COUPLED THERMOMECHANICAL, THERMAL

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COUPLED THERMOMECHANICAL, THERMAL TRANSPORT AND
SEGREGATION ANALYSIS OF ALUMINUM ALLOYS
SOLIDIFYING ON UNEVEN SURFACES
Lijian Tan, Deep Samanta and Nicholas
Zabaras Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//mpdc.mae.cornell.edu/zabaras/
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RESEARCH SPONSORS
DEPARTMENT OF ENERGY (DOE) Industry
partnerships for aluminum industry of the future
- Office of Industrial Technologies
ALUMINUM CORPORATION OF AMERICA (ALCOA)
Ingot and Solidification Platform
Alcoa Technical Center
CORNELL THEORY CENTER
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OUTLINE OF THE PRESENTATION

• Brief introduction and motivation of the current
  study
• Numerical model to study deformation of
  solidifying alloys
• Closure criteria
• Computational strategies for solving the coupled
  numerical system
• Numerical examples
• preliminary studies of deformation of
  solidifying alloys
• parametric investigations of
  solidification from molds with uneven
• mold topography (coupled thermal,
  solutal and momentum transport)
• Conclusions
• Future Work

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Introduction and motivation of the current study
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INTRODUCTION
Surface defects in casting (Ref. ALCOA Corp.)
(a)
(b)
(a) Sub-surface liquation and crack formation on
top surface of a cast (b) Non-uniform front
and undesirable growth with non-uniform shell
thickness
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INTRODUCTION

• Aluminum industry relies on direct chill casting
  for aluminum ingots
• Aluminum ingots are often characterized by
  defects in surface due to non-uniform heat
  extraction, improper contact at metal/mold
  interface, inverse segregation, air-gap formation
  and meniscus freezing etc
• These surface defects are often removed by post
  casting process such as scalping/milling
• Post-processing leads to substantial increase of
  cost , waste of material and energy.
• The purpose of this work is to reduce
  scalp-depth in castings
• Detailed understanding of the highly coupled
  phenomenon in
• the early stages of solidification is required

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INTRODUCTION
Engineered mold surface (Ref. ALCOA Corp.)

• In industry, the mold surface is pre-machines to
  control heat extraction in directional
  solidification
• This periodic groove surface topography allows
  multi-directional heat flow on the metal-mold
  interface
• However, the wavelengths should be with the
  appropriate value to obtain anticipated benefits.

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Numerical model of deformation of
solidifying alloys
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SHEMATIC OF THE PROBLEM DEFINITION

• An Aluminum-copper alloy is solidified on an
  sinusoidal uneven surface.
• With growth of solid shell, air gaps form
  between the solid shell and mold due to imperfect
  contact which further leads to variation in
  boundary conditions.
• The solid shell undergoes plastic deformation
  and development of thermal and plastic strain
  occurs in the mushy zone also.
• Inverse segregation caused by shrinkage driven
  flow causes variation in air gap sizes, front
  unevenness and stresses developing in the casting.

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SCHEMATIC OF THE HIGHLY COUPLED SYSTEM
Mold
Phase change and mushy zone evolution
Heat transfer
Solute transport
Casting domain
Fluid flow
Deformation or mold non-deformable
Heat transfer

Contact pressure or air gap criterion
Inelastic deformation

• There are heat transfer and deformation in both
  mold and casting region interacting with the
  contact pressure or air gap size between mold and
  casting.
• The solidification, solute transport, fluid flow
  will also play important roles.

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PREVIOUS WORK

• Zabaras and Richmond (1990,91) used a
  hypoelastic rate-dependent small deformation
  model to study the deformation of solidifying
  body
• Rappaz (1999), Mo (2004) modeled the deformation
  in mushy zone with a volume averaing model
  Continuum model for deformation of mushy zone in
  a solidifying alloy and development of a hot
  tearing criterion Rappaz (1999), Mo et al (2004).
  Surface segregation and air gap formation in DC
  cast Aluminum alloys Mo et al. (1995-98)
• Hector and Yigit (2000) did a semi-analytical
  studies of air gap nucleation during
  solidification of pure metals using a hypoelastic
  perturbation theory Effect of strain rate
  relaxation on the stability of solid front growth
  morphology during solidification of pure metals
  Hector and Barber (1994,95)
• The inverse segregation and macro-segregation
  have also been studied by Chen, Heinrich, Samanta
  and Zabaras etc Inverse segregation caused by
  shrinkage driven flows during solidification of
  alloys Chen et al. (1991 93), Heinrich et al.
  (1993,97) Effect of uneven surface topography on
  fluid flow and macrosegregation during
  solidification of Al-Cu alloys Samanta and
  Zabaras (2005)
• A thermo-mechanical study of the effects of mold
  topography on the solidification of Al alloys
• - Tan and Zabaras (2005)

