Progress Toward an Integration of Process–Structure ... · Progress Toward an Integration of Process–Structure– ... process–structure–property–performance relation-ship

Progress Toward an Integration of Process–Structure–Property–Performance Models for ‘‘Three-Dimensional(3-D) Printing’’ of Titanium Alloys

P. C. COLLINS,1,4 C. V. HADEN,2 I. GHAMARIAN,1 B. J. HAYES,1

T. ALES,1 G. PENSO,1 V. DIXIT,3 and G. HARLOW2

1.—Department of Materials Science and Engineering and the Center for Advanced Research andTesting, University of North Texas, Denton, TX 76203, USA. 2.—Mechanical Engineering andMechanics, Lehigh University, Bethlehem, PA 18015, USA. 3.—Present address: Intel Corpora-tion, Hillsboro, OR, USA. 4.—e-mail: [email protected]

Electron beam direct manufacturing, synonymously known as electron beamadditive manufacturing, along with other additive ‘‘3-D printing’’ manufac-turing processes, are receiving widespread attention as a means of producingnet-shape (or near-net-shape) components, owing to potential manufacturingbenefits. Yet, materials scientists know that differences in manufacturingprocesses often significantly influence the microstructure of even widelyaccepted materials and, thus, impact the properties and performance of amaterial in service. It is important to accelerate the understanding of theprocessing–structure–property relationship of materials being produced viathese novel approaches in a framework that considers the performance in astatistically rigorous way. This article describes the development of a processmodel, the assessment of key microstructural features to be incorporated intoa microstructure simulation model, a novel approach to extract a constitutiveequation to predict tensile properties in Ti-6Al-4V (Ti-64), and a probabilisticapproach to measure the fidelity of the property model against real data. Thisintegrated approach will provide designers a tool to vary process parametersand understand the influence on performance, enabling design and optimi-zation for these highly visible manufacturing approaches.

INTRODUCTION

Since about the mid-1990s, there have existedvarious additive manufacturing approaches for thedirect manufacture of components from structuralmetallic materials. While there were certainly earlyindustrial applications that were developed toexploit the advantages of additive manufacturing,they tended to be somewhat limited or were con-ducted in a highly proprietary or secret fashion.Only recently have the various additive manufac-turing approaches received the attention typicallyassociated with paradigm-changing technologies.Indeed, some expect the economic output of theadditive manufacturing industries to be at $3.1billion in 2016 and $5.2 billion in 2020.1,2 Invest-ments being made by industry are well aligned withthese projections. Thus, in the not-to-distant future,

various products will be produced by additivemanufacturing. Arguably, these projections of theeconomic impact are largely due to the benefitsrelated to manufacturing (e.g., near-net shapes,reduced machining, higher material yield, shortertime to insertion, increased design flexibility, andincreased part complexity). However, materialsscientists know that manufacturing processes, atthe end of the day, must manufacture a productwith a material that must perform in service. It isthen necessary to ask the question, ‘‘What is theeffect of the novel processing on the materials per-formance?’’ The materials science community alsoknows that the process can have a significantinfluence on the microstructure and properties of amaterial. Thus, process development should proceedin parallel with materials development, or at thevery least, an effort aimed at understanding of the

JOM, Vol. 66, No. 7, 2014

DOI: 10.1007/s11837-014-1007-y� 2014 The Minerals, Metals & Materials Society

(Published online June 3, 2014) 1299

process–structure–property–performance relation-ship. If additive manufacturing is going to see use inapplications where performance is required, thisrelationship needs to be established rapidly.

To accelerate the establishment of the requisitescientific and technical understanding of theseprocesses and (I) their influence on microstructuralevolution; (II) the microstructure–property rela-tionships; and (III) the expected statistical perfor-mance of material, a team consisting of industrialmembers (Boeing and Sciaky) and researchers atthe University of North Texas, Lehigh University,and The Ohio State University have adopted anintegrated approach. The approach seeks to providedesigners a means of quantifying the confidence of apredicted property as they consider the productforms where additive manufacturing achieves anadvantage (technical or economic) without an un-known risk of material or component failure duringservice. To accomplish this, the team is focused onthe following tasks:

1. The development of a process model that will beable to predict the local thermal histories andmaterials compositions

2. The development of models to predict the micro-structure, given a starting local composition andthermal history

3. The development of physically based strengthmodels that incorporate composition and micro-structure

4. The development of stochastic models to accountfor the natural probability variations that existin a given material (database)

The specific additive manufacturing process beingconsidered is the Sciaky electron beam, a wire-basedadditive manufacturing approach for large-scalestructural applications. It is being applied to thewidely used Ti-6Al-4V. Selected highlights of eachactivity are provided below, and a key linkage isdemonstrated between tasks 3 and 4.

