Modeling of macrosegregation caused by volumetric deformation in a coherent mushy zone

Modeling of Macrosegregation Caused by VolumetricDeformation in a Coherent Mushy Zone

LILIA C. NICOLLI, ASBJØRN MO, and MOHAMMED M’HAMDI

A two-phase volume-averaged continuum model is presented that quantifies macrosegregation for-mation during solidification of metallic alloys caused by deformation of the dendritic network andassociated melt flow in the coherent part of the mushy zone. Also, the macrosegregation formationassociated with the solidification shrinkage (inverse segregation) is taken into account. Based onexperimental evidence established elsewhere, volumetric viscoplastic deformation (densification/dilatation)of the coherent dendritic network is included in the model. While the thermomechanical model pre-viously outlined (M. M’Hamdi, A. Mo, and C.L. Martin: Metall. Mater. Trans. A, 2002, vol. 33A,pp. 2081-93) has been used to calculate the temperature and velocity fields associated with the ther-mally induced deformations and shrinkage driven melt flow, the solute conservation equation includ-ing both the liquid and a solid volume-averaged velocity is solved in the present study. In modelingexamples, the macrosegregation formation caused by mechanically imposed as well as by thermallyinduced deformations has been calculated. The modeling results for an Al-4 wt pct Cu alloy indicatethat even quite small volumetric strains (�2 pct), which can be associated with thermally induceddeformations, can lead to a macroscopic composition variation in the final casting comparable tothat resulting from the solidification shrinkage induced melt flow. These results can be explained bythe relatively large volumetric viscoplastic deformation in the coherent mush resulting from the appliedconstitutive model, as well as the relatively large difference in composition for the studied Al-Cualloy in the solid and liquid phases at high solid fractions at which the deformation takes place.

I. INTRODUCTION

MACROSEGREGATION, i.e., composition inhomo-geneities at the scale of the ingot, occurs in a wide range ofcasting processes. It develops in the solidification interval, andis due to relative motion between liquid and solid phases hav-ing different compositions. Such relative motion can be causedby the motion of free crystals in the melt,[2–6] solidificationshrinkage,[7,8] free convection,[9–13] and exudation.[14,15,16]

Another possible driving force for macrosegregation for-mation is deformation, or more specifically densification ordilatation, of the dendritic network in the coherent mushy zoneat relatively high solid fractions. Deformation can be externallyimposed or thermally induced. Examples from the first can befound in continuous casting of steel as well as in aluminumtwin roll casting. In both these processes, deformation imposedby rolls leads to centerline macrosegregation in the finalproducts.[17–21] In aluminum direct chill casting, on the otherhand, there is no externally imposed deformation. Thermallyinduced deformations might, however, cause dilatation/densi-fication of the mushy zone.[22] Models for macrosegregationdue to deformations in continuous casting of steel were devel-oped by Janssen et al.,[23] Miyazawa and Schwerdtfeger,[17] andKajitani et al.[24] using quite ad hoc expressions for the defor-mations. Addressing the same phenomenon, Lesoult et al.[25]

recently suggested a Lagrangian model for the heat and thesolute conservation equations. No mechanical model was, how-

ever, included in order to relate the volume change of the den-dritic network to the forces of the rolls. This phenomenon hasbeen experimentally studied in organic alloys.[26]

This report presents a mathematical model that quanti-fies macrosegregation formation during solidification ofmetallic alloys caused by mechanically imposed as well asthermally induced deformation of the dendritic network inthe coherent part of the mushy zone. Based on experimen-tal evidence established elsewhere,[27] volumetric viscoplasticdeformation (densification/dilatation) of the coherent den-dritic network is included in the model. Also, the melt flowand corresponding macrosegregation associated with thesolidification shrinkage is taken into account, and it willbecome apparent that the macrosegregation caused by thedensification/dilatation is in some cases of the same order ofmagnitude as that caused by the shrinkage. Macrosegrega-tion caused by free convection is quite small at high solidfractions[10] and not accounted for in the model. Forcedconvection on the other hand, e.g., that associated withexudation in situations in which (a part of) the coherentmush is not in direct contact with a mold,[16] is taken intoaccount in the modeling equations, but this phenomenonwill not be further discussed here. The model is based uponvolume averaged conservation equations,[28] which meansthat the liquid velocity as well as the solid deformation andassociated stresses are treated quite formally withoutintroducing any ad hoc expressions for the deformationmechanisms.

