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Prediction of the Fatigue Life of Cast Steel ContainingShrinkage
Porosity
RICHARD A. HARDIN and CHRISTOPH BECKERMANN
A simulation methodology for predicting the fatigue life of cast
steel components with shrinkageporosity is developed and validated
through comparison with previously performed measure-ments. A X-ray
tomography technique is used to reconstruct the porosity
distribution in 25 testspecimens with average porosities ranging
from 8 to 21 pct. The porosity eld is imported intonite element
analysis (FEA) software to determine the complex stress eld
resulting from theporosity. In the stress simulation, the elastic
mechanical properties are made a function of thelocal porosity
volume fraction. A multiaxial strain-life simulation is then
performed to deter-mine the fatigue life. An adaptive subgrid model
is developed to reduce the dependence of thefatigue life
predictions on the numerical mesh chosen and to account for the
eects of porositythat is too small to be resolved in the
simulations. The subgrid model employs a spatiallyvariable fatigue
notch factor that is dependent on the local pore radius relative to
the niteelement node spacing. A probabilistic pore size
distribution model is used to estimate the radiusof the largest
pore as a function of the local pore volume fraction. It is found
that, with theadaptive subgrid model and the addition of a uniform
background microporosity eld with amaximum pore radius of 100 lm,
the measured and predicted fatigue lives for nearly all 25
testspecimens fall within one decade. Because the fatigue lives of
the specimens vary by more thanfour orders of magnitude for the
same nominal stress amplitude and for similar average
porosityfractions, the results demonstrate the importance of taking
into account in the simulations thedistribution of the porosity in
the specimens.
DOI: 10.1007/s11661-008-9755-3 The Minerals, Metals &
Materials Society and ASM International 2009
I. INTRODUCTION
INHOMOGENEITIES due to porosity are currentlynot considered in
the design of structural componentsmade from metal castings.
Instead, ad hoc safety factorsare used to address a designers
uncertainty in how thecasting will perform in service. These safety
factors arebased on the assumption that castings perform
unpre-dictably, if not poorly. Applying such safety factors tothe
entire cast material might do little for the robustnessof the
design other than increase the casting weight.Many part designers
become frustrated by castingsdesigned with very large safety
factors that fail inservice; they are hesitant to use castings.
Such frustra-tions could be avoided if the quality of the cast
metalthroughout the casting could be known ahead of timeand
incorporated into the design.The present study extends our recently
developed
method of modeling the eects of porosity on stinessand stress
redistribution[1] to the prediction of thefatigue life of steel
castings containing porosity. Forease of use in standard design
practice, commonly usedcommercial software is employed in the
present stress
and fatigue life simulations. The part design can bemade safe by
assuring that the casting process results inthe best possible
quality (i.e., lowest porosity) steel athighly stressed locations.
Should porosity form, thecasting rigging or process parameters can
be changed sothat the porosity will not aect the service
performanceof the part. If designers wish to use lighter-weight
andmore thinly walled steel castings, understanding theeects of
porosity becomes especially critical. Anintegrated design process
is emerging in which a castingprocess simulation that predicts the
location, amount,and size of the porosity is directly coupled with
themechanical simulation of the part performance thattakes into
account the eects of porosity.[2] It isanticipated that such a
design process will also helpguide and improve casting inspection
procedures, bylinking acceptance criteria with expected
performance.Fatigue life analysis can be divided into two
parts:[3]
(1) the stage of life of the component up to the initiationof a
crack on the order of 1 mm in size and (2) the stageof life of the
component undergoing the growth of acrack and its propagation to
failure. The combination ofthe two gives the total life of a
component. Perhaps themost often used approach to predicting crack
initiationis the strain-life method,[36] in which specimen
fatiguelife test data to failure is tted to the strain-life curve
andthe curve is used in life estimation. Because fatigue
testspecimens are small compared to most components,once cracks
initiate in them, propagation is very rapid.The total life from the
strain-life testing of specimens is
RICHARD A. HARDIN, Research Engineer, and CHRISTOPHBECKERMANN,
Professor, are with the Department of Mechanicaland Industrial
Engineering, University of Iowa, Iowa City, IA 52242.Contact
e-mail: [email protected]
Manuscript submitted January 30, 2008.Article published online
January 21, 2009
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the crack initiation life, because the second stage of lifeis
extremely short relative to the rst, for these testconditions.
Multiaxial strain-life models[712] have beenimplemented in
commercial software packages (fe-safe,for example[13]), to predict
the durability of componentsfrom nite element analyses (FEAs). The
second stageof fatigue life, the fatigue crack growth stage,
isdescribed using fracture mechanics, in particular, linearelastic
fracture mechanics (LEFM), in which crackgrowth is assumed to
follow the ParisErdogan equa-tion.[14] Here, a crack pre-exists,
formed either fromfatigue crack initiation or during manufacturing
of thepart; the crack growth is determined by the stress eldand
material properties until failure.[3,14] Because thelargest
possible nonpropagating crack can be predictedfor a given material
and an alternating stress eld usingfracture mechanics, it can be
used in damage-tolerantdesign life prediction in
castings.[1517]
The eects of porosity on the fatigue behavior of castmetals have
been measured in steels,[1824] in castirons,[16,17,25,26] and in
aluminum alloys.[2736] Also,Murakami[37] gives a good overview of
the eects ofsmall defects and inclusions on metal fatigue.
Jayet-Gendrot et al.[20] have reviewed the fatigue and
fractureliterature for steel and other metals and showed that
thefatigue behavior depends on the pore size and volumefraction.
For example, in stainless steel, they report thatreductions in
fatigue strength of 35 and 50 pct areobserved for areas of crack
initiating defects of less thanand greater than 3 mm2,
respectively, in a specimen testsection 70 9 22 mm in size.[21]
Based on the square rootof the defect area as a representation of
the dimension ofthe porosity,[37] this size is approximately 55 lm.
Also,for low-alloy steel, fatigue strength reductions from 8 to30
pct were reported when shrinkage porosity cavitiescovered 3 to 7
pct of the fracture surface.[22] Recently,for an aluminum alloy,
Linder et al.[29] measured a15 pct decrease in fatigue strength for
smooth specimenswhen the porosity level was increased from 0.7
to4.1 pct, while notched specimens gave no dependenceon porosity in
the same range. Extreme valuestatistics[2830,34,37] have been used
to show the strongdependency of fatigue life on the maximum likely
poresize from measured pore size distributions.Crack initiation
life analyses of cast components in
the presence of porosity have been performed by usingthe
strain-life approach and modeling the pores asequivalent
notches.[1824,26,3133] Applying a strain-lifemodel alone assumes
that crack nucleation encompassesthe majority of the life of the
component and that thetime for fatigue crack propagation to
fracture isinsignicant relative to crack initiation. The use
ofstrain-life models requires that pore geometry informa-tion,
primarily the minimum notch radius and the majoraxes of the
ellipsoidal notch, be known or be determinedfrom fracture surfaces,
to determine a stress concentra-tion factor. Researchers studying
aluminum castingshave treated pores in castings as notches for
predictingthe eect of porosity on crack initiation life[31] and
themechanism of crack formation and growth frompores.[33] For the
second stage of component life,fatigue life estimation of cast
metals by fracture
mechanics approaches are used: for nodular
castiron,[16,17,25,26] cast aluminum alloys,[27,3236,38] and
steelalloys.[15,19,20,23] One diculty in applying fracturemechanics
concepts to cast parts is determining thenal, or failure, crack
length. For example, failure ofcomponents has been assumed when the
remaining netsection area stress is at or is greater than the
yieldstrength[19] and when the crack depth propagates to thesection
wall thickness at the defect location.[15] Acomparison between the
measured fatigue life of caststeel test specimens containing
porosity and inclusionsand the fatigue life obtained by modeling
the specimenusing crack initiation (local stain-life concepts)
andLEFM approaches has been presented.[23] In the crackinitiation
model, the defects were considered to be three-dimensional notches;
in the crack growth model, thedefects were treated as
two-dimensional elliptical crackshaving an envelope around the
defect. It was found[23]
that the crack initiation estimate of life was moreaccurate than
the fracture mechanics approach, andthat interpreting the porosity
as pre-existing cracksresulted in too conservative an estimate of
the fatiguelife. Dabayeh and Topper[32] came to a somewhatdierent
conclusion for cast aluminum, in which thelocal strain approach
gave quite nonconservative esti-mates of fatigue life and the crack
growth method gavebetter agreement. The current authors have found
thatlocal strain-life modeling of porosity as a notch gavebetter
agreement with measurements than did LEFM incast 8630
steel.[19]
In the present study, a simulation method is devel-oped to
predict the measured fatigue lives of 25 cast steelspecimens
containing up to 21 pct shrinkage porosityover the gage length.