METHODOLOGIES AND DISCUSSION

Multiphysics Process Model

Historically, the efforts to predict the thermalhistory of additive manufacturing process have beenthe most robust modeling activities related toadditive manufacturing.3–10 The team is currentlyusing COMSOL Multiphysics 4.3b (COMSOL, Inc.,Burlington, MA), a commercially available anduser-community backed multiphysics modeling andsimulation software package, to predict the localthermal history for every node of the model. Some ofthe earliest efforts at developing finite-elementmodels to predict the thermal history of additivelymanufactured components were based on the mod-eling of laser-based additive manufacturing of Ti-64by S. Kelly,6 Kelly and Kampe,11,12 and Kobryn andcolleagues.3,13–16 Their simulations demonstratedthe complex thermal histories that are expected in

additively manufactured materials, where eachthermal excursion is followed by a decay in tem-perature. In general, additive manufacturing pre-sents some nontrivial challenges, including thebasic fact that the part geometry is constantlychanging as material is being added. In addition,the environment under which additive manufac-turing is conducted may result in a spatially varyingcompositional modulation.

There are two strategies for the addition of newmaterial in these models. The first one is to imple-ment a scheme where new nodes can be added to themodel once a condition has been satisfied, such asthe time at which the heat source (and thus, newmaterial) will be present at a given x–y–z coordi-nate. The new nodes are added, and according toconventional finite-element strategies, the entiremodel is meshed (or, remeshed in this case). Obvi-ously, this introduces a nontrivial time step thatbecomes increasingly burdensome for large geome-tries where the size of the melt pool is relativelysmall. The second one is based on a strategydescribed elsewhere,17,18 in which the component ismeshed one time only, and a number of nodes are‘‘inactive’’ until the condition is met where thematerial would be added, at which point the nodesare ‘‘activated.’’ A reasonable way to accomplishsuch activation is by modifying the thermal physicalproperties of the material of the inactive node sothat they act as if the material described by the nodeis not present. For example, it is possible to set thethermal conductivity of the inactive node to be avery small fraction of the thermal conductivity ofthe active node (e.g., kinactive = 0.0001 9 kactive). Inthis manner, the inactive nodes do not conduct heataway from the melt pool or component. As this wire-fed electron beam-based additive manufacturingapproach is conducted under vacuum, this is areasonable approximation and is easier to imple-ment than powder bed based approaches. In addi-tion, because the electron beam-based additivemanufacturing process occurs under vacuum, onlyconduction and radiation are considered in solvingfor the states of heat as a function of time andlocation. The general heat equation, given in Eq. 1,

qCp@T

@tþ qCp u!� rT ¼ r � ðkrTÞ þQ (1)

can be solved using COMSOL, where q is the den-sity of the material, Cp is the specific heat capacityat constant pressure, u! is the heat flux vector,rT isthe gradient of temperature, k is the thermal con-ductivity, and Q is the internal heat energy per unitvolume added the system. For this model, the heatflux vector can be coupled with the moving heatsource. The density, specific heat capacity, andthermal conductivity are functions of temperature(and composition) and are expressed in Table I.6

Ultimately, the dependencies for compositionshould also be incorporated.

Collins, Haden, Ghamarian, Hayes, Ales, Penso, Dixit, and Harlow1300

Another complexity that arises in additivelydeposited material is the possibility that the com-position of the additively manufactured materialmay exhibit fluctuations as a function of composi-tion. This is due to two characteristics of additivemanufacturing approaches. First, there can be fluxof elemental species from the liquid into the vacuum(for electron beam processes, e.g., Al in Ti-6-4) oreither from or to the liquid from the atmosphere forother additive manufacturing processes (e.g., laser-based under an inert gas, for example, gettering oftrace oxygen from argon). Second, given the shortresidence time in the liquid phase and high solidi-fication rates, there is often insufficient time formost elemental species to redistribute homoge-neously within the liquid phase prior to solidifica-tion. For Ti-64, this effort is aimed at predicting theloss of aluminum to the vacuum. Fortunately, it ispossible to leverage the work of Semiatin et al.,19

who developed a general and simple modeling ap-proach for melt losses in electron-beam, cold hearthmelting operations that is readily extended to elec-tron beam direct manufacturing (EBDM). Theyconsider the possibility that the composition can beinterface mediated (by incorporating the Langmuirequation20) or liquid diffusion mediated. For thedetails of the model, including equations for diffu-sivity with rational activation energies and vaporpressure, the reader is referred to the work ofSemiatin et al.19 Figure 1a and b provides snap-shots of the model for a single pass for both tem-perature and solute variation as a function of heightfor a single pass. The latter shows that the alumi-num varies over 200 lm. It is noted that thesepredicted results are early results and require val-idation and calibration. Nonetheless, the strategythat is being adopted in this integrated effort isclearly demonstrated. For every location in a build,it will be possible to obtain a predicted thermalprofile and composition. These two variables willthen be directly passed to the efforts involved incharacterizing and modeling the microstructure(modeling activity 2). The composition is also arequisite input parameter to understand the mate-rial properties (modeling activity 3). The variabilityof the process and composition will also be used to

ultimately inform the natural variability that ismodeled by the stochastic models (modelingactivity 4).