In the work by Lesoult et al.,[25] a set of analytical expres-sions for local micro- and macrosegregation rates duringdeformation of a solidifying mushy zone was presentedemploying a Lagrangian approach. Similar expressions canbe derived from the equations of the present study. In addition,the present model also includes liquid and solid momentum

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 36A, FEBRUARY 2005—433

LILIA C. NICOLLI, Doctorate Student, and ASBJØRN MO, Professor,are with SINTEF Materials Technology, N-0314 Oslo, Norway, and theDepartment of Physics, University of Oslo, Oslo, Norway, N-0316. Contacte-mail: [email protected]. MOHAMMED M’HAMDI, Research Scientist,is with SINTEF Materials Technology.

Manuscript submitted April 19, 2004.

434—VOLUME 36A, FEBRUARY 2005 METALLURGICAL AND MATERIALS TRANSACTIONS A

Table I. Nomenclature (Except for Material and BoundaryParameters, Presented in Tables III through V)

aii � tr (a), [a]t trace, transpose of tensor aC cohesion factorcp specific heat capacitycs, cl, c average mass concentration of the solid,

of the liquid, and of the mixtureE elasticity tensor in Hookes’ lawg acceleration due to gravitygs, gl average volume fraction of the solid and

of the liquid~gl unique solidification path, i.e., the liquid

fraction as a unique function of the temperature

hs, hl, h average specific enthalpy of the solid, ofthe liquid, and of the mixture

I the identity tensorJ interfacial solute transferK permeability of the mushy zoneL specific latent heatps, pl average pressure in the solid and in the

liquidP–

s effective pressure on the solid skeletonR molar gas constantSs solid phase deviatoric effective stress

tensort timeT temperatureTliq, Tsol liquidus, solidus temperaturevs, vl, v average solid, liquid, and mixture

velocity vectorsX triaxiality of the stressGreek letters� interfacial mass transfer

solid strain rate tensorthermal, elastic, and viscoplastic parts of

the strain rate�s, �l, � heat conductivity of the solid, of the

liquid, and of the mixture�s, �l, � density of solid, liquid, and mixture�c average mass concentration�s solid stress tensor�̂s effective solid stress tensor�–s von Mises stress�s

m mean stress�s deviatoric part of solid stress tensor �s

�#sT, �# S

r , �# sp�#s

equations by which the deformation rate is related to processconditions in the casting process.

The model and its governing equations are presented inSection II. Section III is devoted to two modeling casesfocusing on mechanically imposed and on thermally induceddeformation of the coherent mushy zone. The conclusionfollows in Section IV.

II. MODEL AND GOVERNING EQUATIONS

The present work combines an already existing modeladdressing the transport of heat, mass, and momentum in anisotropic and coherent mushy zone[1,29] with a model for themacroscopic solute transport. The existing model was orig-inally developed in order to quantify the main transport phe-nomena involved in hot tearing formation, namely, melt flowcaused by solidification shrinkage and thermally induceddeformation. The present model also accounts for the meltflow into or out of the solidifying two-phase volume ele-ments due to possible densification or dilatation of the den-dritic network. Such volume changes, which could be dueto thermally induced or mechanically imposed deformations,require a constitutive description in which the solid part ofthe two-phase mixture can densify (if under compression)or dilate (if under tensile stress conditions). Experimentalevidence shows that a metallic mush behaves as a com-pressible porous medium saturated with liquid.[30–33] Thisrheology is incorporated into the constitutive equations[32]

that are applied in the present study.With reference to the Nomenclature given in Table I, the

equations expressing the conservation of mass and momen-tum in an isotropic, coherent mushy zone are

[1]

[2]

[3]

[4]

where inertial and viscous forces have been neglected. Furtherdetails on the assumptions and simplifications on which theseequations are based are thoroughly outlined in Reference29. A permeability, K, has been introduced in the interfa-cial momentum transfer term in both momentum conserva-tion equations. A relation between the permeability and theliquid fraction, and closing relations reflecting the constitu-tive behavior of the solid phase in the coherent mush as wellas in the fully solid material, will be introduced along withthe modeling example in Section III–B.

