Defects die casting

8/19/2019 Defects die casting

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Modeling of Hot Tearing and

Other Defects in Casting ProcessesBrian G. Thomas, University of Illinois (UIUC)

AS COMPUTATIONAL MODELSMATURE, their practical benefit to improvingcasting processes is growing. Accurate calcula-tion of fluid velocities, temperature, microstruc-ture, and stress evolution is just the first step.Achieving tangible improvements to castingprocesses requires the accurate prediction of actual casting defects and product properties.Defects that form during solidification areimportant not only to the casting engineer butalso to engineers involved in subsequentmanufacturing processes and evaluation. Solid-ification defects are responsible for many of thedefects in final manufactured products andfailures in service. They originate from inclu-sion entrapment, segregation, shrinkage cav-ities, porosity, mold-wall interactions, cracks,and many other sources that are process-specific. Casting defects can be modeled byextending the results of casting simulations

through postprocessing and/or by solving fur-ther coupled equations that govern these phe-nomena. The prediction of defect formation ismade difficult by the staggering complexity of the phenomena that arise during commercialcasting processes. This article introduces someof the concepts involved in modeling some of these solidification defects and focuses in moredetail on hot tearing.

Inclusions

Inclusions are responsible for many serioussurface defects and internal quality problemsin cast products. They arise from foreign parti-cles, such as eroded sand particles, and impuri-ties remaining in the liquid metal after upstreamrefining (Ref 1). Nonmetallic inclusion particlesact as sites of stress concentration and hydrogengas nucleation, leading to lower fatigue life,hydrogen embrittlement, surface defects, andother problems in the final product. Predictingtheir damage requires knowledge of the num-ber, size distribution, composition, and mor-phology of the inclusions coming from

upstream processing prior to casting. Obtainingthis knowledge ideally involves modeling themultiphase fluid flow, turbulent mixing and dif-fusion, species transport, chemical reactions,and particle interactions that create the inclu-sions in upstream processes.

Considerable modeling of these phenomenahas been addressed in previous simulations of vacuum degassers, R-H degassers (a type of recirculating degasser), ladles, tundishes, andother refining vessels and transfer operationsused in metallurgical processing (Ref 2). Thesemodels solve the multiphase Navier-Stokesequations for turbulent fluid flow, using softwaresuch as FLUENT (Ref 3), and provide the flowfield for subsequent simulation of inclusion par-ticle transport. The first challenge is to properlyincorporate the phenomena that drive the flow,which usually include the buoyancy of injectedgas bubbles (Ref 4), which depends on the shape

of the bubbles, ranging from spherical caps tospheres. Other effects important to accuratelycomputing the flow field may include naturalconvection, which requires a coupled heat-trans-port solution for the temperature field. Whenelectromagnetic stirring is used, these forcesrequire modeling the applied magnetic field.Another challenge is to incorporate the effectsof turbulence. Computationally-efficient choicesinclude simple “mixing-length” models, the two-equation modelssuch as k-e to simulate thetime-average flow pattern. Large-eddy simulation(LES) models can simulate the details of thetime-evolving turbulent vortices, but at greatcomputational expense. These methods havebeen compared with each other and with mea-surements of fluid flow in continuous casting(Ref 5–7).

Formation. Modeling the thermodynamicsand kinetics of particle formation, transport,collisions, and removal or entrapment in themolten metal during upstream refining pro-cesses is the next crucial step. Thermodynamicreactions to quantify the precipitates that formin these multicomponent alloy systems can bepredicted by simultaneous solution of chemical

equilibrium equations, where the biggest chalenge is to find accurate activity coefficienEquilibrium compositions can also be fouby comparing free-energy functions, such used in Thermo-Calc (Ref 8), FACT-Sa(Ref 9), MTDATA (Ref 10), Gemini (Ref 11and other thermodynamic modeling softwarThe kinetics of nonmetallic inclusion formatioare generally controlled by species transport the liquid and at reaction interfaces, such the slag-metal surface, where droplets of tdifferent liquid phases, solid particles, and gbubbles interact. The physical entrainment slag particles into the molten metal is anothimportant source of inclusions (Ref 12), whicrequires transient multiphase modeling of tfree surface, considering its breakup into drplets and surface tension effects, and pushcurrent modeling capabilities to their limit.

Another important source of inclusions

reoxidation of the molten metal by exposuto air. Oxygen absorbs rapidly from the atmsphere into any exposed molten metal and combines to form precipitates, which has bepredicted in molten steel from the alloy conte(Ref 13). Predictions are limited by understaning of the entrainment of oxygen from tatmosphere, the turbulent flow of the liqusteel during pouring, which determines tgas-metal interface shape, and the interntransport and reactions of chemical species the molten metal.

Transport. The transport of particles througthe flowing metal is the next crucial step determine the inclusion distribution in the finproduct and can be modeled in several wa(Ref 14). Although the effect of bubbles on thflow pattern can be modeled effectively usiEulerian-Eulerian multiphase models, the faof inclusion distributions is best modeled vLagrangian particle tracking. In this methothe trajectories of many particles are integratfrom the local velocity field, based on previosolution of the fluid velocities of a mold-fillinsimulation. The effect of turbulence on the chotic particle paths is very important and is be

ASM Handbook, Volume 22A: Fundamentals of Modeling for Metals Processing

D.U. Furrer and S.L. Semiatin, editors, p 362-374

Copyright © 2009 ASM Internationa

All rights reserv

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modeled with the transient turbulent velocityfield, using LES (Ref 15). In more computa-tionally-efficient time-averaged simulations of the turbulent flow field, the effect of turbu-lence on particle motion can be approximatedusing methods such as “random walk,” wherethe velocity at each time increment is given arandomly-generated component with magni-

tude proportional to the local turbulence level(Ref 3). This method has been applied success-fully to simulate particle motion in continuouscasting molds (Ref 16).

Inclusion particle size distributions evolveduring transport due to collisions with eachother and by their attachment to the surfaceof bubbles. Collisions can be modeled bytracking the evolution in the number distribu-tion of particles in each size range, includingthe local effects of Brownian motion, turbu-lence, and diffusion, which is aided by size-grouping models to cover the large range of particle sizes (Ref 17). Attachment andremoval by bubbles can be modeled by com-puting the attachment rates of different parti-

cle sizes to different bubble sizes and shapesin computational models of these microscalephenomena (Ref 16, 18). These attachmentrates can then be incorporated into the macro-scale models of fluid and particle trajectories(Ref 16, 18). In the extreme, inclusions mayagglomerate into large clogs, which canrestrict the flow of molten metal, cause detri-mental changes in the downstream flow pat-tern, and can lead to catastrophic defects inthe final product. Modeling and analysis of clogging is a complex subject that has beenreviewed elsewhere (Ref 19).

