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Multiaxial Fatigue of 16MnR Steel

Article in Journal of Pressure Vessel Technology · April 2009

DOI: 10.1115/1.3008041

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 Zengliang Gao
Multiaxial Fatigue of 16MnR
College of Mechanical Engineering,
Zhejiang University of Technology,
Hangzhou 310014, China
Steel
Uniaxial, torsion, and axial-torsion fatigue experiments were conducted on a pressure
Tianwen Zhao vessel steel, 16MnR, in ambient air. The uniaxial experiments were conducted using solid
Department of Mechanical Engineering (312), cylindrical specimens. Axial-torsion experiments employed thin-walled tubular specimens
University of Nevada, Reno, subjected to proportional and nonproportional loading. The true fracture stress and
Reno, NV 89557 strain were obtained by testing solid shafts under monotonic torsion. Experimental re-
sults reveal that the material under investigation does not display significant nonpropor-
Xiaogui Wang tional hardening. The material was found to display shear cracking under pure shear
College of Mechanical Engineering, loading but tensile cracking under tension-compression loading. Two critical plane mul-
Zhejiang University of Technology, tiaxial fatigue criteria, namely, the Fatemi–Socie criterion and the Jiang criterion, were
Hangzhou 310014, China evaluated based on the experimental results. The Fatemi–Socie criterion combines the
maximum shear strain amplitude with a consideration of the normal stress on the critical
Yanyao Jiang1 plane. The Jiang criterion makes use of the plastic strain energy on a material plane as
Department of Mechanical Engineering (312), the major contributor to the fatigue damage. Both criteria were found to correlate well
University of Nevada, Reno, with the experiments in terms of fatigue life. The predicted cracking directions by the
Reno, NV 89557 criteria were less satisfactory when comparing with the experimentally observed cracking
e-mail: yjiang@unr.edu behavior under different loading conditions. 关DOI: 10.1115/1.3008041兴

Introduction materials that developed tensile cracks. To consider the different

cracking behaviors, a model combining shear and normal compo-
Fatigue and fracture are of great concern in the design of a
nents was developed 关11,12兴.
load-bearing component. This is particularly true for the pressure A common characteristic of most fatigue models is the use of
vessels. Fatigue should be considered in the design stage. Once in stress/strain quantities, such as strain amplitude and maximum
service, evaluation should be made to assess the durability of a stress, that are based on the definition of a loading cycle or rever-
pressure vessel, especially after cracks and defects are detected. sal. As a result, a cycle-counting method is needed for variable
Fatigue design and evaluation heavily rely on the experimentally amplitude loading. Currently, the only accepted cycle-counting
obtained fatigue properties of a material. Although theoretical and method is the rain-flow method that is designed to find the closed
numerical simulations are widely available and are used in the stress-strain hysteresis loops in a loading history. For uniaxial
design and evaluation of a load-bearing component, the baseline loading, the use of the rain-flow method to identify a loading
data for any models are obtained experimentally. Also, experimen- cycle is straightforward. However, when the rain-flow counting
tal investigations are essential for the validation of a fatigue method is used for nonproportional multiaxial loading, difficulties
model. arise because the stress and strain quantities on a material plane
One important development in understanding fatigue crack ini- may not form consistent stress-strain hysteresis loops for the nor-
tiation concerns a critical plane in a given material for a known mal and shear components. On a given material plane, both the
stress state on which cracks nucleate. The notion is that cracking magnitude and the direction of the shear components may vary
behavior is material and loading magnitude dependent. Some ma- with time. In addition, most of the existing fatigue criteria cannot
terials display shear cracking where fatigue cracks are observed in account for the well-documented loading sequence effect without
the direction of maximum shear amplitude and some materials a separate model. For variable amplitude loading, a fatigue dam-
exhibit tensile cracking where cracks are initiated on the plane of age accumulation law is needed to assess the total fatigue damage.
maximum normal stress. Other materials display a mixed cracking The current investigation experimentally studied the fatigue
behavior where cracks are initiated on the maximum normal crack initiation properties of a pressure vessel steel. In addition,
planes for tension-compression loading, but on the maximum two critical plane multiaxial fatigue criteria are evaluated using
the experimental data.
shear plane for torsion loading. Careful observations on cracking
behavior were made by Socie and co-workers 关1–3兴 and Kurath
and Fatemi 关4兴 on a number of materials. Experiments and Results
Several critical plane fatigue criteria have been developed. To The material under investigation was 16MnR, a widely used
consider the mean stress influence on crack initiation, Socie and pressure vessel material in China. The chemical compositions of
co-workers 关2,5兴 modified a multiaxial fatigue parameter devel- the material are summarized in Table 1. In terms of the mechani-
oped earlier by Brown and Miller 关6兴 and Kandil et al. 关7兴. In a cal properties and the chemical composition, the 16MnR steel is
subsequent work, Fatemi and Socie 关8兴 developed a shear-based similar to St52-3 in Germany, ASME SA302B, and JIS SM490A.
parameter. These fatigue criteria were designed for materials that The testing specimens were taken from a hot rolled plate with a
display of shear cracking. It was shown that the parameter by thickness of 35 mm. No additional heat treatment was made be-
Smith et al. 关9兴 with a critical plane interpretation 关10兴 applied to fore the fabrication of the testing specimens. Three types of speci-
multiaxial loading resulted in a good fatigue life correction of mens were used in the experimental program: solid cylindrical
specimens for uniaxial loading 共Fig. 1共a兲兲, solid shafts for torsion
共Fig. 1共b兲兲, and tubular specimens for combined axial-torsion
1
Corresponding author. loading 共Fig. 1共c兲兲. The axis of the testing specimen was perpen-
Contributed by the Pressure Vessel and Piping Division of ASME for publication
in the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received February 27,
dicular to the rolling direction of the plate material, as shown in
2007; final manuscript received July 18, 2007; published online December 11, 2008. Fig. 1共d兲. The microstructure of the material is shown in Fig. 2.
Review conducted by Poh-Sang Lam. The microstructure obtained from the optical microscope 共Fig.