This is a challenging task, not onlybecause the measured fatigue
lives of the specimens aremuch below the values corresponding to
sound steel, butalso because the measured lives vary by more than
fourorders of magnitude at the same applied stress ampli-tude.[19]
These variations can only be attributed to thedierent amounts and
distributions of porosity withinthe specimens. The fatigue tests
and the tomographicreconstruction of the porosity elds in the
specimens[1]
are briey reviewed in Section II. In Section III, theprocedures
and results of the fatigue life simulations arepresented. An FEA is
performed to determine thecomplex stress eld resulting from the
porosity;[1]
multiaxial strain-life analysis is used to compute thefatigue
life distribution in the specimens. The strain-lifeapproach is
adopted here, because the preliminarycalculations in Reference 19
revealed that LEFM doesnot predict the measured fatigue lives of
the specimens.An adaptive subgrid fatigue notch factor model,
whichapproximately accounts for the eect of porosity that
isunder-resolved by the computational mesh used in thestress and
fatigue life simulations, is presented in SectionIV. As part of
this subgrid model, a probabilistic poresize distribution model for
determining the maximumpore radius as a function of the local pore
volumefraction is developed. Detailed comparisons are madebetween
the measured and predicted fatigue lives. Theconclusions of the
present study are summarized inSection V.
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II. EXPERIMENTS
Fatigue tests were performed using cast and heattreated AISI
8630 steel specimens containing a range ofshrinkage porosity. All
the details of these experiments,together with the results of
fractography and prelimi-nary fatigue life calculations, were
presented in Refer-ence 19. Prior to the fatigue testing, X-ray
tomographywas performed on the machined specimens, in order
toobtain a measured three-dimensional porosity eldwithin each
specimen. The details of the tomographicreconstruction procedures
were recently presented inReference 1. For completeness, a brief
overview of thesetests and measurements is provided here.
A. Cast Specimens
Specimen blanks were cast from AISI 8630 steel. Themold geometry
was designed using computer modeling,to obtain a range of shrinkage
porosity levels.[19] Thecast blanks were 152-mm-long cylinders
having anominal 14.3 mm diameter. To produce shrinkageporosity, a
cylindrical disk 25.5 mm in diameter waspositioned at the midlength
of the blanks, as shown inFigure 1. This design concentrated the
porosity at thecenterline and midlength of the cast blanks, so that
theporosity could be located in the gage section of the
testspecimens. The severity of the porosity was controlledby
varying the disk thickness (dimension along thecasting length);
disk thicknesses of 5, 7.5, and 10 mmwere cast. Generally, a
smaller disk thickness resulted in
a lower porosity level, but there was overlap ofthe porosity
levels among the three disk thicknessgroups. Cut surfaces of
specimens from castings havingdisk thicknesses of 5, 7.5, and 10 mm
are shown inFigures 2(a), (b), and (c), respectively. It can be
seenthat the porosity level ranges from dispersed macropo-rosity to
holes and gross section loss. Radiographs areshown to the right of
each cut specimen surface with a
X 152.0
14.25 25.5
Fig. 1Dimensions of cast porous specimen blanks in
millimeters.Dimension X = 5, 7.5, and 10 mm for the least, middle,
andmost porosity specimen groups, respectively.
Fig. 2Cut and polished surfaces of three specimens cast with
dif-ferent porosity levels ranging from (a) specimen 22
representing theleast, to (b) specimen 3 representing the middle
range, to (c) speci-men 13 representing the most porosity.
Radiographs of the specimengage sections are given to the right of
each surface, with the longitu-dinal position of the cut
indicated.
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
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white mark that indicates the approximate location ofthe cut
along the length of the machined specimen.Comparing these
radiographs qualitatively, the porosityappears to be increasingly
severe proceeding fromFigure 2(a) to Figure 2(b), and to Figure
2(c). In somespecimens, as shown, for example, in Figure
2(c),porosity extended to the specimen surface.The cast blanks
received identical heat treatment: they
were normalized at 900 C, austenized at 885 C, waterquenched,
and nally tempered for 1 1/2 hours at510 C. This heat treatment
resulted in a temperedmartensitic structure with a Rockwell C
hardness of 34.The blanks were then machined to the dimensionsshown
in Figure 3. More than 50 specimens wereproduced in this
manner.[19]
B. Tomography for Porosity Measurement
The X-ray tomography of the machined specimenswas performed in
order to obtain a measured three-dimensional porosity eld within
each specimen. Thismethod is described in detail in Reference 1. In
brief,lm radiography of the specimens was performed usinga
sensitivity of 2 pct of the gage section diameter, whichcorresponds
to a resolution of approximately 100 lm.Three examples of
radiographs are shown in Figure 2.Orthogonal radiographic views of
each specimen wereobtained. The radiographic lms were digitized
usingan X-ray scanner resulting in 8 bit gray level, 1200 dpi(~21
lm pixel side length) images. The gray levelinformation was
converted to porosity percentage usingan in-situ calibration
method. A tomography algo-rithm[1,39] was then employed to
reconstruct a three-dimensional porosity eld from the orthogonal
views.Figure 4 shows two examples of porosity elds
obtained in this manner. The X-ray tomography tech-nique used
cannot resolve the exact geometry of theshrinkage pores on a
microscopic scale. This wouldrequire a resolution on the order of 1
lm over aspecimen gage length and diameter of 17.5 and 5
mm,respectively, which is not possible given present daycomputing
resources. The porosity volume fractionsobtained in the tomographic
reconstruction correspondto voxels that have a side length of ~21
lm. If a voxelfalls entirely within a pore (or sound steel), the
porosityvolume fraction is equal to unity (or zero). In
thesimulations presented here, the measured porosity eldsare mapped
onto computational meshes with nodespacings that are one to two
orders of magnitude larger
than the voxel side length, i.e., between 100 lm and1 mm. Thus,
the accuracy and resolution of the presenttomographic
reconstruction technique were deemed tobe sucient (also Reference
1).Table I lists the average porosity volume fraction
measured for the 25 specimens for which a
tomographicreconstruction was performed. The average
porosityfractions listed in the table correspond to a 12 mm
longcenter portion of the gage section, rather than the entire17.5
mm gage length (Figure 3) over which the porosityeld was
reconstructed; this is related to the location ofthe extensometer
with which the strain was measured(Section C). It can be seen that
the average porosityvaries from approximately 8 to 21 pct. Table I
also liststhe maximum cross-sectional porosity fraction for
eachspecimen. This fraction was obtained by averaging theporosity
over one voxel thick cross-sectional slices andselecting the
maximum value along the gage length. Itvaries from approximately 15
to 59 pct. It is shown inReference 1 that this maximum
cross-sectional porosity,rather than the average volume fraction
over the gagelength, correlates well with the measured eective
elasticmodulus, E, of the specimens.
