Multiaxial Fatigue Damage Models
D. Socie
Department of Mechanical and
Industrial Engineering,
University of Illinois at
Urbana-Champaign,
Urbana,IL 61801
Two multiaxial fatigue damage models are proposed: a shear strain model for
failures that are primarily mode II crack growth and a tensile strain model for
failures that are primarily mode I crack growth. The failure mode is shown to be
dependent on material, strain range and hydrostatic stress state. Tests to support
these models were conducted with Inconel 718, SAE 1045, and AISI Type 304
stainless steel tubular specimens in strain control. Both proportional and nonproportional loading histories were considered. It is shown that the additional cyclic
hardening that accompanies out of phase loading cannot be neglected in the fatigue
damage model.
Introduction
A large number of multiaxial fatigue damage models have
been proposed. Stress, strain, and energy have all been used to
correlate test data. Despite much research in multiaxial
fatigue, no single theory has been able to correlate the data for
a wide variety of materials and loading conditions. Early investigators (Gaugh, 1933; Stulen, 1954; Sines and Waisman,
1959) were interested in long life problems and proposed shear
stress based theories for ductile materials and principle stress
based theories for brittle materials. In the low cycle fatigue
region, equivalent strain (Taira et al. 1967; Pascoe and
DeVilliers, 1967; Yokobori et al. 1965), plastic work (Garud,
1979), plastic strain energy, (Ellyin and Valaire, 1982) and
critical plane approaches (Brown and Miller, 1973; Lohr and
Ellison, 1980) have been used to correlate the data. Since
plastic shear strains play an important role in crack nucleation
and early growth, it is not surprising that all of these approaches are essentially shear stress and strain theories with
the exception of plastic work. This approach includes a contribution from the tensile stresses and strains. Material
dependency is introduced only in the constants required to fit
each set of data. By adjusting the constants some shear strain
theories can be made to be numerically equivalent to a maximum principal strain theory. Observations of crack formation and early growth for a number of materials by Bannantine and Socie (1985) show two types of behavior. Materials
such as AISI Type 304 stainless steel fail by the growth of a
mode I tensile crack or a mode II shear crack depending on the
stress state and cyclic strain amplitude. Mode I failures were
observed for tensile loading at all strain amplitudes. In torsion, mode II, shear failures were observed at high strain
amplitude and mode I failures at low strain amplitudes. Other
ductile materials such as Inconel 718 fail by the growth of a
mode II shear crack in both tension and torsion loading.
A multiaxial fatigue damage parameter has been developed
for Inconel 718 at room temperature. Extensive observations
have shown that the majority of the fatigue life in tubular
specimens of this material is spent in growing cracks from
about 0.01 mm to 1.0 mm on planes of maximum shear strain
amplitude. Stresses and strains perpendicular to the shear
crack influence the rate of crack growth. A schematic illustration of the damage model is shown in Fig. 1. The crack surfaces are irregularly shaped as the crack advances through the
individual grains. Mechanical interlocking occurs during shear
loading and high frictional forces are developed. This reduces
the stresses and strains at the crack tip and is responsible for a
lower growth rate. Stresses and strains perpendicular to the
shear crack can be developed for loading situations other than
torsion. This opens the crack surfaces and reduces or
eliminates the friction forces. Higher stresses and strains will
be developed at the crack tip and higher growth rates are
observed. Fractographic evidence has been presented by Socie
et al. (1985a), where the individual slip bands are clearly visible on the fracture surfaces of the tension test while the torsion
test has a fracture surface that has been burnished by the rubbing of the two crack surfaces.
A three parameter description of this damage process has
been proposed by Socie et al. (1985b).
y + e„+^ = y}(2Ny + T}/G{2N)b
Torsion
(1)
y-
-y
CT,€
Tension
y-»-
-*~Y
Contributed by the Materials Division for publication in the JOURNAL OF
ENGINEERING
MATERIALS AND TECHNOLOGY.
Manuscript
received
by
the
Or, 6
Fig. 1 Shear damage model
Materials Division July 22, 1987.