Capture. Particle capture into the solidifica-tion front is a critical step during the modeling

of inclusion transport. Small particles flowbetween the dendrites, so they can be modeledas entrapped when they touch a domain wall.Larger particles may be pushed by the inter-face or engulfed by a fast-moving planar front(Ref 20). More often, they are entrapped whenthey are suspended in front of the solidificationfront long enough for the dendrites to surroundthem. Entrapment is greatly lessened by tan-gential flow across the solidification front(Ref 21). A criterion for entrapment has beendeveloped based on balancing the many forcesthat act on a particle suspended at the interface(Ref 21). Particles that never touch the inter-face, or escape capture, eventually may beremoved if the flow pattern transports them to

the casting boundaries, such as the top surfaceof some processes, where they can enter theslag layer.

The final step is to predict the propertychanges caused by the entrapped inclusions,which is a challenging modeling task anddepends on downstream processing, such asrolling and heat treatment. Even with simplycooling to ambient, precipitation continues inthe solid state, where the inclusion distribution

is greatly affected by kinetic delays due tonucleation and solid-state diffusion (Ref 22).This is further complicated by preferential pre-cipitation at grain boundaries and compatibleexisting inclusions and is affected by strains,local microsegregation, and many other phe-nomena. Clearly, the modeling of inclusions isa challenging task.

Segregation

Segregation is caused by the partitioning of alloying elements between the liquid and solidphases during solidification. Because species dif-fusion in the solid is very slow, this phenomenonis usually manifested by small-scale compositiondifferences, called microsegregation, whichexplains how the spaces between dendrites areenriched in alloy relative to the dendrite centers.Although it contributes greatly to macrosegrega-tion, porosity, inclusions, and other defects,microsegregation alone is not usually considereda defect, and it can be removed by homogeniza-

tion heat treatment. When fluid flow is present,however, large-scale species transport leads tomacrosegregation, where the composition differ-ences arise over large distances, such as betweenthe center and surface of a casting. This seriousdefect cannot be removed. It is extremely diffi-cult to predict, because it involves so many dif-ferent coupled phenomena, and at vastlydifferent length and time scales. In addition topredicting fluidflow, species transport, and solid-ification, segregation requires prediction of thedendrite morphology and microstructure andthe complete stress state, including deformationof the spongy mushy zone (Ref 23) and mechan-ical bulging and bending of the casting surface

(Ref 24). Moreover, the fluid flow must be accu-rately characterized at boththe microscopic scalebetween dendrite arms and at the macroscopicscale of the entire casting. Each of these model-ing tasks is a large discipline that has receivedsignificant effort over several decades.

Segregation is the main phenomenon respon-sible for many different kinds of special defectsthat only affect particular casting processes. For example, “freckle” defects can arise during thedirectional solidification of turbine blades whenbuoyancy-driven flow allows winding verticalchannels to penetrate between dendrites andbecome filled with segregated liquid near theend of solidification (Ref 25). Inverse segrega-tion or surface exudation in direct-chill continu-

ous casting of aluminum ingots arises duringthe initial stages of solidification when thermalstress pushes out droplets of enriched interden-dritic liquid through pores in the spongy mushyzone where it extends to the ingot surface (Ref 26). A comprehensive summary of the model-ing of this important class of defects is beyondthe scope of this article, and reviews of variousaspects of this complex subject can be foundelsewhere (Ref 23, 27–29).

Shrinkage Cavities, Gas Porosity,and Casting Shape

Shrinkage cavities are voids in a casting tform due to the thermal contraction of liqpockets after they become surrounded by sothat prevents the feeding of additional liquPorosity is the name for small voids that fo

due to the evolution and entrapment of gas bubles. These two important classes of defects related. Both involve the entrapment of liqupockets, a criterion for the nucleation of bubbles, and depend on the overall shrinkof the casting, which requires a complete thmomechanical stress calculation, in additionaccurate prediction of fluid flow and solidifition. A rough estimate of shrinkage cavpotential is possible from postprocessing anasis of the results of a simple solidification hetransfer analysis, looking for regions whsolid surrounds the liquid. This simple analycan be automated by tracking parameters trepresent shrinkage potential, such as

Niyama criterion (Ref 30, 31) More accurprediction of shrinkage requires complete meling of fluid flow, heat transfer, and thermstress analysis. The fluid flow analysis is furtcomplicated by the need for accurate charactization of the permeability of the porous dedritic network, which also depends on microstructure and alloy segregation. The stranalysis depends on the evolving strength of solid, in addition to the mushy zone, interactwith the mold, and other phenomena that discussed further in the section on hot tearin

In addition to the phenomena that govshrinkage cavity formation, gas porosity predtion also requires modeling the transport of dsolved gases, the nucleation of bubbles or

pockets, and their possible transport after thform. This modeling also involves the sacomplications discussed in the predictioninclusions, including nonequilibrium thermonamics, chemical reactions, nucleation, precitate formation, and growth kinetics. Indeprecipitation reactions are alternative ways the dissolved gases to be consumed. Finagas bubbles that float during solidification ccollide and coalesce, depending on surface tsion. When combined with improper ventithis can lead to the creation of a defect fouat the top of foundry castings, known as a sface blow hole.

Shrinkage and porosity defects are related

the final shape of the casting. When the sometal shell is strong enough to resist shrinkaand retain its external dimensions, intershrinkage and porosity may be more probleatic. In contrast, practices that lessen shrinkaand porosity may involve more external shrinage of the exterior. Inaccurate final dimensiois another casting defect. Because comprehsive modeling of these defects requires simultaneous solution of so many differ

Modeling of Hot Tearing and Other Defects in Casting Processes / 3


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equation systems, with so many uncertain fun-damental properties, this class of defects is dif-ficult to predict and is the subject of intenseongoing research. The art of modeling thesedefects involves how to make simplifyingassumptions with the least loss of accuracy.Further details on the current state of the artin modeling of this important class of defects

is given elsewhere (Ref 28, 32, 33).