C. Fatigue Testing
The specimen preparation and fatigue testing werecarried out
according to the ASTM E606 standard.[4] Allfatigue tests were
performed under fully reversed,R = 1, loading conditions. During
fatigue testing,the mechanical behavior of the specimens was found
tobe elastic; therefore, load-controlled testing at 10 to20 Hz was
used. This permitted accurate strain ampli-tude measurements while
using the faster testing capa-bility of load control. All fatigue
tests were performeduntil fracture of the specimen occurred or a
runout lifewas achieved at 5 9 106 cycles. Four nominal
stressamplitudes, DS/2, were applied and held constant
duringtesting of the specimens: 126, 96, 66, and 53 MPa;
thesestress levels were selected to obtain a large range ofmeasured
fatigue lives while avoiding runouts. Anextensometer with ends at 6
mm above and below themidpoint of the specimen length was used to
measurethe strain amplitude. The eective (or apparent)
elasticmodulus E was determined from the measured strainand load
data, using the sound specimen gage cross-sectional area to
determine the nominal stress. Themeasured elastic moduli and
fatigue lives of the 25specimens for which the measured fatigue
live was lessthan the runout life are given in Table I; reference
willbe made to individual specimens in the table using thespecimen
numbers.The monotonic and cyclic material properties for
sound 8630 steel were measured in Reference 40 and areprovided
in Tables II and III, respectively. The cyclicstress-strain curve
is given by
De2 DS2E
DS2K0
1n0
1
where DS/2 and De/2 are the nominal stress and strainamplitudes,
respectively; K is the cyclic strength
116.5
38.5 11 17.5
5 .025
12R 19
Gage section
Fig. 3Dimensions of test specimens in millimeters.
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coecient; and n is the cyclic strain hardening expo-nent. Note
from Tables II and III that the 661 MPavalue for the cyclic yield
strength, S0y, is much less
than the monotonic yield strength, Sy, value of985 MPa,
indicating the presence of considerablecyclic softening. The
strain-life curve (often referred to
Fig. 4Example of internal porosity distributions reconstructed
from X-ray tomography in the test section of fatigue test specimens
8 and 22.Slices at three axial positions are shown for each
specimen.
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
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as the ConManson relationship[5,6]) for sound 8630steel is given
by
De2 Dee
2 Dep
2 r
0f
E2Nf b e0f 2Nf c 2
where De/2, Dee/2, and Dep/2 are the total, elastic, andplastic
strain amplitudes, respectively; 2Nf is the numberof reversals to
failure; r0f is the fatigue strength coe-cient; b is the fatigue
strength exponent; e0f is the fatigueductility coecient; and c is
the fatigue ductilityexponent. The sound 8630 steel cyclic
properties listedin Table III were obtained in Reference 40 by
tting testdata to Eqs. [1] and [2]. For sound 8630 steel, the
fatiguestrength (or fatigue limit), Sf, at 5 9 10
6 cycles is equalto 293 MPa.The measured fatigue lives (as
cycles to failure, Nf) of
the porous specimens are plotted in Figure 5 as afunction of the
applied stress amplitude, DS/2. InFigure 5(a), the measurements are
compared to thestress-life curve for sound 8630 steel. The sound
stress-life curve was obtained from Eq. [2] by converting thestrain
to stress using the elastic modulus from Table II;plasticity was
found to be negligibly small.[19] It can beseen that the fatigue
lives of the porous specimens fallfar below the sound material
curve. Depending on thenumber of cycles to failure, the stress
amplitudes in thepresent tests are between approximately 170 and900
MPa below the stress amplitudes for sound 8630
Table I. Summary of Measurements for 25 Cast Steel Specimens
Containing Porosity:[1,19] Test Stress Level, Porosity
fromRadiographic Analysis, Strain, Elastic Modulus, and Fatigue
Life
SpecimenNumber
AppliedStress Levelof Test(MPa)
AveragePorosity
Volume Fractionin Gage Section
MaximumCross-Sectional
PorosityFraction Strain 9104
ElasticModulus(GPa)
FatigueLife
(Cycles)
1 96 0.104 0.432 7.01 137 13652 96 0.101 0.208 6.44 149 79,9083
126 0.101 0.256 8.81 143 24,3204 53 0.128 0.297 3.84 138 851,2755
126 0.097 0.185 8.24 153 29,0236 66 0.134 0.275 4.55 145 216,5167
66 0.095 0.201 4.68 141 4,053,8008 66 0.185 0.363 4.89 135 57,5669
53 0.146 0.587 6.09 87 10,81210 126 0.213 0.326 9.33 135 37,08911
66 0.117 0.250 4.85 136 113,50312 66 0.148 0.415 5.84 113 15,41913
96 0.185 0.538 12.47 77 604214 126 0.139 0.507 10.50 120 16015 53
0.187 0.551 6.09 87 15,86816 66 0.076 0.160 3.98 166 1,681,01817 96
0.096 0.254 8.65 111 439218 53 0.085 0.148 3.66 145 1,342,21819 126
0.093 0.225 8.87 142 13,01320 53 0.099 0.182 3.71 143 249,75221 96
0.122 0.300 7.68 125 41,06622 66 0.117 0.262 4.37 151 769,07423 126
0.121 0.205 8.13 155 40,89624 96 0.144 0.288 6.76 142 333,02525 126
0.129 0.280 8.51 148 7456
Table II. 8630 Steel Monotonic Properties[40]
Property Sound Material
Su (MPa) 1,144Sy (MPa) 985E0 (GPa) 207Pct EL not measuredPct RA
29rf (MPa) 1,268ef 0.35K (MPa) not measuredN not measured
Note: EL = elongation, RA = reduction in area.
Table III. 8630 Steel Cyclic Properties[40]
Property Sound Material
Sf (MPa) 293Sf/Su 0.26K (MPa) 2,267n 0.195S0y (MPa) 661b 0.121c
0.693r0f (MPa) 1,936e0f 0.42
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steel. If the specimens had been sound, they would allhave a
life greater than the runout value (>5 9 106
cycles). The porous specimen data is examined in moredetail in
Figure 5(b). For all applied stress levels, largevariations in the
measured fatigue lives can be observed.The lives of the porous
specimens range from 160 cyclesto a runout. Clearly, the fatigue
behavior of thesespecimens is primarily controlled by porosity.
However,the variation in the measured fatigue life of more thanfour
orders of magnitude cannot be explained by thecomparably small
dierences in the average porosity ofthe specimens (between 8 and 21
pct (Table I)). Theseresults demonstrate that each specimen has a
uniqueporosity eld; this, in combination with the loading,results
in a complex stress eld that must be accuratelysimulated in order
to predict the fatigue life of a givenspecimen.
III. FATIGUE LIFE PREDICTION
A. Simulation Procedure
The present procedure for predicting the fatigue life ofthe
porous specimens can be summarized as follows.
(a) Map the measured porosity eld from the tomo-graphy onto the
nodes of the FEA mesh.
(b) Degrade the elastic properties at each node accord-ing to
the local porosity fraction, /.
(c) Perform a nite element elastic stress analysis
thatcorresponds to the loading in the fatigue tests.
(d) Import the predicted stress elds correspondingto tension and
compression into the life predic-tion software and perform a
multiaxial strain-lifeanalysis.
The rst three steps in this procedure were developedpreviously
by the present authors; all details can befound in Reference 1. In
brief, the three-dimensionalquadratic interpolation subroutine
QD3VL from theInternational Mathematics and Statistics
Library(IMSL)[41] is used to map the porosity data onto theFEA
mesh. Ten-node quadratic tetrahedral elements areused to perform
the stress analysis. Elastic mechanicalproperties are assigned at
each node as a function of theporosity fraction at that location.
