Modeling and Simulation of Press and Sinter Powder Metallurgy · Modeling and Simulation of Press...

Modeling and Simulation of Press andSinter Powder MetallurgySuk Hwan Chung, Hyundai Steel CompanyYoung-Sam Kwon, CetaTech, Inc.Seong Jin Park, Pohang University of Science and TechnologyRandall M. German, San Diego State University

EFFECTIVE COMPUTER SIMULATIONSof metal powder compaction and sintering areat the top of the powder metallurgy industry’swish list. There is much anticipated advantageto such efforts, yet there are problems that willinhibit widespread implementation. Press-sinterpowder metallurgy computer simulations cur-rently focus on the use of minimal input datato help with process setup. Although the simula-tions are reasonably accurate, a large data arrayis required to hone in on current industrial prac-tice. For example, final dimensions for automo-tive transmission gears are required to be heldwithin 10 mm, but the simulations are not capa-ble of such accuracy. Simple factors such asfrictional tool heating are missing from thesimulations. Additionally, powders vary in par-ticle size distribution between production lots,but the simulations assume a nominally uniformpowder. Because it is expensive to test eachpowder lot, the logic is to assume a nominalset of characteristics. In production, such pro-cess and powder variations are handled by con-stant adaptive control techniques. As anexample, when an outside door is opened onthe press room, it is common that press adjust-ments will be required to hold sintered dimen-sions. The press-sinter powder metallurgysimulations have not advanced to such levelsof sophistication. Instead, the press-sinter pow-der metallurgy simulations are used to help setup production operations, with heavy relianceon experienced operators to make final trial-and-error adjustments.In practice, the variations in powder, press,

tooling, and other process variables are handledthrough skilled technicians, quality charts, andadaptive process control that relies on frequentsampling and periodic equipment adjustments.The gap between press-sinter practice and mod-eling may close if more rapid data-generationroutes were developed. For example, a studyon modeling the press-sinter production of amain bearing cap required 10,000 measures toisolate the behavior. It is not economically

feasible to repeat this testing for each 20 tonlot of powder. Even so, a great benefit comesfrom the fact that the computer simulationshave forced the technical community to orga-nize our knowledge and determine where thereare problems.Computer simulations of press-sinter opera-

tions trace to the 1960s (Ref 1 to 20). Theearly simulations were generally unstable andtwo-dimensional (for example, the sinteringof aligned wires). By 1975, a variety of two-dimensional sintering approaches existed.With the expansion in computer power, theimplementation of three-dimensional simula-tions arose to provide realistic outputs. Inmore recent times, the simulations haveprovided valuable three-dimensional treat-ments to predict the final component size andshape after sintering. Because the pressedgreen body is not homogeneous, backwardsolutions are desired to select the powder,compaction, and sintering attributes requiredto deliver the target properties with differenttool designs, compaction presses, and sinteringfurnaces. In building toward this goal, varioussimulation types have been evaluated: MonteCarlo, finite difference, discrete element, finiteelement, fluid mechanics, continuum mechan-ics, neural network, and adaptive learning.Unfortunately, the input data and some of thebasic relations are not well developed; accu-rate data are missing for most materials underthe relevant conditions. For example, rarely isthe strength measured for a steel alloy at thetypical 1120 �C sintering temperature. Further,constitutive models do not exist for the condi-tions relevant to sintering; for example, fric-tion in die compaction changes during thesplit-second pressure stroke, because lubricant(polymer) particles deform and undergo vis-cous flow to the die wall, effectively changingfriction constantly during compaction. Thus,the simulations are approximations usingextrapolated data and simplified relations. Forthis reason, computer simulations of press-

sinter routes work best in the setup mode.The simulations help define the processingwindow and set initial operating parameters.Presented here is information relevant to com-puter simulation of first-article production,what is best termed setup calculations, realiz-ing that practice relies heavily on adaptiveprocess control to keep the product in specifi-cation after the initial setup is accomplished.

Brief History

The first major publication on computer sim-ulation of sintering came out in 1965 (Ref 1).Early simulations were two-dimensional (sin-tering two wires) with a single diffusion mech-anism. These simulations were slow, requiringten times more computer time than the actualphysical sintering time. Most damaging, theseearly models were unstable, because they lostvolume and increased energy. However, within20 years the concept was extended to includemultiple transport mechanisms, multiple sinter-ing stages, and even pressure-assisted sintering(Ref 6, 9 to 11). These simulations predicteddensity versus compaction pressure, sinteringtime, peak temperature, heating rate, green den-sity, and particle size.One of the first realizations was the limita-

tions arising from the assumed isothermalconditions and simplistic microstructure coars-ening. Dilatometry experiments show that mostsintering occurs on the way to the peak temper-ature, so isothermal models poorly reflect actualbehavior (Ref 20). Indeed, production powdermetallurgy often simply “kisses” the peak tem-perature, a situation far from what is assumedin the simulations. Also, the assumed homoge-neous and ideal microstructure unrealisticallylimits the models. Today (2009), the sinteringbody first treated with a compaction or shapingsimulation to predict the green microstructuregradients, and subsequent sintering simulations,use those density gradients, via finite-element

ASM Handbook, Volume 22B, Metals Process Simulation

D.U. Furrer and S.L. Semiatin, editors

Copyright # 2010, ASM International®

All rights reserved.

www.asminternational.org

analysis, to predict the final size, shape, andproperties (Ref 16 to 19).

