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Effect of Porosity on Deformation, Damage,and Fracture of Cast Steel

R.A. HARDIN and C. BECKERMANN

A combined experimental and computational study is performed to investigate the effect ofcenterline shrinkage porosity on deformation, damage, and fracture of cast steel under tensiletesting. Steel plates containing shrinkage porosity are cast in sand molds, machined into testcoupons, and tensile tested to fracture. The average volumetric porosity in the gage section ofthe specimens with porosity ranges from 0.10 to 0.27 pct. Ductility in the test castings withporosity is markedly reduced with the percent elongation data ranging from 12.8 to 19.6 pct; vs22 pct elongation for the sound material. Radiographic imaging is used to measure andreconstruct the porosity field in the test specimens. The reconstructed porosity field is then usedin a finite-element stress analysis simulating the tensile testing. Local elastic properties arereduced according to the porosity fraction present. Porous metal plasticity theory is used tomodel the damage due to porosity and the fracture. Good agreement is obtained between themeasured and predicted stress–strain curves and fracture behaviors. The reduction in ductility ispredicted well by comparing the measured and the simulated elongations. The computationalmodeling approach used in this study allows for a detailed evaluation of the effect of porosity,including its size, shape, and location, on the fracture behavior of steel castings.

DOI: 10.1007/s11661-013-1669-z� The Minerals, Metals & Materials Society and ASM International 2013

I. INTRODUCTION

STEEL castings are under-utilized because of uncer-tainties in their performance and lack of expertise incasting design. Discontinuities in castings, like porosity,play an important role in casting underutilization.Porosity creates uncertainty in a design’s robustness,since there are no methodologies for including itspresence in the design. As a result, designers employoverly large safety factors to ensure reliability leading toheavier components than necessary. Contributing to theissue, the processes of designing and producing castingsare usually uncoupled except for the specification ofnondestructive evaluation (NDE) requirements. Unlessdesign engineers have test data or experience for a part,they call for NDE requirements without knowing howthis relates to part performance. By predicting porosityaccurately from casting simulation and realisticallymodeling its effects on the part performance, engineerscan develop robust designs that are tolerant of theporosity and reliable. In the current study, engineeringapproaches have been applied to simulate the effect ofporosity on deformation, damage, and fracture for acast steel during tensile testing, and the simulationresults are compared with measurements.

The material used in this study is ASTM A216 GradeWCB steel. It is a cast carbon steel having a combina-tion of good ductility and strength. It has the followingchemical composition (maximum wt pct): C 0.3; Mn 1.0;P 0.035; S 0.035; Si 0.6; Cu 0.3; Ni 0.5; Cr 0.5; Mo 0.2;and V 0.03; and the total of Cu, Ni, Cr, Mo, and Vcannot exceed 1.0 wt pct. At room temperature, GradeWCB steel has a yield strength and a tensile strength of248 and 485 MPa, respectively, and 22 pct elongation asminimum tensile requirements in ASTM A216. Failureof such ductile metals occurs on the microscopic scale bymechanisms of void nucleation, growth, and coales-cence.[1] Voids can pre-exist as microporosity and canalso nucleate from imperfections like second-phaseparticles. After nucleation, voids grow with increasinghydrostatic stress and local plastic straining. As voidsnucleate and grow, the void (or porosity) volumefraction increases. The voids begin to interact, and theporosity fraction at which interactions between voidsbegins is the critical porosity volume fraction fc. Asplastic strain continues to increase, local necking andcoalescence occur in the material between voids until aconnected chain of voids forms and failure occurs. Theporosity fraction at which fracture occurs is the failureporosity volume fraction fF.The effects of porosity on the structural performance

of carbon and low alloy steel castings on the macro-scopic scale are not as clearly defined as they are on themicroscopic scale. In previous studies, the effects oflarge amounts of porosity on stiffness and fatigue lifewere investigated.[2,3] Porosity less than a few percentdoes not result in a measurable loss of stiffness, or largestress concentrations, or stress redistribution, but it

R.A. HARDIN, Research Engineer, and C. BECKERMANN,Professor, are with the Department of Mechanical and IndustrialEngineering, University of Iowa, Iowa City, IA 52242. Contact e-mail:[email protected]

Manuscript submitted August 31, 2012.Article published online February 28, 2013

5316—VOLUME 44A, DECEMBER 2013 METALLURGICAL AND MATERIALS TRANSACTIONS A

greatly affects fatigue resistance.[4,5] Also, the presenceof low-level porosity will reduce the ductility of metalssince microvoids pre-exist before any stress is appliedand the nucleation stage is bypassed. Porosity largerthan a few percent in metals causes gross section loss,and locally reduces their effective stiffness.[6–8] Thishigher-level porosity is not uniformly distributedthroughout the entire cast part, and the materialproperties in the casting are heterogeneous. As a result,stress redistribution occurs in parts because of macrop-ores, and stress concentrations occur near them, whichlead to localized plastic deformation and the develop-ment of microcracks causing failure. Generally speak-ing, the performance of a casting with macroporositydepends on the amount, size, and location of porosityrelative to the cast component’s geometry and loading.All these factors must be considered together. Deter-mining the effects of large levels of porosity on partperformance is typically more case-by-case dependentthan for lower levels.

It has been proposed that the stiffness and strengthbehaviors of porous materials can be categorized intothree groups based on porosity amounts[9]: less than 10,10 through 70 pct, and materials with greater than 70pct. This division is promoted because the materials atthe extremes (<10, and>70 pct) behave quite differently.The highest porosity group is not applicable to caststeels; these are foams and cellular structures. Theelastic–plastic behavior of porous materials in the 10through 70 pct porosity range exhibit a nonlineardependence on the amount of porosity[2,6–9] The behav-ior of materials in the lowest range demonstrate a morelinear dependence on porosity amount, assuming thatvoids do not interact[10] and by considering isolatedpores,[11] or a uniform distribution of pores.[12] Applyingthe ductile failure micromechanical mechanisms de-scribed previously, one such micromechanics-basedmodel is the porous metal plasticity model. It isavailable in the finite element analysis (FEA) softwareABAQUS.[11–15] In the model, the volume fraction ofporosity is a primary state variable, and the inelasticflow of the material is modeled as voids grow andcoalesce at higher strains until failure occurs. Porousmetal plasticity was developed assuming voids arespherical and grow as spheres. It also assumes thematerial’s plastic behavior is dependent on hydrostaticpressure because of the porosity, and therefore neglectseffects of shear stresses on porous material behavior.Considering this and the limitations of porous metalplasticity, there have been numerous developments inmodeling of ductile fracture[16] addressing other voidgeometries and stress states. While not state-of-the-art,porous metal plasticity is a constitutive model readilyavailable to designers of cast components, and can be auseful tool to investigate the effect of porosity on acasting’s fracture behavior. It is a common constitutivemodel found in many commercial finite element pack-ages. This article uses the porous metal plasticity modelin FEA to predict the ductile fracture of cast steel withcenterline porosity. The primary goal of the currentstudy tests whether or not this commonly availablemodel can predict the fracture of steel with relatively

large amounts of porosity detectable through typicalindustrial radiography.Here, castings with porosity were produced, made

into plate specimens, and radiographed. The porositywas quantitatively determined from the radiographs.The castings underwent tensile testing. Using porositydata from the radiographs, finite element stress modelsof the test specimens with porosity were created, and thetensile testing simulated using an elastic–plastic materialmodel. Results of the simulated fracture are comparedhere with the measured fracture to test the model’scapabilities in predicting the fracture behavior of steelwith centerline shrinkage porosity.