Mold-Wall Erosion

Feeding molten metal into the casting cavityis a critical operation where defects may arise.Excessive turbulence and velocity impingementof the molten metal can erode the surface of themold wall, especially near the in-gate. In sandmolds, this can dislodge sand particles to actas another source of inclusions in the final cast-ing. Even with permanent metal molds such asused in pressure die casting, excessive velocityagainst the metal walls can locally erode themetal, enlarging the casting cavity and creating

surface defects.Erosion rate has been related to the metal

velocity and other parameters in a few previousstudies, based mainly on empirical correlations(Ref 34). For example, in die casting, erosionstrength has been characterized by integratingthe instantaneous velocity over the time of theinjection cycle for each local portion of themold-wall surface (Ref 35). The resulting con-tours over the mold surface can be correlatedwith erosion damage.

Erosion of the mold wall due to fluid flow alsomayremove protective surfacecoatings andallowchemical reactions between the mold and theexposed mold metal. Thus, the mechanical ero-

sion may be accompanied by chemical erosionand/or metallurgical corrosion, which often acttogether to wear away the mold surface. Analysisof the chemical component requires considerationof the thermodynamic reactions and their kinetics.Theinterdiffusion of elements in themolten metalto contaminate the mold walls can lower the localmelting temperature. This is responsible for theproblem of soldering in aluminum die casting insteel molds (Ref 36, 37).

Mold-Wall Cracks

Cracks in the mold wall are another source of defects in the casting, in addition to lowering

the lifetime of permanent molds. Mold cracksdecrease the local heat-transfer rate, allowinglocal strain concentration in the adjacent solidi-fying metal and causing hot-tear cracks at thecasting surface that mirror those in the mold.In water-cooled molds, mold cracks also posea safety hazard, from the chance of moltenmetal contacting the cooling water. Moldcracks, or heat checks, are caused by repeatedrapid and severe fluctuations in the mold sur-face temperature. They can be predicted from

the results of a transient thermal-stress analysisof the mold itself, by combining the calculatedinelastic strain (due to plasticity and creep) withmeasurements of cycles to failure from thermal-fatigue experiments. For example, surface cracksin copper molds used from continuous castingwere predicted by comparing the results of tran-sient three-dimensional finite-element analysis

of the copper mold and its support structure dur-ing cyclic loading with measured fatigue cycle-to-failure data (Ref 38). In addition to adoptingpractices to lower the maximum surface temper-ature, the mold lifetime was predicted toincrease by lessening constraint of the mold byloosening bolts (Ref 38, 39). Often, the predic-tion of mold cracks requires consideration of the chemical interaction of the liquid metal withthe mold, such as formation of brass in copper molds by the preferential absorption of zinc fromthe molten metal.

Other Defects

Many other casting defects arise due to pro-blems specific to individual processes. Graindefects, such as unwanted grain boundaries,are important in directional solidification pro-cesses, such as the casting of single-crystal tur-bine blades, where high-temporature creepresistance is the most important property. Inthe Czochralski process, where single crystalsare slowly pulled from doped melts to cast rodsfor making semiconductor wafers, even disloca-tions are serious defects that must be mini-mized. Examples in foundry sand castinginclude cold shut, blow holes, liquid metal pen-etration into the sand grains, and other surfacedefects. Some insight into these defects can be

found from the results of a solidification heat-transfer analysis. For example, problems relatedto cold shut can be estimated from a simulationif the molten metal freezes before the castingcavity is completely filled, leaving voids or seams at the junction where two streams met.Crystal defects depend on the temperature gra-dient across the solidification front. Further insight can be gained from direct modeling of the microstructure (Ref 40) and molecular dynamics or quantum-mechanics models of dis-locations and other phenomena at the atomicscale (Ref 41). Many important process-specificdefects have received little attention by themodeling community.

A final category of defects may be termed

“goofups” because their cause is so obvious,and the solution involves, at most, only basiccalculations. For example, a short pour occurswhen the volume of metal poured is less thanthe volume of the casting cavity. Unsightly mis-match seams arise when the two halves of thefoundry casting mold are not aligned due to poor maintenance of the hinges and pins. Althoughobvious, avoiding such defects requires carefuland diligent operations. Here, expert-system-type software may help, aided in these examples

by embedding simple volume calculations antracking of maintenance schedules. The rest this article focuses on the important defect hot-tear crack formation.

Hot-Tear Cracks

Crack formation is caused by a combination tensile stress and metallurgical embrittlemeAlthough solidifying metal is subject to embrittment due to a number of different mechanisms different temperature ranges, hot-tear cracks fornear the solidus temperature. Embrittlement is severe near this temperature that hot-tear cracform at strains on the order of only 1%, makinthem responsible for most of the cracks observin cast products.Hot-tearcracks formbecausethliquid films between the dendrites at grain bounaries are susceptible to strain concentration, cauing separation of the dendrites and intergranulcracks. The prediction of these cracks presentformidable challenge to modellers, owing to thmany complex, interacting phenomena that go

ern stress and embrittlement, some of which anot yet fully understood:

Predicting temperature, strain, and stress duing solidification requires calculation of thhistory of thecastproductand itsenvironmeover huge temperature intervals. Characteriing the heat-transfer coefficients at the bounaries and interfaces is one of manydifficultie

The mechanical problem is highly nonlineinvolving liquid-solid interaction and complconstitutive equations. Stress arises primarfrom the mismatch of strains caused by largtemperature gradients and depends on ttime- and microstructure-dependent inelast

flow of the material. Even identifying tnumerous metallurgical parameters involvin these relationships is a daunting task.

The coupling between the thermal and tmechanical problems is an additional difculty. This coupling comes from tmechanical interaction between the castinand the mold components, through gap fomation or the buildup of contact pressurlocally modifying the heat exchange.

Accounting for the mold and its interactiowith the casting makes the problem multidmain, usually involving numerous deformable components with coupled interactioand contact analysis.

Cast parts usually have very complex thre

dimensional shapes, which puts gredemands on the interface between computeaided design and the mechanical solvers anon computational resources.

The main cause of embrittlement is the seregation of solute impurities and alloyielements to the interdendritic liquid betweeprimary grains, which lowers the solidtemperature locally. Segregation is mosevere, and thus most important, at tgrain boundaries, owing to the greater loc

364 / Fundamentals of the Modeling of Damage Evolution and Defect Generation


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interdenritic spacing locally, which allowsthe liquid to persist longer between grainboundaries.

Larger primary grain size increases strainconcentration and embrittlement, so it mustalso be predicted. Because the grain sizeevolves with time, the grain size in the finalcooled microstructure differs from the pri-

mary grain size, so grain size measurementsfor model validation should be inferred fromanalysis of the microsegregation pattern.