Microstructural Characterizationand Modeling

Materials subject to far-from-equilibrium pro-cessing conditions or complex thermal histories,

Table I. Temperature dependencies of q, Cp, and k

Property Phase Temperature range, K Polynomial

Density, kg/m3 a + b T £ 1268 4461.1 � 0.1419*Tb 1268< T £ 1923 4462.6 � 0.1425*T

Liquid T > 1923 5227.6 � 0.688*TThermal conductivity, W/(m K) a + b T £ 1268 1.2595 + 0.0157*T

b 1268< T £ 1923 3.5127 + 0.0127*TLiquid T > 1923 �12.752 + 0.024*T

Heat capacity, J/(kg K) a + b T £ 1268 483.04 + 0.215*Tb 1268< T £ 1923 412.7 + 0.1801*T

Liquid T > 1923 831

Fig. 1. Results of preliminary thermal model showing (a) the tem-perature distribution along the first pass and (b) the predicted alu-minum concentration in a single layer due to evaporative losses fromthe molten pool.

Progress Toward an Integration of Process–Structure–Property–Performance Modelsfor ‘‘Three-Dimensional (3-D) Printing’’ of Titanium Alloys

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such as additive manufacturing, invariably containmicrostructural features that are atypical whencompared with conventionally processed materials.The effort underway by the team will ultimatelyincorporate phase-field modeling, which previouslyhas been used to predict microstructural evolutionin conventionally processed Ti-6421–24 in order topredict the microstructure as a function of compo-sition and local thermal history, provided by themultiphysics process model (modeling activity 1). Inaddition, in order to validate the phase-field modelsand provide a database of process–microstructure–property relationships in additive manufacturedTi-64 for subsequent extraction of constitutive pre-dictive models for yield strength, it is necessary toaccurately characterize the microstructure of theas-deposited material.

Figures 2a, c and 3 show microstructural detailsof Ti-64 of an electron beam additively manufac-tured build. In the very low magnification opticalmicrograph shown in Fig. 2a, it is clear that the as-deposited microstructure is highly anisotropic.There are a series of horizontal bands that are theresult of residual stress, as well as elongated priorb grains that span across multiple layer passes. Thepresence of the residual stress has been confirmedthrough both microhardness indentations withinthe banded structure and orientation maps (seeFig. 2b) where the color and contrast correspond toorientation and residual stress, respectively. Fig-ure 2c shows a strong 001 body-centered cubic (bcc)b growth direction and the corresponding hexagonalclose-packed (hcp) a texture. Thus, in additivelymanufactured materials, it is reasonable to expectspatially varying composition, residual stress, andanisotropic and highly textured b grains. In addi-tion, the variation in the thermal history, specifi-cally the last thermal excursion and subsequentcooling from above the b-transus, will result in dif-ferent distributions of the a-laths in the priorb grains. The variations can be significant over arelatively small length scale. Consider Fig. 3, whichshows such variation along a single prior b grainboundary. In the top half of the micrograph, thereare coarse a-laths along the grain boundary as wellas a coarse basketweave microstructure within thegrains. In the bottom part of the micrograph, thea-laths along the grain boundary are finer, and thegrain interior is more colony like. In addition, itappears as if the volume fraction of the two phasesis also different, although careful analysis is neededto determine whether the b regions contain veryfine-scale secondary a-laths. These observations areconsistent with spatially varying thermal historyand materials composition, though the preciseinterrelationships have yet to be rigorously estab-lished. While the parts are ultimately subjected to a

Fig. 2. Ti-64 produced via additive manufacturing. (a) Optical micrograph showing elongated vertical grains and horizontal bands of residualstress originating from the AM process. (b) Orientation map, where color and contrast correspond to orientation and residual stress, respectively.(c) Strong 001-bcc growth direction and corresponding hcp texture.

Fig. 3. Scanning electron micrograph showing the significant varia-tion in the scale and morphology of the microstructural featureswithin an electron beam additively manufactured Ti-64 deposit.


stress-relief anneal, the starting microstructure andits variability is remembered. Although the scale ofthe microstructural features is coarser after heattreatment, the variation in size and type (i.e., bas-ketweave versus colony) is retained. The quantifiedmicrostructure will be used to model the materialproperties (modeling activity 3). The variability ofmicrostructure will also be used to ultimately in-form the natural variability that is modeled by thestochastic models (modeling activity 4).