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SALIENT FEATURES OF OUR NUMERICAL MODEL

• Volume averaging with a single domain and single
  set of transport equations for mass, momentum,
  energy and species transport
• Individual phase boundaries are not explicitly
  tracked
• Complex geometrical modeling of interfaces
  avoided
• Single grid used with a single set of boundary
  conditions
• A rate dependent hypo-elastic visco-plastic
  model is used for deformation of solid shell and
  mushy zone
• Dynamic air gap contact pressure coupling at
  the mold metal interface
• On the whole, a highly coupled model combining
  solidification and deformation in the
• casting is used.

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GOVERNING TRANSPORT EQUATIONS FOR SOLIDIFICATION
(Ref Shyll and Udaykumar, 1996) (Ref C.
Beckermann et al., explicit modeling of
Interfacial terms) (Ref Incropera, 1987-2000
mixture theory)
Initial conditions
Isotropic permeability
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CLOSURE RELATIONSHIPS FOR FINDING CONCENTRATION
AND FRACTION
Lever Rule (Infinite back-diffusion)
T
Scheil Rule (Zero back-diffusion)
Cl
C
(assumed constant for all problems)
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MODELING DEFORMATION IN MUSHY ZONE

• Low solid fractions usually accompanied
• by melt feeding and no deformation due to
• weak or non existent dendrites ?
• leads to zero thermal strain.
• With increase in solid fraction, there is an
  increase in strength and bonding ability of
• dendrites ? to non zero thermal strain.
• The presence of a critical solid volume fraction
  is observed in experiment and varies for
  different alloys.

The parameter w is defined as

• Liquid or low solid fraction mush
• - any deformation induced by thermal expansion
  is permanent. (Without any strength)
• Solid or high solid fraction mush
• - plastic deformation is developed only
  gradually.

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MODEL FOR DEFORMATION OF SOLIDIFYING ALLOY

• For deformation, we assume the total strain can
  be decomposed into three parts
• elastic strain, thermal strain and plastic
  strain.
• Elastic strain rate is related with stress rate
  through an hypo-elastic constitutive law
• Plastic strain evolution satisfy this creep law
  with its parameters determined from experiments
  (Strangeland et al. (2004)).
• The thermal strain evolution is determined from
  temperature decrease and shrinkage.

Strain measure
Elastic strain
Thermal strain
Plastic strain
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Parameters for simulation of deformation in mushy
zone
Critical solid fraction for different copper
concentrations in aluminum-copper alloy
Ref Mo et al.(2004)
Creep law for plastic deformation Ref.
Strangeland et al. (2004)
Strain-rate scaling factor
Stress scaling factor
Activation energy
Creep law exponent
Volumetric thermal expansion coefficient
Mushy zone softening parameter
Volumetric shrinkage coefficient
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THERMAL RESISTANCE AT THE METAL-MOLD INTERFACE
Contact resistance

• At the very early stages, the solid shell is in
  contact with the mold and the thermal
• resistance between the shell and the mold is
  determined by the contact conditions

• Before gap nucleation, the thermal resistance
• is determined by pressure

• After gap nucleation, the thermal resistance
• is determined by the size of the gap

Example Aluminum-Ceramic Contact
Heat transfer retarded due to gap formation

• Uneven contact condition generates an uneven
  thermal stress development and may accelerates
  distortion or warping of the casting shell.

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MOLD METAL BOUNDARY CONDITIONS
Consequently, heat flux at the mold metal
interface is a function of air gap size or
contact pressure
Air-gap size at the interface
Contact pressure at the interface

• The actual air gap sizes or contact pressure
  are determined from the contact sub problem.
• This modeling of heat transfer mechanism due to
  imperfect contact very crucial for studying the
  non-uniform growth at early stages of
  solidification.