Stress on the liquid films depends on theability of liquid to flow through the dendriticstructure to feed the volumetric shrinkage,relative to the strength of the surroundingdendritic skeleton. Thus, accurate perme-ability models are required for the mushyzone, which, in turn, require accurate predic-tion of the microstructure, including the den-drite arm shapes, especially at the grainboundaries.

Crack prediction requires modeling the distri-bution of supersaturated dissolved gas and itsnucleation into pores or crack surfaces.

The formation of solid precipitates tends topin the primary grain boundaries, enhancingstrain concentration. The interfering precipi-tates also act as nucleation sites for both gasbubbles and voids, both of which increaseembrittlement. Modeling precipitation is dif-ficult, owing to the multicomponent natureof commercial alloys and the importance of kinetic delays.

The subsequent refilling of hot tears withsegregated liquid alloy can cause internaldefects that are just as serious as exposedsurface cracks, which oxidize. This againrequires accurate prediction of both inter-dendritic and intergranular solute flow.

The most important parameters to hot tear-ing— the stress-tensor field, which acts toconcentrate tensile strain in liquid regions of the mushy zone, and the fluid-velocity vector field, which acts to fill the voids— are both

three-dimensional time-varying quantitiesthat depend greatly on the orientation andshape of the microstructure. Thus, evenempirical criteria to predict hot tears dependon conducting experiments with the proper load orientation, rates, and microstructures.

The important length scales range frommicrometers (dendrite arm shapes) to tens

of meters (metallurgical length of a continu-ous caster), with a similar huge order-of-magnitude range in time scales.

Heat-Transfer Modeling

Accurate calculation of the evolving temper-ature distribution during the casting process isthe first and most important step in the analysisof hot tears. In addition to solving the transientheat-transport equation with phase change, thiscritical task usually requires coupling with tur-bulent fluid flow during mold filling and inter-action with the mold walls, with particular

attention to the interfacial gap.Heat transfer across the mold-casting inter-

face depends on the size of the gap (if open)or the contact pressure (if closed), so couplingwith results from a mechanical analysis is oftenneeded. Figure 1 shows the changes in interfa-cial heat transfer for these two cases. When agap opens between the casting and the molddue to their relative deformation, the heat trans-fer drops in proportion to the size of the gap.Heat flows across the interface, qgap, by con-duction through the gas within the gap and byradiation between the two parallel surfaces:

q gap ¼ kgas

g

ðT c T mÞ þsðT 4c T

4m Þ

1

ec þ 1

em 1

(Eq 1)

where k gas (T ) is the thermal conductivity of thegap; g is the gap thickness; T c and T m are thelocal surface temperature of the casting and

mold, respectively; ec and em are emissivitiand s is the Stefan-Boltzmann constant. avoid numerical problems at small gap sizthis function should be truncated to a finvalue, h0, which corresponds to the closed-gcase and depends on the average roughneMore sophisticated functions can be applto account for mold coating layers, differ

material layers, radiation conduction, contresistances to incorporate surface roughneand other phenomena. Specific examples these gap heat-transfer laws are provided elwhere for continuous casting with oil lubrition (Ref 42) and mold flux (Ref 43).

When contact between the mold and castis good, the interfacial heat flux increases wcontact pressure according to a power l(Ref 44), such as:

q contact ¼ ðh0 þ ApB c ÞðT c T mÞ (Eq

where pc is the contact pressure, and A andare fitting parameters that depend on materials, lubricants, roughnesses, and tempe

ture. After removal from the mold, htransfer is given by uncoupled surface convtion coefficients. Accurate characterizationthe surface heat flux for all of these conditiorequires careful calibration and validation wexperimental measurements and is a critical sin modeling.

Thermomechanical Modeling

Prediction of the displacements, strains, astresses during the casting process is the nstep in predicting residual stress, the distorshape, and crack defects, including hot te

As previously mentioned, stress analysis is aimportant in the prediction of porosity and sregation. The modeling of mechanical behavrequires solution of the equilibrium or momtum equations relating force and stress, compatibility equations relating strain and dplacement, and the constitutive equations reling stress and strain. This is because boundary conditions specify either force or dplacement at different regions of the domboundaries.

Governing Equations. The conservationforce (steady-state equilibrium) or moment(transient conditions) can be expressed by:

r

@ v

@ t þ v

rv

¼ r s

þ rg (Eq

where s is the stress tensor, r is the densg is the gravitational acceleration, v is velocity field, r is the gradient operator amatrices (vectors and tensors) are denotedbold. Once solidified, the velocity terms tcomprise the left side of Eq 3 can be neglect

The strains that dominate thermomechanibehavior during solidification are on the orof only a few percent, prior to crack formatiWith small gradients of spatial displacemeFig. 1 Modeling heat-transfer coefficient across the mold-casting gap



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ru ¼ @ u=@ x, the compatibility equationssimplify to (Ref 45):

« ¼ 1

2 ru þ ðruÞT

(Eq 4)

where « is the strain tensor, u is the displace-ment vector, and T denotes transpose. Thissmall-strain assumption simplifies the analysis

considerably. The compatibility equations canalso be expressed as a rate formulation, wherestrains become strain rates and displacementsbecome velocities. This formulation is moreconvenient for a transient computation withtime integration involving fluid flow and/or large deformation.

Choosing constitutive models to relate stressand strain is a very challenging aspect of stressanalysis of solidification, because it depends onaccurately capturing the highly nonlinear evolu-tion of the material microstructure with numericalparameters. Traditionally, this is accomplishedwith a family of elastic-plastic stress-strain curvesat the appropriate temperatures and strain rate(s)

and perhaps by adding a separate strain-rate func-tion of temperature, stress, and time to account for the time-dependent softening effects of creep.