II. MATERIAL PREPARATION,MEASUREMENT, AND ANALYSIS

A. Cast Specimens and Mechanical Testing

For the tensile fracture study presented in this article,WCB steel specimens were produced from 2.5 cmthick 9 12.7 cm wide vertically cast plates of two lengths(38.1 cm and 45.7 cm) as shown in Figure 1(a). Theplates were designed through casting simulation withMAGMAsoft[17] to contain centerline shrinkage as shownin Figures 1(b) and (c). Five cast plates were producedand tested from the longer length castings, and four wereproduced and tested for the shorter-length castings. Theletters ‘‘D’’ and ‘‘E’’ were used to identify the shorter andlonger plates, respectively; followed by numbers 1through 5. The cast plates were normalized and tempered,and machined into 19-mm-thick tensile test coupons witha gage section width of 86 mm as shown in the front andside views for the specimen in Figure 2(a). The tensilespecimen dimensions for the plates with centerline poros-ity were determined from the ASTM E8 tensile teststandard.[18] The positioning of the extensometer on thenarrow/thickness face of the specimen is indicated inFigure 2(a). InFigure 2(b), a smaller tensile test specimenis shown, which was machined from the porosity-free (or‘‘sound’’) section of a cast plate. The specimen shown inFigure 2(b) was tested to characterize the steel’s fracturebehavior in the absence of porosity. The sound steelspecimen was taken from the top end of the plate, and itsdimensions shown in front and end views in Figure 2(b)were taken from ASTM E8. Parameters in the porousmetal plasticitymodelwere determined by achieving goodagreement between the measured and predicted soundsteel tensile curves. Tensile testing of the sound materialwas performed at the University of Iowa, and the steelplates with porosity were tested at the SSAB AmericasMontpelier, Iowa Steelworks. Examining the fracturesurfaces of the plates with porosity, the porosity wasfound to lie about the mid-thickness of the plates asexpected from centerline porosity.

B. Radiographic Porosity Measurement, Testing,and Fracture Analysis of the Specimens

Film radiography was performed on the as-cast platesat the foundry where they were produced. All plates

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 44A, DECEMBER 2013—5317

were found to contain shrinkage porosity of levels 4 or 5according to the radiographic testing standard ASTME446.[19] In the standard, level 5 signifies the worst level

of porosity defect, and level 1 signifies the least defective.Since the cast plates had porosity of levels 4 or 5, theywould typically be scrapped at the foundry. After

(a) (b) (c)

12.7 cm

2.5 cm

38.1 or 45.7 cm

10.2 cm

Riser

Casting

CenterlinePorosity

Porosity (%)

Sound Material

Fig. 1—(a) Top and front view diagrams of casting geometry (not to scale), (b) Casting simulation model for 45.7-cm-long plate, (c) centerlineporosity prediction at mid-width using MAGMAsoft.[16]

Extensometerposition

19

38143

114

153

R57

76

57

Front View Side View Front View

127

32

6

516

6

End View

10

(a) (b)

Fig. 2—Tensile testing specimens with dimensions in mm. (a) Front and side views of tensile test plate used for measurements with steel contain-ing centerline shrinkage having a 19-mm-thick by 86-mm-wide gage section, and (b) front and end views of smaller test specimen used in tensiletesting of the steel without porosity having a 6 mm square gage section.


machining and before testing, digital radiographs of theplate tensile test coupons with centerline shrinkage andthe sound specimen were taken at 16-bit gray scale and10 pixels per mm at Alloyweld Inspection Co. inBensenville, Illinois. The radiograph of the soundspecimen was free of any indications of shrinkage orother defects. Stepped gage blocks were placed in theradiographs of the plate specimens with centerlineshrinkage as shown in Figure 3(a). Using the gageblocks and image analysis, a calibration curve ofradiographic gray level vs plate thickness was deter-mined for each radiograph. All curves were found to bewithin the same margin of error, but the individualradiograph calibration curves were used in analyzing theporosity level in the plates. Filtering and backgroundcorrections were performed, and only indications iden-tified by image analysis were analyzed for porosity levelas seen Figure 3(b). The thickness of the steel, as shownin Figure 3(c), was determined from the gray level vsthickness calibration curve. Finally, the measuredporosity fraction through the plate thickness at eachpixel of the radiograph was calculated by dividing thethickness of steel measured from the radiograph by thenominal machined plate thickness tplate (19.1 mm) whichis shown in Figure 3(d). Given the radiography equip-ment used, and the image analysis, the detectablethreshold porosity fraction for the current study wasestimated to be 1.5 pct. A radiograph for the specimenD3 is shown in Figure 4(a), where the fracture region isindicated by the area in the box. The fracture region onthe radiograph was determined from the fracturedspecimen by measuring from the gripped plate endsand using punch marks made before testing on thenarrow/edge faces of the specimens. A box is used toidentify the fracture region due to the irregularity of the

fracture surface and the uncertainty in the measure-ments. Since the tensile testing force was applied alongin the length direction of the specimen, the cross sectionporosity normal to the force was determined as plottedin Figure 4(b) along the specimen length. Note here thatpositive distances are above the plate mid-length andnegative values below. The overall average porosity inthe gage section of each test plate was determined byaveraging all the pixel measurements within the gagesection, and as indicated in Figure 4(b) it is about 0.19pct for specimen D3. The maximum cross sectionporosity for specimen D3 in Figure 4(b) is seen to beabout 0.8 pct, and it falls within the observed fractureregion along the length. For specimen D3, the fractureregion is seen to correspond with a region of relativelyhigh cross sectional porosity along the gage length, andthis correspondence was observed in the other specimensas well. As shown in Figure 5, the average gage sectionporosity for all specimens was determined, ranging from0.07 to 0.23 pct.The tensile testing results for the sound WCB steel

and the steel containing centerline porosity are shown inFigure 6. Note the sound material has the greatestelongation at fracture. The ‘‘D’’ centerline shrinkageporosity specimens (the shorter-length plates) generallyhave greater elongations than the ‘‘E’’ family of plates,but the most ductile of the plates with porosity was plateE1. In addition, the tensile curve for plate E1 appearsquite different from the other curves and has muchlower yield and ultimate strengths.Unfortunately, X-ray tomography was not performed

on the steel specimens because of their relatively largesize. Because of this, an assumption about how theporosity measured from the radiographs was distributedthrough the plate thickness had to be made. Regions

