Wang CH, Rose LRF. Transient and steady state deformation at ing, vol. 4. Warrendale, PA: Society of Automotive Engineers,
notch root under cyclic loading. Mech Mater 1998;30:229–41. 1968:21–9.
[5] Jones R, Molent L, Paul J, Saunders T, Chiu WK. Development [18] Manson SS, Halford GR. Practical implementation of the double
of a composite repair and the associated inspection intervals for linear damage rule and damage curve approach for treating cumu-
the F111C stiffener runout region. FAA/NASA International lative fatigue damage. Int J Fract 1981;17(2):169–92.
Symposium on Advanced Structural Integrity Methods for Air- [19] Smith KN, Watson P, Topper TH. A stress strain function for
frames Durability and Damage Tolerance, NASA Conference the fatigue of metals. J Mater 1970;5(4):767–78.
Publication 3274, Part 1, 1994:339–51. [20] Jones R, Alesi H, Mileshkin N. Australian developments in the
[6] Kuruppu MD, Williams JF, Bridgford N, Jones R, Stouffer DC. analysis of composite structures with material and geometric non-
Constitutive modelling of the elastic–plastic behaviour of 7050- linearities. J Comp Struct 1998;41:197–214.
T7451 aluminium alloy. J Strain Anal 1992;27(2):85–92. [21] Chiu WK, Jones R, Chao M. A visco-plastic analysis of bonded
[7] Shin CS. Fatigue crack growth from stress concentrations and joints under complex loading. J Polym Polym Comp
fatigue life prediction in notched components. Handbook of 1998;6:15–24.