Theoretical Background andGoverning Equations

The methodologies used to model the pressand sinter powder metallurgy include contin-uum, micromechanical, multiparticle, andmolecular dynamics approaches. These differin length scales. Among the methodologies,continuum models have the benefit of shortestcomputing time, with an ability to predict rele-vant attributes such as the component density,grain size, and shape.Mass, volume, and momentum conservation

are evoked in the continuum approach.Although such assumptions may seem obvious,powder metallurgy processes are ill-behavedand difficult to properly simulate. For example,polymers are added to the powder for toollubrication, but the polymers are pyrolyzedduring sintering, resulting in 0.5 to 1.5 wt%mass loss. Likewise, pore space is not con-served during compaction and sintering, sobulk volume is not conserved. Even so, massconservation equations are invoked to trackdensification, while momentum conservationis used to follow force equilibrium, includingthe distortion effect from gravity. Energy con-servation is also essential in the continuumapproach. However, it is typical to assume tem-perature is uniform in the compact—set toroom temperature during compaction and fol-lowing an idealized thermal cycle (often iso-thermal) during sintering. Both are incorrect,because tool heating occurs with repeated com-paction strokes, and compact position in thesintering furnace gives a lagging thermal his-tory that depends on location. Indeed, becausedimensional precision is the key to powdermetallurgy, statistical audits have repeatedlyfound that subtle factors such as position inthe furnace are root causes of dimensional scat-ter. For example, fluid flow and heat-transportcalculations show considerable temperature dif-ferences associated with atmosphere flow andcomponent shadowing within a furnace.Because such details are not embraced by themodels, the typical assumption is to ignoretemperature distribution within the component,yet such factors are known to cause part distor-tion during production.Additionally, constitutive relations are

required to describe the response of the com-pact to mechanical force during compactionand sintering. Many powder metallurgy materi-als are formed by mixing powders that melt,react, diffuse, and alloy during sintering. Thisrequires sophistication in the models to addphase transformations, alloying, and other fac-tors, many of which depend on particle sizeand other variations (Ref 15). From conserva-tion laws and constitutive relations, a systemof partial differential equations is created that

includes the initial and boundary conditions.These must be integrated with microstructureand property models so that final compact prop-erties can be predicted. Because the constitutiverelations for compaction and sintering arecompletely different, they are described herein two separate sections.

Constitutive Relation duringCompaction

Continuum plasticity models are frequentlyused to describe the mechanical response ofmetal powders during compaction. These phe-nomenological models, originally developed insoil mechanics, are characterized by a yield cri-terion, a hardening function, and a flow rule.Representative models include those known asthe Cam-Clay (Ref 21), Drucker-Prager-Cap(Ref 22), and Shima-Oyane (Ref 23) models.Of these, the most successful for metal powdershas been the Shima-Oyane model, although forceramics, soils, and minerals, other relations aregenerally more successful.The typical initial and boundary conditions

during compaction are as follows:

� Initial condition for the powder: Tap density� Boundary conditions: Velocity prescribed in

upper and bottom punches and friction con-dition in the tooling side wall; usuallyassumed the same for all tool surfaces inde-pendent of wear and independent of lubri-cant flow during the compaction stroke

During compaction and ejection, a damagemodel, such as the Drucker-Prager failure sur-face (Ref 22) and failure separation length(FSL) idea (Ref 24), is required. To predict crackformation, the FSL assumes there is an accumu-lated separation length from the Drucker-Pragerfailure surface, which provides the possibility

of crack formation, as shown in Fig. 1. The equa-tion for the FSL is expressed as:

FS ¼ q þ p tan b� d (Eq 1)

where q and p are the effective stress and hydro-static pressure, respectively. Note that d and bare the offset stress and slope, respectively, forthe Drucker-Prager failure surface shown inFig. 1. Because the models predict green den-sity versus location, defect sensitivity is possi-ble. For example, elastic relaxation occurs onejection, and if the stress exceeds the greenstrength, then green cracking occurs. It is in thisarea that the compaction models are mosteffective.

Constitutive Relation during Sintering

Continuum modeling is the most relevantapproach to modeling grain growth, densifica-tion, and deformation during sintering. Key con-tributions were by Ashby (Ref 6, 9), McMeekingand Kuhn (Ref 25), Olevsky et al. (Ref 17, 19),Riedel et al. (Ref 13, 26, 27), Bouvard and Meis-ter (28), Cocks (29), Kwon et al. (Ref 30, 31), andBordia and Scherer (Ref 32 to 34) based on a sin-tering mechanism such as surface diffusion,grain-boundary diffusion, volume diffusion, vis-cous flow (for amorphous materials), plastic flow(for crystalline materials), evaporation conden-sation, and rearrangement. For industrial appli-cation, the phenomenological models are usedfor sintering simulations with the following keyphysical parameters:

� Sintering stress (Ref 20) is a driving force ofsintering due to interfacial energy of poresand grain boundaries. Sintering stress dependson the material surface energy, density, andgeometric parameters such as grain size whenall pores are closed in the final stage.