Journal of Pressure Vessel Technology Copyright © 2009 by ASME APRIL 2009, Vol. 131 / 021403-1

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 Table 1 Chemical compositions of 16MnR „wt %…

C 0.203 Cr 0.008 Nb 0.039

Mn 1.535 V 0.003 Zr 0.001
Si 0.236 Cu 0.010 Bi 0.003
S ⬍10−3 Ti 0.002 Zn 0.002
P 0.009 Co 0.004 La 0.001
Ni 0.003 Al 0.035 Fe Balance

2共a兲兲 clearly shows the laminated nature of the material. The ma-
terial consists mainly of pearlites and ferrites 共Fig. 2共b兲兲.
An Instron hydraulic tension-torsion load frame was used for
the fatigue experiments. The testing system was equipped with the
Instron 8800 electronic control, computer control, and data acqui-
1
sition. A 2 in. gauge length uniaxial extensometer was used for
the measurement of the strain in the gauge section of the uniaxial
specimen. The extensometer had a range of ⫾5% strain. For the
solid shaft for torsion and the tubular specimens, a biaxial exten-
someter was attached to the gauge section of the specimen to
measure the axial, shear, and diametral strains. The extensometer
had a range of ⫾5% in the axial strain, a range of ⫾3 deg in the
torsion deformation, and a 0.25 mm range in the diametral direc-
tion. All the experiments were conducted in ambient air.
Two uniaxial specimens were tested under monotonic tension,
and the stress-strain curves are shown in Fig. 3. The material
displayed distinct lower and upper yield stresses typical of low
and medium carbon steels. This is an indication that the material
involves the Lüders band propagation in the plateau of the stress-
strain curve. As a common practice, engineering stress and engi-
neering strain were used in Fig. 3 for the monotonic stress-strain
curve.
Two solid torsion specimens 共Fig. 1共b兲兲 were tested under
monotonic torsion, and the shear stress-shear strain curves are Fig. 1 Specimens and orientation of the specimens taken from
shown in Fig. 4. The surface shear stress was determined by using the plate „all dimensions in millimeters…: „a… solid specimen for
Nadai’s formula 关13兴. Due to the limited range in measuring the uniaxial loading, „b… solid shaft for torsion, „c… tubular speci-
men for axial-torsion loading, and „d… orientation of the speci-
shear strain using the extensometer, the shear strain was obtained men taken from the plate
through the calibration of the relationship between the rotation
angle measured from the rotary variable differential transformer
共RVDT兲 in the testing machine and the periodical extensometer
measurement of the surface strain. Similar to the monotonic stress-shear strain curve. The advantage in using a torsion test is
stress-stain curve, there exists a plateau after yielding in the shear that the torsion specimen does not neck during the experiment

Fig. 2 Microstructure of 16MnR

021403-2 / Vol. 131, APRIL 2009 Transactions of the ASME

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 Table 3 Fully reversed uniaxial fatigue experiments

Loading Strain Stress

frequency amplitude amplitude Fatigue life
Specimen f 共Hz兲 ⌬␧ / 2 共%兲 ⌬␴ / 2 共MPa兲 N f 共cycles兲