The local elasticmodulus is calculated from[1]
E / E0 1 /0:5
2:53
where E0 is the elastic modulus of the sound materialand / is
the porosity volume fraction. The Poissonratio, m, as a function of
/ is obtained from a relation-ship developed by Roberts and
Garboczi:[42]
m / mS //1m1 mS 4
with m = 0.14, / = 0.472, and mS = 0.3. Signicantplasticity was
not detected during testing of the speci-mens; therefore, plastic
eects are ignored in the FEAsimulations.[19] Mechanical simulation
of the specimenswith porosity is performed using ten-node
quadratictetrahedral stress/displacement elements (type C3D10)with
the commercial FEA package Abaqus/Standard(version 6.6.1).[43] The
FEA simulation boundary con-ditions are chosen to closely match the
test conditions.During testing, the specimens were held xed at
theirupper grip, and the loading was applied to the lower(ram) end,
which was free to move vertically, placing thespecimen in tension
and compression during fatiguetesting. The simulated specimen
geometry considersonly the initial 5 mm of the length of the grips
in theFEA mesh. This is a distance away from the llet andtest
section that is more than sucient for producingstress-strain
results that are insensitive to the locationsat which the boundary
and loading conditions areapplied. Shorter execution times are the
only dierencefrom simulations using more of the grip length. At
theupper grip face, a clamped boundary condition isapplied (having
no translations or rotations); at thelower grip, translations are
allowed only in the axialdirection, with no axial rotation allowed.
A uniformdistributed loading is applied over the face at the
lowergrip end, to produce the total load corresponding to
thetesting conditions. Using this procedure, it is shown
inReference 1 that the measured strains (Table I) arepredicted to
within approximately 10 pct. This goodagreement is achieved for an
FEA node spacing of
Runout at 5106 cycles
Fatigue Life, Nf (Cycles to Failure)
Runout at 5106 cycles
Fatigue Life, Nf (Cycles to Failure) 101 102 103 104 105 106
107
101 102 103 104 105 106 107
900Sound keel block data [19]Test Data
800
700
600
500
400
300
200
100
0
0
20
40
60
Stre
ss A
mpl
itude
(MPa
)St
ress
Am
plitu
de (M
Pa)
80
100
120
140
160
180
200
(a)
(b)
Fig. 5(a) Fatigue life measurements of specimens with
porositycompared with sound data from Ref. 40. (b) Measured fatigue
livesof specimens with porosity tested at four stress levels: 126,
96, 66,and 53 MPa. Note the single runout specimen.
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
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0.25 mm, but the results are relatively independent ofthe mesh
for node spacings up to 2 mm.[1] These resultsshow that, by using
porosity fractions that are denedover a volume that is small
compared to the (macro-scopic) specimen geometry but large compared
to the(microscopic) pore geometry, and by degrading theelastic
properties locally in accordance with the porosityfraction, the
overall stiness response of the specimenscan be simulated
accurately.After running ABAQUS, the resulting stress elds
corresponding to the tension and compression steps ofthe fully
reversed loading (R = 1) are imported intothe fatigue life
prediction software. In the present study,the software package
fe-safe[13] is used for the fatigue lifecalculations. The loading
cycle and the fatigue proper-ties for the sound material (Table
III) are entered intothe fe-safe model. The nodal stress tensors
from theAbaqus simulations are converted to strains within
thefe-safe software, using a sound material elastic modulusof 207
GPa (Table II). As long as this elastic modulus isused throughout
the fatigue calculations, the use of avariable E is not necessary
as the alternating strainamplitude and the elastic term in the
strain-life equationscale with whatever modulus is used. In the
presentstudy, fe-safes multiaxial BrownMiller algorithm withthe
Morrow mean stress correction is used to calculatethe fatigue life
as cycles to failure. For steel, fe-saferecommends using the
multiaxial BrownMiller algo-rithm for ductile steel and the
principle strain algorithmfor brittle steels, both with the Morrow
mean stresscorrection.[13] The BrownMiller algorithm is said to bea
more conservative method; it uses a critical planeanalysis to
determine the life in reversals to failure, 2Nf,by solving
Dcmax2
Den2
1:65 r0f rmE
2Nf b 1:75e0f 2Nf c 5
at each node, where Dcmax/2 is the maximum shearstrain
amplitude, Den/2 is the strain amplitude normalto the shear stress
plane, and rm is the mean stress. Thecritical plane is dened as the
plane having themaximum value of Dcmax/2+Den/2. In the
criticalplane analysis, the calculated strain tensor at a
niteelement node (having three direct and three shearcomponents) is
resolved onto a number of planes wherethe damage associated with
the strain is evaluated oneach plane. The plane with the most
damage is thenselected for use in the strain-life calculations. In
aCartesian x-y-z coordinate system, unique planes can bedened by
the orientation the normal of the planesurface makes with respect
to the coordinate system.This orientation can be dened by one angle
from thex-axis toward the y-axis, and a second angle from thez-axis
toward the x-y plane.[13] The software fe-safesearches for the
critical plane with the worst damage(shortest life) in 10 deg
increments over the 180 degrange of the rst angle and the 90 deg
range of thesecond angle. Direction cosines are used to project
thestrains onto the calculation plane. Additional detailsabout
multiaxial fatigue analysis can be found in thebook by Socie and
Marquis.[7]
B. Results of Fatigue Life Simulations Using
MultiaxialStrain-Life Method
Figure 6 shows a series of simulation results forspecimen 19 on
a longitudinal center section, startingwith the measured porosity
eld mapped onto the FEAmesh and followed by the predicted maximum
principlestress and strain elds and, nally, the predicted cyclesto
failure eld. The latter is also shown on the surface ofthe
specimen. In these baseline simulations, a nodespacing of 0.25 mm
is used. As mentioned previously,this grid of approximately 20
nodes across the specimendiameter was found to give good results
for the overallstiness.[1] It can be seen that the porosity causes
acomplex three-dimensional stress eld to develop in thespecimen
during loading. Larger stresses are observedadjacent to regions
with a high porosity fraction. On theother hand, large strains
coincide with high porosityfractions. The shortest lives are
predicted in regions ofhigh stress concentration. For this
specimen, the node
Fig. 6Results in a longitudinal section for specimen 19. (a)
Poros-ity distribution from tomography. Results from Abaqus stress
analy-sis for (b) maximum principal stress and (c) maximum
principalstrain. Fatigue life prediction from fe-safe using Abaqus
results (d)in the section and (e) on the surface with minimum
predicted life of15,847 cycles indicated.
588VOLUME 40A, MARCH 2009 METALLURGICAL AND MATERIALS
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with the smallest predicted number of cycles to failure
islocated on the surface of the specimen. In the following,the
fatigue life prediction for a specimen is always takenas the
shortest life resulting at any node in the fe-safecalculations.The
measured and predicted fatigue lives for all 25
porous specimens are compared in Figure 7. A line ofperfect
correspondence is provided in the gure, to helpdetermine whether a
prediction is conservative (belowthe line) or nonconservative
(above the line). It can beseen that 18 of the predicted fatigue
lives are within afactor of 10 of the test results, which can be
regarded asgood agreement. Of these, only ve are conservative.Seven
data points are in poor agreement with themeasurements (i.e., by
more than a factor of 10) and arenonconservative, and two of these
have predicted livesthat are approximately three orders of
magnitude longerthan the measurements. While these results are
generallyencouraging, the overall nonconservative nature of
thepredictions warrants further investigation.It is well known that
fatigue life predictions are very
sensitive to local stress concentrations. This issue is
ofparticular importance in the present study, because
thesimulations rely on pore fractions dened over avolume that is
large compared to the microscopic poregeometry. It is unlikely that
a node spacing of 0.25 mmresolves the local stress concentrations
around smallshrinkage pores, which can result in
nonconservativefatigue life predictions if the pores are located in
ahighly stressed region. Therefore, a mesh sensitivitystudy was
performed. Specimen 19 was simulated usingthree FEA meshes that are
ner than the baseline grid(0.25 mm node spacing). The three ner
meshes have
node spacings of 0.16, 0.12, and 0.09 mm. Figure 8shows the
predicted stress elds on the surface ofspecimen 19, for all four
meshes. The predictedmaximum stress increases from 400 MPa for
thebaseline mesh to 467, 488, and 602 MPa for the threener meshes.