Fig. 1 Definition of failure separation length (FSL) based on Shima-Oyane yield model and Drucker-Prager failuresurface. d, offset stress; b, slope

324 / Simulation of Powder Metallurgy Processes

� Effective bulk viscosity is a resistance todensification during sintering and is a func-tion of the material, porosity, grain size,and temperature. The model of the effectivebulk viscosity has various forms accordingto the assumed dominant sinteringmechanism.

� Effective shear viscosity is a resistance todeformation during sintering and is also afunction of the material, porosity, grain size,and temperature. Several rheological modelsfor the effective bulk viscosity are available.

The preceding parameters are a function ofgrain size. Therefore, a grain-growth model isneeded for accurate prediction of densificationand deformation during sintering.Typical initial and boundary conditions for

the sintering simulations include:

� Initial condition: Mean particle size andgrain size of the green compact for graingrowth and initial green density distributionfor densification obtained from compactionsimulations

� Boundary conditions: Surface energy condi-tion imposed on the free surface and frictioncondition of the component depending on itssize, shape, and contact with the supportsubstrate

The initial green density distribution within thepressed body raises the necessity of starting thesintering simulation with the output from anaccurate compaction simulation, because diecompaction induces green density gradients thatdepend on the material, pressure, rate of pres-surization, tool motions, and lubrication. Theinitial and boundary conditions help determinethe shape distortion during sintering from grav-ity, nonuniform heating, and the green bodydensity gradients.Numerical Simulation. Even though many

numerical methods have been developed, thefinite-element method (FEM) is most popularfor continuum models of the press and sinterprocess. The FEM approach is a numericalcomputational method for solving a system ofdifferential equations through approximationfunctions applied to each element, calleddomain-wise approximation. This method isvery powerful for the typical complex geome-tries encountered in powder metallurgy. This isone of the earliest techniques applied to materi-als modeling and is used throughout industrytoday (2009). Many powerful commercial soft-ware packages are available for calculatingtwo- and three-dimensional thermomechanicalprocesses such as found in press and sinter pow-der metallurgy.To increase the accuracy and convergence

speed for the press and sinter simulations,developers of the simulation tools have selectedexplicit and implicit algorithms for timeadvancement, numerical contact algorithms forproblems such as surface separation, andremeshing algorithms as required for large

deformations such as seen in some sinteredmaterials, where up to 25% dimensional con-traction is possible.Figure 2 shows the typical procedure for

computer simulation for the press-sinterprocess, which consists of five components:simulation tool, pre- and postprocessors, opti-mization algorithm, and experimental capabil-ity. Pre- and postprocessors are important forusing the simulation tools efficiently. The pre-processor is a software tool to prepare inputdata for the simulation tool, including computa-tional domain preparation such as geometrymodeling and mesh generator. Figure 3 is anexample of the component, compaction, andsintering models, in this case for an oxygensensor housing. Executing this model requiresconsiderable input, including a material prop-erty database (including strain effects duringcompaction and temperature effects during sin-tering) and a processing condition database(loading schedule of punches and dies for com-paction simulation and heating cycle for sinter-ing simulation). A postprocessor is a softwaretool to visualize and analyze the simulationresults, which enhances the usefulness of thesimulations. From the standpoint of processsetup calculations, the optimization algorithmis essential to maximize computer simulationcapability providing the optimum part, die,and process condition design. Experimentalcapability is very important in computer simu-lation, providing a means to evaluate changesin materials, powders, compaction schedules,heating cycles, and generally to provide verifi-cation of the simulation results.

Experimental Determination ofMaterial Properties and SimulationVerification

Material Properties and Verification forCompaction

One of the first needs is to measure the pow-der density as a function of applied pressure togenerate the material parameters in the consti-tutive model for compaction, including theCoulomb friction coefficient between the pow-der and die. Note that these factors vary withthe powder lot, lubricant, tool material, andeven tool temperature. The procedure to obtainthe material properties based on the generalizedShima-Oyane model is as follows (Ref 24):

� Measure the pycnometer and tap densities ofthe powder.

� Conduct a series of uniaxial compressiontests with die wall lubrication to minimizethe die wall friction effect. The tap densityis considered the starting point (after particlerearrangement) corresponding to zero com-paction pressure. By curve fitting, six mate-rial parameters (a, g, m, a, b, and n) aredetermined for the yield surface, F:

F ¼ q

sm

� �2

þa 1�Dð Þg p

sm

� ��Dm (Eq 2)

where q and p are the effective stress andhydrostatic stress or pressure, respectively;D is the relative or fractional density; and

Fig. 2 Typical procedure for computer simulation for press and sinter process

Modeling and Simulation of Press and Sinter Powder Metallurgy / 325

sm is the flow stress of the matrix material,which is expressed as:

sm ¼ aþ b�enm (Eq 3)

where em is the effective strain of the matrixmaterial.

� A series of uniaxial compression tests areperformed without wall lubricant, then theCoulomb frictional coefficient is obtainedby FEM simulation.