19.018.8518.718.5518.418.2518.117.9517.817.6517.5

(a) (b) (c) (d)

Stepped blocks used

for thickness-gray level calibration

Thickness(mm)

Porosity Fraction

Fig. 3—(a) Digital radiograph of a test plate taken at 16-bit gray scale and 10 pixels per mm, (b) binary image of indications detected by analy-sis where porosity level is measured, (c) thickness measured at pixels with porosity and (d) the through-section porosity fraction measured in thetest plate from the digital radiograph.


with porosity on the fracture surfaces of the specimenswere examined and measured to establish a reasonableassumption for the distribution of porosity in the plates.An example of a fracture surface having a region ofporosity is shown in Figure 7(a), where the porosityappears to be centerline type located near the mid-thickness of the plate. Measurements of the thickness of

the porosity regions were performed on the fracturesurfaces to determine a reasonable thickness of porosityto use in simulating fracture of the plates. Since theseregions lie on the fracture surface, they are representa-tive of the porosity that determines the fracture behaviorof the specimen. For a given region with porosity on thefracture surface, the maximum thickness dimension

Mid -Length

+

-

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

-80 -60 -40 -20 0 20 40 60 80

Distance from Mid-Length of Plate (mm)

Cro

ss S

ectio

n P

oros

ity

Average Gage

Section Porosity

Fracture Region

(a) (b)

Fig. 4—(a) Radiograph of specimen D3 with the location of the fracture region indicated by the box. Note plate mid-length and sign conventionused for distance from mid-length. (b) Cross section porosity along gage length of plate vs distance from mid-length. Note the average porositylevel in the plate (0.19 pct) is indicated in addition to the region where the plate fractured.

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

D2 D3 D4 D5 E1 E2 E3 E4 E5

Specimen Identifier

Ave

rage

Por

osity

in th

e T

est S

peci

men

Gag

e Se

ctio

n

Fig. 5—Average porosity fraction measurements in the volume ofthe test plate gage sections determined from averaging the cross sec-tional porosity measurements as shown in Fig. 4(b).

0

100

200

300

400

500

600

0 0.05 0.10 0.15 0.20 0.25

Stre

ss (

MPa

)

D1

D2

D3

D4

D5

E1

E2

E3

E4

E5

Sound Data

Strain (mm/mm)

Fig. 6—Tensile testing results for sound WCB steel and the steelspecimens containing centerline porosity.


tregion of a given region of porosity was measured asshown for example in Figure 7(a). A frequency histo-gram of the fraction of plate thickness data correspond-ing to the tregion measurements is shown in Figure 7(b).Note that eight regions of porosity were found to fallinto both the 20 to 25 pct, and the 40 to 45 pct, ranges offraction of plate thickness. These measurements do notappear the follow a normal distribution. In Figure 7(b),the average tregion for all the fracture surface regionswith porosity is indicated at 33 pct of the plate thickness.However, note that the bin range for the averagecontains relatively few observations, and might not bethe best choice as a representative thickness dimensionfor the centerline porosity. A constant thickness ofcenterline porosity was assumed when reconstructingthe porosity in the simulated plates. For this constantthickness, the average fraction of plate thickness withporosity from Figure 7(b) was used as a starting point incomparing predicted and measured tensile curves toestablish a simulated centerline porosity thickness thatwould produce the best agreement. Iterative trial anderror simulations were performed, varying the thicknessof the simulated porosity region until the best combinedagreement between measured and predicted tensilecurves was found. Specimens D4 and E4 (representingthe smallest measured elongation), and specimens D3and D5 (representing the largest measured elongation)were used in these comparisons. The fracture behaviorof specimen E1, which has the greatest elongation of theplates with porosity, appeared anomalous and was notused to establish the thickness of the porosity region.After these trial and error comparisons, a centerlineregion with porosity having 25 pct of the plate thickness

was found to give the best overall agreement betweenthe measured and predicted fracture behavior for theplates.

C. Mapping of Measured Porosity onto Stress AnalysisMesh

Finite element meshes of the tensile test plates weredeveloped with mesh refinement at the plate mid-thickness to better resolve the centerline porosity asshown in Figure 8(a). The extensometer position duringtesting is shown in Figure 8(a). The edge of the platewhere the extensometer was positioned during testing,relative to the porosity distribution in the plate, wascarefully followed because of the nonuniform strainingof the plate. FEA of the test plate specimens withporosity required development of a method to transferthe through-section porosity measurements determinedfrom the radiographs onto the nodes of the FEA mesh.The porosity from the radiograph (Figure 3(d)) wasmapped to the mesh assuming that it lies symmetricallyabout the mid-thickness of the plate in a region 25 pct ofthe plate thickness. A computer code was written totransfer the measurements from the radiographs to theFEA nodes where the average porosity about a nodalposition in the plane of the radiograph is determined,conserving porosity. The averaging volume used abouteach node had dimensions of half the nodal spacing.Since this averaged porosity mapped to the FEA nodesis assumed to lie in a region 25 pct of the plate thickness,it must be scaled up by a factor (0.25)�1 from theaveraged values determined from the radiograph, whereit is distributed through the entire plate thickness. Oncethe averaged/scaled-up porosity was determined at anodal position, it was mapped to all nodes within theregion centered at the plate mid-thickness. An exampleof the porosity mapping is shown for a simulated testplate on the mid-thickness plane and for a slice throughthe plate thickness in Figure 8(b). Comparing theporosity determined from the radiograph in Figure 3(d)to the centerline region of Figure 8(b), the porosity atthe FEA nodes are diffused (caused by the averagingprocess about the nodes) and then amplified (due to thescaling-up at the centerline region).

III. STRESS MODELLING ANDCOMPUTATIONAL METHODS

A. Material and Failure Models

The constitutive material model used in the currentstudy is a standard material model available in thecommercial stress analysis software ABAQUS.[14] Itassumes linear small strain theory decomposing the totalstrain tensor e into elastic eel and plastic epl componentsso that e = eel+ epl. The recoverable elastic strains ofthe material are calculated from

r ¼ Deleel ½1�

where r is the stress tensor, Del is the fourth orderelasticity tensor, and eel is the elastic strain tensor.

(a)

(b)

0

1

2

3

4

5

6

7

8

9

Fraction of Plate Thickness with Porosity (%)

15 20 25 30 35 40 45 50 55

Freq

uenc

y of

Mea

sure

men

ts o

n

F

ract

ure

Surf

ace

Average Thickness of

Porosity Regions is 33%

tregion

Fig. 7—(a) Image of fracture surface showing region with centerlineporosity and measured thickness of the region, (b) histogram offrequencies of measured fraction of plate thickness with porosity(tregion/tplate) on the fracture surfaces.