UN06N 0.1 1.999 514.6 280

UN11S 0.125 1.501 487.3 400
UN03N 0.2 1.000 459.2 880
UN03S 0.4 0.700 418.2 2,220
UN08S 0.5 0.500 380.0 4,850
UN05S 0.8 0.380 351.0 9,200
UN04S 1.0 0.300 322.9 17,300
UN01S 1.5 0.243 303.7 37,600
UN10S 2.0 0.199 285.1 79,000
Fig. 3 Monotonic stress-strain curve UN02S 3.0 0.180 274.5 148,400
UN06S 5.0 0.160 261.9 388,500
UN09S 10.0 0.146 256.0 1,400,000
before fracture. The torsion experiment can provide a true fracture
stress and a true fracture strain or ductility 关13兴. The static mate-
rial properties obtained from monotonic tension and monotonic
torsion are summarized in Table 2. The reported value is the av-
erage of the stress-strain curves obtained from the two tested
specimens.
The uniaxial fatigue experiments were conducted under the
fully reversed strain-controlled condition. The detailed control
condition and fatigue lives are summarized in Table 3. The stress
amplitude was taken at a stabilized stress-strain hysteresis loop
around half of the fatigue life. The reported fatigue life corre-
sponded to the moment when the stress amplitude was reduced by
5% from the stabilized value, or when a visible crack was found
on the surface of the testing specimen. For any specimen when the
testing was terminated, a visible crack can be found on the speci-
men surface.
The strain-life curve obtained from the fully reversed uniaxial Fig. 5 Strain-life fatigue curve from fully reversed uniaxial
loading is shown in Fig. 5. The experimental data can be de- experiments
scribed using the following three-parameter equation:

冉 ⌬␧
2
冊␯
− ␧0 N f = C 共1兲
␧0 in the cyclic stress-strain curve. It should be noted that gener-
ally ␴0 ⫽ E␧0, where E is the elasticity modulus of the material.
where ⌬␧ / 2 is the strain amplitude, N f is the number of cycles to
For the 16MnR steel under investigation, ␧0 = 0.00132, ␯ = 1.74,
failure, and the remaining three parameters are material-related
and C = 0.2551. Corresponding to ␧0 = 0.00132, the endurance
constants that are obtained by best fitting the experimental strain-
limit, ␴0, was found to be 240.0 MPa from the cyclic stress-strain
life data. ␧0 can be viewed as the fatigue endurance limit in terms
curve. Equation 共1兲 was first proposed by Manson 关14兴 and can be
of the strain amplitude below which fatigue damage is minimal.
used for most engineering metallic materials for the description of
The endurance limit, ␴0, is the stress amplitude corresponding to
the strain-life curve.
To better understand the stress-strain behavior of the material,
several companion uniaxial specimens were tested in an incre-
mental step loading condition. An incremental step experiment
consists of several constant amplitude tests using a single speci-
men. In each test, constant amplitude loading is applied for long
loading cycles until a stabilization of the stress-strain response is
reached. Figure 6 summarizes the relationship between the stress
amplitude and the strain amplitude from the fatigue experiments
and the incremental step experiments. The early part of the mono-
tonic stress-strain curve is also presented together with the cyclic
stress-strain curve. It is clear that the cyclic stress-strain curve is
very different from the monotonic stress-strain curve. It should be
noted that the fatigue endurance limit, ␴0, reported in the last
paragraph was obtained from the cyclic stress-strain curve, Fig. 6,
corresponding to a strain amplitude of ␧0 = 0.00132.
Fig. 4 Monotonic shear stress-shear strain curve

Table 2 Static material properties

Elasticity modulus, E 共GPa兲 212.5 Ultimate strength, Su 共MPa兲 544.5

Shear modulus, G 共GPa兲 81.1 Engineering fracture strain, e f 0.55
Poisson’s ratio, ␮ 0.31 Reduction in area, RA 共%兲 73.8
Upper yield stress, ␴yu 共MPa兲 351.4 True fracture shear stress, ␶ f 共MPa兲 365.0
Lower yield stress, ␴yl 共MPa兲 324.4 True fracture shear strain, ␥ f 2.64