However, the state variables of strain andtime are not enough to quantify the strength of the material, especially during loading rever-sals. Furthermore, the effects of plastic strainand creep-strain rate are not independent. Thus,unified models have been developed that com-bine the different microstructural mechanismstogether in terms of state variables that relatemore closely to fundamental microstructuralparameters such as dislocation density. Manymodels of different complexity can be foundin the literature (Ref 46, 47). In their simplestform, these constitutive equations for metals

are often expressed in terms of the state vari-ables of temperature and inelastic strain:

_« ¼ _«el þ _«in þ _«th (Eq 5)

_«el ¼

1 þ n

E _s

n

E tr ð _sÞI þ _T

@

@T

1 þ n

E

s

_T @

@T

n

E

tr ðsÞI

(Eq 6)

_«in ¼ f ðs; T; structureÞ (Eq 7)

_«th ¼

ffiffiffiffiffiffiffiffiffiffiffiffir T 0ð Þ

r

3

s 1

" #I (Eq 8)

These tensor equations are expressed interms of rates, where an overdot is the timederivative, tr is the trace of a matrix, I is theidentity tensor, and every variable shoulddepend on temperature, T . The strain-rate ten-sor, _«, is split into an elastic component _«el

,

an inelastic (nonreversible) component _«in

,and a thermal component _«th

. Equation 6 is

the hypoelastic Hooke’s law, where E isYoung’s modulus, n is the Poisson’s coefficient,and _s is the time derivative of the stress tensor s. Equation 7 gives a framework for evolvingthe inelastic strain tensor, «in, which is oftenused as the only parameter to characterize

material structure. The thermal strains (Eq 8)include the solidification shrinkage and arebased on the temperature field solved with theheat-transfer model. Care should be taken inchoosing a consistent reference temperature,T 0, and in differentiating to extract the thermalstrain rate, which can be accomplished numeri-cally. Finding suitable constitutive equations tocharacterize the material mechanical responsefor the wide range of conditions experiencedduring solidification is a formidable task thatrequires careful experiments under differentloading conditions, a reasonable form for thetheoretical model, and advanced fitting proce-dures to extract the model coefficients.

Solution Strategies. Thermomechanical

analysis of casting processes poses special diffi-culties due to the simultaneous presence of liq-uid, mushy, and solid regions that move withtime as solidification progresses, the highlynonlinear constitutive equations, complexthree-dimensional geometries, coupling withthe thermal analysis, interaction with the mold,and many other reasons. Several different stra-tegies have been developed, according to theprocess and model objectives:

A first strategy is to perform a small-strainthermomechanical analysis on just the solidi-fied portion of the casting domain, extractedfrom the thermal analysis results. This strat-egy is convenient when the solidificationfront is stationary, such as the continuouscasting of aluminum (Ref 48) and steel (Ref 42, 49). For transient problems, such as theprediction of residual stress and shape (butt-curl) during startup of the direct-chill andelectromagnetic continuous casting processesfor aluminum ingots, the domain can beextended in time by adding layers (Ref 48).

A second popular strategy considers theentire casting as a continuum, modifyingthe parameters in the constitutive equationsfor the liquid, mushy, and solid regionsaccording to the temperature and phase frac-tion. For example, liquid can be treated bysetting the strains to zero when the tempera-

ture is above the solidus temperature. Theprimary unknowns are the displacements or displacement increments. To facilitate thetracking of state variables, a Lagrangian for-mulation is adopted, where the domain fol-lows the material. This popular approachcan be used with structural finite-elementcodes, such as MARC (Ref 50) or ABAQUS(Ref 51), and with commercial solidificationcodes or special-purpose software, such asALSIM (Ref 52)/ALSPEN (Ref 53), CASTS(Ref 54), CON2D (Ref 55, 56), Magmasoft

(Ref 57), and Procast (Ref 58, 59). It hbeen applied successfully to simulate defomation and residual stress in shape castin(Ref 60, 61), direct chill casting of alumnum (Ref 48, 52, 53, 60, 62, 63), and continuous casting of steel (Ref 55, 64). Timintegration of the highly nonlinear constittive equations can benefit from spec

local-global integration numerical metho(Ref 56) or recent explicit metho(Ref 65). Assuming small strain and avoiing Poisson’s ratio close to 0.5 for stabilireasons (Ref 66, 67) means that the liquphase is not modeled accurately. Thus, somphenomena must be incorporated from othmodels, such as heat transfer from impining liquid jets (Ref 68) and fluid feeding inthe mushy zone (Ref 55).

A third strategy simulates the entire castintreating the mass and momentum equatioof the liquid and mushy regions with mixed velocity-pressure formulation. Tprimary unknowns are the velocity (timderivative of displacement) and pressu

fields, which makes it easier to impose thincompressibility constraints and to handhydrostatic pressure loading. Indeed, tvelocity-pressure formulation is also applito the equilibrium of the solid regions, order to provide a single continuum framwork for the global numerical solution. Thstrategy has been implemented into coddedicated to casting analysis, such as THERCAST (Ref 64, 69, 70) and VULCA(Ref 71). If stress prediction is not importaso that elastic strains can be ignored, thethis formulation simplifies to a standafluid-flow analysis, which is useful in tprediction of bulging and shape in larg

strain processes. For problems involvilarge strain, such as squeeze casting, thstrategy is suited to an arbitrary LagrangiEulerian (ALE) formulation. In a Euleriformulation, material moves through tcomputational grid, which remains statioary in the laboratory frame of reference anrequires careful updating of the state vaables. In ALE, mesh updating is partialindependent of the material velocity maintain the quality of the computationgrid. Further details are provided elsewhe(Ref 69, 72).

Hot-Tearing Criteria

The next step is to quantify embrittlemeand to incorporate it with the thermal-streanalysis to predict hot-tear cracks. Hot-tearinphenomena are too complex, too small-scaand insufficiently understood to model in detaas part of the macroscale thermomechanicanalysis. Thus, several different criteria anapproaches have been developed to predict htears from the results of such analyses. Thtopic is the focus of many ongoing researefforts, and although many of these criter



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reproduce observed trends, much more work isneeded before quantitative predictions arereliable.

Different approaches are needed for differentmicrostructures and metals, according to themost important phenomena that govern crackformation. Hot-tear cracks forming within largenetworks of mushy, equiaxed grains require

accurate constitutive models to quantify therheology of the mushy region. Cracks betweencolumnar grains require models that incorporatethe balance between liquid feeding betweendendrites and tensile deformation perpendicular to the direction of dendrite growth. Thehot tearing of aluminum alloys additionallydepends on the critical stress to nucleate a gasbubble. In steel, dissolved gas contents are usu-ally low, so hot tears usually refill withsegregated liquid without opening into cracks.This macrosegregation is very damaging, so itbecomes a very important phenomenon tomodel accurately. Every criterion depends onexperimental measurements and how best toincorporate them.

Thermal-Analysis-Based Criteria. The resultsof the solidification heat-transfer analysis alonecan provide some important insights into hottearing. As illustrated in Fig. 2, the location of hot-tear cracks observed in a casting can be

related to their time of formation. Cracks tendto initiate near the casting surface (x1) andpropagate toward the center of casting (x2) assolidification progresses. Figure 2(b) showsthe progress of the mushy zone and importantisotherms with time, based on the results of asolidification heat-transfer model. In the caseof continuous casting, the time axis also corre-

sponds to distance in the casting direction, sothe figure depicts the actual shape of the solidi-fication fronts in the real caster.