Note that the applied stress amplitude forspecimen 19 is 126 MPa.
Obviously, higher maximumstresses result in lower fatigue lives.
The predictedfatigue lives for the four meshes are indicated
assquares in Figure 7. The fatigue life predictions varyfrom
approximately 160,000 cycles for the coarsestmesh to 8700 for the
nest mesh. For specimen 19, theresult for the nest mesh is in good
agreement with themeasured fatigue life of 13,013 cycles to
failure. Asimilar mesh renement study was performed forspecimen 20,
where the baseline mesh gives a fatiguelife prediction that is
three orders of magnitude abovethe measured life. In addition to
the four meshes usedfor specimen 19, simulations were also
performed withtwo meshes that are coarser than the baseline
mesh(node spacings of 0.42 and 0.58 mm). The predictedfatigue lives
for specimen 20, shown in Figure 7 assquares, strongly decrease
with increasing mesh ne-ness. The coarsest mesh yields a fatigue
life predictionof approximately 3.4 9 109 cycles to failure. The
threenest meshes give approximately the same fatigue
life,indicating that the predictions converge to a constantvalue
for a suciently ne mesh. However, theconverged value of
approximately 4 9 106 cycles tofailure does not agree well with the
measured life of250,000 cycles for specimen 20. This disagreement
maybe related to the limited sensitivity of the lm radio-graphs
from which the porosity eld was reconstructed,as is investigated in
more detail in Section IV. Becausethe resolution of the radiography
was approximately100 lm, a node spacing that is smaller than
thisvalue does not change or improve the fatigue
lifepredictions.Mesh renement studies conducted for other
speci-
mens (such as shown in Figure 7 for specimen 17)showed the same
trend. Generally, for those specimensfor which the prediction for a
coarse mesh is alreadyclose to the measurement, further mesh
renement doesnot decrease the predicted fatigue life substantially.
Forthose specimens for which the baseline prediction isvastly
nonconservative (e.g., specimen 20), the meshrenement has a much
stronger eect. These dierencescan be attributed to the nature of
the porosity distribu-tion in the specimens. Computational meshes
with nodespacing on the order of 100 lm are generally notpractical
in the structural analysis of entire metalcastings. Even for the
relatively small specimens of thepresent study, such a ne mesh
pushes the limits ofcurrent computational capabilities.
Furthermore, itappears from the mesh renement study for specimen20
that porosity features smaller than 100 lm can playan important
role in the fatigue behavior of thespecimens. Therefore, in the
following, an approximatesimulation method is devised that reduces
the strongmesh dependence of the results shown in Figure 7
andpredicts the measured fatigue lives more closely, even fora
relatively coarse mesh.
Arrows indicates direction ofincreasing mesh fineness
Specimen 20
Specimen 19
Specimen 17
Measured Fatigue Life, Nf (Cycles to Failure) 102
102
103
104
105
106
107
108
109
1010
103 104 105 106 107
Pred
icte
d Fa
tigue
Life
, Nf (C
ycles
to Fa
ilure)
Fig. 7Comparison between measured and predicted fatigue lives
ofspecimens with additional runs made for specimen 19 at three
nergrids, and at two coarser and three ner nite element mesh
gridsfor specimens 17 and 20.
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
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IV. ADAPTIVE SUBGRID MODELFOR FATIGUE LIFE PREDICTION
The adaptive subgrid model developed in the presentstudy is
based on the idea of using a local fatigue notchfactor at each node
in the fatigue life calculations. Thisnotch factor is intended to
account for the eect ofporosity that is not at all or only
partially resolved in the
simulations. The lack of resolution can stem from eithera nite
element mesh that is too coarse or a porosity thatis too small to
be detected by standard lm radiography(such porosity is referred to
as microporosity in thediscussion that follows). The use of a local
fatigue notchfactor is dierent from the common practice of
applyingthe same notch factor to an entire part or the entire
Fig. 8Predicted axial stress distribution on surface for four
grids used to predict the fatigue life of specimen 19 shown in Fig.
7. Finer gridsgive higher stresses from more detailed porosity
eld.
590VOLUME 40A, MARCH 2009 METALLURGICAL AND MATERIALS
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surface of a part.[3,13] In the present model, a fatiguenotch
factor is only applied at those locations of the partat which the
stress concentrations due to porosity arenot fully resolved. The
magnitude of the fatigue notchfactor is calculated as a function of
the local nodespacing and the local pore size. If the node spacing
issmall enough that the stress concentration around apore of a
certain size is fully resolved, the local fatiguenotch factor would
be equal to unity. If the pore is sosmall relative to the node
spacing that the FEA does notpredict any stress concentration, the
local fatigue notchfactor would be equal to some maximum value
thatcorresponds to that pore. This aspect is what makes thepresent
subgrid model adaptive. The development of theadaptive subgrid
model involves numerous consider-ations that are discussed
separately in Sections Athrough E.
A. Fatigue Notch Factor
A fatigue notch factor, Kf, is commonly used infatigue life
calculations to account for the eects ofnotches or
discontinuities.[3] The fatigue notch factorrelates the local notch
root stress and strain amplitudesto the nominal true stress and
strain amplitudes througha relationship known as Neubers rule;
i.e.,
De2
Dr2 K2f
De2
DS2
6
where Dr/2 and De/2 are the local axial stress andstrain
amplitudes at the notch root, respectively, andDS/2 and De/2 are
the nominal true stress and strainamplitudes, respectively. The
multiaxial version ofNeubers rule uses the stresses and strains in
the criti-cal plane, as explained previously in connection withthe
BrownMiller algorithm. Equation [6] is solvedtogether with the
cyclic stress-strain curve for thenotch stress and strain
amplitudes; i.e.,
De2 Dr2E
Dr2K0
1n0
7
The notch strain amplitude resulting from the simulta-neous
solution of Eqs. [6] and [7] is then used in thestrain-life
equation (i.e., Eq. [5]) to calculate the fati-gue life. The
fatigue notch factor, Kf, in Eq. [6] isobtained from[3]
Kf 1 Kt 11 a=r 8
where Kt is the stress concentration factor and r is thenotch
root radius in millimeters. The material constanta is obtained from
the following relation originallydeveloped for wrought
steel:[44]
a 0:0254 2070Su
1:89
where Su is the ultimate strength in MPa (Table II). Forr a,
which for Su = 1144 MPa is approximately thecase for r> 1 mm,
Eq. [8] yields that Kf = Kt and thematerial is said to be fully
notch sensitive.[3]
The use of the fatigue notch factor concept outlinedhere in a
strain-life calculation results in accuratepredictions of the eect
of microporosity on fatigue ofcast steel.[19] In Reference 19,
additional fatigue tests arepresented with cast steel specimens
that have onlymicroporosity. Microporosity is generally thought of
asporosity that is not detected in standard lm radiogra-phy. In the
context of the present measurements, it isdened as porosity with a
pore volume fraction less thanapproximately 1 pct and with pore
radii no larger thanapproximately 100 lm. Such microporosity should
beopposed to the macroporosity in the present specimens(Table I).