Journal of Engineering Materials and Technology
OCTOBER 1987, Vol. 109 / 293
Copyright © 1987 by ASME
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TORSION
TENS la,
101, - - - - - - , - - - - - - , - - - - - - - , - - - - - - ,
Inconel 718
;:<>I'H."l\lI'"
,,-~L~
"l".'tf.""Ufl AI~t.1lW I>W1,
__• ,.1 J. • ff';:"
•
. .i'J.«tJ
~
163L.;,-----~----...l...;------l.::-----...J
~
d
~
Nl.O
~
~.US.:
~
Correlalion of Inconel 718 Test Results with Shear
Strain Parameter
Fig. 2
Correlation of Inconel 718 test data
The right-hand side is the description of the strain life curve
generated from torsion testing with the following
nomenclature:
'Yj shear fatigue ductility coefficient
c fatigue ductility exponent
shear fatigue strength exponent
b fatigue strength exponent
G shear modulus
2N reversals to the formation of a 1.0 mm surface crack.
7j
The terms on the left-hand side represent the loading
parameters defined on the plane experiencing the largest range
of cyclic shear strain and have the following definitions:
r
En
anD
E
maximum shear strain amplitude
tensile strain perpendicular to the maximum shear
strain amplitude
mean stress perpendicular to the maximum shear
strain amplitude
elastic modulus.
Fig. 3
Table 1 Material properties, 304 stainless steel
MONOTONIC TENSILE PROPERTIES
E,
Elastic Modulus
183 GP.
ai%'
0.2% Offset Yield Strength
325 MPa
au'
of'
E ,
f
% RA,
Ult imate Strength
True Fracture Strength
True Fracture Strain
% Reduction in Area
Strength Coefficient
Strain Hardening Exponent
K,
Fifteen proportional and nonproportional loading histories
have been used to evaluate the model. Fifty-five tests were performed on tubular specimens using these loading histories. An
additional twenty tests were performed on solid axial
specimens to evaluate mean stress effects in uniaxial tension.
Correlation of all of the test data with the shear parameter
given in equation (1) is shown in Fig. 2. The solid line
represents the best fit through the torsion data that was used
to establish the material constants.
Tension and torsion tests were also conducted on tubular
AISI Type 304 stainless steel specimens and reported by Bannantine and Socie (1985). Although both materials have considerable ductility, the stainless steel tends to crack on planes
of maximum principal strain except for the high strain torsion
tests. The cracking behavior of both materials is shown in Fig.
3 for both tension and torsion tests. Note the extensive shear
cracking in the Inconel 718 tension and torsion tests. All of
these photos were taken from tests where the elastic strains
were approximately equal to the plastic strains. The fracture
surfaces of the AISI Type 304 stainless steel tests were examined and showed no evidence of shear cracking in the tension
tests. Shear nucleation was observed in the AISI Type 304
stainless steel torsion tests but the majority of the life is consumed in mode I tensile growth at this strain amplitude. These
observations suggest that the shear parameter used to correlate
the Inconel 718 data would be inappropriate for AISI Type
304 stainless steel. This paper reports results, observations and
correlations for both proportional and nonproportional
loading tests on AISI Type 304 stainless steel specimens and
makes comparisons between these two materials and SAE
1045 steel.
294/VoI.109, OCTOBER 1987
Cracking behavior of Inconel 718 and 304 stainless steel
n,
650 MPa
1400 MPa
1. 731
80%
1210 MPa
0.193
AXIAL CYCLIC PROPERTIES (R = -I)
E
E,
Elastic Modulus
185 GPa
Of'
b
Fatigue Strength Coefficient
Fatigue Strength Exponent
1000 MP.
-0.114
<f'
c
K'
n'
Fatigue Ductility Coefficient
Fatigue Ductil ity Exponent
Cycl ic Strength Coefficient
Cyclic Strain Hardening Exponent
0.171
-0.402
TORSIONAL CYCLIC PROPERTIES (R y = -I)
G,
Torsional Modulus
f,
1660 MP.
0.287
82.8 GPa
Fatigue Strength Coefficient
Fatigue Strength Exponent
709 MP.
-0.121
c,
Fatigue Ductility Coefficient
Fatigue Ductil ity Exponent
0.413
-0.353
K',
n',
Cyclic Strength Coefficient
Cyclic Strain Hardening Exponent
T
b
Yf'