Casting conditions that produce faster solidi-fication and alloys with wider freezing rangesare more prone to hot tears. Thus, many criteriato indicate hot-tear cracking susceptibility(HCS) are solely based on thermal analysis.One (Ref 73) simply compares the local timespent between two critical solid fractions, gs1and gs2 (typically 0.9 and 0.99, respectively),with the total local solidification time (or areference solidification time):

HCSclyne ¼ t0:99 t0:90t0:90 t0:40

(Eq 9)

Mechanical-Analysis-Based Criteria. Manydifferent criteria have been developed to predicthot-tear cracks from the results of a mechanicalanalysis. Regardless of the model formulation,developing an accurate criterion function topredict hot tears relies on measurements, suchas the submerged split-chill tensile test (Ref 74–76). This experiment applies and measuresa tensile load on the solidifying shell, perpen-dicular to the growth direction, so it matchesthe conditions present in hot tearing betweencolumnar grains. Other experiments, such asthe Gleeble, apply a tensile load to remeltedmetal that is held in place by surface tension.Care must be taken in the interpretation of suchmeasurements because the load is generallyapplied in the same direction as solidification-front growth. Proper interpretation of any hot-tearing experiment requires detailed modelingof the experiment itself, because conditionsare never constant, and, at best, only raw datasuch as temperature, displacement, and forcecan be measured. The parameters of greatestinterest must be extracted using models.

Criteria based on classical mechanics oftenassume cracks will form when a critical stressis exceeded, and they are popular for predictingcracks at lower temperatures (Ref 77–80).Tensile stress is also a requirement for hot-tear formation (Ref 81). This critical stress depends

greatly on the local temperature and strain rate.The maximum tensile stress occurs just beforeformation of a critical flaw (Ref 82).

Measurements often correlate hot-tear forma-tion with the accumulation of a critical level of mechanical strain while applying tensile load-ing within a critical solid fraction where liquidfeeding is difficult. This has formed the basisfor many hot-tearing criteria. One such model(Ref 81) accumulates inelastic deformationover a brittleness temperature range, which isdefined, for example, as g s 2 0:85; 0:99½ for a

Fe-0.15wt%C steel grade. The local conditfor fracture initiation is then:Xg s2

g s1ein ecr (Eq

in which the critical strain, ecr , is 1.6% atypical strain rate of 3 104 s1. Caremeasurements during bending of solidify

steel ingots have revealed critical strains raing from 1 to 3.8% (Ref 81, 83). The lowvalues were found at high strain rate andcrack-sensitive grades (e.g., high-sulfur perittic steel) (Ref 81). In aluminum-rich aluminucopper alloys, critical strains were reporfrom 0.09 to 1.6% and were relatively indepdent of strain rate (Ref 82).

The critical strain decreases with increasstrain rate, presumably because less timeavailable for liquid feeding, and also decreafor alloys with wider freezing ranges. Tfollowing empirical equation for the critistrain in steel, ecr , was based on fitting measuments from many bend tests (Ref 84):

ecr ¼ 0:02821_e0:3131T 0:8638B

(Eq

where _e is the strain rate (s1) and DT B is brittle temperature range (C) defined betwethe temperatures corresponding to solid frtions of 0.9 and 0.99.

An elegant analytical-criterion model been derived to predict hot tearing, based when the local liquid feeding rate along interdendritic spaces between the primcolumnar dendrites is insufficient to balanthe rate of tensile strain increase in the ppendicular direction across the mushy z(Ref 85, 86). Specifically, gas pores cavitateseparate the residual liquid film between dendrites when the tensile strain rate excea critical value:

_e 1

R

l22 rT k k

180ml

rLrS

pm pCð Þ vTrS rL

rSH

(Eq

in which ml is the dynamic liquid viscosity, lthe secondary dendrite arm spacing, pm is local pressure in the liquid ahead of the muszone, pC is the cavitation pressure, vT is velocity of the solidification front, and jjris the magnitude of the temperature gradiacross the mushy zone. The quantities R a H depend on the solidification path of the allo

R ¼

ð T 1T 2

g s2F ðT Þ

g l 3 dT H ¼

ð T 1T 2

g s2

g 2ldT

F ðT Þ ¼ 1

rT k k

ð T T 2

g sdT

(Eq

where the integration limits are calibratparameters (Ref 87). The upper limit T 1 mbe the liquidus or the coherency temperatuwhile the lower limit T 2 typically is within solid fraction range of 0.95 to 0.99 (Ref 8

Fig. 2 Relating the location of hot-tear crack formationto results of a transient thermal simulation. (a)

Measure crack location in casting. (b) Predict shellthickness history



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This criterion model has been applied to hottearing of aluminum microstructures (Ref 87).

For hot tearing within large mushy regions,typically equiaxed microstructures, constitutivebehavior of the mushy zone to predict the localfluid flow and deformation of the dendriticnetwork presents an important additional chal-lenge. Other criterion models that focus more

on this aspect of hot tearing have recently beendeveloped (Ref 88–90). Further details on hottearing of aluminum alloys are reviewed else-where (Ref 91).

Microscale Model-Based Criteria. Detailedcomputational models can be developed based ontemperature, fluid flow, stress, and strain in themushy zone during solidification. For example, afinite-element model of an equiaxed mushy zoneof aluminum has been applied to investigate con-stitutive behavior and to quantify strain concentra-tion in the liquid films for a few specific sets of conditions (Ref 92). Once such models are moremature, their results can be incorporated into bet-ter criteria for hot tearing. A final difficult task isextracting results from the macroscale model

results to compare with the criterion models,owing to the sensitivity of numerical estimates of parameters suchas strain rate to numericaloscilla-tions and mesh refinement effects. Thus, couplingdifficulties between the macro- and microscalemodels is another reason that hot-tear crack pre-diction is an ongoing challenge.