As in Reference 1, the coherent mushy zone is definedas the part of the solid-liquid mixture in which the solidphase can sustain stresses different from those in the liquidphase. Consequently, the solid momentum equation, Eq. [3],is solved in the coherent mush and in the fully solidifiedmaterial, while the liquid momentum equation, Eq. [4],applies in the coherent mush.

�gl§pl � glrlg �gl

2m

K(�l � �s)

§ # (gs�s) � gsrsg � �gl

2m

K(�l � �s) � pl§gs

�(glrl)

�t� § # (gl rl�l) � �G

�(gsrs)

�t� § # (gs rs�s) � G

In metallic systems, there is approximately thermal equi-librium within the averaging volume elements. It is thenconvenient to add the enthalpy conservation equations forthe solid and liquid phases. Introducing a constant latentheat of fusion, L, assuming for simplicity that the heat con-ductivities and capacities be constant, and the last ones thesame in each phase, i.e.,

[5]

and using Eqs. [1] and [2] along with the introduction of amixture density, � � gs�s � gl�l, a mixture heat conductiv-ity l � gsls � glll, and a mixture velocity, �� gs�s�s �gl�l�l, the heat-transfer equation can be written in the form

[6]rcp a�T

�t� � # §Tb � LG � § # (l§T )

hl � hs � L � cpT � L


Fig. 1—Schematic description of the isothermal compression experiment.[27]

Equations for the transport of solute in the solid and liq-uid phases of the mush are required in order to couple themodel represented by Eqs. [1] through [4] and [6] tomacrosegregation formation. These equations can be takendirectly from Reference 28, and reads for a binary alloy.

[7]

[8]

Global diffusion is very small in metallic systems, and hastherefore been neglected. In the present study, it is conve-nient to add these two equations to yield

[9]

by which the change with time of the mixture concentra-tion, �c � �sgscs � �lglcl, is related to the convective termson the right-hand side of the equation.

In most models addressing macrosegregation in a coher-ent mushy zone, the solid phase is considered stagnant (or,in continuous casting, moving with a constant casting speed).This approximation is not made in the present model inwhich there is a nonzero solid velocity field, �s, in Eq. [9]as well as a separate Eq. [3] for the conservation of momen-tum in the solid phase. Also, Eq. [4] expressing liquidmomentum conservation contains the solid velocity vector.

In order to relate the local interfacial mass and solutetransfers, � and J, respectively, in a volume element to otherquantities in that element, closing relations are needed, andwill be presented in Section III. Normally, the temperatureenters into such a relation, but also the total solute concen-tration is part of the relation due to the transport of soluteinto or out of the so-called open volume elements.[34] Thismeans that the governing conservation Eqs. [1] through[4], [6], and [9] are coupled, even when the convective termin Eq. [6] is neglected.

III. MODELING EXAMPLES

The governing equations are solved for two modelingproblems. The first addresses a simple “directional” mechan-ical densification of an Al-Cu alloy sample in order to estimatethe order of magnitude of the macrosegregation resultingeven from a quite small mushy zone deformation. The sec-ond model problem addresses an axisymmetric mold cast-ing of an Al-Cu alloy in order to estimate the contributionof thermally induced deformation to macrosegregation in areal casting situation.

A. Macrosegregation in Densification Test

A modeling problem reflecting “directional” mechanicaldensification is shown in Figure 1. It is here assumed thata binary Al-4 wt pct Cu alloy sample solidifies in a moldsmall enough to make sure that homogeneous temperatureand solid fraction conditions can be assumed. Then, whenthe mush is coherent and at constant temperature, the sam-ple is densified by a grid through which the melt can easily

�(rc)

�t � �§ # (rsgscs�s) � § # (rlglcl�l)

�(glrlcl)

�t� § # (rlglcl�l) � �J

�(gsrscs)

�t� § # (rs gscs�s) � J

evacuate. This model problem reflects the rheological testscarried out by Martin and co-workers[30–33] as a part of theirdevelopment of constitutive equations for the rheology inpartially solidified aluminum alloys.