Determination of a Constitutive Equationfor the Prediction of Yield Strength

Ultimately, the microstructural features, includ-ing specifically the volume fraction of the a and bphases, the thickness of the Widmanstatten a-laths,the colony scale factor, the percent colony, and thethickness of the grain boundary a will be quantifiedusing rigorously derived stereological procedures25

across a wide range of specimens, resulting in adatabase consisting of thermal history, composition,microstructure, and properties. As the materialscontain a natural distribution in the size of thefeatures, it is necessary to develop distributions offeature sizes for inclusion in the database. Thisdatabase will then be interrogated using a novelapproach in which artificial neural networks havebeen linked with genetic algorithms to deduce awell-hidden constitutive equation for yieldstrength.26

Until quite recently, the prediction of many de-sign properties (e.g., yield strength, ductility, frac-ture toughness, and fatigue) for multicomponent,multiphase engineering alloys, including specifi-cally titanium alloys, has not been possible. This isdue to the complex interrelationships that existamong composition, microstructure, and properties.As shown previously by Kar et al.27 and Collinset al.,28 it is quite difficult to systematically vary asingle microstructural variable while keeping allothers fixed at some value and thus determine thedependency of a property on a given feature. How-ever, these efforts did demonstrate that artificialneural networks could be exercised on high-fidelitydatabases to capture the hidden interrelationshipsand unknown physics through a summation ofweighted and biased hyperbolic tangent functions.In addition, these previous efforts demonstratedthat it was possible to conduct virtual experimentsto understand the influence of microstructure onproperties. Unfortunately, the forms of the equa-tions for yield strength that are the outcome of anoptimized neural network architecture are quitecomplex, with forms that do not readily show theinterrelationships between the compositional andmicrostructural details and the attending mechan-ical properties in any form that represents ‘‘legacyforms’’ (e.g., solid solution strengthening, theHall–Petch relationship, etc.29). In addition, as theneural network approach is interpolative and

the hyperbolic tangent functions are quite flexible,there is a risk when making predictions beyond theranges of the database against which the optimizedequation has been extracted.

The works of Ghamarian et al.26,30 demonstratethat a new, state-of-the-art approach is possible.This approach, shown in Fig. 4, is based on thepremise that the optimized flexible hyperbolic tan-gent functions have captured the physics and can berewritten in an equivalent form that captures the-ories related to the property of interest, in this casestrengthening mechanisms. Thus, this approachseeks to extract a physically based equation for yieldstrength that most closely approximates the com-plex hyperbolic tangent function deduced by theartificial neural network. To accomplish this, adatabase containing input variables (e.g., composi-tion, microstructure) and outcomes (e.g., yieldstrength) is interrogated using artificial neuralnetworks (labeled #1: ANN in Fig. 4), and from thisa series of virtual experiments is conducted toassess the dependence of the outcome, yieldstrength, on the composition and microstructure.These virtual experiments then point to potentialstrengthening mechanisms, and a constitutiveequation is postulated, based on the rather generalstrengthening equation given in Eq. 2

rys ¼ ro þ rss þ rppt þ rdisp þ rgrain size þ rinterface þ . . .

(2)

where ro is the intrinsic flow strength of the mate-rial without any other contribution to strength, rSS

is the contribution from solid solution strengthen-ing, rppt is the contribution from precipitatestrengthening, rdisp is the contribution from dis-persion strengthening, rgrain size is the contributionfrom grain size effects (i.e., the Hall–Petch effect),and rinterface is the contribution from interfacialstrengthening. Other strengthening mechanisms,such as forest hardening, are not included in theexample equation above. This equation can beappropriately modified for multicomponent andmultiphase materials.26,29 For example, based onboth the virtual dependencies extracted from theartificial neural network and well-established formsof equations for strengthening mechanisms, Eq. 3 ispostulated for a high-fidelity a + b wrought Ti-64database.* In its current form, there are manyunknowns. The unknowns of this equation are thenoptimized using genetic algorithms (labeled #2:Genetic Algorithms in Fig. 4), which iterativelysolves for the solution that leads to the optimumresult (see Eq. 4).

*Currently, a database for electron beam additively manufac-tured Ti-64 does not exist. This database is underdevelopment. Todemonstrate the integration of the tool, another database hasbeen used, and another one can be used to make predictions ofstrength in the a + b wrought substrate onto which the Ti-64 isadditively deposited.