Figure 4 is an illustration of the compactioncurve for an iron-base powder (Distaloy AE,Hoganas) in a simple cylindrical geometry.Uniaxial compression tests are provided fortwo samples: the smaller sample of 4.5 g in a12 mm diameter die with die wall lubrication,and the larger sample of 9.3 g without walllubrication. By curve fitting, the six materialparameters are (Ref 24):

a ¼ 6:20; g ¼ 1:03; m ¼ 7:40; a ¼ 184 MPa;

b ¼ 200 MPa; and n ¼ 0:240

By FEM simulation, the Coulomb frictionalcoefficient was obtained as 0.1. Table 1 showsthe example of a complete set of material prop-erties of an iron-base powder as input data forthe compaction simulation.Verification of the predicted density gradi-

ents in the green compact has been approachedby many techniques. The most reliable, direct,and sensitive comes from taking hardness ormicrohardness traces on a polished cross sec-tion. Thus, to verify the compaction simulationresults, the relationship between hardness andgreen density is conducted according to the fol-lowing procedure:

� Use the same samples as used for obtainingthe material parameters.

� Presinter the compacts at a temperature suf-ficient to bond the particles but below thetemperature range where dimensionalchange or chemical reactions occur.

� Carefully prepare a metallographic crosssection of the presintered samples and treatwith a vacuum annealing cycle to minimizeany hardness change induced by the cuttingprocess.

� Measure the hardness of each sample with aknown green density, and from that developa correlation between density and hardness.

� Apply the same procedure and hardness tracesto real components, and from precise measure-ments of hardness and location develop a con-tour plot of the green density distribution forcomparison with the computer simulation.

As an example, Fig. 5 is a plot of the correla-tion between green density and hardness forthe WC-Co system. For this plot, the presinter-ing cycle of the WC-Co system was at 790 oCfor 30 min, and the annealing cycle was at520 oC for 60 min; in this case, a Rockwell15T hardness scale was used. The obtained cor-relation is (Ref 35):

D¼0:638þ 1:67�10�3H� 5:44�10�7H2 (Eq 4)

where H is the 15T Rockwell hardness number,and D is the fractional density. Figure 6 com-pares the simulation results taken from a com-mercial software package (PMsolver) with theexperimental results for a cutting tool geometryformed from a cemented carbide powder, basedon Eq 4.

Material Properties and Verification forSintering

In the development of a constitutive modelfor sintering simulation, a wide variety of testsare required, including data on grain growth,densification (or swelling), and distortion.These are approached as follows:

� Grain growth: Quenching tests are con-ducted from various points in the heatingcycle, and the mounted cross sections areanalyzed to obtain grain-size data to imple-ment grain-growth models. A verticalquench furnace is used to sinter the com-pacts to various points in the sintering cycleand then to quench those compacts in water.This gives density, chemical dissolution (forexample, diffusion of one constituent intoanother), and grain size as instantaneousfunctions of temperature and time. Thequenched samples are sectioned, mounted,and polished prior to optical or scanningelectron microscopy (SEM). Today (2009),automated quantitative image analysis pro-vides rapid determination of density, grainsize, and phase content versus location inthe compact. Usually during sintering, themean grain size, G, varies from the startingmean grain size, G0 (determined on the

XZ

Y

(a)

Z

Y

X

(b)

(c)

Y

XZ

Fig. 3 Modeling and mesh-generation example showing the simulation of compaction and sintering for an oxygensensor housing. (a) Mesh generation for the compact. (b) Modeling of punches and dies during compaction

for the press simulation. (c) Modeling of compact in contact with the substrate during sintering simulation


green compact). A new master sinteringcurve concept is applied to fit the experi-mental grain-size data to an integral workof sintering (Ref 36), because actual cyclesare a complex combination of heats andholds. The resulting material parameters

trace to an apparent activation energy asthe only adjustable parameter. Figure 7shows SEM micrographs after a quenchingtest and grain-growth modeling for aW-8.4wt%Ni-3.6wt%Fe mixed powder com-pact during liquid-phase sintering.

� Densification: To obtain material parametersfor densification, constant heating rate dila-tometry is used for in situ measurement ofshrinkage, shrinkage rate, and temperature.By fitting the experimental data to modelsthat include the sintering stress, ss, and bulkviscosity, K, as functions of density andgrain size, again relying on the master sin-tering curve concept (Ref 37), the fewunknown material parameters are extracted.Figure 8 shows the dilatometry data andmodel curve-fitting results used to obtainthe missing material parameters during sin-tering of a 316L stainless steel (Ref 38).

� Distortion: Powder metallurgy compactsreach very low strength levels during sinter-ing. Accordingly, weak forces such as grav-ity, substrate friction, and nonuniformheating will induce distortion and evencracking. To obtain the material parametersrelated to distortion, three-point bending orsinter forging experiments are used for insitu measurement of distortion (Ref 39). Byfitting the experimental data with FEMsimulations for shear viscosity, m, with graingrowth, the parameters such as apparentactivation energy and reference shear viscos-ity are extracted. Figure 9 shows an in situbending test and FEM results for obtainingmaterial parameters in shear viscosity for a

Fig. 4 Example compaction curve for an iron-base powder showing the relative density model results versuscompaction pressure with no friction and with a friction coefficient of 0.1 for two compact samples.

Source: Ref 24

Table 1 Complete set of materialproperties for die compaction of an iron-base powder

Densities Pycnometer density 7.8 g/cm3

Fractional tap density 0.45Yield function(Shima-Oyane)

a 6.2g 1.03m 7.4

Flow stress of matrixmaterials (workhardening)

a 184 MPab 200 MPan 0.24

Friction coefficient 0.1Failure surface d 0.01

tan b 3.41

Source: Ref 24

Fig. 5 Plot of relative density and Rockwell 15Thardness scale for the die compaction of a

WC-Co powder. Source: Ref 35

Fig. 6 Comparison between the computer-simulated green density gradients (PMsolver) and the experimental resultstaken from hardness tests on a cross-sectioned green compact of a cutting tool formed from WC-Co powder.