In this modeling problem, the mush is assumed to becoherent at a temperature corresponding to a solid fractionof � 0.7, or higher. This means that the compressionis carried out at constant temperature below 628 °C for anAl-4 wt pct Cu alloy in thermodynamical equilibrium. Free-slip boundary conditions are imposed on both the liquid andsolid velocities along the vertical walls of the sample suchthat the “densification rate,” �s, can be considered uni-form in the sample, and known, given by the velocity of thegrid at the top boundary. Please note that neither the momen-tum conservation equations nor any constitutive equationsneed to be solved in this case, as long as the forces associatedwith the densification are not to be quantified.

Because the compression is isothermal, there is no phasechange and therefore no interfacial mass transfer from theliquid to the solid. Since also the solid and liquid densitiesare constant, and keeping in mind that gs is uniform, Equa-tion [1] reduces to

[10]

Assuming small strains and linear kinematics, the integra-tion of this equation yields

[11]

by which (the homogeneous) gs is related to the (homoge-neous and known) accumulated strain, , and is the solidfraction before compression. Furthermore, using Eqs. [1]and [2], and assuming the densities to be not only constantbut also equal, Eq. [9] reduces to

[12]

where the use of Eq. [10] leads to the second equality. Asexpected, this equation shows that the concentration evolves

�c

�t� �(cl � cs)

�gs

�t� gs (cl � cs) � ii

s

gs0�ii

s

gs � gs0 exp (�� ii

s ) � gs0 [1 � � ii

s ]

�gs

�t� �gs§ # �s � �gs� ii

s

gscoh


with the densification, and that the evolution is directly pro-portional to the difference in concentrations between the solidand liquid phases. This means that the higher the solid frac-tion is, or the lower the temperature is, the more extensivethe macrosegregation formation. Because of the thermody-namical equilibrium assumption, the compositions of the solidand liquid phases depend solely on the temperature and aretherefore constant during the isothermal compression. Inte-grating Eq. [12] in combination with Eq. [11] then yields

[13]

Figure 2 shows the evolution of the alloy composition, c, vsaccumulated strain, , for an alloy compressed from the ini-tial solid fraction , assuming a partition coefficientk � 0.17. It is seen that even at very small strains, the effectof deformations on the macrosegregation is considerable. Fora volume change of only 2 pct, the concentration was reducedfrom 4.00 wt pct Cu to about 3.76 wt pct Cu, i.e., a relativemacrosegregation of almost 6 pct. Please note that a volumetricviscoplastic strain of 2 pct is on the same order of magnitudeas that being thermally induced during casting of aluminumalloys. This motivates the model problem in Section B.

B. Macrosegregation Caused by Thermally InducedDeformations

The second model problem reflects a cylindrical castingof an Al-4 wt pct Cu alloy ingot, for which the driving forcefor dilatation/densification are the thermally induced defor-mations. In order to quantify these thermally induced defor-mations, the momentum conservation equations, Eqs. [3]and [4], have to be solved along with constitutive equa-tions for the mushy zone permeability and the mechanicalbehavior of the alloy. The development of the temperature

gs0 � 0.9�ii

s

c � c0 � (cl � cs)gs0 � ii

s � c0

1 � (k � 1)gs0(1 � �ii

s )

1 � (k � 1)gs0

field is input to these equations. In order to mathematicallyhandle the rheology of the porous solid skeleton saturatedwith liquid,[35] it is convenient to introduce the so-calledeffective solid stress tensor:[30]

[14]

and to rewrite the momentum equations to yield[1]

[15]

[16]

Here, the last equation stems from a combination of theliquid momentum (Eq. [4]) and the solid and liquid massconservation equations (Eqs. [1] and [2]).

The constitutive equations are summarized in Table II, andthe material parameters are presented in Table III. Brieflydescribed (refer to References 1, 32, 36, and 37 for details),the permeability K is assumed to follow the Kozeny–Carmanrelation. The solid strain and strain rate are subdivided into anelastic, a viscoplastic, and a thermal part. The linear kinemat-ics assumption is furthermore introduced, due to the smallstrains involved with thermally induced deformations in metal-lic alloys. The constitutive model for the viscoplastic part ofthe solid phase deformation in the mushy zone takes intoaccount the possibility of the compression/dilatation of themush, and the effect of the dendrites interlocking, representedby the internal variable C, whose evolution equation, due to lackof experimental data, is approximated by an expression forpure shear, i.e., considering X � 0. The fully solid metal is, on theother hand, assumed to be an incompressible strain-hardeningmaterial, as formulated in Reference 38, with properties depen-dent on the temperature (interpolated from Reference 39).