Journal of Pressure Vessel Technology APRIL 2009, Vol. 131 / 021403-3

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 to a loading path that the increment in the shear strain is linearly
proportional to the increment in the axial strain 共Figs. 7共b兲–7共d兲兲.
It should be noted that proportional loading 共Fig. 7共b兲兲 is a type of
linear axial-torsion loading.
The fatigue results obtained from testing the tubular specimens
are summarized in Table 4 together with the loading conditions.
The stress amplitudes listed in Table 4 were taken from the stabi-
lized stress-strain hysteresis loops generally corresponding to half
of the fatigue life. The reported fatigue life corresponds to the
moment when either the axial stress amplitude or the shear stress
amplitude was reduced by 5% from its stabilized value or when a
visible crack was found on the specimen surface. The axial stress
Fig. 6 Cyclic stress-stain curve and the shear stress in a tubular specimen were determined by
using the elastic formulas from the traditional strength of materi-
Tubular specimens were used for cyclic torsion and axial- als. All the stress and strain quantities listed in Table 4 are on the
torsion experiments. All the experiments were tested under the specimen surface within the gauge section. It should be noted that
strain-controlled condition. Three loading paths were used for tu- neither the axial stress nor the shear stress is uniformly or linearly
bular specimens: pure torsion, linear axial-torsion, and 90 deg distributed along the wall thickness of the tubular specimen due to
out-of-phase axial-torsion 共Fig. 7共a兲兲. Linear axial-torsion refers plastic deformation. However, these stress and strain values ob-

Fig. 7 Loading paths: „a… 90 deg out-of-phase axial-torsion, „b… proportional axial-torsion, and „c… and „d… linear
axial-torsion

Table 4 Strain-controlled axial-torsion fatigue results: ⌬εx / 2 = axial strain amplitude; ⌬␥xy
s
/ 2 = shear strain amplitude on specimen
surface; ⌬␴x / 2 = axial stress amplitude; ⌬␶xy
s
/ 2 = shear stress amplitude on specimen surface; εm = mean axial strain; ␴m = mean
axial stress; ␥m = mean shear strain; ␶m = mean shear stress

Testing Specimen Loading ⌬␧x / 2 ␧m ⌬␥xy

s
/2 ␥m ⌬␴x / 2 ␴m ⌬␶xy
s
/2 ␶m Nf
type ID path 共%兲 共%兲 共%兲 共%兲 共MPa兲 共MPa兲 共MPa兲 共MPa兲 共cycles兲

Pure TU13N — — — 0.277 — 178.1 435,350

torsion TU10N — — — 0.312 — 188.5 92,000
TU08S — — — 0.362 — 194.5 184,660
TU03S — — — 0.432 — 204.3 85,450
TU05S — — — 0.606 — 224.3 24,360
TU11N — — — 0.658 — 215.0 11,880
TU01S — — — 0.693 — 205.4 4,720
TU04S — — — 0.693 — 229.0 6,200
TU02S — — — 1.092 — 258.5 2,680
TU01N — — — 1.732 — 286.3 800

Linear TU20N 共b兲 0.106 0 0.185 0 181.1 −18.6 115.3 −2.2 496,000
axial- TU21N 共b兲 0.136 0 0.243 0 196.2 −10.5 123.8 −2.3 73,100
torsion TU09N 共b兲 0.707 0 1.225 0 356.6 −6.1 224.3 1.4 700
TU05N 共c兲 0.133 −0.100 0.346 0 186.6 −26.9 164.6 −5.0 64,300
TU22N 共d兲 0.100 −0.100 0.260 0 150.0 −22.3 141.4 1.5 242,810
TU12N 共d兲 0.160 −0.100 0.416 0 201.6 −10.9 176.0 2.0 22,230
TU16N 共d兲 0.300 −0.100 0.780 0 254.1 −12.5 223.9 2.8 3,650

90 deg TU15N 共a兲 0.100 0 0.173 0 218.9 −3.3 142.7 0.8 839,520
out-of- TU06S 共a兲 0.110 0 0.191 0 250.7 −10.2 156.4 1.0 185,600
phase TU09S 共a兲 0.138 0 0.242 0 298.3 −5.0 183.2 2.5 35,300
axial- TU07S 共a兲 0.200 0 0.346 0 378.9 −4.6 225.7 2.5 6,900
torsion TU08N 共a兲 0.277 0 0.484 0 404.6 −8.3 236.5 2.0 2,960
TU06N 共a兲 0.400 0 0.692 0 439.8 −10.7 254.4 2.2 1,500
TU03N 共a兲 0.397 0 0.693 0 451.4 −6.4 260.7 1.3 930
TU14N 共a兲 0.600 0 1.039 0 479.9 −4.9 290.3 0 390

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 tained from using simple equations can serve as a good approxi-
mation due to the small wall thickness.