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Importantly, the same virtual experiments can beconducted on the optimized equation from the ge-netic algorithm. The premise of this hybrid artificialneural network + genetic algorithm approach isthat if the virtual experiments from both the arti-ficial neural network and the genetic algorithmclosely match each other, then the constitutiveequation is an approximation of the more complex

equation containing a summation of weighted andbiased hyperbolic tangents. Monte Carlo simula-tions have confirmed that, given the individualstatistical uncertainties of the microstructuralmeasurements for every data point in the database,the two equations are statistically equivalent.

The compositional and microstructural virtualdependencies determined by the artificial neural

rysðMPaÞ ¼ ðra0 � Fa

VÞ þ ðrb0 � Fb

VÞþ Two - phase composite of intrinsic strengthFa

V � ðAAl � CnAl

Al þ AO � CnO

O Þþ Solid - solution strengthening - hcp alpha

FbV � ððAV � CnV

V Þn1 þ ðAFe � CnFe

Fe Þn2Þn3þ Potential synergistic solid - solution strengthening - bcc beta

kequiaxed ay � Fequiaxed a

V � Equiaxed size�n4þ Hall - Petch effect - Equiaxed alpha particles

ð1� Fequiaxed aV Þ � Colony

100 � ka lathy � ðLWÞ�n5 � ðRTÞn5þ Hall - Petch effect - Alpha lath

ð1� Fequiaxed aV Þ � 100�Colony

100 � B � SSS Basketweave factor

(3)

rysðMPaÞ ¼ ð89 � FaVÞ þ ð45 � Fb

VÞþ Two - phase composite of intrinsic strengthFa

V � ð149:5 � C0:667Al þ 745 � C0:667

O Þþ Solid - solution strengthening - hcp alpha

FbV � ðð34 � C0:765

V Þ0:5 þ ð245 � C0:765Fe Þ0:5Þ2:15þ Potential synergistic solid - solution strengthening - bcc beta

110 � Fequiaxed aV � Equiaxed size�0:5þ Hall - Petch effect - Equiaxed alpha particles

ð1� Fequiaxed aV Þ � Colony

100 � 126 � ðLWÞ�0:26 � ðRTÞ0:26þ Hall - Petch effect - Alpha lath

ð1� Fequiaxed aV Þ � 100�Colony

100 � 0:25 � SSS Basketweave factor

(4)

Fig. 4. The integration approach is shown. The combination of neural networks, genetic algorithm, and Monte Carlo simulation leads to derivinga phenomenological equation from a database.


network and genetic algorithm are given in Fig. 5a–d and 6a–d, respectively. Clearly, the predicted re-sults using both the hyperbolic tangent functions ofthe neural network and the phenomenological are invery close agreement. Some of these terms, oncededuced, are consistent with the limited legacydata. For example, the intrinsic strength of hcp a inthe model is 89 MPa, which is consistent with thevarious literature sources on very high-purity, well-annealed titanium. Others, such as the intrinsicstrength of pure bcc b, are not available in the lit-erature. For the case of the intrinsic strength ofpure bcc b, this is due to the obvious fact that it isnot a thermodynamically stable at room tempera-ture. Although currently under review, such anequation has not been presented in the literature.26

Key Linkage

Although this equation is useful for a materialsscientist and obviously can be directly coupled to thecomposition output of the process model (modelingactivity 1) and the microstructures predicted by thephase-field model (modeling activity 2), it is lessuseful for a designer. Ultimately, a designer mustknow the level of confidence associated with aproperty, whether experimentally measured orpredicted. Thus, it is important that this equation

be assessed using stochastic approaches, as isshown in the next section.

Determination of Stochastic Modelsto Account for Uncertainty of the ConstitutiveEquation for the Prediction of Yield Strength

Such constitutive equations (see Eq. 4), whilechallenging to deduce and thus relatively rare forreal multiphase and multicomponent engineeringmaterials, are deterministic in nature and do notaccount for the large variability that is known tooccur naturally for each variable.25,27 For this rea-son, an additional and innovative stochastic ana-lysis was performed on the constitutive model givenin the previous section. For this approach, variablestates in the constitutive equation were describedusing the most appropriate probability distributionsof Weibull, bimodal Weibull, normal, and uniformdistributions to describe the inherent uncertainty inthe composition and microstructure variables. Thefull approach is described in more detail below andallows for a confidence interval prediction on theyield stress prediction of the model. To the best ofknowledge of the authors, such a confidence pre-diction on the predicted material strength of a/bprocessed Ti-6Al-4V alloys has never been produced.

Fig. 5. Compositional virtual dependencies calculated by neural network and genetic algorithm for (a) Al, (b) Fe, (c) O, and (d) V are shown. Yieldstrength considerably increases by increasing the Al and O concentration. In contrast, the concentration of V has a negligible influence on yieldstrength.