Because the velocities associated with solidification shrink-age and dendritic network densification/dilatation in the coher-ent mush are quite small, the convective heat transport canbe neglected. The interfacial mass transfer, �, in Eq. [6] canthen be approximated by the first term on the left-hand sideof Eq. [2] (convective term neglected), leading to

[17]

where the liquid density has been approximated by a constantvalue. In order to solve this equation, closing relations betweenthe liquid fraction in a volume element and other quantitiesin that element are required. Normally, the temperature andthe total solute concentration[34] enter into such a relation.However, in the case of relatively small macrosegregationformation, as the thermally induced one, these closing rela-tions can simply be replaced by a unique solidification pathdetermined for a “representative,” closed volume element

[18]

that is assumed to be valid everywhere in the solution domainfor the heat, mass, and momentum equations.[40] As a con-sequence, Eq. [17] can be further simplified, to yield

[19]arcp � Lrl �g�l

�Tb

�T

�t� § # (l§T )

gl � g�l (T)

rcp �T

�t� Lrl

�gl

�t� § # (l§T )

�r

�t� § # (r�s) � § # c rlK

m (§pl � rlg) d � 0

§ # �̂s � rg � §pl

�̂s � (gs�s � glpl I ) � plI � gs�s � gs (ps � pl) I

Fig. 2—Evolution of the Cu concentration during the compression experi-ment for an initial solid fraction and initial concentration c0 �4 wt pct Cu.

gs0 � 0.9


Table II. Closing Relations

Mush viscoplasticity

Fully solid viscoplasticity

cl � cglk�1gl � g�l � aT � Tm

c0mb1/(k�1)

rs � rl (1 � bs) exp (�bT (T � Tliq))K(gl) � K0

gl3

(1 � gl)2

z(T) � Po � P1T � P2T2 � P3T

3 � P4T4

g#

� �# p if T � 423 °C, and 0 otherwise

h(T) � ah a1 � bh tanh aT � dh

ch

bbs � A3

2 �s : �s

v(T) � ava1 � bv tanh aT � dv

cv

bb�# p

� A2

3 �# p :�# p

s � v(T ) (g � g0)h(T) (�

# p)z(T)�# p �

32

�# p

s�s

C*(gs , X � 0) �1 � (1 � gs)

p

1 � exp [(gscoh � gs) /�gs]

a (gs , X � 0) �

a0 � a1

gs1/3

1 � gs1/3

1 � exp [(gscoh � gs) /�gs]

X �Ps

ssss � A3

2 Ss : Ss

�#eq � A2

3 �

#sp :�

#sp Ps � �

1

3 tr (�̂s)

A3 � a1 �2

3 (1 � gs)b(gs)

�2n

n�1Ss � �̂s � Ps I

A2 �9

4 an a(1 � gs)

�1n � 1bb�

2n

n�1�#

0 � A exp a� Q

RTb

dC

dt� a(gs,X) a1 �

C

C*(gs,X)b�# eq�

#sp �

�#0

(Cs0)n a�A2

3 Ps I �

3

2A3 Ssb aA2 Ps

2 � A3s2sb n�1

2

�s � E�se�

#sT �

1

3bT

dT

dt I�

#s �

1

2 1§vs � 3§vs 4 l2 � �

# Ts � �

# es � �

# ps

The unique path could be determined by the lever rule, byScheil’s equation, or by a more elaborate solidification pathmodel based on thermodynamics and local kinetics thatrelates gl to T and possibly other relevant quantities. In thepresent model problem, Scheil’s equation is used (Table II).

The governing equations can now be solved sequentially:first, Eq. [19] with given in Table II is solved providingthe temperature input to the combined mass and momentumEqs. [15] and [16]. Then, after solving these equations, thevolume-averaged velocity fields, �l and �s, are taken as inputto Eq. [9] by which the final macrosegregation is quantified.