Microsegregation Modeling

Quantifying the relationship between temper-ature and phase fractions is an essential part of each model involved in thepredictionof hottear-ing, including the heat transfer, the mechanical,

and the hot-tear criterion models. This relation-ship determines how latent heat is evolved inthe heat-transfer model and how to switchbetween constitutive models in the mechanicalmodel. Although simple linear, lever-rule, or Scheil-based relations are usually sufficient for these macroscale models, microsegregation isan essential aspect of embrittlement and greatlyaffects the phase-fraction temperature relation-ship involved in any hot-tearing criterion. Better relationships use the results of microsegregationmodels that consider partial diffusion of multiplesolute elements in the solid phase, using simpleanalytical solutions (Ref 93), or one-dimensionalmodels of a single secondary dendrite arm (Ref

94). More advanced models couple this calcula-tion together with the macroscale models andallow the relationship to evolve to incorporatenucleation undercooling and other phenomena(Ref 27). Ideally, the relationships appliedbetween dendrites and at grain boundariesshould be different, and they should vary withlocation in the casting, to account for macrose-gregation and other phenomena. An importantconcept, which is often overlooked, is that thesame (or very close) relationship must be usedin each model of the analysis. Inconsistency

between microsegregation models is one of themain reasons why different researchers have pro-posed different critical temperatures in their hot-tear criteria. Experiments conducted to quantifythe parameters in hot-tearing models should fullyreport both the raw data and the models used toextract hot-tearing parameters, including themicrosegregation model.

Model Validation

Model validation is a crucial step in anycomputational analysis. Analytical solutionsare needed to prove internal consistency of themodel and to control discretization errors.Comparison with experiments is needed toprove the model assumptions, property data,and boundary conditions. Weiner and Boley(Ref 95) derived an analytical solution for uni-directional solidification of an unconstrainedplate, which serves as an ideal benchmark prob-lem to validate thermal and mechanical models.

The plate is subjected to sudden surface quenchfrom a uniform initial temperature to a constantmold temperature, with a unique solidificationtemperature, an elastic-perfectly-plastic consti-tutive law, and constant properties.

This benchmark problem can be solved with asimple mesh of one row of elements extendingfrom the casting surface into the liquid, as shownin Fig.3. Numericalpredictions shouldmatch withacceptable precision using the same element type,mesh refinement, and time steps planned for thereal problem. For example, the solidification stressanalysis code CON2D (Ref 55) and the commer-cial code ABAQUS (Ref 51) were applied for typ-ical conditions of steel casting (Ref 56).

Figures 4 and 5 compare the temperature anstress profiles in the plate at two times. Ttemperature profile through the solidifyinshell is almost linear. Because the intericools relative to the fixed surface temperaturits shrinkage generates internal tensile strewhich induces compressive stress at the surfacWith no applied external pressure, the averag

stress through the thickness must naturally equzero, and stress must decrease to zero in the liuid. Stresses and strains in both transverse diretions are equal for this symmetrical problem. Tclose agreement demonstrates that the computtional model is numerically consistent and han acceptable mesh resolution. Such studireveal that a relatively fine mesh is needed achieve reasonable accuracy, and that resufrom many thermomechanical models reportin previous literature had insufficient merefinement. Comparison with experimental mesurements is also required, to validate that thmodeling assumptions and input data areasonable.

Fig. 3 One-dimensional slice domain for modelsolidifying plate

Fig. 4 Temperatures through solidifying plate at different times, comparing analytical solution and numerpredictions



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Case Study—Billet Casting SpeedOptimization

A Lagrangian model of temperature, distor-tion, strain, stress, and hot tearing has beenapplied to predict the maximum speed for con-tinuous casting of steel billets without formingoff-corner internal hot-tear cracks. The two-dimensional transient finite-element thermome-

chanical model, CON2D (Ref 55, 56), hasbeen used to track a transverse slice throughthe solidifying steel strand as it moves downwardat the casting speed to reveal the entire three-dimensional stress state. The two-dimensionalassumption produces reasonable temperaturepredictions, because axial (z-direction) conduc-tion is negligible relative to axial advection(Ref 43). In-plane mechanical predictions arealso reasonable, because bulging effects aresmall, and the undiscretized casting directionis modeled with the appropriate condition of generalized plain strain. Other applicationswith this model include the prediction of ideal taper of the mold walls (Ref 96)and quantifying the effect of steel grade on

oscillation mark severity during level fluctua-tions (Ref 97).

The model domain is an L-shaped region of atwo-dimensional transverse section, shown inFig. 6. Removing the central liquid region savescomputation and lessens stability problemsrelated to element locking. Physically, this“trick” is important in two-dimensionaldomains because it allows the liquid volume

to change without generating stress, whichmimics the effect of fluid flow into and out of the domain that occurs in the actual open-topped casting process. Simulations start at themeniscus, 100 mm below the mold top, andextend through the 800 mm long mold andbelow, for a caster with no submold support.The instantaneous heat flux, given in Eq 14,was based on plant measurements (Ref 98). Itwas assumed to be uniform around the perime-

ter of the billet surface in order to simulateideal taper and perfect contact between the shelland mold. Below the mold, the billet surfacetemperature was kept constantat its circumferen-tial profile at mold exit. This eliminates theeffectof spray cooling practice imperfections on sub-mold reheating or cooling and the associatedcomplication for the stress/strain development.A typical plain carbon steel was studied (0.27%C, 1.52% Mn, 0.34% Si) with 1500.7 Cliquidus temperature and 1411.8 C solidustemperature:

q MW=m2

¼ 5 0:2444t sð Þ t 1:0 s

4:7556t sð Þ0:504 t > 1:0 s

(Eq 14)

Different constitutive models were used for each phaseof the solidifying steel. The followingelastic-visco-plastic constitutive equation wasdeveloped for the austenite phase (Ref 99) asa function of percent carbon content (%C)by fitting constant strain-rate tensile tests(Ref 100, 101) and constant-load creep tests(Ref 102) to the form in Eq 5 and 7:

_eeq ¼ f %C seq s0 1=m

exp 4:465104

T

where

f %C ¼4:655 104 þ 7:14 104ð%CÞ

þ 1:2 104ð%CÞ2

s0 ¼ ð130:5 5:128 103T Þef 2eq

f 2 ¼ 0:6289 þ 1:114 103T

1=m ¼ 8:132 1:54 103T

with T in kelvin; and seq and s0 in MPa

(Eq

Further equations, such as the associated flrule, are needed to transform this scalar eqtion into tensor form and to account reversals in loading conditions. Equation and a similar one for delta-ferrite have beimplemented into the finite-element coCON2D (Ref 55) and THERCAST (Ref 1and applied to investigate several probleinvolving mechanical behavior during contious casting.