Source: Ref 35


316L stainless steel powder doped with0.2% B to induce improved sintering.

Such data-extraction techniques have been alliedto several materials, including tungsten alloys,molybdenum, zirconia, cemented carbides, nio-bium, steel, stainless steel, and alumina. Table2 is an example set of material properties for W-8.4 wt%Ni-3.6wt%Fe as used as input data forthe sintering simulation (Ref 40). The precedingexperiment techniques can be used for verificationof sintering simulation results.

Demonstration of System Use

Time Advance Algorithm andCompaction Simulation Accuracy

Time advance in the simulation models is aconcern with respect to the balance between

accuracy and computational speed. The explicitmethod is fast but sometimes exhibits conver-gence and accuracy problems, while the implicitmethod is accurate but slow. Figure 10 illustratesthis case for a simple cylindrical geometry in theWC-Co system (Ref 35). As shown in Fig. 10(b), the implicit method is more accurate forthis case.

Gravitational Distorting in Sintering

The rheological data for the sintering systemallow the system to respond to the internal sin-tering stress that drives densification and anyexternal stress, such as gravity, that drives dis-tortion. When a compact is sintered to highdensity, it is also necessary to induce a lowstrength. (The material is thermally softenedto a point where the internal sintering stresscan induce densification.) Figure 11 shows

sintering simulation results for a tungsten heavyalloy, relying on test data taken on Earth andunder microgravity conditions, to then predictthe expected shapes for various gravitationalconditions: Earth, Moon, Mars, and in space.The results show that gravity affects shape dis-tortion during sintering (Ref 40). Accordingly,the computer simulations can be used to reverseengineer the green component geometry toanticipate the distortion to achieve the desiredsintered part design.

Compaction Optimization

There are two different simulationapproaches used to optimize die compaction.One is based on the concept of design ofexperiment, and the other is a derivative-basedoptimization scheme. The process designermust first select a reasonable initial guess, an

Fig. 7 (a) Scanning electron micrograph of a liquid-phase sintered tungsten heavy alloy (W-8.4wt%Ni-3.6wt%Fe) after quenching. (b) Grain-size model results taken from anintegral work of sintering concept that includes only the thermal cycle (time-temperature path) to predict grain size for any point in a heating path for three different

tungsten contents. Source: Ref 36

Fig. 8 Dilatometry data showing in situ shrinkage data during constant heating rate experiments and the curve-fitting results used to obtain the material parameters to predictdensification of a 316L stainless steel powder. (a) Shrinkage with time. (b) Shrinkage with temperature. Source: Ref 38


objective function that needs to be minimized,and design variables for both approaches.Figure 12 shows the first approach used to

optimize the loading schedule to generate auniform green density during die compaction(Ref 35). In this case, the target is a cuttingtool formed from WC-Co. The displacementof the upper punch was set as the design vari-able, and the lower punch displacement was

automatically calculated because the finaldimension was fixed. The objective functionwas set to the standard deviation in green den-sity, which is called the nonuniformity, and thiswas to be minimized. Figure 12(a) plots thedensity histograms for five different processingconditions. The first one (black) is the initialcompaction process design, and the fourth one(yellow) shows the optimum design for

maximum uniformity with only a change inthe upper punch motion. The density distribu-tions of those two cases are shown in Fig. 12(b). For optimization in more complicated sys-tems, the Taguchi method with an orthogonalarray can be used to create simulation experi-ments to efficiently isolate solutions.In using a derivative-based optimization

scheme, it is important to define the searching

Fig. 9 (a) Video image taken during in situ bending test for a 316L stainless steel sample doped with 0.2 wt% B. (b) Finite-element modeling results used to verify the shear viscosityproperty as a function of time, temperature, grain size, and density during heating. Source: Ref 39

Table 2 Complete set of material properties for simulating the sintering of W-8.4wt%Ni-3.6wt%Fe

Pycnometer density 16.75 g/cm3

Initial density distribution, r0 Results from press simulationInitial mean grain size 0.30 mmSurface energy, g 2.5 J/m2

Transition temperature from solid state to liquid phase 1460 �CFriction coefficient 0.3

Sintering stress

ss ¼ 6gG

r2 2r� r0ð Þy0

for r < 0:85

ss ¼ 2gG

6ry

� �1=3

for rW > 0:95

ss ¼ ðr2 � rÞðr2 � r1Þ

ssi þ ðr� r1Þðr2 � r1Þ

ssf for 0:85 � rW � 0:95

State Solid Liquid

Grain growthdG

dt¼ k0 expð�QG=RT Þ

Glk0 (m

n+1/s) 2.8�10�13 1.1�10�15

QG (kJ/mol) 241 105l 2.0 2.0

Bulk viscosity

Ki ¼ rðr� r0Þ28y20

TG3

ai expð�QD=RT Þ for r � 0:92

Kf ¼ r

8y1=20

TG3

af expð�QD=RT Þ for r > 0:92

with af ¼ y20ffiffiffiffiffiffiffiffiffi0:08

p ð0:92� r0Þ2ai

QD (kJ/mol) 250 250ai (m

6�K/s) 1.3�10�17 5.0�10�17

Shear viscosity

mi ¼r2ðr� r0Þ

8y0

TG3

bi expð�QD=RT Þ for r � 0:92

Kf ¼ r8

TG3

bf expð�QD=RT Þ for r > 0:92

with bf ¼y0

0:92ð0:92� r0Þbi

bi (m6�K/s) 1.3�10�17 1.3�10�12

Note that R is the universal gas constant, r is the density, y (= 1 � r) is the porosity, y0 (= 1 � r0) is the initial porosity, and rW is the density of the tungsten skeleton. Source: Ref 40