In the present work, T, �l, and �s are calculated by a pre-viously developed model for aluminum direct chill castingthat solves the same equations as those outlined in the pre-sent work,[41] namely, Eqs. [17] and [18]. The energy equa-tion is solved by the standard Galerkin method, and theenthalpy is regarded as the primary unknown. The temperaturein the diffusive term is linearized using the method proposedby Amiez and Gremaud.[42] The finite-element formulation ofthe momentum equation for the solid phase is developed sothat it states the principle of virtual work.[46] By using theconstitutive relations, the solid stress is expressed by the totalstrain tensor. The solid momentum equation is then solvedwith the displacements as unknowns, as follows: the solidstress tensor is first computed for a given total strain, whichis estimated by entering into the expression for the total strain

g�l

Table III. Material Parameters for the Closing Relations

Mushy zonePermeability relation

constant K0 � 5.55 � 10�11 m2

Dynamic liquid viscosity � 0.003 kg/m sCooling contraction

coefficient �T � 1.0 � 10�4 K�1

Solidification shrinkage factor �S � 0.06Young’s modulus Em � 10 GPaPoisson’s modulus �m � 0.3Thermal activation energy

of the dense solid phase Q � 154 kJ mol�1

Power-law coefficient n � 3.831Constant in Arhenius law A � 9 � 105 s�1

Coherency solid fractionReference stress for the

dense solid phase s0 � 4.77 MPa�0 � 4.45 �1 � 1.07 � 10�2

p � 0.11 �gs � 1.8 � 10�3

Fully solidYoung’s modulus Es � 70 GPaPoisson’s modulus �s � 0.28�0 � 0.001 P0 � �5.17 � 10�5

a� � 200 MPa a� � 0.155 P1 � 2.70 � 10�4 K�1

b� � 0.83 b� � 1.0 P2 � �2.57 � 10�6 K�2

c� � 132 K c� � 135 K P3 � 6.78 � 10�9 K�3

d� � 540 K d� � 500 K P4 � �4.38 � 10�12 K�4

gscoh � 0.70


a guess for the viscoplastic strain, the known thermal strain,and Hookes’ law for elasticity. The liquid pressure is knownfrom the last time-step. Then, the nonlinear set of constitu-tive equations is solved for each element. These local itera-tions are performed on the increments of the viscoplastic strain,until they converge everywhere. This provides a new guessfor the viscoplastic strain that is again introduced in the solidmomentum equation, until the convergence criterium is ful-filled. The momentum equation for the liquid (Eq. [16]) issolved by the standard Galerkin method with the liquid pres-sure as unknown. The relative velocity �l � �s is calculateda posteriori using Eq. [4].

For the purpose of numerical stability,[43] a small diffu-sive term is introduced in Eq. [9], which is rewritten to yield

[20]

where . This solute conservation equationis solved in the present study by a finite-element methodusing a streamline upwind interpolation function in orderto avoid numerical oscillations.[44] The algebraic equationsresulting from the discretization are solved by the diagonalpreconditioning conjugate gradient method, except for themomentum equation for the solid phase, which is solvedby the incomplete Cholesky-conjugated gradient method.[45]

Due to the cylindrical symmetry, the mathematical prob-lem reduces to two dimensions and the domain is constitutedby a radial cross section of the ingot, as indicated in Figure 3.The ingot is cooled from the bottom and from the vertical walls,and the heat flux through these boundaries is related to thetemperature by a given heat-transfer coefficient, �h. The topboundary is thermally isolated; i.e., the heat flux through thisboundary is zero. Furthermore, liquid with the nominal com-

D � 10�12 m2/s

�(rc)

�t� D §2 (rc) � �§ # (rc�s) � § # (rlgl (�l � �s )cl)

position Al-4 wt pct Cu flows into the domain through the topboundary to compensate for solidification shrinkage and ther-mal contraction. No solute flow is allowed through the otherboundaries. The solid momentum equation, Eq. [15], is solvedonly for the coherent mush and the fully solid material. Thebottom and vertical walls are allowed to contract, whereas nodisplacement through the top boundary is allowed. At thecoherency boundary, given by the coherency solid fraction

*, the stresses undergone by the solid phase are equal

*As shown in Reference 1, this model is not sensitive to the choice ofthe coherency solid fraction because most of the stress buildup in the mushyzone takes place at the end of the solidification interval, at high values ofthe solid fraction.

to those in the liquid phase; i.e., shear stresses are zero and thesolid pressure is equal to metallostatic head. The left boundary

gscoh � 0.7

(a) (b)

Fig. 3—(a) Casting of a cylindrical ingot (radial cross section). The gray area between the liquid and solid indicates the solid-liquid mixture that is subdi-vided into a coherent (darker gray) and a noncoherent part (lighter gray). For symmetry reasons, only half of the ingot is shown. (b) Evolution of the solidfraction during casting.