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SOLUTION ALGORITHM AT EACH TIME STEP
Convergence criteria based on gap sizes or
contact pressure in iterations
All fields known at time tn
n n 1
Check if convergence satisfied
Advance the time to tn1
Contact pressure or air gap obtained from Contact
sub-problem
Solve for displacement and stresses in the
casting (Deformation problem)
Solve for the temperature field
(energy equation)
Decoupled momentum solver
Solve for velocity and pressure
fields (momentum equation)
Inner iteration loop
Solve for the concentration field
(solute equation)
(Ref Heinrich, et al.)
Yes
Is the error in
liquid concentration and liquid mass
fraction less than tolerance
Solve for liquid concentration, mass
fraction and density (Thermodynamic relations)
Segregation model (Scheil rule)
No
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COMPUTATIONAL STRATEGY AND NUMERICAL TECHNIQUES

• The thermal problem is solved in a region
  consisting of both mold and casting to account
  for non-linear (contact pressure/air gap
  dependent) boundary conditions at the mold
  metal interface.
• Deformation problem is solved in both casting
  and mold (if mold deformable) or only the casting
  (if mold rigid, for most of our numerical
  studies).
• Solute and momentum transport equations is only
  solved in casting with multistep predictor
  Corrector method for solute problems, and
  Newton-Raphson method for solving heat transfer,
  fluid flow and deformation problems.
• Backward Euler fully implicit method is
  utilized for time discretization to make the
  numerical scheme unconditionally stable.
• The contact sub-problem is solved using
  augmentations (using the scheme introduced by
  Larsen in 2002).
• All the matrix computations for individual
  problems are performed using the parallel
  iterative Krylov solvers based on the PETSc
  library.

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Numerical examples
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SOLIDIFICATION OF Al ON UNEVEN SURFACES
Hypoelastic model without plastic deformation
(Hector et al. 2000)

• Heat transfer in the mold, solid shell and melt.
• Heat transfer causes deformation (thermal
  stress).
• Gaps or contact pressure affect heat transfer.
• Solidification after air-gap nucleation not
  modeled.

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GAP NUCLEATION TIME EFFECTS OF WAVELENGTH

• At the very early stages of aluminum
  solidification, contact pressure between mold and
  solid shell will drop at the trough due to
  thermal stress development. When this contact
  pressure drops to zero, gap nucleation is assumed
  to take place. This study compares very well with
  Hectors semi-analytical study. It shows that gap
  nucleation is faster for smaller wavelength,
  smaller liquid pressure and better heat
  conductivity of the mold.
• .

• For rigid mold (with an topography
• amplitude1 µm, wavelength1-5 mm), under liquid
  pressure 8000 Pa, the gap nucleation time
• is in the order of seconds.
• Physical Conditions
• Liquid pressure P8000 Pa
• Thermal resistance at mold-shell interface R10-5
  m2 oC sec J-1

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GAP NUCLEATION TIME EFFECTS OF MOLD CONDUCTIVITY

• Mold conductivity affects gap nucleation time
• The higher the conductivity, the quicker the gaps
  nucleate from the mold surface

In this calculations, the deformation of the mold
is neglected to illustrate the effects of mold
conductivity. Physical conditions Liquid
pressure P10000 Pa Mold thickness h0.5
mm Thermal resistance at mold-shell interface
R10-5 m2 oC sec J-1 Wavelength2 mm
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GAP NUCLEATION TIME EFFECTS OF MOLD MATERIAL
(deformable mold)

• When the wavelength is relatively small, the
  evolution of the contact pressure at the trough
  is mainly affected by the conductivity of the
  mold, i.e. the deformation of the mold does not
  play a crucial role.

Physical Conditions Liquid pressure P10000
Pa Mold thickness h0.5 mm Thermal resistance at
mold-shell interface R10-5 m2 oC sec
J-1 Wavelength10 mm, (20 mm, 30 mm in the next
two slides)
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GAP NUCLEATION TIME EFFECTS OF MOLD MATERIAL
(deformable mold)

• When the wavelength increases, the Ptr-t line is
  about to show a turn-around pattern when pressure
  reaches zero. This is defined as the critical
  wavelength in the analytical studies of L.
  Hector.