direction and stepping size. The searchingdirection is decided by the direct differentiationor adjoint variable methods, and the steppingsize is usually selected by polynomial curve fit-ting. Using the results of a finite-element simu-lation with a mesh system generated by thegiven design variables, design sensitivities arecalculated by the algorithm of the searchingdirection. The searching direction is selectedby the conjugate gradient method, and theproper stepping size is selected by the polyno-mial curve fitting, with the objective functionsobtained by additional finite-element simula-tions. The design parameters are iterativelyupdated until the convergence criteria are satis-fied. Figure 13 shows the procedure for opti-mizing the loading schedule to have uniformdensity distribution during die compaction fora hub part formed using a steel alloy powder(Ref 24). The design variables considered inthis example were the loading schedules ofupper, inner lower, core rod, and die, and theobjective function is the nonuniformity afterdie compaction. The goal was to minimize thenonuniformity. Figure 13(a) shows the compac-tion tool set and analysis domain. Figure 13(b)plots the objective function during optimiza-tion, and Fig. 13(c) plots the density distribu-tions of the initial and optimum designs.

Sintering Optimization

Usually, a small grain size is desired toimprove properties for a given sinter density.In this illustration, the design variable is thesintering cycle. To obtain maximum densityand minimum grain size, the following objec-tive function, F, is proposed (Ref 41):

F ¼ aDrr

� �þ 1� að Þ DG

G

� �(Eq 5)

where a is an adjustable parameter. Figure 14(a)shows an example for maximum density andmin-imum grain size for a 17-4 PH stainless steel pow-der. For example, the minimum grain size will be21.9 mm if the specified sintered density is 95% ortheoretical. Figure 14(b) shows the corresponding

Fig. 10 Comparison of explicit and implicit methods for compaction simulation in the WC-Co system. (a) Meshedgeometry. (b) Left, half-axis experimental result; center, full-axis explicit method result; and right, implicit

method result. Source: Ref 35

Fig. 11 Final distorted shape by sintering under various gravitational environments for complicated test geometries. (a) T-shape. (b) Joint part. Source: Ref 40


sintering cycle bymatching the value of the adjus-table parameter a.

Coupled Press and Sinter Optimization

Coupled simulations are necessary to predictthe quality of the final component after the com-bined press and sinter operations. Figure 15 illus-trates this for the case for a shaped cutting toolformed from WC-Co. In practice, the final com-ponent exhibited cracking along the corners,but there was no sign of cracking after die

compaction, as shown in Fig. 15(b). From thesimulation results for the initial design, the den-sity distribution is found to vary from a fractionaldensity of 0.583 to 0.814. After optimization ofthe tool loading schedule with the objectionfunction to make the green density as uniformas possible, the green density distribution rangedfrom 0.638 to 0.675. Implementation of thisloading cycle resulted in elimination of crackingin the sintered component.Figure 16 illustrates the distortion in final

shape for oxygen sensor components. After

optimization of the tool loading schedule andthe sintering cycle, the final distortion fromthe target shape is significantly reduced for boththe holder and sleeve.

Conclusion

Computer simulations of the press-sintercycle in powder metallurgy have advanced con-siderably and, in combination with standardfinite-element techniques, show a tremendous

Fig. 12 Optimization for uniform density distribution during die compaction for a cutting tool fabricated from WC-Co. (a) Histogram for various processing conditions. (b) Greendensity distributions in the initial and optimum designs. Source: Ref 35


ability to guide process setup. Illustrated hereare the compaction and sintering conceptsrequired to perform process optimization, typi-cally via selection of appropriate compactiontool motion. Although the models are onlyapproximations to reality, they are still of valuein forcing a careful inspection of what is under-stood about the press-sinter process. In thisregard, the greatest value of modeling is in theforced organization of process knowledge.There remain several barriers to widespread

implementation. The largest is that traditionalpowder metallurgy is largely dependent onadaptive process control, because many of theimportant factors responsible for dimensionalor quality variations are not measured. Thevariations in particle size, composition, toolwear, furnace location, and other factors, suchas reactions between particles during heating,impact the important dimensional-controlaspects of press-sinter powder metallurgy.However, nominal properties, such as strength,hardness, or fatigue life, are dominated by theaverage component density. In that regard,especially with respect to the initial processsetup, the computer simulations are of greatvalue. Nevertheless, important attributes suchas dimensional tolerances and internal cracksor other defects are outside the cost-benefit cap-abilities of existing simulations. Further, thevery large number of materials, processes, toolmaterials, sintering furnaces, and process cyclesmakes it difficult to generalize; significant datacollection is required to reach the tipping pointwhere the simulations are off-the-shelf. Thus,much more research and training is required tomove the simulations into a mode where theyare widely applied in practice. Even so, com-mercial software is available and shows greatvalue in the initial process definition to set upa new component.

