Both analytical and computational methods for solidification problems are introduced. First, the inward solidification process in a spherical vessel is studied. Expressions of the stress, displacement in the solid phase, and the liquid pressure are deduced based on the solidification interface position. A phase-change expansion orientation factor is introduced to characterize the nonisotropic expansion behavior at the freezing interface. Then, an efficient coupled thermomechanical finite-element method is proposed to evaluate the thermal stress, strain, displacement, and pressure in solidification problems with highly nonlinear constitutive relations. Two particular methods for treating the liquid phase with fixed-grid approaches are introduced. The thermal stress is computed at each integration point by integrating the elastoviscoplastic constitutive equations. Then, the boundary value problem described by the global finite-element equations is solved using the full Newton–Raphson method. This procedure is implemented into the finite-element package abaqus via a FORTRAN subroutine UMAT. Detailed implementation steps and the solution procedures are presented. The numerical model is validated first by the analytical solutions and then by a series of benchmark tests. Finally, an example of solidification in an open reservoir with a free liquid surface is introduced. Potential industrial applications of the numerical model are presented.

References

References
1.
Wiesche
,
S.
,
2007
, “
Numerical Heat Transfer and Thermal Engineering of Adblue (scr) Tanks for Combustion Engine Emission Reduction
,”
Appl. Therm. Eng.
,
27
(11–12)
, pp.
1790
1798
.
2.
Morgan
,
K.
,
1981
, “
A Numerical Analysis of Freezing and Melting With Convection
,”
Comput. Methods Appl. Mech. Eng.
,
28
(
3
), pp.
275
284
.
3.
Nithiarasu
,
P.
,
2000
, “
An Adaptive Finite Element Procedure for Solidification Problems
,”
Heat Mass Transfer
,
36
(3)
, pp.
223
229
.
4.
Voller
,
V. R.
,
Cross
,
M.
, and
Markatos
,
N. C.
,
1987
, “
An Enthalpy Method for Convection/Diffusion Phase Change
,”
Int. J. Numer. Methods Eng.
,
24
(1)
, pp.
271
284
.
5.
Voller
,
V. R.
,
1990
, “
Fast Implicit Finite-Difference Method for the Analysis of Phase Change Problems
,”
Heat Transfer
,
17
(2)
, pp.
155
169
.
6.
Kurtze
,
D. A.
,
1991
, “
A Fixed-Grid Finite Element Method for Solidification
,”
Numer. Methods Free Bound. Problems
,
99
(
1
), pp.
235
241
.
7.
Usmani
,
A. S.
,
Lewis
,
R. W.
, and
Seetharamu
,
K. N.
,
1992
, “
Finite Element Modelling of Natural Convection Controlled Change of Phase
,”
Int. J. Numer. Methods Fluids
,
14
(9)
, pp.
1019
1036
.
8.
Wintruff
,
I.
,
Gunther
,
C.
, and
Class
,
G.
,
2001
, “
An Interface-Tracking Control-Volume Finite Element Method for Melting and Solidification Problems
,”
Numer. Heat Transfer B
,
39
(2)
, pp.
101
125
.
9.
Michalek
,
T.
, and
Kowalewski
,
T. A.
,
2003
, “
Simulations of the Water Freezing Process - Numerical Benchmarks
,” ,
7
(
3
), pp.
389
408
.
10.
Bellet
,
M.
,
Decultieux
,
F.
,
Menai
,
M.
,
Bay
,
F.
,
Levaillant
,
C.
, and
Svensson
,
I. L.
,
1996
, “
Thermomechanics of the Cooling Stage in Casting Processes: Three-Dimensional Finite Element Analysis and Experimental validation
,”
Metallurgical Mater. Trans. B
,
27
(
1
), pp.
81
99
.
11.
Bellet
,
M.
,
Jaouen
,
O.
, and
Poitrault
,
I.
,
2005
, “
An ALE-FEM Approach to the Thermomechanics of Solidification Processes With Application to Prediction of Pipe Shrinkage
,”
J. Numer. Methods Heat Fluid Flow
,
15
(
2
), pp.
120
142
.
12.
de Saracibar
,
C. A.
,
Cervera
,
M.
, and
Chiumenti
,
M.
,
2001
, “
On the Constitutive Modeling of Coupled Thermomechanical Phase-Change Problems
,”
Int. J. Plast.
,
17
(
12
), pp.
1565
1622
.
13.
Cervera
,
M.
,
de Saracibar
,
C. A.
, and
Chiumenti
,
M.
,
1999
, “
Thermo–Mechanical Analysis of Industrial Solidification Processes
,”
Int. J. Numer. Methods Eng.
,
46
, pp.
1575
1591
.
14.
Chiumenti
,
M.
,
Cervera
,
M.
, and
de Saracibar
,
C. A.
,
2006
, “
Coupled Thermomechanical Simulation of Solidification and Cooling Phases in Casting Processes
,”
MCWASP 2006
,
Opio, France
,
4-6 June 2006
M. Schlesinger, ed., TMS 2006.
15.
Koric
,
S.
, and
Thomas
,
B. G.
,
2006
, “
Efficient Thermo-Mechanical Model for Soldification Processes
,”
Int. J. Numer. Methods Eng.
66
(
12
), pp.
1955
1989
.
16.
Koric
,
S.