From this figure, we can say that the critical
wavelength is slightly above 20 mm. In Hectors
analytical study, the critical wavelength is
16.60 mm, for iron mold and 14.03 mm for lead
mold under the same conditions.
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GAP NUCLEATION TIME EFFECTS OF MOLD MATERIAL
(deformable mold)

• Notice that when the wavelength is greater than
  the critical value, the pressure-time curve shows
  a turn- around pattern before the contact
  pressure reaches zero.

• This implies that a large wavelength is
  preferred since the contact pressure wont
  decrease to zero to generate gap nucleation.
• But in practice, we can never get a such a
  smooth mold topography with amplitude 1 µm and
  wavelength 30 mm as in these examples. Gap
  nucleation occurs for most casting processes.

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SOLIDIFICATION OF Al-Cu ALLOY ON UNEVEN SURFACES

• Combined thermal, solutal and
• momentum transport in casting.
• Assume the mold is rigid.
• Imperfect contact and air gap
• formation at metal mold interface

Solidification problem
We carried out a parametric analysis by
change these four parameters 1) Wavelength of
surfaces (?) 2) Solute concentration (CCu) 3)
Melt superheat (?Tmelt) 4) Mold material (Cu, Fe
and Pb)
Heat Transfer (Mold is rigid and non-deformable)
Deformation problem
Both the domain sizes are on the mm scale
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SOLIDIFICATION COUPLED WITH DEFORMATION AND
AIR-GAP FORMATION
Important parameters 1) Mold material - Cu 2) CCu
8 wt. 3) ?Tmelt 0 oC Air gap is
magnified 200 times.

• Preferential formation of solid occurs at the
  crests and air gap formation occurs at the
  trough, which in turn causes re-melting.

• Because of plastic deformation, the gap formed
  initially will gradually decrease.
• As shown in the movies, a 1mm wavelength mold
  would lead to more uniform growth and less fluid
  flow.

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TRANSIENT EVOLUTION OF IMPORTANT FIELDS (? 5 mm)

• Temperature
• Solute concentration
• Equivalent stress
• (d) Liquid mass fraction

Important parameters 1) Mold material - Cu 2) CCu
5 wt. 3) ?Tmelt 0 oC
(b)
(a)

• We take into account solute transport and the
  densities of solid and liquid phases are assumed
  to be different.
• Inverse segregation, caused by shrinkage driven
  flow, occurs at the casting bottom.This is
  observed in (b).

(d)
(c)
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TRANSIENT EVOLUTION OF IMPORTANT FIELDS (? 3 mm)

• Temperature
• Solute concentration
• Equivalent stress
• (d) Liquid mass fraction

(b)
(a)

• For smaller wavelengths, similar result is
  observed (1) preferential formation of solid
  occurs at the crests (2) remelting at the trough
  due to the formation of air gap.
• For wavelength 3mm, the solid shell unevenness
  decreases faster than the case of 5mm wavelength.

(d)
(c)
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VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT
STRESS
? 5 mm, CCu 5 wt., mold material Cu

• Air-gap sizes increase with time
• Increasing melt superheat leads to
• some suppression of air gaps

• Initially, stresses higher for lower superheat
• At later times, the difference is small

Increasing melt superheat leads to some
suppression of air gaps and a smaller stress at
beginning stages. At later times, the difference
of equivalent stresses is however small.
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EFFECT OF WAVELENGTH ON AIR-GAP SIZES AND MAX
EQUIVALENT STRESS
?Tmelt 0 oC, CCu 5 wt., mold material Cu

• Max. equivalent stress seq variation with ?
• seq first increases and then decreases
• Initially, seq is higher for greater ?
• Later (t100 ms), stress is lowest for
• 5 mm wavelength.

• Air-gap size variation with wavelength ?
• Initially, air-gap sizes nearly same for
• different ?
• At later times, air-gap sizes increase
• with increasing ?

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VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT
STRESS
?Tmelt 0 oC, ? 5 mm, mold material Cu
Increase of solute concentration leads to
increase in air-gap sizes, but its effect on
stresses are small.

• seq first increases and then decreases
• Variation of seq with Cu concentration
• is negligible after initial times

• Air-gap sizes increase with time
• Increasing Cu concentration leads to
• increase in air-gap sizes

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VARIATION OF AIR-GAP SIZES AND MAX. EQUIVALENT
STRESS
?Tmelt 0 oC, ? 5 mm, CCu 5 wt.
Gap nucleation and stress development are
prominent for a mold of higher thermal
conductivity like Cu. For Fe or Pb molds, heat
removal is inhibited due to their lower thermal
conductivity. This in turn inhibits air-gap
formation and development of stresses..