Elastic modulus is a crucial property tdecreases with increasing temperature. It is dficult to measure at the high temperatuimportant to casting, owing to the susceptibil

of the material to creep and thermal strain ding a standard tensile test, which resultsexcessively low values. Higher values obtained from high-strain-rate tests, suchultrasonic measurements (Ref 104). Elamodulus measurements in steels near the sodus temperature range from 1 (Ref 105)44 GPa (Ref 106), with typical modulus valuof 10 GPa near the solidus (Ref 98, 107, 10

Fig. 5 Lateral (y and z ) stress through solidifying plate at different times, comparing analytical solution and numericalpredictions

Fig. 6 Model domain



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The density needed to compute thermalstrain in Eq 8 can be found from a weightedaverage of the values of the different solid andliquid phases, based on the local phase frac-tions. For the example of plain low-carbonsteel, the following equations were compiled(Ref 55) based on the phase fractions of alpha-ferrite ( f a), austenite ( f g), delta-ferrite

( f d) (Ref 109, 110), and liquid ( f l) measure-ments (Ref 111):

r kg=m3

¼raf a þ rgf g þ rdf d þ rlf l

ra ¼7881 0:324T Cð Þ 3 105T Cð Þ2

rg ¼ 100 8106 0:51T Cð Þ½

100 %Cð Þ½ 1 þ 0:008 %Cð Þ½ 3

rd ¼ 100 8011 0:47T Cð Þ½

100 %Cð Þ½ 1 þ 0:013 %Cð Þ½ 3

rl ¼7100 73 %Cð Þ 0:8 0:09 %Cð Þ½

T Cð Þ 1550½

(Eq 16)

Sample results are presented here for one-

quarter of a 120 mm square billet cast at speedsof 2.0 and 5.0 m/min. The latter is the criticalspeed at which hot-tear crack failure of the shellis just predicted to occur. The temperature andaxial (z) stress distributions in a typical sectionthrough the wideface of the steel shell cast at2.0 m/min are shown in Fig. 7 and Fig. 8 at four different times during cooling in the mold.Unlike the analytical solution in Fig. 4, the sur-face temperature drops as time progresses. Thecorresponding stress distributions are qualita-tively similar to the analytical solution (Fig. 5).The stresses increase with time, however,as solidification progresses. The realistic consti-tutive equations produce a large region of tension near the solidification front. The magni-

tude of these stresses (and the correspondingstrains) are not predicted to be enough to causehot tearing in the mold, however. The resultsfrom two different codes reasonably match,demonstrating that the formulations are accu-rately implemented, convergence has been

achieved, and that the mesh and time-step refine-ment are sufficient.

Figure 9(a) shows the distorted temperaturecontours near the strand corner at 200 mmbelow the mold exit for a casting speed of 5.0 m/min. The corner region is coldest, owingto two-dimensional cooling. The shell becomeshotter and thinner with increasing casting

speed, owing to less time in the mold. Thisweakens the shell, allowing it to bulge moreunder the ferrostatic pressure below the mold.

Figure 9(b) shows contours of hoop stressconstructed by taking the stress component act-ing perpendicular to the dendrite growth direc-tion, which simplifies to s x in the lower rightportion of the domain and s y in the upper leftportion. High values appear at the off-corner subsurface region, due to a hinging effect thatthe ferrostatic pressure over the entire faceexerts around the corner. This bends the shellaround the corner and generates high subsur-face tensile stress at the weak solidificationfront in the off-corner subsurface location. Thistensile stress peak increases slightly and moves

toward the surface at higher casting speed.

Stress concentration is less and the surfahoop stress is compressive at the lower castinspeed. This indicates no possibility of surfacracking. However, tensile surface hoop streis generated below the mold at high speed Fig. 9(b) at the face center due to excessibulging. This tensile stress, and the accompaning hot-tear strain, may contribute to longitud

nal cracks that penetrate the surface.Hot tearing was predicted using the criterioin Eq 10 with the critical strain given in Eq 1Inelastic strain was accumulated for the compnent oriented normal to the dendrite growdirection, because that is the weakest directioand corresponds to the measurements used obtain Eq 11. Figure 9(c) shows contours hot-tear strain in the hoop direction. The higest values appear at the off-corner subsurfaregion in the hoop direction. Moreover, signicantly higher values are found at higher castinspeeds. For this particular example, hot-testrain exceeds the threshold at 12 nodes, located near the off-corner subsurface regioThis is caused by the hinging mechanis

Fig. 7 Temperature distribution along the solidifyingslice in continuous casting mold

Fig. 8 Lateral (y and z ) stress distribution along tsolidifying slice in continuous casting mold

Fig. 9 Distorted contours at 200 mm below mold exit. (a) Temperature. (b) Hoop stress. (c) Hot-tear strain



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around the corner. No nodes fail at the center surface, in spite of the high tensile stress there.The predicted hot-tearing region matches thelocation of off-corner longitudinal cracksobserved in sections through real solidifyingshells, such as the one pictured in Fig. 10. The

bulged shape is also similar.Results from many computations were used

to find the critical speed to avoid hot-tear cracks as a function of section size and workingmold length, as presented in Fig. 11 (Ref 108).These predictions slightly exceed plant prac-tice, which is generally chosen by empiricaltrial and error. This suggests that plant condi-tions such as mold taper are less than ideal, thatother factors limit casting speed, or thosespeeds in practice could be increased. The qual-itative trends are the same.

This quantitative model of hot tearing pro-vides many useful insights into the continuouscasting process. Larger section sizes are moresusceptible to bending around the corner andthus have a lower critical speed, resulting inless productivity increase than expected. Thetrend toward longer molds over the past threedecades enables a higher casting speed withoutcracks by producing a thicker, stronger shell atmold exit.

Conclusions

The prediction of defects represents the cul-mination of solidification modeling. It enablesmodels to make practical contributions to realcommercial processes, but it requires incorpor-ating together and augmenting the models of

almost every other aspect of casting simulation.Hot-tear crack prediction requires accurate ther-mal and mechanical analysis, combined withcriteria for embrittlement. As computing power and software tools for computational mechanicsadvance, it is becoming increasingly possible toperform useful analysis of fluid flow, tempera-ture, deformation, strain, stress, and relatedphenomena in real casting processes. Computa-tions are still hampered by the limits of meshresolution and computational speed, especially

for realistic three-dimensional geometries anddefect analysis. The modeling of defects suchas hot tears is still in its infancy, and there ismuch work to be done.

ACKNOWLEDGMENTS

The author wishes to thank the member com-panies of the Continuous Casting Consortium,

the National Center for Supercomputing Appli-cations at the University of Illinois, and theNational Science Foundation for support of thiswork. Thanks are also extended to Michel Bel-let, Ecole des Mines de Paris, for coauthoring aprevious version of this article, which appearedin Casting, Volume 15 ASM Handbook , 2008(Ref 72), upon which this article is based.

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