In the microporosity fatigue life calculationsof Reference 19, the
notch root radius, r, was takenequal to the radius of the pore, Rp,
found on the fracturesurface to be responsible for failure, i.e., r
= Rp. Thestress concentration factor, Kt, was taken as a
rstapproximation to be equal to 2.045, which is the
valuecorresponding to a single spherical hole in an
essentiallyinnite body.[45]
Because the fatigue notch factor model discussed herewas found
to work well for microporosity, it forms thebasis for the present
subgrid model. As is shown inSection IVB, pores with a radius less
than 100 lm arenot at all resolved for nite element node
spacingsgreater than approximately 0.2 mm (in fact, nodespacings
less than 10 lm would be necessary to correctlycalculate the stress
concentration for a 100-lm radiuspore). For such pores and meshes,
the full fatigue notchfactor as given by Eq. [8] should thus be
applied in thefatigue life calculations. For fully unresolved
pores, as inthe limiting case of microporosity, the simulations
canthen be expected to yield the correct fatigue life.
B. Adaptive Stress Concentration Factor
As discussed in Section A, when performing a niteelement stress
analysis, and the mesh is so coarse relativeto the pore size that
the predicted stresses are notaected by the presence of the pore
(as would beexpected for microporosity), the stresses at the
locationof the pore should be enhanced by a factor that gives
thecorrect stress concentration corresponding to the pore.This
factor is referred to as a stress concentration factor,Kt, and
values for holes of various shapes and sizes canbe found in
handbooks (e.g., Reference 45). However, ifthe node spacing is of
the same order as or smaller thanthe pore size, a nite element
stress analysis wouldpredict some or all of the enhanced stresses
due to thepresence of the pore. In that case, the factor used
toenhance the stresses should be less than the full
stressconcentration factor from a handbook or even equal tounity.
Thus, the objective of this section is to determinea relation for
an adaptive stress concentration factor,Kt,a, that is a function of
the node spacing, Ln, relative tothe pore size. For simplicity,
pores are approximated assingle spherical holes of a certain
eective radius, Rp, inan essentially innite body, so that the
maximum valuefor Kt,a is equal to 2.045.
[45]
A nite element elastic stress analysis was performedof a single
spherical hole with Rp = 0.1 mm inside of a5-mm-diameter axially
loaded cylinder made of 8630
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
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steel. This cylinder diameter is large enough that thepredicted
stresses around the hole are not inuenced bythe boundaries. The
hole is modeled as a spherical eldof nodes having 100 pct porosity
(/ = 1); the hole isnot modeled as a feature in the nite element
mesh.The elastic properties at each node are calculated fromEqs.
[3] and [4], such that the elastic modulus, E, isessentially equal
to zero at the nodes corresponding tothe hole. The applied load on
the cylinder ends is takento be 207.4 MPa. Because Kt = 2.045 for a
singlespherical hole,[45] the maximum stress at the hole shouldbe
equal to 424 MPa. Figure 9 shows the predictedstress elds at a
midsection cut of the hole for threedierent node spacings: 0.025,
0.1, and 0.2 mm. For thesmallest node spacing, the stresses around
the hole arewell resolved and the predicted maximum stress is
equalto 411 MPa. This stress is very close to the handbook
value of 424 MPa, so that Kt,a 1. For the interme-diate node
spacing, the hole is poorly resolved and themaximum stress from the
FEA is equal to only264 MPa. Hence, the corresponding adaptive
stressconcentration factor is Kt,a = 424/264 = 1.606. Forthe
coarsest mesh, only a single node is present withinthe hole and the
predicted maximum stress from theFEA of 219 MPa is close to the
nominal applied stressof 207.4 MPa; therefore, Kt,a = 424/219 =
1.94.These results can be generalized by realizing that, for
a single hole in an innite body, the full stressconcentration
factor is independent of the radius ofthe hole. Figure 10 shows the
computed adaptive stressconcentration factor, Kt,a, as a function
of the nodespacing to the pore radius ratio, Ln/Rp. It was
veriedthat the same result is obtained for hole radii other than0.1
mm. It can be seen from Figure 10 that, forLn/Rp< 0.08, the
adaptive stress concentration factoris equal to unity, implying
that the stresses around thehole are fully resolved. For Ln/Rp>
4, the mesh is socoarse that the full stress concentration factor
of 2.045must be applied. For use in the present simulations,
thefollowing curve was t through the computed datapoints
Kt;a 2:045 1 2:045 exp0:08 Ln
Rp
1:286
10
for Ln/Rp> 0.08. Equation [10] then provides the
stressconcentration factor for use in Eq. [8]. For discontinu-ities
other than a single spherical hole, a value that isdierent from
2.045 could be used in Eq. [10]. Becausethe use of Kt = 2.045
yields accurate fatigue life predic-tions for the microporosity
specimens in Reference 19,this value is kept here.
C. Implementation in fe-safe
The software fe-safe only allows for the application ofa
constant fatigue notch factor to the entire surface of
Fig. 9Predicted maximum principal stress distribution for a100
lm radius spherical hole in a 5 mm diameter cylinder under aloading
of 207 MPa, with increasing mesh neness from (a) through(c). Node
spacing and maximum stress for each case provided.
1
1.2
1.4
1.6
1.8
2
2.2
0 1 2 3 4 5 6Node Spacing/Hole Radius, L n /R p
100 m radius hole
Ada
ptiv
e St
ress
Con
cent
ratio
n Fa
ctor
, Kt,a
Curve fit: Equation (11)
Fig. 10Adaptive stress concentration factor Kt,a required
toachieve correct fatigue life at ve node spacing to hole radius
ratiosfor a 100 lm radius spherical hole modeled using a porosity
eld.
592VOLUME 40A, MARCH 2009 METALLURGICAL AND MATERIALS
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the part or the entire part. However, the use of thepresent
adaptive subgrid model necessitates the use of afatigue notch
factor that varies from node to node. Thisissue can be resolved by
making use of the fact thatfe-safe allows for temperature dependent
fatigue prop-erties (r0f; e
0f; b, and c) in the strain-life equation, Eq. [5].
That feature is utilized in the present implementation ofthe
adaptive subgrid model, by importing the calculatedfatigue notch
factors at each node as a temperature eldinto fe-safe, and
evaluating the fatigue properties as afunction of the fatigue notch
factor. The functionaldependence between the fatigue properties and
thefatigue notch factor must be such that the strain-lifeequation
gives the same life with the modied(Kf dependent) fatigue
properties as with the true fatigue
properties but the strains being the notch strains fromNeubers
rule, Eq. [6].To determine the fatigue properties as functions
of
Kf, software was developed that minimizes the errorbetween a
strain-life curve generated for a given Kfvalue and the true
fatigue properties from Table III,and a strain-life curve with Kf =
1 but modied fatigueproperties. For a given Kf, the fatigue
properties r0f, e
0f,
b and c are iterated upon and solved for until the
twostrain-life curves match. The resulting dependencies ofe0f, b,
and c on Kf are plotted in Figures 11(a), (b), and(c),
respectively. The present minimization proce-dure revealed that r0f
should not be made a functionof Kf (and thus be kept at the true
value given inTable III).
D. Pore Size Model
The sole remaining unknown in the adaptive subgridmodel is the
pore radius, Rp. It is needed in Eq. [8] forthe fatigue notch
factor and in Eq. [10] for the adaptivestress concentration factor.
Porosity is always charac-terized by a distribution of pore sizes,
rather than asingle pore radius, in the volume element over
whichthe porosity volume fraction is dened (e.g., References27
through 29). For the purpose of fatigue lifecalculations it is
important to know the radius of thelargest pore in the
distribution, because it is that porethat initiates
failure.[19,24,2729,34] Thus, the present poresize model is
concerned primarily with predicting themaximum pore radius, Rp,max.