REFERENCES

1. F.A. Nichols and W.W. Mullins, Morpho-logical Changes of a Surface of Revolut-ion due to Capillarity-Induced SurfaceDiffusion, J. Appl. Phys., Vol 36, 1965,p 1826–1835

2. F.A. Nichols, Theory of Sintering of Wiresby Surface Diffusion, Acta Metall., Vol 16,1968, p 103–113

3. K.E. Easterling and A.R. Tholen,Computer-Simulated Models of the Sinter-ing of Metal Powders, Z. Metallkd., Vol61, 1970, p 928–934

4. A.J. Markworth and W. Oldfield, ComputerSimulation Studies of Pore Behavior inSolids, Sintering and Related Phenomena,G.C. Kuczynski, Ed., Plenum Press, NewYork, NY, 1973, p 209–216

5. K. Breitkreutz and D. Amthor, Monte-Carlo-Simulation des Sinterns durch Volu-men-und Oberflachendiffusion, Metallur-gie, Vol 29, 1975, p 990–993

Fig. 13 Optimization to minimize the green density gradients during die compaction of a steel hub component. (a)Compaction tool set and analysis domain. (b) Variation of objective function during optimization iteration.(c) Green density distributions in the initial and optimum designs. Source: Ref 24


6. M.F. Ashby, A First Report on SinteringDiagrams, Acta Metall., Vol 22, 1974,p 275–289

7. R.M. German and J.F. Lathrop, Simulationof Spherical Powder Sintering by SurfaceDiffusion, J. Mater. Sci., Vol 13, 1978,p 921–929

8. P. Bross and H.E. Exner, Computer Simu-lation of Sintering Processes, Acta Metall.,Vol 27, 1979, p 1013–1020

9. F.B. Swinkels and M.F. Ashby, A SecondReport on Sintering Diagrams, ActaMetall., Vol 29, 1981, p 259–281

10. C.M. Sierra and D. Lee, Modeling ofShrinkage During Sintering of InjectionMolded Powder Metal Compacts, PowderMetall. Int., Vol 20 (No. 5), 1988, p 28–33

11. K.S. Hwang, R.M. German, and F.V.Lenel, Analysis of Initial Stage SinteringThrough Computer Simulation, PowderMetall. Int., Vol 23 (No. 2), 1991, p 86–91

12. R.M. German, Overview of Key Directionsand Problems in Computational andNumerical Techniques in Powder Metal-lurgy, Computational and Numerical Tech-niques in Powder Metallurgy, D.S. Madan,

I.E. Anderson, W.E. Frazier, P. Kumar, andM.G. McKimpson, Ed., The Minerals,Metals, Materials Society, Warrendale,PA, 1993, p 1–15

13. H. Riedel, D. Meyer, J. Svoboda, and H.Zipse, Numerical Simulation of DiePressing and Sintering—Development ofConstitutive Equations, Int. J. Refract.Met. Hard Mater., Vol 12, 1994,p 55–60

14. K.Y. Sanliturk, I. Aydin, and B.J. Briscoe,A Finite Element Approach for the ShapePrediction of Ceramic Compacts During

Fig. 15 Effect of density distribution after die compaction on sintering and the formation of corner cracks. (a) Simulation result of green density gradients. (b) Experimental result ofgreen compact. (c) Experimental result of sintered compact.

Fig. 14 (a) Minimum grain size for a given final sinter density and (b) the corresponding sintering cycle for achieving this goal in a 17-4 PH stainless steel. Source: Ref 41


Sintering, J. Am. Ceram. Soc., Vol. 82,1999, p 1748–1756

15. R. Raman, T.F. Zahrah, T.J. Weaver, andR.M. German, Predicting DimensionalChange During Sintering of FC-0208 Parts,Advances in Powder Metallurgy and Par-ticulate Materials—1999, Vol 1, MetalPowder Industries Federation, Princeton,NJ, 1999, p 3.115–3.122

16. O. Gillia and D. Bouvard, Phenomenologi-cal Analysis and Numerical Simulation ofSintering Application to WC-Co System,Proceedings of the 2000 Powder Metal-lurgy World Congress, Part 1, K. Kosugeand H. Nagai, Ed., Japan Society of Pow-der and Powder Metallurgy, Kyoto, Japan,2000, p 82–87

17. V. Tikare, M.V. Braginsky, E.A. Olevsky,and R.T. Dehoff, A Combined Statistical-Microstructural Model for Simulation of Sin-tering, Sintering Science and Technology,R.M. German, G.L. Messing, and R.G. Corn-wall, Ed., Pennsylvania State University,State College, PA, 2000, p 405–409

18. A. Zavaliangos and D. Bouvard, NumericalSimulation of Anisotropy in Sintering dueto Prior Compaction, Int. J. PowderMetall., Vol 36 (No. 7), 2000, p 58–65

19. V. Tikare, E.A. Olevsky, and M.V. Brag-insky, Combined Macro-Meso Scale Mod-eling of Sintering, Part II: MesoscaleSimulations, Recent Developments in Com-puter Modeling of Powder Metallurgy Pro-cesses, A. Zavaliangos and A. Laptev, Ed.,ISO Press, Ohmsha, Sweden, 2001, p 94–104