Note that in the case of uniaxial tension, Del reducesto E the elastic modulus, and Eq [1] reverts to Hooke’sLaw. Two properties are required to define the isotro-pic elasticity tensor Del; the elastic modulus E, and thePoisson ratio m. Here the elastic material propertiesdepend on the local porosity fraction f at the FEAnodes. The porosity fraction is f ¼ Vpore

�V0 where

Vpore is the volume of porosity in the averaging vol-ume about a finite element node (described in Sec-tion II–C above), and V0 is the total averaging volumecomprising voids and the steel matrix. The relationshipbetween the elastic modulus and porosity fraction usednode-by-node in the FEA analysis is

E fð Þ ¼ E0 1� f=0:5ð Þ2:5 ½2�

where E0 = 198 GPa is the elastic modulus for WCBsteel without porosity.[2] The Poisson ratio m is depen-dent on f using

v fð Þ ¼ v0 þf

f1v1 � v0ð Þ ½3�

with v¥ = 0.14, f¥ = 0.472 and the Possion ratio forthe sound metal v0 was 0.3. Equation [2] was determinedfrom comparing, and achieving excellent agreementbetween, measured and predicted strains for cylindricaltest specimens having a wide range of porosity.[2]

Equation [3] was determined through simulations ofcomputer generated representative elemental volumes ofporous structures.[20] The effect of porosity on stiffness[2]

was studied using locally mapped porosity at FEAnodes determined from radiography.

Here, a standard plasticity material model was used insimulating ductile plasticity and failure. A completepresentation of the model is found in Gurson et al.,[11–13]

the software manual,[14] and the numerical integrationprocedure used in the software was developed byAravas.[15] The model requires the user to define thehardening behavior of the metal matrix without porositythrough true stress–strain data. This was determinedfrom tensile tests for the sound WCB steel. The yieldcondition used in the model is the plastic potential /given by

/ ¼ q

ry

� �2

þ2q1 f cosh � 3

2

q2p

ry

� �� 1þ q3f

2� �

¼ 0 ½4�

where f is the porosity fraction; q is the effective VonMises Stress; p is the hydrostatic stress; ry is the yieldstress of the sound steel as function of plastic straingiven in Figure 9; and q1, q2, and q3 are three materialmodel parameters. In Figure 9, the plastic true stress-strain curve was derived from the measured sound steeltensile test data given in Figure 6. Note from Eq. [4] thatwhen f = 0 (for porosity free material), the yieldcondition becomes q = ry, or the Von Mises yieldcondition, giving a yield surface that is independent ofpressure. The hydrostatic stress or pressure p and theVon Mises stress q are the two stress invariants;p ¼ �ð1=3Þ r : I and q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3=2ÞS : S

p, where S is the

deviatoric stress tensor S = pI+ r.The material parameters in Eq. [4] q1, q2, and q3 were

added to Gurson’s original model by Tvergaard[12], andthe original Gurson model corresponds to setting all the

(b)

Porosity Fraction f

Slice at mid-thickness

Slice at line through thickness

(a)

Extensometerposition

Encastre BC

Displacement BC

Porosity Thickness =0.25*Plate Thickness

Fig. 8—(a) Example of a 12 element thick finite element mesh biased about the mid-thickness of a plate and (b) example mapping a measuredporosity fraction field to the mid-thickness plane of the finite element mesh.


material parameters to 1. The material parameters wereadded by Tvergaard to consider the interactionsbetween voids and improved the Gurson model’saccuracy. In the current study, values for these param-eters are used, which appear to be used often in theliterature when applying the model to ductile metals;[14]

q1 = 1.5, q2 = 1.0, and q3 = 2.25, where q3 = q12.

Plastic flow is assumed to be normal to the yieldsurfaces formed by Eq. [4] for a given value of porosityfraction f. Assuming this, the yield condition of Eq. [4] isused to determine the plastic strain to grow and nucleateporosity, starting from a initial porosity fraction. Asplastic strain increases, so does porosity volume fractionas will be described shortly. The flow rule determinedfrom Eq. [4] for the plastic strain rate _epl tensor is

_epl ¼ _k@/@r¼ _k �1

3

@/@p

Iþ 3

2q

@/@q

S

� �½5�

where _k is a nonnegative scalar constant of propor-tionality for the plastic flow rate. Given that the plas-tic strain in Eq. [5] causes porosity growth andnucleation as it increases, the equation describing thegrowth rate of porosity voids by growth and nucle-ation is

_f ¼ 1� fð Þ_eplkk þ A_eplm ½6�

where the first term on the right hand side denotesgrowth of existing voids from current porosity fractionf and _eplkk the total plastic strain rate (trace of the strainrate tensor), and the second term denotes the growthrate due to nucleation. In the nucleation term, theequivalent plastic strain rate _eplm is multiplied by ascaling coefficient

A ¼ fN

sNffiffiffi2p

pexp � 1

2

eplm � eNsN

� �2" #

½7�

based on the assumption that the nucleation function(A/fN) follows a normal distribution depending on theplastic strain range about a mean value eN, a standarddeviation sN, and a volume fraction of nucleated

porosity voids fN. Using the void nucleation modelrequires fitting these three parameters for a givenmaterial’s tensile curve. Here values in the rangerecommended for metals were used:[14] eN = 0.3,sN = 0.1, and fN = 0.03 and these have good agree-ment with the sound WCB steel tensile curve. The effectof varying fN on the predicted tensile curve will bepresented shortly.Coalescence and failure modeling extends the porous

metal plasticity model to higher levels of porosity thanthe range of Gurson’s original model, which is less than10 pct. The coalescence and failure criteria used herewere developed by Needleman and Tvergaard[13] wherethe porosity fraction f in Eq. [4] is replaced by aneffective porosity volume fraction due to coalescence f*.In Eq. [4], f* takes on the actual porosity volumefraction f when it is less than the critical value fc, wherecoalescence begins. Therefore, for values of f< fc,f* = f. When f* > fc the effective porosity fractionincreases more rapidly than f because of the coalescence.This coalescence is modeled by increasing f* by a factorof (�fF � fc)/(fF � fc) relative to the increase in f. Thematerial has no load carrying capacity when f ‡ fF,where fF is the porosity fraction at failure. The equa-tions describing the dependence of f* on f where f* isused in place of f in Eq. [4] are

f� ¼f if f � fcfc þ

�fF�fcfF�fc f� fcð Þ if fc<f � fF;

�fF if f � fF;

8<

:½8�

The value of �fF in Eq. [8] is set using the ‘‘q’’ porousmetal plasticity model material parameters by

�fF ¼q1 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq21 � q3

q

q3½9�

There are two additional parameters that must bedetermined when using the coalescence and failuremodel: fc and fF, the critical and failure porosityfractions, respectively. Here, these values were deter-mined by fitting the model fracture curve to themeasured curve for sound WCB steel. The values givingthe best agreement were fc = 0.05 and fF = 0.2, and theeffect of varying fF on the predicted tensile curve will bediscussed below. To summarize, the nine model param-eters used in the porous metal plasticity model here aregiven in Table I, and these give the best agreementfound between the predicted and measured tensilecurves. The model described here is intended to givethe most important model features and the parametersused. Additional details on the model and the solutionscheme are given in References 11 through 16.