Multiaxial Fatigue Criteria

A multiaxial fatigue criterion is designed to correlate fatigue for
general multiaxial stress state. Fatemi and Socie 关8兴 proposed a
shear-based multiaxial fatigue criterion that can be expressed with
the following mathematical form:

FP =
⌬␥
2
冉
1+K
␴n max
␴y
冊 共2兲

where FP denotes fatigue parameter. The symbol ⌬␥ / 2 represents

the shear stain amplitude and ␴n max is the maximum normal stress Fig. 8 Base line experimental data for determining the fatigue
on the critical plane. ␴y is the yield stress of the material and K is constants in the Fatemi–Socie criterion
a material constant.
The original fatigue criterion proposed by Fatemi and Socie 关8兴
defined the critical plane as the plane associated with the maxi-
mum shear strain amplitude. In the current investigation, the criti- damage accumulation first reaches a critical value, D0.
cal plane using the criterion is defined as the material plane where One noticeable characteristic of the criterion is its capability to
the fatigue parameter 共FP兲 expressed by Eq. 共2兲 is a maximum. deal with different cracking behavior. The material constant b in
Arguments can be made with regard to the reasonableness of the Eq. 共5兲 reflects the cracking behavior of the material under differ-
choice. A difficulty associated with the original definition of the ent stress states. The constant b ranges from 0.0 to 1.0. A low
critical plane is in the determination of the range of material value b characterizes shear cracking and a high value b is for
planes with similar fatigue damage, as will be discussed in a later tensile or normal cracking. A detailed derivation 关15兴 shows that a
section. It is found in the current investigation that the fatigue life b value between zero and 0.375 characterizes a material display-
predictions made by the Fatemi–Socie criterion is not practically ing shear cracking behavior. When b is equal to or larger than 0.5,
influenced by the definition of the critical plane. the material exhibits normal or tensile cracking. If a material dis-
Often, the material constants in a fatigue model are determined plays shear cracking under shear loading but normal cracking un-
based on the experimental data under tension-compression and der tension-compression loading, 0.375ⱕ b ⱕ 0.5. A b value be-
torsion loading. For the Fatemi–Socie criterion, the fatigue param- tween 0.375 and 0.5 can characterize this mixed cracking
eter, FP, expressed using Eq. 共2兲 can be obtained for any tension- behavior.
compression and torsion experiments. ␴y in Eq. 共2兲 is the yield The incremental form of the fatigue criterion eliminates the
stress obtained from the monotonic loading experiment. For the need for a cycle-counting method for any general loading condi-
16MnR steel under investigation, the lower yield stress obtained tions. The linear accumulation of fatigue damage is assumed using
from the monotonic tension experiment is used for ␴y. According the criterion. In this way, the definition of a loading cycle or
to Table 2, ␴y = 324.4 MPa. The material constant, K, in Eq. 共2兲, is reversal is not needed in order to assess fatigue damage under
obtained by the trial-and-error method. By using K = 1.0, the ex- random and multiaxial loading conditions. This also eliminates the
perimental data from fully reversed tension-compression and tor- need for a damage accumulation rule for random loading condi-
sion come to fit into one curve 共Fig. 8兲. This FP-N f curve can be tions. Also, the criterion considers the mean stress effect and the
best described by using a three-parameter equation mathemati- loading sequence effect.
cally identical to Eq. 共1兲 for the 16MnR steel The material constants used in the fatigue criterion are deter-
共FP − 0.0022兲1.75N f = 1.0 共3兲 mined from the fatigue results under fully reversed uniaxial load-
ing and torsion loading. There are a total of five material con-
For any given loading case, the fatigue parameter, FP, can be stants: ␴ f , ␴0, m, b, and D0. The true fracture stress ␴ f is a static
obtained following Eq. 共2兲 through stress and strain transforma- material property that can be determined by a monotonic torsion
tions. Once FP is known, Eq. 共3兲 is used to predict fatigue life. experiment. From Table 2, the true fracture shear stress is
A recently developed multiaxial fatigue criterion is a critical 365.0 MPa. The equivalent true fracture stress is 632.2 MPa. The
plane approach that combines an energy concept and the material fatigue limit, ␴0, is obtained by using Eq. 共1兲 for the fully reversed
memory 关15兴. The criterion, as will be referred to as the Jiang uniaxial loading. It was found that ␴0 = 240.0 MPa. The other
criterion, takes the following form: three material constants, m, b, and D0, are obtained by comparing
dD = 具␴mr/␴0 − 1典m共1 + ␴/␴ f 兲dY 共4兲 fully reversed uniaxial fatigue with the fully reversed torsion fa-
tigue. Such a comparison is based on the principle that the same
where fatigue life should correspond to identical fatigue damage regard-
1−b less of the loading mode. A detailed description can be found in
dY = b␴d␧ p + ␶d␥ p 共5兲 Ref. 关15兴. A rule of thumb can be used to assist the determination
2
of the material constants m and b. The constant b is designed to
In the equations, ␴ is the normal stress and ␶ is the shear stress on consider the fatigue cracking behavior of a material. The value of
a material plane. The quantities ␧ p and ␥ p represent plastic strains b should be consistent with the experimental observation of fa-
corresponding to ␴ and ␶, respectively. b and m are material con- tigue cracking behavior for tension-compression and pure torsion
stants, and the prefix d denotes an increment. The braces 具 典 de- loading. While it is difficult to determine experimentally the fa-
note the Macauley bracket 共i.e., 具x典 = 0.5共x + 兩x兩兲兲, and ␴ f is the true tigue cracking direction precisely, it is possible to determine
fracture stress of the material. The quantity ␴mr is a material whether the material displays shear cracking 共0 ⱕ b ⱕ 0.375兲, ten-
memory parameter. The letter Y represents the plastic strain en- sile cracking 共b ⱖ 0.5兲, or mixed cracking 共0.375ⱕ b ⱕ 0.5兲 from
ergy 共density兲 on a material plane. The symbol D denotes fatigue the tension-compression and pure torsion tests 关15兴. This observa-
damage and ␴0 is the endurance limit of the material. The endur- tion of fatigue cracking behavior will narrow down the selection
ance limit corresponds to the critical strain amplitude below of the value of b. The material constants for 16MnR are ␴0
which persistent slip bands will not develop in the material. The = 240.0 MPa, b = 0.38, m = 1.1, ␴ f = 632.2 MPa, and D0
critical plane is defined as the material plane where the fatigue = 4490 MJ/ m3.