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It is necessary to first introduce the variability ofthe compositional and microstructural inputs. Forexample, the variability in material composition(composition of elements O, Fe, Al, and V) wasmodeled using a uniform cumulative distributionfunction (CDF) that was fit to the data obtainedfrom the parameter space discussed above andbased on prior literature.31 Other distributions,including normal, Weibull, and bimodal Weibull

CDFs, are potential probability distribution func-tions. In general, the ‘‘best’’ CDF can be deduced byplotting the raw data in both a normal and Weibullreference frame. The plot that results in the databeing presented as a straight line is the ‘‘best’’ CDF(see, for example, Figs. 7 and 8). Using this ap-proach, the volume fractions of alpha and betaparticles were best approximated by normal CDFs,whereas Weibull and bimodal Weibull CDFs were

Fig. 6. Microstructural virtual dependencies calculated by neural networks and genetic algorithm for (a) mean equiaxed alpha size, (b) volumefraction of equiaxed alpha, (c) volume fraction of total alpha, and (d) lath thickness are shown.

Fig. 7. Curve fit (line) and experimental data (red circles) for thevolume fraction equiaxed alpha. Plotted as a normal probability.

Fig. 8. Equiaxed alpha particle size data (red circles) and regressionline (black line) as plotted on a Weibull probability plot.


used to estimate the equiaxed size and colony vari-ables, respectively. These variable estimations werethen used to produce a Monte Carlo simulation ofthe phenomenological equation described above,producing a prediction range on the yield strengthcalculation for the wrought a + b Ti-64. Ultimately,a similar approach will be used for electron beamadditively manufactured material. While the fulldetails will be presented soon by Collins et al.,32 twoexamples of the distribution formalisms are givenbelow. These relate to the equiaxed alpha volumefraction and particle size, distinctive features ofwrought a + b Ti-64.

An assumption of normality was made for thedistribution of volume fractions of the equiaxed al-pha particles in the wrought a + b Ti-64. The nor-mal distribution of the data was confirmed by thelinear relationship of the volume fraction variableswhen plotted on normal probability paper, as shownin Fig. 7. The parameters l and r for the normalCDF (Eq. 5)

FxðxÞ ¼Zx

�1

1ffiffiffiffiffiffiffiffiffiffiffi2pr2p e

12

x�lrð Þ

2

dx (5)

were estimated from the curve fit through the data(also shown in Fig. 7) according to Eqs. 6 and 7

r ¼ 1

m(6)

l ¼ �b

m(7)

where m and b are the slope and intercept of thecurve fit to the data.

To simulate the volume fraction variables, theBox–Muller transform method was used to generatenormally distributed random numbers between 0and 1 (uN = N(0,1)). The normal CDF was then setequal to these random numbers and solved for xi asfollows (Eqs. 8 and 9):

FxðxiÞ ¼ uNi(8)

xi ¼ F�1 uNið Þ (9)

where x represents either volume fraction variables,Fv

a or Fveq_a. The normally distributed variables of

volume fraction are then simulated by the followingequation (Eq. 10),

xi ¼ r � uNiþ l (10)

where r and l are the estimates for the mean andthe standard deviation found from the curve fit tothe raw data of the volume fraction of equiaxed aparticles.

In a similar way, it is possible to describe theWeibull probability distribution function for theequiaxed a particle size (see Fig. 8). The fact thatthe data distribution is approximately linear on thisWeibull plot indicates the distribution of the equi-axed a particle size follows a two-parameter Weibulldistribution, and it is therefore suitably modeledusing such a distribution (Eq. 11).

FxðxÞ ¼ 1� e�xbð Þ

a

(11)

where the parameters a and b are approximated bythe curve fit through the data as shown in Fig. 8and according to the following equations (Eqs. 12and 13)

a ¼ m (12)

b ¼ e�ba (13)

To simulate a Weibull distribution for the equi-axed a particle size variable, the two-parameterWeibull CDF (Eq. 11) was then set equal to uni-formly distributed random numbers and solved forx, as given below (Eqs. 14 and 15).

Table II. List of all independent variables in the constitutive equation with their respective CDFs used tomodel their uncertainty

Variable name CDF Range l r2 CoV

CO Uniform [0.0639, 0.2718] 0.1353 0.0033 0.4218CAl Uniform [4.284, 7.205] 5.66 0.6830 0.1461CV Uniform [2.967, 4.895] 3.84 0.2350 0.1262CFe Uniform [0.0963, 0.4477] 0.2522 0.0185 0.5399FV

total alpha Normal – 90.24 2.8527 0.0187FV

equiaxed alpha Normal – 55.24 152.0289 0.2221Equiaxed a size Weibull – 6.9064 0.7189 0.1228Colony size Bimodal Weibull – – – –

Data include (when appropriate or available) the range on which these data were generated, as well as the mean, variance, and coefficientof variation for their distributions


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FxðxiÞ ¼ ui (14)

xi ¼ F�1 uið Þ (15)

where u = U(0,1). Similar approaches were takenfor the other microstructural variables.