0

100

200

300

400

500

600

700

800

0.00 0.05 0.10 0.15 0.20 0.25 0.30

True Plastic Strain (mm/mm)

Tru

e S

tres

s (M

Pa)

Fig. 9—True plastic stress–strain curve for ASTM A216 GradeWCB steel without porosity.

Table I. Parameters Used in the Porous Metal Plasticity

Model for WCB Steel in ABAQUS Simulations

q1 q2 q3 f0 fc fF eN sN fN

1.5 1.0 2.25 0.005 0.05 0.2 0.3 0.1 0.03


B. Finite Element Modeling

The boundary conditions used in the simulations ofthe tensile testing of the plates were an encastrecondition on the base of the specimen and a normaldisplacement applied to the top surface of the specimenas shown in Figure 8(a). The simulated testing load wasgiven by the total force on the wire element used toapply the displacement boundary condition. As men-tioned earlier, the edge of the specimen where theextensometer was positioned during testing was markedon each specimen, and that edge was used to extract thesimulated testing strains. This was done in case theporosity field resulted in differences between the twoedges of the specimen. Simulated extensometer strainspresented in the stress–strain plots that follow weredetermined from the displacements of node sets definedat the extensometer knife-edge positions as shown inFigure 8(a).

In order to use the failure criterion in the porousmetal plasticity model, ABAQUS Explicit was used toperform the fracture simulations. Two considerationsare important when applying an explicit procedure to aquasi-static process such as the tensile testing simula-tions. The first is the time period of the tensile test event.While the actual testing might require on the order of aminute, the simulated event can be modeled over a muchsmaller time scale without the results changing. Thesecond consideration is the stable time increment whichis dependent on the smallest characteristic elementlength in the mesh and the dilatational wave speed ofthe material. The stable time increment issue wasexamined first, and then the event time period wasdetermined, so that the simulations could be executed ina reasonable number of increments without the resultsbeing affected. The wave speed in steel cd is about5,000 m/s, given by

ffiffiffiffiffiffiffiffiffiE=q

pwhere q is the density. The

largest stable time step Dtstable that can be taken in anexplicit dynamics calculation is given by Dtstable ¼ Le=cd,where Le is the smallest characteristic element length.For characteristic element lengths ranging from 0.7 to4 mm, the largest stable time step range is 1.4 to8.0 9 10�7 s. If the tensile testing event were to besimulated on its actual time scale, then the simulationswould require 10 s of millions of time steps and wouldbe impractical. A practical time scale for the testingevent was established by considering the time the stresswave transmits through the test specimen and multiply-ing it by a safety factor that is sufficiently large. Thespecimen is 318 mm long, and the time to transmit astress wave through the specimen is about 6 9 10�5 s. Amultiplying safety factor between 10 and 100 times thetransmission time would be probably be an adequateevent length, but a multiplying factor of over 1000 timeswas used. The time period used for the tensile testingevent in the simulations was 0.085 s. Experimenting withsmaller and larger event periods confirmed the eventduration did not affect the results. Another check on therates of the simulated of loading performed was acomparison between the velocity of the displacement ofthe testing and the wave speed. Based on the tensile

event being 0.085 s long, the test specimen end displace-ment speed is about 0.53 m/s. While this displacementspeed seems high, it is only 0.01 pct of the material’swave speed. As another final check, the kinetic andinternal energies of the model were stored duringsimulation runs to confirm that the kinetic energy didnot exceed approximately 5 pct of the internal energy asper the software’s guidelines for simulating quasi-staticprocesses. It was found that the kinetic energy did notexceed 0.6 pct of the internal energy.Unless otherwise noted in the simulations in the

computational results section presented here, the FEAmesh employed in the length and width directions of thegage section had a uniform nodal mesh spacing of1 mm. Nonuniform meshes through the plate speci-mens’ thickness were used in simulating the final resultsfor all the specimens, where the mesh had smallerelements biased toward the centerline as seen forexample in Figure 8(a). In that figure, the mesh isshown for demonstration purposes and has 12 elementsacross the plate thickness. In computational modelsused to generate results reported here, 20 elements wereused across the plate thickness with a bias factor of 3.The FEA model used to generate final results presentedhere had 379,681 C3D8 elements, 403,878 nodes, andthe total number of variables was 1,213,081. Thesesimulations required about 663,212 increments andapproximately 4 days and of computer time using eightcores of a Linux workstation with 16GB ram, and twoIntel Xeon CPU E5472 CPUs running at 3.00 GHz. Forthe finest mesh case used in the grid sensitivity study tobe described later, the ‘‘0.7-mm mesh spacing’’ resultshaving 30 elements across the plate thickness and usingcenterline-biased elements, very small characteristicelement lengths required the use of the mass scalingoption, ‘‘*FIXED MASS SCALING,’’ which increasesthe density of elements if their stable time increment issmaller than a specified increment. The specified timeincrement for those elements was chosen such that thetotal number of increments was limited to about2.8 million.

IV. COMPUTATIONAL RESULTS

A. Results for Sound Material and Determinationof Constitutive Material Model Parameters

Simulations were first run using ‘‘sound’’ steel todetermine the nine porous metal plasticity modelparameters given in Table I. After these model param-eters were determined by matching the predicted andmeasured fracture behavior of the sound material, theplates with porosity were simulated. The porous metalplasticity parameters were not changed when simulatingthe plates with porosity; only the mapped porosity fieldfor each plate was added to the simulation input whensimulating the plates with porosity.The porous metal plasticity model material parame-

ters given in Table I were determined by matching themeasured and predicted tensile curves and fracture