Journal of Pressure Vessel Technology APRIL 2009, Vol. 131 / 021403-5

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 Fig. 9 Comparison of observed fatigue life and prediction ob- Fig. 10 Comparison of the observed fatigue life and prediction
tained by using the Fatemi–Socie criterion „Eq. „2…… obtained from using the Jiang criterion „Eqs. „4… and „5……

ing axis for the uniaxial loading cases. The Fatemi–Socie predic-
For constant amplitude loading or block loading, the fatigue tion for uniaxial loading is different from the experimental obser-
damage per loading cycle or per loading block can be determined vation for the material.
by integrating Eqs. 共4兲 and 共5兲 over one cycle or block For an axial-torsion tubular specimen, the material plane can be
represented by its normal direction using one angle, ␪, measured
⌬D = 冖
cycle
冉
具␴mr/␴0 − 1典m共1 + ␴/␴ f 兲 b␴d␧ p +
1−b
2
␶d␥ p 冊 共6兲
counterclockwise from the axial direction 共refer to the upper-right
insert in Fig. 11兲. For a given loading condition, it is often found
that several material planes or a range of material planes may
where ⌬D is the fatigue damage per loading cycle or block. Fa- experience very similar fatigue damage according to a given fa-
tigue failure occurs when the accumulated fatigue damage on one tigue criterion. Due to the inherent data scatter in fatigue experi-
of the material planes, the critical plane, first reaches a critical ments, it would be preferable to identify the material planes with
value, D0, similar fatigue damage. In the current investigation, a range of
10% from the maximum fatigue damage is used for the discussion
⌬DN f = D0 共7兲 of possible cracking material plane predicted by a fatigue crite-
rion. As shown in Fig. 11 for the tubular specimen TU05N sub-
Due to the linear accumulation of fatigue damage, a cycle or block jected to linear axial-torsion loading 共refer to Table 4兲, the distri-
can be defined as any event that is repeatable in a loading history. bution of the fatigue damage per loading cycle over the material
For all the tubular specimens tested, the stabilized stress-strain plane orientation can be determined according to the Jiang fatigue
hysteresis loops were recorded from the experiments. These loops criterion. The fatigue damage per loading cycle is the reciprocal of
were used in the determination of the fatigue damage according to the predicted fatigue life under constant amplitude loading. The
a given fatigue criterion. fatigue damage shown in Fig. 11 is normalized so that the maxi-
Figures 9 and 10 show the comparisons of the experimentally mum fatigue damage with respect to all the possible material
observed fatigue lives and the predictions made by using the planes is unity. Theoretically, the maximum peak points are pre-
Fatemi–Socie criterion and the Jiang criterion, respectively. The dicted to be the critical planes, and cracks are predicted to poten-
solid thick diagonal lines signify a perfect prediction. The dotted tially form on these particular material planes. By considering a
lines are the boundaries of factor of 2 difference. An arrow by a range of 10% from the maximum fatigue damage, a range of the
data point denotes run-out in the experiment. Clearly, both multi- cracking directions can be obtained. It should be noted that a 10%
axial fatigue criteria correlate well with the experiments. range was arbitrarily selected without physical consideration.
An important notion behind a critical plane multiaxial fatigue Figures 12 and 13 summarize the comparisons of the observed
criterion is that fatigue damage is accumulated on a material cracking behavior and the predicted cracking orientations by the
plane, and the critical plane is the plane where fatigue crack is two fatigue criteria. The filled circles represent the observed crack
initiated. Therefore, a critical plane multiaxial fatigue criterion orientations in the figures. The range bars in the figures are the
should also predict the cracking behavior. For all the uniaxial predicted ranges of the cracking directions. Observations on the
specimens tested, the cracking planes were found to be approxi- experimental cracking behavior were made in the millimeter scale
mately perpendicular to the loading axis. When a tubular speci- of the crack length. Such a treatment is consistent with the mac-
men was subjected to pure torsion loading, the cracking planes roscopic continuum assumption adopted in the current investiga-
were found to be either perpendicular to or parallel to the speci- tion for the stress and strain 关16兴. The results shown in Figs. 12
men axis. As a basic feature with b = 0.38, the Jiang fatigue crite- and 13 suggest that both criteria predict fatigue crack planes ap-
rion predicts exactly the same cracking behavior as that observed proximately perpendicular to the axial loading axis 共the normal of
in the experiments on the uniaxial and torsion specimens. The the plane along the axial loading direction兲 for the 90 deg out-of
Fatemi–Socie criterion predicts correct cracking directions for the phase axial-torsion loading cases. The prediction is in agreement
pure torsion cases. However, the Fatemi–Socie criterion predicts a with the general experimental observation. For the linear axial-
critical plane that has its normal making ⫾45 deg from the load- torsion loading cases, the Jiang criterion can provide cracking