• Air gap sizes higher for Cu molds than
• Fe or Pb molds

• Equivalent stress far lower for Cu molds
• than Fe or Pb molds

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EFFECT OF INVERSE SEGREGATION AIR GAP SIZES
(a) With inverse segregation
(b) Without inverse segregation
By comparing the result with modeling inverse
segregation and without modeling inverse
segregation, we can find that inverse segregation
actually plays an important role in air-gap
evolution.

• Differences in air-gap sizes for different solute
  concentrations are more pronounced in the
  presence of inverse segregation.

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VARIATION OF EQUIVALENT STRESSES AND FRONT
UNEVENNESS
Time t 100 ms

• Value of front unevenness and maximum equivalent
  stress for various wavelengths
• one cannot simultaneously reduce both stress and
  front unevenness
• when the wavelength greater than 5mm, both
  unevenness and stress increase-gt implies
  wavelength less than 5 mm is optimum

• Equivalent stress at dendrite roots
• The highest stress observed for 1.8 copper
  alloy suggest that aluminum copper alloy with
  1.8 copper is most susceptible to hot tearing
• Phenomenon is also observed experi-mentally
  Rappaz(99), Strangehold(04)

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EFFECTS OF SURFACE ROUGHNESS AND MOLD COATINGS

• Effect of uneven surface topography and non
  uniform contact on
• microstructure evolution.
• Incorporating the effects of surface tension and
  surface coatings to study
• solidification on microscale.
• Studying the effects of surface roughness on
  solidification on microscale.
• Optimal design of a mold surface topography to
  minimize surface
• defects.

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PRELIMINARY STUDY OF EFFECTS OF SURFACE TENSION

• In the macro-scale, the liquid pressure exerted
  by the droplet can overcome surface tension and
  causes the molten Aluminum droplet to contact the
  bottom of the cavity.

Materials Process Design and Control Laboratory
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EFFECT OF SURFACE TENSION

• However, in the microscale, a change of surface
  tension could drastically change the
  solidification speed at very early stages of
  solidification.
• This suggests taking account of surface tension
  in our future study is very important.

Materials Process Design and Control Laboratory
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CURRENT AND FUTURE RESEARCH

• Shell growth kinetics
• uneven growth
• distortion

Metal/mold interaction
Air gap formation (non uniform contact and
shell remelting)
Meniscus instability
Varying stresses in solid
Lap marks, ripples, cold shuts
Interfacial heat transfer
Inverse segregation
Microstructure evolution
Surface parameters and mold topography in
transport processes
Macrosegregation
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CONCLUSIONS

• Early stage solidification of Al-Cu alloys
  significantly affected by non uniform boundary
• conditions at the metal mold interface.
• Variation in surface topography leads to
  variation in transport phenomena, air-gap sizes
• and equivalent stresses in the solidifying
  alloy.
• Air-gap nucleation and growth significantly
  affects heat transfer between metal and mold.
• Distribution of solute primarily caused by
  shrinkage driven flows and leads to inverse
• segregation at the casting bottom.
• Presence of inverse segregation leads to an
  increase in gap sizes and front unevenness.
• Effect of melt pressure on solidification beyond
  gap nucleation was found to be negligible.
• Effects of surface topography more pronounced
  for a mold with higher thermal conductivity
• Computation results suggests that aluminum
  copper alloy with 1.8 copper is most susceptible
  for hot tearing defects. An optimum mold
  wavelength should be less than 5mm.

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RELEVANT PUBLICATIONS

• D. Samanta and N. Zabaras, A numerical study of
  macrosegregation in Aluminum alloys
• solidifying on uneven surfaces, in press in
  International Journal of Heat and
• Mass Transfer.
• L. Tan and N. Zabaras, A thermomechanical study
  of the effects of mold topography on the
• solidification of Aluminum alloys, in press in
  Materials Science and Engineering A.
• D. Samanta and N. Zabaras, A coupled
  thermomechanical, thermal transport and
  segregation
• analysis of the solidification of Aluminum
  alloys on molds of uneven topographies ,
  submitted
• for publication in the Materials Science and
  Engineering A.

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