785 MP.
0.296
Experimental Procedure
Strain controlled tension and torsion tests were performed
on AISI Type 304 stainless steel tubular specimens with an internal diameter of 25 mm and a wall thickness of 3.8 mm. A
complete set of material properties obtained from monotonic
and strain controlled tension, and torsion fatigue tests are
given in Table 1. Additional materials description is given by
Transactions of the ASME
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Biaxial Loading Paths for 3 0 4 Stainless Steel
Table 2 Fatigue test data, 304 stainless steel
CONTROL PARAMETERS
LOADING
SPEC
fiE/2
EQ
AH/2
STEADY STATE STRESSES (HPa)
XQ
to/Z
aQ
AT/2
H
TQ
HISTORY
Axial
1 0
(cycles)
SS-06
0.0035
SS-15
0.00-16
0
A-03
0.0100
2.0
A-09
0.0100
-1.4
0
0
A-06
0.0060
-21.9
0
0
A-01
0.0035
5.8
0
0
A-10
0.0035
-9.1
0
0
A-11
0.0035
8.4
0
0
0
0
240
-2.2
0
37,600
38,500
0
0
10,000
10,300
0
0
'- .
"
1,070
A-12
0.0020
16.5
0
0
A-20
0.0020
29
0
0
0
248
SS-08
0.017
"y/Ji
0
1.2
1,370
1,167
30,700
33,530
29,000
286,400
333,100
0.008
0
191
1.9
42,000
SS-21
0.0079
0
156
0.4
7,000
32,100
SS-22
0.0080
0
157
0
5,000
33,900
SS-07
0.006
0
157
0
SS-16
0.0061
0
140
0
1,000
SS-09
0.0034
0
138
0
883,000
>1.0 X 10
SS-20
0.0034
0
125
0.6
650,000
824,200
-0.6
940,000
>1.1 x 10
0.0034
0
127
SS-02
0.0025
0.0043
2.7
109
SS-12
0.3
44,300
1
48,500
133,000
83,400
53,000
0.0025
0.0043
-3.7
101
SS-25
0.00145
0.0023
0.6
89
SS-26
0.00145
0.0023
0.8
90
SS-10
0.0035
0.0061
2.6
267
SS-13
0.0035
0.0061
-4.5
256
1.3
3,600
SS-13
0.0020
0.0035
18.6
168
-
2.1
38,600
50,000
SS-29
0.0020
0.0033
29.3
176
-
3.5
35,300
45,000
SS-03
0.0025
0.0043
0
226
0
5,110
5,110
SS-11
0.0025
0.0043
0
218
0
6,000
6,200
SS-30
0.0014
0.0024
20.6
139
1.8
82,100
89,312
SS-31
0.0014
0.0024
17.1
137
2.2
90,100
100,000
SS-01
0.0025
0.0043
32
199
-13.2
SS-04
0.0025
0.0043
35
209
-21.8
7,340
SS-32
0.0014
0.0024
47.7
114.7
-17.0
182,000
200,000
SS-33
0.0014
0.0024
44.5
115.6
-15.7
192,200
205,000
0.7
44,000
52,900
1.2
420,000
440,000
2.9
340,000
356,000
- 4.4
3,340
3,560
-
P
I
4,090
i
101,000
Proportional
6,080
SS-05
SS-28
Torsion
A
•
<
t
1
,
90° out of Phase
Box
Two Box
Fig. 4 Loading histories for 304 stainless steel tests
lOOOr
3,730
500
11,080
9,800
Bannantine (1986). Tests were performed with a computer
controlled axial-torsion servo-hydraulic testing system that
was automated for test control and data acquisition. The
strains were measured with an internal extensometer that is
described by Socie et al. (1985b). This technique allows the
outside surface of the specimen to be monitored for crack formation and growth. Most of the axial tests were conducted on
standard 6 mm diameter solid tensile specimens and are
denoted in Table 2 with specimen numbers beginning with A.
All tests were conducted at room temperature. Six loading
histories were used for these tests and are shown in Fig. 4. Two
strain levels were used for each multiaxial loading condition
resulting in fatigue lives ranging from 3 x 103 to 2 x 105 cycles.