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 Fig. 11 Determination of the predicted cracking orientation

Fig. 12 Comparison of the experimentally observed cracking orientation with the pre-
dictions based on the Fatemi–Socie criterion for the tubular specimens under combined
axial-torsion loading

Fig. 13 Comparison of the experimentally observed cracking orientation with the pre-
dictions based on the Jiang criterion for the tubular specimens under combined axial-
torsion loading

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 Fig. 14 Cyclic stress-plastic strain curve for 16MnR steel

direction prediction in reasonable agreement with the experimen- simplicity in form which enables an easy application to the con-
tal observation. The Fatemi–Socie criterion predicts two cases stant amplitude loading conditions where a loading cycle can be
correctly for the seven specimens tested under the linear axial- identified. However, the criterion requires a loading cycle-
torsion loading conditions. counting method and fatigue damage accumulation rule in order
to predict the fatigue life of a material subjected to general load-
Discussion ing conditions. Often the rain-flow counting method is modified to
The distinguishable upper and lower yield stresses shown in accommodate the nonproportional multiaxial loading cases
Figs. 3 and 4 indicate that the material, 16MnR, displays a similar 关20–22兴. For variable amplitude loading, the linear fatigue damage
monotonic stress-strain behavior to that of low and medium car- accumulation law is used to assess the total fatigue damage. As a
bon steels. Lüders band propagation is associated with the plateau result, the well-documented loading sequence effect cannot be ac-
in the monotonic stress-strain curve. Under strain-controlled cy- counted for.
clic loading, the material experiences a short transient period, The Jiang fatigue criterion is based on the cyclic plasticity of a
which is accompanied with the local strain inhomogeneity and material and, therefore, is limited to ductile materials. The crite-
variation of stress amplitude 关17,18兴. After the transient period, rion predicts no fatigue damage when the deformation is purely
the stress-strain response stabilizes until fatigue occurs. elastic. It has always been argued or assumed that in the high-
The material exhibits slight nonproportional hardening behav- cycle fatigue 共HCF兲 regime plastic deformation is minimal or
ior. A comparison of the cyclic stress-plastic strain curve under zero. This argument is not true for a number of engineering ma-
fully reversed uniaxial loading and the curve under 90 deg out-of- terials. The elastic deformation assumption in the HCF regime is
phase axial-torsion loading can be used to show whether or not a mainly based on the monotonic stress-strain curve of a material
material displays nonproportional hardening. A cyclic stress- and the traditional definition of the yield stress. For example, a
plastic strain curve is the relationship between the saturated stress 0.2% offset in the monotonic stress-strain curve is usually taken as
amplitude and the plastic strain amplitude. For the nonpropor- the yield stress for a material that does not display distinct yield-
tional axial-torsion cases, the equivalent stress amplitude and the ing. However, the 0.2% offset is actually the stress corresponding
equivalent plastic strain amplitude for cyclic loading followed the to 0.2% plastic strain. For most engineering materials, a plastic
definitions made by Jiang and Kurath 关19兴. Figure 14 shows the strain amplitude of 0.2% in fatigue is significantly large. For the
cyclic stress-plastic strain curves under fully reversed uniaxial 16MnR steel under investigation, the fatigue life corresponding to
loading and that under the 90 deg out-of-phase axial-torsion load- a plastic strain amplitude of 0.2% is approximately 10, 000 cycles.
ing conditions. When the plastic strain amplitude is less than The other cause of the minimal plastic deformation conception