Table II summarizes the list of nine variablessimulated for use in the constitutive equation givenin the previous section.

The type of cumulative frequency distributionused to represent each variable is shown, as well asthe range, mean, variance, and coefficient of varia-tion for their distributions when appropriate.

The variables defined above were used to producea Monte Carlo simulation of the yield stress usingEq. 4. As seen in Fig. 9, the distribution of theexperimental data obtained for the yield strengthcorrespond well to the simulated values in the upperrange, above approximately 780 MPa. A bifurcationfrom the simulation values in the lower regimeindicates a possible bimodal distribution in thesedata. Fortunately, the Monte Carlo simulation al-lows the prediction of a confidence interval aroundthe predicted values of yield stress, with one andtwo standard deviations shown in blue. As seen inthis figure, the data lay closely within this confi-dence interval, except for a few data around700 MPa, a relatively low yield point for Ti-64.

Further refinement of the stochastic model will bepossible with a wider range of parameters once theybecome available. However, this first approximationon the yield strength of wrought a + b Ti-6-4 alloysis valuable in drawing a confidence range on thecomplex relationship between the variables andoutput generated by the phenomenological equation(Eq. 1).

State of the Art

As noted in the text, the establishment of a con-stitutive equation for the prediction of tensileproperties given a certain composition/micro-structure and the assessment of the uncertainty ofthe predictions represent the current state of theart. Rarely are such constitutive equations producedfor multicomponent and multiphase engineeringalloys. When such equations do exist, invariablythey lack an assessment of the quality of the equa-tions/model. In principle, the stochastic modelingapproach provides an initial method of validatingthe constitutive equations and gives an assessmentof ‘‘confidence’’ associated with an equation. Thedistributions of both the predicted data and exper-imentally measured data should be equivalent.However, these are for well-populated data sets.Thus, in practice for the foreseeable future, thereremains the need to apply these predictive tools tomaterials that are not part of the original database.Only then will there be true confidence in the vali-dation efforts. As applied here for EBDM, the effortsare to validate the models. Once validated, thesenew methods promise to shorten the developmentcycle and ultimately result in fewer tests beingconducted when assessing such new manufacturingapproaches.

CONCLUSION

As this article is presented as an integrated effortto develop and couple process, structure, property,and performance models of electron beam additive(or direct) manufacturing, the conclusions are pre-sented as ‘‘the path forward.’’ It summarizes wherethe team efforts are, as well as discusses some nextsteps.

The Path Forward

The various activities, as described above, eachcontains breakthroughs that have not yet beenpresented in the literature. While many are workingon developing thermal modeling for additive man-ufacturing, the authors are not aware of any whohave integrated a means of predicting variation inthe solute content within the part, which will ulti-mately significantly impact the local properties, asshown by the constitutive equation (Eq. 4). Addi-tionally, efforts are underway to accelerate theprocess model. New techniques based on spectral orfast Fourier transform (FFT)-based finite elementmodeling (FEM) models offer a potentially attrac-tive means to accomplish such acceleration. Thecharacterization clearly points to a significant spa-tial variation in the as-deposited microstructure,along with spatially varying residual stress. Thesevariations will be better understood by coupling inphase-field modeling. The spatial variation willultimately be quantified and will be a key input in

Fig. 9. Probability of experimental yield strength data (red circles)compared with the constitutive model prediction (black squares) andMonte Carlo simulations (black line). A confidence interval of oneand two standard deviations is also shown (blue dashed lines).


this integrated strategy. Among the most excitingdevelopments of this program are the provision of amodeling strategy to deduce a well-hidden consti-tutive equation of a property (e.g., yield strength)and the subsequent interrogation of the constitutiveequation against a database to understand how wellit predicts the given property, given the naturalvariation that occurs for each compositional andmicrostructural variable. Such an integration hasnot been presented for structural titanium alloys.However, it is a means of providing a statisticalframework for subsequent uncertainty analysis,which is important for the designer and provides the‘‘E’’ in ICME (Integrated Computational MaterialsEngineering). While this initial effort to extract aconstitutive equation and model its predictive abil-ity using stochastic models has been demonstratedfor wrought a + b Ti-64, it is expected to be partic-ularly well suited for an additively manufacturedelectron beam which exhibits natural variations inboth composition and microstructure that should bequantified. It is important to also consider what ismissing from this integrated strategy. This currentstrategy does not seek to understand the physicsassociated with solidification process, although it isobvious that such an effort is ultimately needed fora broader understanding of additive manufacturing.In addition, this approach does not include a costmodel, which would also be a requisite tool for endusers.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the supportof DARPA Contract HR0011-12-C-0035, ‘‘An OpenManufacturing Environment for Titanium Fabrica-tion,’’ the support of Boeing Research and Technol-ogy and the technical lead (David Bowden), Sciaky,and The Ohio State University. In addition, theauthors are grateful to Lehigh University, theUniversity of North Texas, and UNT’s Center forAdvanced Research and Technology (CART).