20. R.M. German, Sintering Theory Practice,John Wiley & Sons, Inc., New York, NY,1996

21. A.N. Schofield, Original Cam-Clay, Pro-ceedings of the International Conferenceon Soft Soil Engineering, Nov 8–11, 1993(Guangzhou)

22. D.C. Drucker and W. Prager, Soil Mechan-ics and Plastic Analysis or Limit Design,Q. Appl. Math, Vol 10, 1952, p 157–164

23. S. Shima and M. Oyane, Plasticity Theoryfor Porous Metals, Int. J. Mech. Sci., Vol18, 1976, p 285–292

24. Y.-S. Kwon, S.-H. Chung, K.T. Kim, R.M.German, and H.I. Sanderow, NumericalAnalysis and Optimization of Die Compac-tion Process, Advances in Powder Metal-lurgy and Particulate Materials, Part 4,2003, p 4-37 to 4-50

25. R.M. McMeeking and L. Kuhn, A Diffu-sional Creep Law for Powder Compacts,Acta Metall. Mater., Vol 40, 1992, p 961–969

26. T. Kraft and H. Riedel, NumericalSimulation of Die Compaction and Sin-tering, Powder Metall., Vol 45, 2002,p 227–231

27. P.E. McHugh and H. Riedel, A LiquidPhase Sintering Model: Application toSi3N4 and WC-Co, Acta Metall. Mater.,Vol 45, 1997, p 2995–3003

28. D. Bouvard and T. Meister, ModellingBulk Viscosity of Powder Aggregate Dur-ing Sintering, Model. Simul. Mater. Sci.Eng., Vol 8, 2000, p 377–388

29. A.C.F. Cocks, The Structure of Constitu-tive Laws for the Sintering of Fine GrainedMaterials, Acta Metall., Vol 45, 1994,p 2191–2210

30. Y.S. Kwon and K.T. Kim, High Tempera-ture Densification Forming of AluminaPowder—Constitutive Model and

Experiments, J. Eng. Mater. Technol., Vol118, 1996, p 448–455

31. Y.S. Kwon, Y. Wu, P. Suri, and R.M. Ger-man, Simulation of the Sintering Densifica-tion and Shrinkage Behavior of Powder-Injection-Molded 17-4 PH Stainless Steel,Metall. Mater. Trans. A, Vol 35, 2004,p 257–263

32. R.K. Bordia and G.W. Scherer, On Con-strained Sintering—I. Constitutive Modelfor a Sintering Body, Acta Metall., Vol36, 1998, p 2393–2397

33. R.K. Bordia and G.W. Scherer, On Con-strained Sintering—II. Comparison of Con-stitutive Models, Acta Metall., Vol 36,1998, p 2399–2409

34. R.K. Bordia and G.W. Scherer, On Con-strained Sintering—III. Rigid Inclusions,Acta Metall., Vol 36, 1998, p 2411–2416

35. S.H. Chung, Y.S. Kwon, C.M. Hyun, K.T.Kim, M.J. Kim, and R.M. German, Analy-sis and Design of a Press and Sinter Pro-cess for Fabrication of Precise TungstenCarbide Cutting Tools, Advances in Pow-der Metallurgy and Particulate Materials,Part 8, 2004, p 8-26 to 8-39

36. S.J. Park, J.M.Martin, J.F. Guo, J.L. Johnson,and R.M.German, GrainGrowthBehavior ofTungstenHeavyAlloys Based onMaster Sin-tering Curve Concept, Metall. Mater. Trans.A, Vol 37, 2006, p 3337–3343

37. S.J. Park, J.M.Martin, J.F. Guo, J.L. Johnson,and R.M. German, Densification Behavior ofTungsten Heavy Alloy Based on Master Sin-tering Curve Concept, Metall. Mater. Trans.A, Vol 37, 2006, p 2837–2848

38. S.H. Chung, Y.S. Kwon, C. Binet, R. Zhang,R.S. Engel, N.J. Salamon, and R.M. German,Application of Optimization Technique inthe Powder Compaction and Sintering Pro-cesses, Advances in Powder Metallurgy andParticulate Materials, Part 9, 2002, p 9–131to 9–146

39. D. Blaine, R. Bollina, S.J. Park, and R.M.German, Critical Use of Video-Imaging toRationalize Computer Sintering SimulationModels, Comput. Ind., Vol 56, 2005,p 867–875

40. S.J. Park, S.H. Chung, J.L. Johnson, andR.M. German, Finite Element Simulationof Liquid Phase Sintering with TungstenHeavy Alloys, Mater. Trans., Vol 47,2006, p 2745–2752

41. S.J. Park, R.M. German, P. Suri, D. Blaine,and S.H. Chung, Master Sintering CurveConstruction Software and Its Application,Advances in Powder Metallurgy and Par-ticulate Materials, Part 1, 2004, p 1-13 to1-24

Fig. 16 Final part profiles based on the combined press-sinter simulation for the optimized design of a holder andsleeve for an oxygen sensor. (a) Initial and optimum holder designs. (b) Initial and optimum sleeve designs


Modeling and Simulation of Press and Sinter Powder Metallurgy · Modeling and Simulation of Press...

Documents

Transcript of Modeling and Simulation of Press and Sinter Powder Metallurgy · Modeling and Simulation of Press...