behavior. It was determined that the ‘‘q’’ model param-eters should not be changed from recommended val-ues.[14] The sensitivity of the fracture curve to theremaining six parameters were examined and varied bytrial and error to achieve the best agreement that couldbe obtained between the measured and predicted frac-ture curves. Examples of the sensitivity of the simulatedtensile curves to two of the parameters are given inFigures 10 and 11 for the nucleation porosity fraction fNand the failure porosity fraction fF, respectively.Increasing the void nucleation fraction from 1 to 5 pctprogressively decreases the predicted elongation andresults in a less ductile material behavior as seen inFigure 10 for the 5 pct curve. A nucleation fraction of0.03 was found to give the best agreement overall for fN.The effect of varying the failure porosity fraction fF isshown in Figure 11, where it is increased from 10 to 40pct. The larger the failure porosity fraction is the greaterthe elongation. A failure porosity fraction of 20 pct wasfound to produce the best agreement between predictedfracture behavior and that observed in testing for fF.The measured and best predicted sound steel curves aregiven in Figure 12 with model material parameters fromTable I used to produce the simulated curve. For thesound material, the maximum elongation at failure,ultimate tensile stress, and yield stress are predictedwithin 2 pct. The agreement between the sound curves isgood and represents best possible agreement that wouldexpected between predicted and measured curves for thespecimens with porosity. Since the elongation at fracturecannot be determined from the simulations by reassem-bling the fractured specimen, a common point ofreference along the tensile curves was chosen to comparethe predicted and measured strain at fracture. For themeasured sound curve in Figure 12, the last stressmeasured in the test during fracture was 348 MPa. Thisalso corresponds to a point of nearly perfect agreementbetween the measured and predicted sound steel curves.Since it provides a good basis for comparison for thesound material, as indicated in Figure 12, the strains atthis 348 MPa stress level were used to compare themeasured and predicted elongation for all the fracturespecimens with porosity as well.

B. Simulations of Steel with CenterlineShrinkage Porosity

The dependence of simulation results on computa-tional mesh density was investigated to confirm con-verged results. Several of the plates with porosity werestudied to determine that the mesh used would be fineenough to achieve grid independency in the mapping ofthe porosity to the computational mesh. Consider plateE4 in Figure 13, where its radiograph is shown inFigure 13(a). In Figures 13(b) trough (e), mid-thicknesssections of the porosity fraction for four nodal meshspacings are shown, having spacings of 4 mm, 2 mm,1 mm, and 0.7 mm, respectively. For the 0.7-mm mesh,30 elements were generated across the specimen thick-ness using biasing of elements at the mid-thickness. Forthe 4-mm, 2-mm, and 1-mm mesh spacings, uniformmeshes in the specimen thickness direction were usedhaving 6, 12, and 20 elements in the thickness direction,respectively. In Figure 13, note the porosity field is morediffuse with coarse meshes and that the maximum nodalporosity value (shown for each field in Figure 13) rangesfrom 34 pct for the coarsest mesh to 52 pct for the finestmesh. A white-boxed area is shown for each mesh,

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)

FEA Simulation: fF = 0.20, fN = 0.05



Measured Data

Simulation with f N = 0.05



Measured DataIncreasing f N

Fig. 10—Effect of varying the nucleation porosity fraction fN on thepredicted tensile curves compared with the measured curve.

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Simulation with f F = 0.4




Measured Data Decreasing f F

Fig. 11—Effect of varying the failure porosity fraction fF on the pre-dicted tensile curves compared with the measured tensile curve.

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)Measured Sound

Simulated Sound

Stress level used to compare measured

and predictedelongation for all

specimens

Measured and predicted strains have nearly perfect

agreement at this stress level

Fig. 12—Simulated and measured stress–strain curves for the soundWCB steel. The stress level at which measured and predicted elonga-tion are compared for all plates is indicated.


which is shown inmore detail in Figure 14, along with thecorresponding area from the radiograph in Figure 14(a).In the detailed view, the porosity fields for the two finemeshes (in Figures 14(d) and (e) are nearly identical, andthe porosity mapping process appears to be meshindependent for the finest meshes. The measured tensilecurve for specimen E4, and the simulated tensile curvesfor the four mesh densities are shown in Figure 15, alongwith two additional simulated mesh spacings of 3 and2.5 mm. For meshes coarser than the 1-mm spacing, it isfound that the simulated curves have greater elongationwith increasing mesh spacing up to 3- and 4-mm meshspacings. There is no great difference between the 3- and4-mm tensile curves. The simulated tensile curves for the0.7- and 1-mm mesh spacings appear to convergetogether and agree well with the measured curve. Basedon this grid convergence study, the 1-mm mesh spacingwas used to simulate the plates. The 0.7-mmmesh spacingwith 30 elements through the plate thickness requiredapproximately 1 week of computational time on themachine referred to earlier, and this was too long acomputational time to use in simulating all the specimens.

The tensile curves for the plate specimens withporosity were simulated using the centerline porosityfields mapped to the FEA nodes at the plates’ mid-thickness. Plates with porosity were simulated using the

model parameters in Table I, and where there is nonodal porosity mapped from the radiograph, the poros-ity is 0.5 pct. As shown in Figure 4, the average fractionof plate thickness with porosity was found to be 33 pct.In general, it was found that using 33 pct of the platethickness as the centerline porosity thickness (as inFigure 7) gave greater elongation predictions thanobserved in testing. If a single value for the centerlineporosity thickness is used for all plates, the bestagreement overall between the measured and predictedfracture behavior was found using 25 pct of the platethickness. As discussed earlier, this choice was based onsimulating the tensile curves for specimens D4, E4, D3,and D5, measured curves of which are given in Figure 6.Similar to the grid dependency study, a centerlineporosity region thickness study was performed usingthe 1-mm uniform mesh which gives a grid independentresult in Figure 13. Because of the days required tosimulate a tensile curve, it was decided to do a thoroughstudy using only plate E4 for the range of porosityregion thickness measurements taken from the fracturesurfaces plotted in Figure 7. Considering that thesmallest thickness dimension measured from the frac-ture surface was about 15 pct, a simulation case for a 10pct plate thickness of centerline porosity was run as wellto test a thickness that was much smaller than what wasobserved. The tensile curve results for the centerlineporosity region thickness study are shown in Figure 16,where the measured curve for specimen E4 is shown forcomparison. The porosity region thicknesses used were10, 20, 25, 30, 40, and 50 pct of the plate thickness. Notethat as the porosity region thickness is increased theporosity fraction within the region decreases and viceversa. The tensile curve for the 10 pct porosity thicknesshas an unrealistically large elongation. Indeed, a casewas run (but not shown here) where the porosity wasassumed to be a complete hole of varying thicknesscorresponding to the measured porosity, and that tensilecurve even more closely approached the curve for thesound material. The porosity thickness study resultsshow that as the thickness increases from 10 to 50 pctthe ductility of the simulated material decreases. Themeasured curve appears to be bracketed by the 20 and30 pct of plate thickness curves, and the 25 pct tensilecurve appears to agree best with the measured one.Based on these results and additional simulations forspecimens, D4, D3, and D5 which showed similaragreement, 25 pct of the plate thickness was chosen asthe porosity region thickness in the simulations. As afinal remark, unrealistically large values of assumedporosity region thickness were found to give unrealis-tically large elongations. A plate was simulated wherethe porosity was taken as uniform through the entireplate thickness and the resulting tensile curve had a largeelongation approaching that of the sound material.In Figure 17, the measured and predicted tensile

curves for plate E4 is shown. The reduction in ductilitydue to the porosity and fracture behavior is wellpredicted when one compares it to the sound curve. InFigure 17, three points along the tensile curve areindicated to demonstrate the progressive behavior ofthe failure process within the plates with porosity. The

(e)Maximum

is 52%

(b)

Maximum is 34%

Porosity Fraction

Maximum is 37%

Maximum is 46%

(c)

(d)

(a)

Fig. 13—(a) Radiographs of plate E4, and porosity fraction at mid-thickness plane for four meshes used in simulation grid dependencystudies for plate E4 having the following mesh spacings in the gagesection: (b) 4 mm, (c) 2 mm, (d) 1 mm, and (e) 0.7 mm. White-dashed boxed areas in gage section are shown for more a detailedcomparison in Fig. 14.