REFERENCES

1. T.T. Wohlers, Wohlers Report (2013).2. T.T. Wohlers, Wohlers Report (2010).3. P. Kobryn and S. Semiatin, J. Mater. Process. Technol. 135,

330 (2003).

4. P. Kobryn and S. Semiatin, JOM 53, 40 (2001).5. S. Kelly, S. Babu, S. David, T. Zacharia, and S. Kampe, Cost-

Affordable Titanium: Symposium Dedicated to ProfessorHarvey Flower, ed. F.H. Froes, M.A. Imam, and D. Fray(Warrendale, PA: TMS, 2004), pp. 45–52.

6. S.M. Kelly (Master’s thesis, Virginia Polytechnic Instituteand State University, 2002).

7. K.T. Makiewicz (Ph.D. Dissertation, The Ohio State Uni-versity, 2013).

8. B. Vrancken, L. Thijs, J.-P. Kruth, and J. Van Humbeeck, J.Alloys Compd. 541, 177 (2012).

9. C. Charles (Licentiate Thesis, Lulea University of Technol-ogy, 2008).

10. N. Shen and Y. Chou, 23rd Annual International SolidFreeform Fabrication Symposium, ed. D.L. Bourell (Austin,TX: University of Texas at Austin, 2012), pp. 774–784.

11. S. Kelly and S. Kampe, Metall. Mater. Trans. A 35, 1861(2004).

12. S. Kelly and S. Kampe, Metall. Mater. Trans. A 35, 1869(2004).

13. S. Bontha, N.W. Klingbeil, P.A. Kobryn, and H.L. Fraser, J.Mater. Process. Technol. 178, 135 (2006).

14. S. Semiatin, P. Kobryn, E. Roush, D. Furrer, T. Howson, R.Boyer, and D. Chellman, Metall. Mater. Trans. A 32, 1801(2001).

15. E. Roush, P. Kobryn, and S. Semiatin, Scr. Mater. 45, 717(2001).

16. S. Bontha, N.W. Klingbeil, P.A. Kobryn, and H.L. Fraser,Mater. Sci. Eng. A 513, 311 (2009).

17. H. Conrad, T. Klose, T. Sandner, D. Jung, H. Schenk, and H.Lakner, Proceedings of COMSOL Conference (Hannover,Germany, 2008).

18. P. Michaleris, private communication, 2013.19. S. Semiatin, V. Ivanchenko, and O. Ivasishin, Metall. Mater.

Trans. B 35, 235 (2004).20. I. Langmuir, Phys. Rev. 2, 329 (1913).21. Q. Chen, N. Ma, K. Wu, and Y. Wang, Scr. Mater. 50, 471

(2004).22. F. Zhang, S.-L. Chen, Y. Chang, N. Ma, Y. Wang, and J.

Phase, Equilib. Diffus. 28, 115 (2007).23. R. Shi, N. Ma, and Y. Wang, Acta Mater. 60, 4172 (2012).24. R. Shi, V. Dixit, H.L. Fraser, and Y. Wang, Acta Mater.

(accepted for publication, 2014).25. P. Collins, B. Welk, T. Searles, J. Tiley, J. Russ, and H.

Fraser, Mater. Sci. Eng. A 508, 174 (2009).26. I. Ghamarian, P. Samimi, V. Dixit, and P.C. Collins, Metall.

Mater. Trans. A (submitted for publication).27. S. Kar, T. Searles, E. Lee, G. Viswanathan, H. Fraser, J.

Tiley, and R. Banerjee, Metall. Mater. Trans. A 37, 559(2006).

28. P.C. Collins, S. Koduri, B. Welk, J. Tiley, and H.L. Fraser,Metall. Mater. Trans. A 44, 1441 (2013).

29. P.C. Collins and H.L. Fraser, ASM Handb. 22A, 377 (2009).30. I. Ghamarian (Master thesis, University ofNorth Texas, 2012).31. W. Nicholson, Biometrika 57, 273 (1970).32. P.C. Collins, C.V. Haden, and D.G. Harlow, unpublished

research, 2014.


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