In order to keep themodel relatively simple, the maximum pore
radius isassumed to be dependent on the pore volume fraction,/,
only. As explained here, results from both aprobabilistic model of
pore growth and merging andfrom experimental measurements are used
to developthe pore size model.The following example illustrates the
importance of
distinguishing between the mean and maximum poreradii. In
Reference 19, the present authors andco-workers performed image
analysis on polished metal-lographic sections of 8630 cast steel
specimens havingmicroporosity. The average pore volume fraction
wasfound to be / = 0.7 pct, and the mean distancebetween the pores,
Lp, was observed to range from 177to 344 lm. The pore number
density, n (i.e., the numberof pores per unit volume) can then be
estimated from therelation n L3p to be between 2.4 9 1010 and1.8 9
1011 pores/m3. Assuming the pores are sphericaland have a uniform
radius, the pore volume fraction isgiven by
/ n 4p3R3p 11
Using these values for / and n, a mean pore radius ofRp = 21 to
41 lm is obtained from Eq. [11], which wasveried by the image
analysis. However, images of thefracture surface of the
microporosity specimens revealedthat failure always initiated from
much larger, isolatedpores with a radius on the order of 100
lm.[19] Thisvalue corresponds to the radius of the largest
pore.
-0.22
-0.20
-0.18
-0.16
-0.14
-0.12
Fatigue Notch Factor
0.050.100.150.200.250.300.350.400.45
Fatigue Notch Factor
-0.78
-0.76
-0.74
-0.72
-0.70
-0.68
1.0 1.5 2.0 2.5
1.0 1.5 2.0 2.5
1.0 1.5 2.0 2.5Fatigue Notch Factor
(a)
(b)
(c)
Fatig
ue D
uctil
ity C
oeffi
cien
t f
Fatig
ue D
uctil
ity E
xpon
ent c
Fa
tigue
Stre
ngth
Exp
onen
t b
Fig. 11Fatigue properties as functions of notch factor for (a)
fati-gue strength exponent b, (b) fatigue ductility coecient ef,
and (c)fatigue ductility exponent c.
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
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Taking Rp = 100 lm and n as estimated earlier, Eq. [11]yields
pore volume fractions, /, ranging from 10 to75 pct, which is
clearly unrealistic. Thus, Eq. [11] is notvalid for the maximum
pore radius.As part of the present study, a probabilistic pore
size
distribution model was developed to better understandthe
phenomena of pore growth and merging and therelationship between
the maximum and mean pore radii.Due to space limitations, it is
only briey described here.The pores are assumed to nucleate
instantaneously witha certain initial volume fraction and number
density anda lognormal size distribution. Based on the
micropo-rosity measurements mentioned earlier, the shape ofthe
initial size distribution is adjusted such that for/ = 0.7 pct and
n = 9.97 9 1010 m3, the mean poreradius, Rp, is equal to 26 lm and
the maximum poreradius, Rp,max, in the distribution is 100 lm.
Eachpore in the distribution is assumed to grow at a
rateproportional to the square of its radius. In addition, apore
merging model is implemented. Pores are assumedto be capable of
merging if their radius exceeds half ofthe mean spacing between the
pores. The frequency ofpore merging is determined by the
probability that twopores will take part in a merging event. This
probabilityis dependent on the product of the number density
ratiosof the merging pores (either within a bin of a single
poresize meeting the merging criterion or between two binsof
dierent pore sizes). The number density ratio is thenumber density
of one pore size bin divided by the totalnumber density of pores.
Merged pores continue to formand grow along with the distribution.
Due to merging,the total number density of pores continually
decreasesas the mean pore radius and total porosity volumefraction
increase.Representative calculated pore size distributions are
shown in Figure 12. In Figure 12(a), the curve corre-sponding to
/ = 0.1 pct represents the initial pore sizedistribution. With an
increasing pore fraction, themaximum of the distribution shifts to
higher pore radiiand the distribution becomes narrower. At
approxi-mately / = 3 pct, signicant pore merging is starting
tooccur. This can be seen in Figure 12(a) by the area underthe
distribution decreasing and, hence, the total porenumber density
decreasing. In addition, the pore sizedistributions develop a tail
at pore radii that are muchlarger than the mean. This can be better
seen inFigure 12(b), where the predicted pore size distributionfor
/ = 10 pct is shown in a log-log plot. At 10 pct, thetotal number
density, mean pore radius, and maximumpore radius are approximately
given by n = 1.17 9106 m3, Rp = 0.73 mm, and Rp,max = 2.61
mm,respectively. The maximum pore radius is dicult todetermine from
the distribution because of its long tail,but for / = 10 pct, the
value of 2.61 mm correspondsto the mean of the merged pores (Figure
12(b)). Thismaximum pore radius agrees with the measurements ofASTM
radiographs for shrinkage porosity in steelcastings presented in
Reference 46.For use in the present adaptive subgrid model, the
maximum pore radius results from the probabilistic poresize
distribution model were t to the following piece-wise function:
Rp;max 2825:2/1:0 25; 846:63/ 275; 154:35/2for/ 0:0386
Rp;max 1:76888 1:66246 exp 21; 223:91/3:98865
for 0:0386
-
provide a rst-order correction to the strong meshdependence of
the fatigue life simulations, the maximumpore radius model can be
considered suciently accu-rate. In summary, the function for Rp,max
given here isused for both the notch root radius, r, in Eq. [8] for
thefatigue notch factor, and the pore radius, Rp, in Eq. [10]for
the adaptive stress concentration factor.
E. Results Using the Adaptive Subgrid Model
The adaptive subgrid fatigue life model is rst testedfor
specimen 20. As previously shown in Figure 7, theoriginal fatigue
life predictions for specimen 20, withoutthe subgrid model, are
highly dependent on the nodespacing. A comparison of those results
with the corre-sponding predictions in which the adaptive
subgridmodel is employed in the simulations is shown inFigure 14.
For the three nest meshes (0.16, 0.12, and0.09 mm node spacing),
the fatigue life predictions arethe same with and without the
subgrid model; further-more, the results are independent of the
node spacingand can thus be considered converged. The
convergedvalue for the predicted fatigue life is ~4 9 106 cycles
tofailure. The agreement illustrates that the presentadaptive
subgrid model correctly reduces to a standardstress/fatigue life
simulation for a suciently ne niteelement mesh. In other words, the
introduction of thefatigue notch factor in the adaptive subgrid
model doesnot reduce the predicted fatigue life to a value below
theone for a fully resolved FEA calculation. For the threecoarser
meshes (0.25, 0.42, and 0.58 mm node spacing),the fatigue life
predictions are substantially above theconverged result of
approximately 4 9 106 cycles tofailure. However, the fatigue lives
obtained with theadaptive subgrid model are consistently closer to
theconverged value and are more mesh independent. Forthe coarsest
mesh, the adaptive subgrid model has thestrongest benecial eect and
reduces the fatigue lifepredicted without the subgrid model by
almost twoorders of magnitude. While these results for specimen
20are encouraging and indicate that the adaptive subgridmodel is
working as intended, it can be seen fromFigure 14 that, for the
three coarser meshes, the fatigue
lives predicted with the subgrid model are still up to 1order of
magnitude larger than the converged value.This nding suggests that
a larger maximum stressconcentration factor in Eq. [10], instead of
the 2.045value, would produce even better results. While that
iscertainly true, the use of a larger maximum stressconcentration
factor in the adaptive subgrid model isnot desirable, because the
value of 2.045 gave accuratefatigue life predictions for the
microporosity specimensin Reference 19. Furthermore, note that even
the nestmesh results are still considerably above the
measuredfatigue life for specimen 20 (~250,000 cycles to
failure(Table I)); thus, factors other than the mesh resolutionare
causing disagreement between the measured andpredicted fatigue
lives.Figure 15 shows the eect of the adaptive subgrid
model on the fatigue life predictions for all 25 spec-imens.