1.4⫻ 10−4, the two cyclic stress-plastic strain curves almost coin- in the HCF regime is due to the difference between the cyclic
cide. When the plastic strain amplitude is higher than 1.4⫻ 10−4, plastic deformation and the monotonic stress-strain curve. Figure
the cyclic stress-plastic strain curve under nonproportional load- 6 compares the monotonic stress-strain curve 共dotted line兲 and
ing is consistently higher than that of the proportional loading. cyclic stress-strain curve 共circular markers兲 for the material under
The difference between the two curves is not significant, indicat- investigation. The cyclic stress-strain curve is the relationship be-
ing that the material does not display significant nonproportional tween the stress amplitude and the strain amplitude usually taken
hardening. It should be noted that the demarcation plastic strain from the stabilized stress-strain hysteresis loops under cyclic load-
amplitude, 1.4⫻ 10−4, corresponds to the fatigue endurance limit, ing. Clearly, these two stress-strain curves are very different.
as shown in Fig. 14. From the monotonic point of view, the material will not yield,
The two fatigue criteria discussed in the current article are criti- thus no plastic deformation, when the stress is less than the lower
cal plan approaches. The Fatemi–Socie criterion is designed for yield stress 共324.4 MPa兲. Under cyclic loading with a stress am-
shear-cracking materials where the shear strain amplitude domi- plitude of 324.4 MPa, the strain amplitude is approximately
nates the fatigue damage. As a result, the criterion predicts crack- 0.32% and the corresponding plastic strain amplitude is 0.16%.
ing planes with the normal making ⫾45 deg from the loading axis The fatigue life under a stress amplitude of 324.4 MPa is
for the uniaxial tension-compression loading. Such a prediction of 15, 000 cycles.
cracking direction is inconsistent with what was observed from Figure 14 shows that the plastic strain amplitude corresponding
testing the 16MnR steel. On the other hand, the Fatemi–Socie to the fatigue limit is 1.4⫻ 10−4 共170 ␮␧兲. Such a strain value is
criterion predicts correctly the cracking behavior for the pure tor- measurable using a strain gauge or extensometer and can be pos-
sion. One advantage of using the Fatemi–Socie criterion is the sibly predicted using a cyclic plasticity model.

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 Most existing critical plane approaches are either for a material ␥f ⫽ true fracture shear strain
displaying tensile cracking or a material displaying shear crack- ⌬␥sxy / 2 ⫽ shear strain amplitude on specimen surface
ing. However, engineering materials commonly exhibit mixed ␮ ⫽ Poisson’s ratio
cracking behavior. The mixed cracking phenomenon is reflected ␯ ⫽ material-related constant in an equation de-
by the tensile cracking under uniaxial tension-compression and scribing the S-N curve
shear cracking under torsion loading. Therefore, most of the ex- ␴ ⫽ normal stress on a material plane
isting critical plane approaches fail to correctly predict the crack- ␴0 ⫽ endurance limit of the material
ing behavior of a material that exhibits mixed cracking behavior. ␴f ⫽ true fracture stress of the material
The 16MnR steel is a typical material that displays mixed crack- ␴mr ⫽ material memory parameter in the fatigue
ing behavior. The Jiang fatigue criterion, with its capability to criterion
predict general cracking behavior, was found to predict correctly ␴yu ⫽ upper yield stress
the cracking behavior of the material under different loading con- ␴yl ⫽ lower yield stress
ditions. ⌬␴x / 2 ⫽ axial stress amplitude
␶ ⫽ shear stress on a material plane
Conclusions ␶f ⫽ true fracture shear stress
The current investigation generated comprehensive fatigue and ⌬␶sxy / 2 ⫽ shear stress amplitude on specimen surface
cyclic plasticity experimental results of a widely used pressure
vessel steel 共16MnR兲. The experimental results 共Fig. 14兲 show References
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