Table IV. Initial and Boundary Conditions

Initial composition c0 � 4 wt pct CuInitial temperature T0 � 660 °CHeat-transfer coefficient at

vertical and bottom walls �h � 800 W/m2 KExternal temperature Text � 20 °C

Table V. Thermophysical Data for Al-4 Wt Pct Cu

Melting point of pure aluminum Tm � 660 °CSlope of the liquidus line m � �3.4 K/wt pctPartition coefficient k � 0.17Liquid density �l � 2400 kg/m3

Liquid conductivity �l � 82.6 W/m KSolid conductivity �s � 192.5 W/m KSpecific heat capacity cp � 1092.8 J/kg KSpecific latent heat L � 397.5 � 103 J/kg


is a symmetry line, and perpendicular to it, all gradients of fieldquantities are zero. The initial and boundary conditions forthe heat and solute transport problems are specified in Table IV,and the thermophysical material data are given in Table V.

IV. RESULTS

Figure 3(b) shows the evolution of the solid fraction in aradial cross section of the ingot. Solidification advances fromthe bottom and vertical walls. In the beginning of solidifi-cation, thermally induced deformations are accommodatedin the central upper part of the ingot by fluid flow; while inthe lower, solidified part of the ingot, tensile stresses buildup (Figure 4(b)). After about 200 seconds, the ingot is exclu-sively constituted by a cylindrical coherent mushy zonesurrounded by a solid cylindrical shell (Figure 3(b)). Thetensile stresses are then accommodated by viscoplastic strains.Because the mush is much softer than the solid, the strainsconcentrate in this region (Figure 4(a)). At the end of solidi-fication, the volumetric viscoplastic strains accumulated inthe upper part of the ingot are almost 2.4 pct, although theaccumulated thermal strains are quite low (�0.75 pct). Thefully solidified material is incompressible and therefore can-not undergo additional volumetric viscoplastic strain.

In Section III–A, it was shown that even quite smallvolumetric strains on the order of 2 pct can lead to signifi-cant macrosegregation. Having an accumulated viscoplas-tic strain of 2.5 pct, a similar magnitude on the finalmacrosegregation would be expected in the present modelproblem. In order to investigate this effect, three case stud-ies have been carried out in which the macrosegregationcaused by the viscoplastic volume change has been com-pared to that associated with solidification shrinkage. Thelatter phenomenon is well known from the scientific litera-ture and is often referred to as inverse segregation. In thefirst study, case A, the mushy zone is assumed to be incom-

pressible by setting the parameter A2 in Table II equalto zero. Hence, the viscoplastic strain rate becomes inde-pendent of the solid pressure and the volumetric viscoplas-tic strain rate becomes zero. Although the governingequations are more complex, case A corresponds in otherwords to a situation in which the right-hand side of Eq. [12]is zero. Consequently, the main mechanism leading tomacrosegregation is solidification shrinkage. Then, in caseB, the viscoplastic volume change in the mush is taken intoaccount, but the densities of the solid and liquid phases areassumed equal, so that there is no solidification shrinkage.In the last case, case C, none of these restrictions areimposed; both macrosegregation mechanisms are taken intoaccount.

The final macrosegregation patterns for these three casesare shown in Figure 5. Case A has a typical inverse segre-gation pattern with high solute content close to the walls.The solute-depleted region at the central part of the ingotresults from the difficult liquid feeding through the deepmush close to the end of the full solidification of the ingot(Figure 3(b), 200 seconds). For the same reason, solute accu-mulates in the neighboring upper region, where it causespositive macrosegregation. Case B shows almost nomacrosegregation, except close to the centerline, where vis-coplastic volumetric strain builds up. The profiles for casesA and C are also quite similar except for the region closeto the centerline where a positive macrosegregation formsdue to the viscoplastic dilatation of the mush in the region.†

†Please note that even though the total mass of solute is not kept con-stant during the solidification due to the inflow through the top boundary,the composition of the mass entering into the domain is kept constant, equalto the nominal value. Thus, the differences in composition between casesA and C indicate the contribution of the volumetric viscoplastic strains tothe final macrosegregation.