Ranges of stress and strain are indicated with the A symbol
and mean values of stress and strain by the subscript 0 in Table
2. The first observation of a surface crack 1.0 mm long is
denoted by 7V,]0 and final failure by Nf. All of the stresses and
strains given in Table 2 were computed at the midsection of
the thin walled tube from the stable hysteresis loop of the
measured axial load and torque, gage section deflection and
angle of twist.
Results and Discussion
Results from the fatigue tests are given in Table 2. The
fatigue test data initially was correlated in terms of the maximum principal strain amplitude. This correlation was
satisfactory for all of the in-phase tests for histories A, B, and
C. The nonproportional tests, however, were more damaging
by about a factor of ten in fatigue life when compared on the
basis of principle-strain range alone. It is now well established
that in many materials, including AISI Type 304 stainless
steel, out-of-phase loading produces additional cyclic hardening that is not found in simple proportional tests. The stable
effective cyclic stress strain curve for both in-phase and 90
Journal of Engineering Materials and Technology
0,01
Strain
Fig. 5 Effective stress strain curve for in and out of phase tests
deg. out-of-phase loading is shown in Fig. 5. The stable curve
is doubled during the nonproportional loading. This increase
in stress must be included in the damage parameter. The
simplest parameter which incorporates both cyclic strain range
and maximum stress was proposed by Smith et al. (1970), and
is commonly referred to as the Smith-Watson-Topper (SWT)
parameter.
<>?
afax Ae, / 2 = a'rt'r (27V) * + c + - £ - (27V)
(2)
The right-hand side is a description of the uniaxial strain life
curve generated from tensile testing with the following
nomenclature:
tensile fatigue ductility coefficient
fatigue ductility exponent
tensile fatigue strength exponent
b fatigue strength exponent
E elastic modulus
2N reversals to the formation of a 1.0 mm surface
crack.
c
The terms on the left-hand side represent the loading
parameters and have the following definitions:
Ae,/2 maximum principal strain amplitude
CTmax maximum stress on the maximum principal strain
plane
A considerable data base exists for this parameter for effects
of mean stress during uniaxial loading. It can be reasoned that
the larger stresses during nonproportional loading will have an
effect similar to a tensile mean stress.
OCTOBER 1987, Vol. 109/295
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The principal stresses and strains continuously rotate during
a nonproportional test. The damage parameter can be interpreted in two ways. Analogous to the critical plane shear
strain theory, the SWT parameter can be determined on the
plane that has the greatest range of principal strain. An alternate definition could be the maximum value of the SWT
parameter. In these tests, both definitions result in nearly the
same damage parameter. This parameter is proportional to
plastic work per cycle for in-phase fully reversed loading. It
differs in concept because the plastic work approach does not
consider a critical direction or plane for crack nucleation and
growth. This parameter also has the capability to directly
model mean stress effects.
Failure cracks and critical planes for histories C, N, P, and
Q are shown in Fig. 6. The cracks have a surface length of approximately 500 /-tm in these photos. The plane that intersects
the surface experiencing the largest range of principal strain is
/'
/'
/'
/'
a
b
c
d
Fig. 6 Failure cracks (a) proportional, (b) 90· out of phase, (e) box, (d)
two box
Axial
Proporlionol Loading
~roo
A~iol
80x Polh
TO/sian
shown as a dashed line. Cracks initially form and grow on this
plane for all of the tests. Observations made during the tests
indicated that the majority of the life is consumed in forming a
1.0 mm surface crack. Cracks continue to grow on the maximum principal strain range plane for in-phase loading. During 90 deg. out-of-phase loading the principal strain plane
continuously rotates and cracks could be expected in any
direction depending on the magnitude of the applied axial and
torsional strain. After nucleation, cracks grow during the axial
strain part of the load cycle for paths P and Q after they
become large enough to generate their own stress fields. During this part of the load cycle the cracks are subjected to an axial strain cycle with a mean torsion stress. The other half of the
cycle can be modeled as a torsion crack subjected to a cyclic
shear stress with a mean axial stress. Thus, the cracks turn and
tend to grow normal to the specimen axis. The observed cracking directions provide physical justification for using a principal strain based theory.