(a) (b)

(c) (d) (e)

Porosity Fraction

Fig. 14—(a) Radiograph of area for detailed comparison of porosity fraction indicated in Fig. 13 for four meshes used in the grid studies.Porosity fraction at mid-thickness in the radiograph area for nodal mesh spacings of (b) 4 mm, (c) 2 mm, (d) 1 mm, and (e) 0.7 mm.

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)

Measured Specimen E4Simulation with 4 mm MeshSimulation with 3 mm MeshSimulation with 2.5 mm MeshSimulation with 2 mm MeshSimulation with 1 mm MeshSimulation with 0.7 mm Mesh and Bias through Thickness

Fig. 15—Tensile curves from grid dependency studies for a platewith porosity using uniform meshes having dimensions 4, 3, 2.5, 2,1, and a 0.7-mm spacings having a biased mesh in the plate thick-ness direction to concentrate nodes at the centerline. Measured curveis also shown. Centerline porosity thickness used is 25 pct of theplate thickness.

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Measured Specimen E4

Porosity Thickness=10%






Fig. 16—Tensile curves for plate E4 for assumed thickness of center-line porosity as fraction of plate thickness in percent. Measuredcurve and curves for centerline porosity region thicknesses of 10, 20,25, 30, 40, and 50 pct of the plate thickness are shown. The gridused is the 1-mm uniform mesh from the grid dependency study inFig. 15.


testing strain is approximately 0.002, 0.045, and 0.104 atPoints 1, 2, and 3, respectively. The plastic strain andporosity fraction results at the mid-thickness of theplates are of particular interest since the porosity locatedthere determines the overall tensile response and thefailure. Note that Point 1 is still in the elastic portion ofthe tensile curve, Point 2 is about midway through the

plastic region, and Point 3 is at the position of ultimatetensile strength just before the onset of failure. Theequivalent plastic strain results at these three points areshown in Figure 18. Note that even in the elastic part ofthe curve at Point 1, there is plastic strain occurringbecause of and near to the porosity. In Figure 19, theporosity fraction at the mid-thickness of the plate isshown where the scale is chosen to demonstrate theporosity fraction increasing until failure occurs. InFigure 19, the results for porosity fraction at Point 1are virtually unchanged from the initial porosity field(not shown). From Point 1 to 3 the porosity fractionincreases because of void growth and nucleation untilfailure occurs. The scale in Figure 19 was set to 0.15maximum porosity fraction to accentuate the areaswhere porosity evolves. Recall though that the materialfailure occurs at 20 pct porosity fraction. Positions ofobserved and predicted failure are noted in Figure 19.The position of predicted final failure corresponded tothe position along the plate length where the testedspecimen failed.White-dashed boxed areas in gage sections of

Figures 18 and 19 are shown for more a detailedcomparison in Figure 20. Figure 20(a) shows the equiv-alent plastic strain field and mesh and 20(b) the porosityfraction. Considering these fields, the material’s behavior

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Simulated Specimen E4

Point 1

Point 2Point 3

Fig. 17—Simulated and measured stress–strain curves for the speci-men plate E4 with porosity. Three points of interest along the tensilecurve are indicated.

Point 2

Equivalent PlasticStrain


Point 3Point 1


Fig. 18—Equivalent plastic strain from simulation at the mid-thickness plane of plate E4 for the three points on the tensile curve indicated inFig. 17. The legend scales are increasing from Point 1 to 3. Note there is local plastic straining in the elastic portion of the curve at Point 1.A white-dashed boxed area in the gage section at Point 3 is indicated for a more detailed examination in Fig. 20.


can be quite different depending on how the porosity isdistributed relative to the load. In Figure 20, the loadingdirection is indicated. In the figure, it is demonstratedthat in some sound areas neighboring the porosity, thereis higher equivalent plastic strain because of that soundmaterial bearing more of the load, and these areas areindicated by the four white ‘‘m’’ symbols. They bearmore load because of the porosity reducing the crosssection normal to the load. However, these areas canonly bear the load provided they have continuousregions without porosity along the loading direction(above and below them). If this occurs, and they aresurrounded by porosity, then the sound regions areisolated from bearing the load and have a low plasticstrain. The white symbol ‘‘d’’ indicates a sound regionthat is isolated from bearing the load by two surround-ing areas of porosity above and below it. At Point 3 inFigures 19 and 20(b), it is apparent why the simulationpredicts failure where it does, since the porosity regionsare becoming interconnected. Note the distorted/failedelements that are seen in detail in Figure 20(b).Figure 21 is provided to show the simulated developmentof average porosity in gage section of test plate E4 andpredicted stress as functions of simulated extensometer

strain during tensile testing. As seen by the solid curve inFigure 21, the initial average porosity in the plate gagesection is about 0.75 pct and increases slowly until theultimate tensile stress is reached, after which it increasesrapidly because of the failure event. These results giveadditional insight into the model, and the complexity ofthe interaction between the porosity and the elastic–plastic model, and the resulting nonuniform stress andstrain fields.Figure 22(a) is provided to show the simulated and

measured tensile curves for a second plate, D2. Hereagain the reduction in ductility and fracture behavior isreasonably well predicted. During testing of specimenD2, the control system stopped the test before completefracture, and thus the appearance of the specimen justbefore complete failure was captured in Figure 22(b).This photo of the plate after testing in Figure 22(b)compares favorably with the predicted appearanceof the final failure in Figure 22(c). In comparingFigures 22(b) and (c), note the circled regions wherethe final material failure occurs during testing. Thisinteresting correspondence might have been missed ifthe plate had completely fractured. If the current studywere to be repeated, then comparing the simulation

Point 1Point 2 Point 3

Porosity Fraction

Actual and predicted

failure here

Fig. 19—Simulation results for porosity fraction at plate E4 mid-thickness at the three points on the tensile curve indicated in Fig. 17. Theporosity fraction damage increases until failure occurs. Position of final failure is indicated. At Point 3 a white-dashed boxed area in gage sectionis indicated for more a detailed comparison with the equivalent plastic strain in Fig. 20.


animations with a video record of the testing and failureevents could provide additional evidence supporting thismodeling approach.