Figure 6 shows for the three cases the concentration andaccumulated viscoplastic volumetric strain profiles along the

� iip

� sp

(a) (b)

Fig. 4—Results for a radial cross section of the ingot: (a) the accumulated volumetric viscoplastic strain at the end of the solidification, and (b) the

evoution of the mean stress at the last stages of solidification.�sm �

1

3 tr (�s) � �ps


Fig. 5—Macrosegregation map in a radial cross section (centerline along the left vertical boundary) for the three case studies described in the body text:no viscoplastic volumetric strain (case A), no solidification shrinkage (case B), and both with viscoplastic volumetric strain and solidification shrinkage(case C). The dashed line corresponds to the profiles shown in Fig. 6(a).

(a) (b)

Fig. 6—Concentration and accumulated viscoplastic volumetric strain: (a) along the radial profile at 24.4 cm height and (b) along the centerline for thethree case studies described in the text body.

centerline and along the radius at a height of 24.4 cm fromthe bottom. In the central region above the height of 15 cm,the volumetric viscoplastic strains are above 1 pct and leadin case B to macrosegregation at the same level as the inversesegregation found in case A. Although small (up to about2.4 pct), the volumetric viscoplastic strains alone cause incase B a relative macrosegregation of 9 pct. In case C, where

both mechanisms are taken into account, the concentrationthus becomes 4.7 wt pct Cu.

Figure 7(a) shows for case B (no shrinkage) the evolu-tion of the accumulated viscoplastic volumetric strain andof the concentration in one selected location, at the center-line and at the height of 24.4 cm. Until the mushy zonebecomes coherent, at the solid fraction of 0.7, the solid phase


(a)

(b)

Fig. 7—Evolution of (a) the accumulated viscoplastic volumetric strain,the mixture concentration, and (b) the concentrations of the liquid and solidphases at the centerline location with height 24.4 cm for the no shrinkagecase (case B).

does not sustain strains and the composition remains almostunchanged. Close to the end of solidification, when the solidfraction is about 0.9, the viscoplastic volumetric strainincreases rapidly due to the vicinity of the harder one-phasesolid that causes strain concentration in the easily deformablemush. Furthermore, the difference between the compositionsof the solid and the liquid phases increases as solidificationadvances. From the definition �c � �s g s cs � �l gl cl, theconcentration of the solid phase can be expressed as cs �(�c � �lgl cl)/�sgs. Figure 7(b) shows the evolution of con-centrations of the liquid and solid phases for this location.The liquid concentration increases fast, while the solid con-centration remains almost constant, until the eutectic reaction

starts. The concentration of the liquid phase remains thenconstant and equal to the eutectic value while the averagecomposition of the solid phase keeps increasing. Note thatclose to the end of the solidification, the composition ofthe liquid phase is more than 10 times higher than the com-position of the solid phase. As indicated in Eq. [12],macrosegregation due to deformation is proportional to boththe difference is composition of the solid and liquid phasesand to the volumetric strain. Thus, both phenomena con-tribute to the macrosegregation formation taking place dur-ing the very last stage of solidification.

V. CONCLUSIONS

A two-phase volume-averaged continuum model has beendeveloped that quantifies macrosegregation formation dur-ing solidification of metallic alloys caused by volumetricviscoplastic deformation of the dendritic network and asso-ciated melt flow in the coherent part of the mushy zone.Modeling results for an Al-4 wt pct Cu alloy indicate thateven the quite small volumetric strains (�2 pct) associatedwith thermally induced deformations can lead to a macro-scopic composition variation in the final casting compara-ble to that resulting from solidification shrinkage.

ACKNOWLEDGMENTS

The authors thank Mr. H. Fjær and Mr. D. Mortensen,Institute for Energy Technology, Kjeller, Norway, for assis-tance in the solution of the thermomechanical problemdescribed in Section III–B. This research has been fundedby Hydro Aluminium, Elkem Aluminium, and the ResearchCouncil of Norway.

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Modeling of macrosegregation caused by volumetric deformation in a coherent mushy zone

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