These simulations were performed with a nodespacing of 0.25 mm. The
vertical lines are used toindicate in the gure the movement of the
predictionsdue to the use of the subgrid model. Relative to
themeasurements, the predictions are much improved,especially for
the highly nonconservative data points.It is interesting that those
predictions that were alreadyclose to the measurements are not
aected by thesubgrid model and the good agreement remains. This
isespecially true for the lower lived specimens with fatiguelives
of less than ~10,000 cycles to failure. For thosespecimens, failure
is controlled by large porosity fea-tures and the stress eld is
already adequately resolvedwith a node spacing of 0.25 mm.Although
the adaptive subgrid model results in a
considerable improvement in the fatigue life predictions,there
are still six data points in Figure 15 for which theprediction is
more than an order of magnitude above themeasurement. This could be
the result of the porosityeld not being suciently resolved in the
X-ray tomo-graphy. In particular, the radiography does not
detectmicroporosity, because the resolution is limited by
theradiographic lm sensitivity to 100 lm, as mentioned
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Max
imum
Por
e Ra
dius
, Rp,
max
(m
m)
Pore Volume Fraction, (-)
Start of significant pore merging
1 mm
1 mm
0.7% Porosity
7% Porosity
10% Porosity
20% Porosity
1 mm
Fig. 13Relationship for maximum pore size vs volume
fractioncompared with observed porosity in specimens (5 mm
diameter).
0.0 0.1 0.2 0.3 0.4 0.5 0.6Node Spacing (mm)
Without adaptive sub-grid model
With adaptive sub-grid model
1010
109
108
107
106
105Pred
icte
d Fa
tigue
Life
,Nf(C
ycles
to Fa
ilure)
Fig. 14Eect of nite element model node spacing on
predictedfatigue life from fe-safe for specimen 20. Original method
withoutsubgrid fatigue model and with subgrid model with Kf,max =
2.045.
METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
2009595
-
earlier. Microporosity was always observed on metallo-graphic
sections, such as those shown in Figure 2, inregions next to larger
pores. This can be important in afatigue life calculation, because
high stress concentra-tions occur next to large pores. In order to
investigatethe sensitivity of the predictions to the presence
ofmicroporosity, simulations were performed with auniform
background microporosity having a maxi-mum pore radius, Rp,max, of
100 lm. This is easilyaccomplished in the present subgrid model by
modifyingthe pore size model, Eq. [12], such that all values
ofRp,max that are less than 100 lm are overwritten with100 lm
(which only occurs for / less than 0.1 pct). Theporosity eld used
in the nite element stress analysis ofthe specimens was not
changed, because microporositycorresponds to very small pore
fractions with a negli-gible eect on the elastic properties.The
eect of the background microporosity on the
fatigue life predictions for all 25 specimens is shown inFigure
16. The simulation results in this gure wereobtained using the
adaptive subgrid model and a nodespacing of 0.25 mm. Here, the
vertical lines indicate themovement of the predictions due to the
addition of thebackground microporosity. As expected, the addition
ofthe background microporosity lowers the fatigue lifepredictions
for all specimens. However, the magnitudeof the eect is not the
same for all specimens. For somespecimens, background microporosity
has a negligibleeect, while for others, the predicted fatigue life
changesby almost two orders of magnitude. These dierencescan be
attributed to the nature of the porosity eld in
the specimens and the resulting stress redistributions.Overall,
with the addition of the background micropo-rosity in the
simulations, the agreement between themeasured and predicted
fatigue lives can now beconsidered satisfactory.
V. CONCLUSIONS
A simulation method is developed for predicting theeect of
shrinkage porosity on the fatigue life of steelcastings. The method
is validated using previouslymeasured fatigue life data for 25 cast
steel test speci-mens containing up to 21 pct porosity in the
gagesection.[19] The simulation method relies on the knowl-edge of
the three-dimensional porosity eld. In thepresent study, the
porosity eld is obtained from X-raytomography with a resolution of
approximately100 lm.[1] After importing the porosity eld into
theFEA software, an elastic stress analysis of the fatiguetests is
conducted via the method developed in Reference1. Using this
method, the local elastic properties arereduced according to the
volume fraction of porositypresent at an FEA node. The computed
stresses areimported into fatigue analysis software to calculate
thefatigue life distribution in the specimens using the strain-life
equation for sound steel. It is found that themeasured fatigue
lives of several of the specimens testedare vastly overpredicted
when a nite element mesh isused that is too coarse to resolve all
of the porosity-induced local stress concentrations. For this
reason, anadaptive subgrid model is developed that employs alocal
fatigue notch factor to approximately account for
Without adaptive sub-grid model With adaptive sub-grid model
Measured Fatigue Life, Nf (Cycles to Failure)
Pred
icte
d Fa
tigue
Life
, Nf (C
ycles
to Fa
ilure)
102 103
103
102
104
105
106
107
108
109
104 105 106 107
Fig. 15Comparison between measured and predicted fatigue livesof
specimens for node spacing 0.25 mm for original method(no Kf) and
for the subgrid fatigue model with Kf,max = 2.045. Pre-diction uses
Abaqus simulated stress eld and subgrid fatigue modelwith fatigue
properties dependent on Kf, as shown in Fig. 11. Multi-axial
BrownMiller algorithm with Morrow mean stress correction isused in
fe-safe.
Pred
icte
d Fa
tigue
Life
, Nf (C
ycles
to Fa
ilure)
With adaptive sub-grid model, no background microporosity
With adaptive sub-grid model, with backgroundmicroporosity, Rp =
100 m
Measured Fatigue Life, Nf (Cycles to Failure) 102
102
103
104
105
106
107
108
109
103 104 105 106 107
Fig. 16Comparison between measured and predicted fatigue livesof
specimens for node spacing 0.25 mm for the subgrid fatiguemodel
with Kf,max = 2.045 (higher values), and the same model butusing a
uniform background pore size of 100 lm radius (even atnodes at
which tomography gives material as 100 pct sound).
596VOLUME 40A, MARCH 2009 METALLURGICAL AND MATERIALS
TRANSACTIONS A
-
the eect of under-resolved porosity. In this subgridmodel, an
adaptive stress concentration factor is calcu-lated that depends on
the size of a shrinkage porerelative to the node spacing of the
nite element mesh.The maximum pore radius is obtained from a
probabi-listic pore size distribution model as a function of
thelocal pore volume fraction. With the adaptive subgridmodel and
uniform background microporosity with amaximum pore radius of 100
lm, the measured andpredicted fatigue lives are found to be in good
overallagreement, even for a relatively coarse FEA mesh. Thepresent
simulation method is suciently general andexible that it can be
used for the fatigue analysis ofcomplex-shaped steel castings
containing porosity. Forsuch production steel castings, X-ray
tomographicreconstructions of the three-dimensional porosity eldare
generally not available. It is anticipated that theporosity eld can
instead be predicted using advancedcasting simulation
software.[47,48] A preliminary casestudy demonstrating such an
integrated simulationmethodology was presented in Reference 2.
ACKNOWLEDGMENTS
This research was undertaken through the AmericanMetalcasting
Consortium (AMC), which is sponsoredby the Defense Supply Center
Philadelphia (DSC)(Philadelphia, PA) and the Defense Logistics
Agency(DLA) (Ft. Belvoir, VA). This work was conductedunder the
auspices of the Steel Founders Society ofAmerica (SFSA) through
substantial in-kind supportand guidance from SFSA member foundries.
Anyopinions, ndings, conclusions, or recommendationsexpressed
herein are those of the authors and do notnecessarily reect the
views of DSC, DLA, or theSFSA or any of its members.
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METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 40A, MARCH
2009597
Outline
placeholderAbs1IntroductionExperimentsExperimentsExperimentsExperiments
Fatigue life predictionFatigue life predictionFatigue life
prediction
Adaptive subgrid model for fatigue life predictionAdaptive
subgrid model for fatigue life predictionAdaptive subgrid model for
fatigue life predictionAdaptive subgrid model for fatigue life
predictionAdaptive subgrid model for fatigue life
predictionAdaptive subgrid model for fatigue life prediction
ConclusionsConclusionsConclusions
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