Axial and torsional response from the four loading histories
are given in Fig. 7 for the stable hysteresis loops. Note that the
maximum stress and strain do not occur at the same point in
time during out-of-phase loading. Axial stress was computed
from the measured loads. Average shear stress was computed
from the measured torque employing thin walled tube assumptions. The hoop strain required to determine the strains on the
critical plane was computed from the measured axial stresses
and strains. Stress strain response on the critical plane for
these- tests was computed from the test data with the results
given in Fig. 8. Here, the critical plane is defined as the plane
experiencing the largest range of normal strain. The additional
cyclic hardening is clearly evident when compared to in-phase
and out-of-phase tests. The amount of hardening is nearly the
same for all of the out-of-phase tests. The proportional
loading test has the largest plastic strain range and has the
longest life for any of the tests. This conflicts with the widely
held notion that plastic strain alone is responsible for fatigue
damage. Cyclic stress plays an important role in the damage
process and cannot be neglected.
The SWT parameter was computed from the hysteresis
loops shown in Fig. 8. The maximum stress in the loop was
used in the analysis even though it did not occur at the maximum strain. Correlation of the test data is given in Fig. 9.
Correlation is good for all tests. This parameter does not require the determination of plastic strains so that visco elastic
and plastic constitutive equations could be used to evaluate the
damage parameter. In plastic strain approaches, small errors
in the stress calculation can result in large errors in estimated
life.
Axial
90" aul of Phose Loading
~"'o
Axial
Two Box Poth
Torsion
~"'"
Torsion
Fig. 7 Axial and torsion stress response (a) proportional, (b) 90 out of
phase, (e) box, (d) two box
29B/Vol. 109, OCTOBER 1987
Transactions of the ASME
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: '
1
1
1
1
1
U
1
i 11 m i
Inconel 718
.
-
1 1 THI|
"1
o
%
o
;
:
o
<&
Critical Plane, Proportional Loading
CO °%0
o
o
-
Cfilical Plane, 90° oul of Phase Loading
o oc?
o
oo
£* 0
b
10
qp
o
o
<&
o _
:
•1
1
1
Mill
10=
10°
Ni
Fig. 10
J-600
Cfilical Plane, Box Palh
-~
j_600
„ . , _,
_
Critical Plane, Two Box Path
Fig. 8 Stress strain response on the critical plane, (a) proportional, (b)
90 out of phase, (c) box, (d) two box
Fig. 9
Correlation of 304 stainless steel test data with SWT parameter
The details of constitutive modeling are beyond the scope of
this paper but it is worth noting that the correlations given in
Fig. 9 for the experimentally measured stress and strain and
analytical estimates of stress and strain are nearly identical.
The constitutive model described by McDowell and Socie
(1985) is based on Mroz kinematic hardening and an out-ofphase isotropic hardening parameter proposed by McDowell
(1983).
The SWT parameter would not be expected to correlate the
test data from the Inconel 718 tests because of the difference
in cracking (Fig. 3). This is indeed the case as shown in Fig. 10
where the data given in Fig. 2 has been replotted in terms of
the SWT parameter. These results show the importance of
choosing a damage parameter that is based on the physical
mechanism for the material and loading of interest.
It has been reported that in-phase loading is more damaging
at low strain amplitudes and out-of-phase loading is more
damaging at higher strain amplitudes (Garud, 1979). In the
case of low strains this is misleading because the comparison is
based on the same axial and torsion strains rather than maximum shear strain ranges. If the comparison is made on the
basis of maximum strain ranges rather than applied tensile and
torsion strain ranges out-of-phase loading is equally or more
damaging at low strain levels. For higher strains, out-of-phase
loading is expected to be more damaging because of the inJournal of Engineering Materials and Technology
Correlation of Inconel 718 test data with SWT parameter
creased cyclic hardening. At small plastic strain levels the additional hardening will also be small and no difference in
fatigue lives should be observed. For larger plastic strains, the
hardening can be as much as a factor of two and a considerable effect on the fatigue life should be observed.
Materials that show small amounts of additional cyclic
hardening for out-of-phase loading would be expected to show
small differences in fatigue life between in-phase and out-ofphase loading. Fatima (1986) tested 1045 Steel in 90 deg. outof-phase loading and found that the additional cyclic hardening was only 10 to 15 percent and the fatigue lives were reduced by about 1/2 when compared to the in-phase tests. In
this investigation the stresses doubled during out-of-phase
loading and the fatigue life is reduced by a factor of 10.