In lieu of presenting all the fracture curves for thenine plates with porosity, the predicted and measured

elongation results from the fracture curves at the348 MPa testing stress level (shown in Figure 12) aregiven for all the specimens tested in Figure 23. Thesound elongation data point is circled and demonstratesnearly perfect agreement since the model parameterswere tuned to achieve this agreement. Ductility in thetest castings with porosity is markedly reduced with thepercent elongation data ranging from 12.8 to 19.6 pct; vs22 pct elongation for the sound material. The results inFigure 23 demonstrate the success of this modelingapproach to predict the effect of porosity on the fracturebehavior despite a key assumption: that the centerlineporosity is positioned at the plate mid-thickness, havinga thickness dimension of 25 pct of the plate thickness.The elongation results demonstrate that the mainoutlying data point is Specimen E1, where the measuredelongation is underpredicted. The measured elongationfor Specimen E1 is 19.6 pct, and as was noted earlier,referring to Figure 6, its measured tensile curve wasquite different from the other specimens. From theporosity thickness study summarized in Figure 16, thebehavior of specimen E1 could be explained if theporosity distribution through the thickness of plate E4 ismore concentrated at the centerline relative to the otherplates. Using a smaller fraction of the plate thickness forthe porosity thickness will improve the comparisonbetween the simulation and measurement for E1, sincethe predicted elongation will be greater. Even so, as


Porosity Fraction

Load Direction(a)

(b)

Fig. 20—Simulation results for (a) equivalent plastic strain and(b) porosity fraction for plate E4 at mid-thickness and Point 3 onthe tensile curve indicated in Fig. 17. Area is the white-dashed boxedareas indicated in Figs. 18 and 19. The white symbol ‘‘d’’ indicatesa sound region that is isolated from the loading by surroundingporosity, and it bears little of the load. The four white ‘‘m’’ symbolsindicate sound regions that are bearing large loads and large plasticstrains. These areas have uninterrupted sound material in the load-ing direction.

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Fig. 21—Simulated development of average porosity in gage sectionof test plate E4 and stress as functions of extensometer strain duringtensile testing.

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(a)

(b) (c)

Fig. 22—Results for plate D2 (a) predicted and measured tensilecurves, (b) photo of plate after testing, and (c) simulated porosity onmid-thickness plane at fracture.


summarized in Figure 23, the methodology presentedhere for predicting the fracture behavior of a castingwith centerline porosity using analysis of radiographsand porous metal plasticity in FEA appears validated.By improving the realism of the porosity reconstructionand mapping in the simulation, through X-ray tomog-raphy, for example, the assumption employed here onthe distribution of centerline porosity can be removed.Upon doing this, the validity of the modeling approachcould be judged with greater certainty.

V. CONCLUSIONS

The current study demonstrates that the tensilefracture of cast steel with porosity can be predictedfrom industrial radiographs using a porous metalplasticity material model commonly implemented inFEA software. Nine model material parameters weredetermined for sound WCB steel from its tensile curve.Using these model parameters and porosity fieldsmapped from the radiographs of the test specimens tonodes of FEA models, the tensile test curves for thecastings with centerline porosity were simulated. Pre-dicted and measured tensile curves compared well, andthe measured and predicted elongation comparisonswere also good. The most noticeable effect of porosity incast steel is reduction in ductility, and the modelpredicted it well. It is difficult to make general state-ments about the effect of porosity on fracture behavior,since it has been demonstrated here that its effects aredependent on the porosity distribution relative to theload. For the seemingly small amounts of porosityshown here, where the average volumetric porosities inthe gage sections of the specimens ranged from 0.10 to0.27 pct, the ductility in the test castings with porosity ismarkedly reduced with the percent elongation data

ranging from 12.8 to 19.6 pct vs 22 pct elongation for thesound material. This range of ductility behavior is dueto the differing porosity distributions in the specimens.The current study demonstrates that the effect ofporosity on ductility and fracture behaviors can bepredicted provided that a reasonably accurate porosityfield is used in the simulations.Some disagreement still remains between prediction

and measurement. The clearest factors contributing tothe disagreement are ascribed to assumptions andlimitations in the porous metal plasticity model, andthe assumptions and deficiencies in determining how theporosity was distributed through the thickness of theplate specimens. The current study can be improved bymore realistically reconstructing the porosity distribu-tion through the plate thickness via tomography, forexample. In addition to improving the fidelity of themeasured porosity distributions, the authors plan toinvestigate whether porosity distributions predictedfrom casting simulation can be used in place ofmeasured porosity distributions to accurately modelthe mechanical performance of castings.

ACKNOWLEDGMENTS

This research was undertaken through the AmericanMetalcasting Consortium’s (AMC) Castings for Im-proved Readiness Research Program. AMC is spon-sored by Defense Supply Center Philadelphia (DSC,Philadelphia, PA) and the Defense Logistics Agency(DLA, Ft. Belvoir, VA). The current study was con-ducted under the auspices of the Steel Founders’Society of America (SFSA) through substantial in-kindsupport and guidance from SFSA member foundries.(Any opinions, findings, conclusions, or recommenda-tions expressed herein are those of the authors and donot necessarily reflect the views of DSC, DLA, or theSFSA and any of its members.)

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pp. 1595–1609.7. A.P. Roberts and E.J. Garboczi: J. Am. Ceram. Soc., 2000, vol. 83,

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56.10. J.M. Dewey: J. Appl. Phys., 1947, vol. 18, pp. 578–81.11. A.L. Gurson: J. Eng. Mater. Tech., 1977, vol. 99, pp. 2–15.12. V. Tvergaard: Int. J. Frac. Mech., 1981, vol. 17, pp. 389–407.

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Fig. 23—Predicted vs measured elongation for sound WCB steel(circled) and for plates with porosity. Result for specimen plateE1 is also circled.


13. A. Needleman and V. Tvergaard: J. Mech. Phys. Solids, 1984,vol. 32, pp. 461–90.

14. ABAQUS Theory Manual, Version 6.10, section 4.3.6 PorousMetal Plasticity, Dassault Systemes, Providence, RI, 2010, pp. 1–7.

15. N. Aravas: Int. J. Num. Meth. Eng., 1987, vol. 24, pp. 1395–1416.16. J. Besson: Int. J. Damage Mech., 2010, vol. 19, pp. 3–52.17. MAGMA GmbH, MAGMAsoft, Kackerstrasse 11, 52072

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18. ‘‘Standard Test Methods for Tension Testing of Metallic Materi-als, Standard Designation: E8/E8M-11,’’ ASTM International,West Conshohocken, PA, 2011, pp. 1–27.

19. ‘‘Standard Reference Radiographs for Steel Castings Up to 2 in.(51 mm) in Thickness, Standard Designation: E446,’’ ASTMInternational, West Conshohocken, PA, 2011, pp. 1–4.

20. A.P. Roberts and E.J. Garboczi: J. Am. Ceram. Soc., 2000, vol. 83,pp. 3041–48.


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