The shear strain based model presented in equation (1)
predicts no effect from the additional cyclic hardening. The
effects could be included by combining the cyclic normal
strain term and the mean stress term into a maximum stress
term (Fatima and Socie, 1986). Materials tested to date in our
laboratory that have shear dominated failures show small
amounts of out-of-phase hardening so the importance of an
out-of-phase hardening term in the shear model cannot be
confirmed. Out-of-phase loading will always be more damaging in the shear model because the normal strain term is increased during this type of loading.
Test data reported here show at least two multiaxial damage
criteria are required: one for shear cracking failure modes and
another for tensile cracking failure modes. Knowledge of the
cracking behavior allows the selection of the most appropriate
damage parameter. It is suggested that fatigue damage in most
finite life situations could be described by one of these two
simple models.
If both tension and torsion test data is available, good life
estimates can be made by considering both tensile and shear
failure modes. The shear model overestimates the life in situations dominated by tensile crack growth and the tensile model
overestimates the life for situations dominated by shear crack
growth. This suggests that a possible engineering approach is
to calculate the two damage parameters and estimate the life
from each one. The expected life will then be the lower of the
two estimates. Results for three materials are shown in Fig. 11.
Correlation of the data for 167 datapoints is good. The dashed
lines represent a factor of 2 in life. Test data for SAE 1045
steel is given by Fatima (1986), Fash et al. (1985), and Leese
and Morrow (1985). The primary drawback to this approach is
that it will require test data from both tension and torsion
testing to evaluate both sets of constants in equations (1) and
(2). Knowledge of the cracking behavior would allow a single
set of material parameters to be used to correlate all of the
data.
OCTOBER 1987, Vol. 109/297
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i-ft-ni
Inconel 718
SAE 1045 Steel
304 Stainless Steel
References
§•
167 Datapoints
Bannantine, J. A., and Socie, D. F., 1985, "Observations of Cracking
Behavior in Tension and Torsion Low Cycle Fatigue," ASTM Symposium on
Low Cycle Fatigue-Directions for the Future.
Bannantine, J. A., 1986, "Observations of Tension and Torsion Fatigue
Cracking Behavior and the Effect on Multiaxial Damage Correlation," M.S.
thesis, Mechanical Engineering Department, University of Illinois at UrbanaChampaign.
Brown, M. W., and Miller, K. J., 1973, " A Theory for Fatigue Failure Under
Multiaxial Stress and Strain Conditions," Proc. Inst. Mechanical Engineers,
Vol. 187, pp. 746-755.
Ellyin, F., and Valaire, B., 1982, "High Strain Multiaxial Fatigue," ASME
o Tensile
aShear
Iff
10
Fig. 11
10 2
stresses can be increased by a factor of two during nonproportional loading.
JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY, Vol. 104, pp. 165-173.
103
ID 4
Estimated
105
Ifj 3
Correlation of test data using tensile and shear parameters
This work has been restricted to the finite life region of less
than 106 cycles. It is likely that a third damage model is required for longer lives where crack nucleation may be the
primary failure mechanism. The possibility of nonpropagating
cracks should also be considered in some materials. Fracture
mechanics approaches could also be used rather than the bulk
stress-strain approaches suggested here. Some success has
been attained for in-phase loading by Socie et al. (1987), using
this approach. The analytic procedures are straight forward
but have some difficulties in applications to shear cracks.
Referring to Fig. 3, it is difficult to estimate the stress intensity
factor for the multiple cracking observed in both the tension
and torsion tests of Inconel 718.
Summary
This paper has shown the importance of considering the
manner in which cracks form and grow in the selection of the
most appropriate multiaxial damage model. Two simple
models based on the critical plane for crack nucleation and
early growth have been presented. A shear strain model is used
for materials and loadings that result in failure by mode II
cracking and a tensile strain model for mode I failure. The importance of considering the additional cyclic hardening
observed during out-of-phase loading was shown.
For equal maximum shear strain ranges, proportional
loading tests may have both larger plastic stain ranges and
longer lives than non-proportional loading tests. Cyclic
298/Vol. 109, OCTOBER 1987
Fash, J. W., Socie, D. F., and McDowell, D. L., 1985, "Fatigue Life
Estimates for a Simple Notched Component under Biaxial Loading," ASTM
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