, and
Thomas
,
B. G.
,
2007
, “
Thermo-Mechanical Models of Steel Solidification Based on Two Elastic Visco-Plastic Constitutive Laws
,”
J. Mater. Process. Technol.
,
197
(
1–3
), pp.
408
418
.
17.
Koric
,
S.
,
Hibbeler
,
L. C.
, and
Thomas
,
B. G.
,
2008
, “
Explicit Coupled Thermo-Mechanical Finite Element Model of Steel Solidification
,”
Int. J. Numer. Methods Eng.
,
78
(
1
), pp.
1
31
.
18.
Zhu
,
H.
,
1993
, “
Coupled Thermal-Mechanical Finite-Element Model With Application to Initial Solidification
,” Ph.D Thesis.
University of Illinois
,
Champaign, IL
.
19.
Li
,
C.
, and
Thomas
,
B. G.
,
2004
, “
Thermo-Mechanical Finite-Element Model of Shell Behaviour in Continuous Casting of Steel
,”
Metallur. Mater. Trans. B
,
35B
(
6
), pp.
1151
1172
.
20.
Weeks
,
W. F.
, and
Wettlaufer
,
J. S.
,
1996
, “
Crystal Orientations in Floating Ice Sheets
,”
The Johannes Weertman Symposium
,
Anaheim, CA
,
Feb. 4–8
, TMS 1996, pp.
337
350
.
21.
Schulson
,
E. M.
, and
Duval
,
P.
,
2009
,
Creep and Fracture of Ice
,
Cambridge University Press
,
Cambridge
.
22.
Weiner
,
J. H.
, and
Boley
,
B. A.
,
1963
, “
Elastic-Plastic Thermal Stresses in a Solidifying Body
,”
J. Mech. Phys. Solids
,
11
(
3
), pp.
145
154
.
23.
Zhekamukhov
,
M. K.
, and
Shokarov
,
K. B.
,
2003
, “
Calculation of Stresses and Strains Developing in Freezing of Water in Closed Vessels
,”
J. Eng. Phys. Thermophys.
,
76
(
1
), pp.
210
221
.
24.
Tao
,
L. C.
,
1967
, “
Generalized Numerical Solutions of Freezing a Saturated Liquid in Cylinders and Spheres
,”
AIChE J.
,
13
(
1
), pp.
165
169
.
25.
McCue
,
S. W.
,
Wu
,
B.
, and
Hill
,
J. M.
,
2008
, “
Classical Two-Phase Stefan Problem for Spheres
,”
Proc. R. Soc. London A
,
464
(
2096
), pp.
2055
2076
.
26.
Bellet
,
M.
, and
Thomas
,
B. G.
,
2007
, “
Solidifcation Macroprocesses–Thermal-Mechanical Modeling of Stress, Distortion and Hot Tearing
,”
Materials Processing Handbook
,
CRC Press, Taylor and Francis
,
Boca Raton
.
27.
Koric
,
S.
,
Hibbeler
,
L. C.
,
Liu
,
R.
, and
Thomas
,
B. G.
,
2010
, “
Multiphysics Model of Metal Solidification on the Continuum Level
,”
Numer. Heat Transfer, Part B: Fundamentals
,
58
(
6
), pp.
371
392
.
28.
Li
,
J.
,
Saharan
,
A.
,
Koric
,
S.
, and
Ostoja-Starzewski
,
M.
,
2012
, “
Elastic-Plastic Transition in Three-Dimensional Random Materials: Massively Parallel Simulations, Fractal morphogenesis and Scaling Functions
,”
Philos. Mag.
,
92
(
22
), pp.
2733
2758
.
29.
abaqus 6.14 documentation
,
2014
, Analysis User's Guide.
30.
Chalmers
,
B.
,
1970
, “
Principles of Solidification
,”
Applied Solid State Physics
,
Springer
,
Boston, MA
.
31.
Budd
,
W. F.
, and
Jacka
,
T. H.
,
1989
, “
A review of Ice Rheology for Ice Sheet Modelling
,”
Cold Regions Sci. Technol.
,
16
(
2
), pp.
107
144
.
32.
Weertman
,
J.
,
1983
, “
Creep Deformation of Ice
,”
Annu. Rev. Earth. Planet. Sci.
,
11
(
1
), pp.
215
240
.
33.
Dunne
,
F.
, and
Petrinic
,
N.
,
2005
,
Introduction to Computational Plasticity
,
Oxford University Press
,
New York
.
34.
de Souza Neto
,
E.
,
Peric
,
D.
, and
Owen
,
D.
,
2008
,
Computational Methods for Plasticity: Theory and Applications
,
John Wiley & Sons Ltd.
,
Chichester
.
35.
Schulson
,
E. M.
,
2001
, “
Brittle Failure of Ice
,”
Eng. Fract. Mech.
,
68
(
17–18
), pp.
1839
1887
.
36.
Lush
,
A. M.
,
Weber
,
G.
, and
Anand
,
L.
,
1989
, “
An Implicit Time-Integration Procedure for a Set of Integral Variable Constitutive Equations for Isotropic Elasto-Viscoplasticity
,”
Int. J. Plast.
,
5
(
5
), pp.
521
549
.
37.
Simo
,
J. C.
, and
Taylor
,
R. L.
,
1985
, “
Consistent Tangent Operator for Rate-Independent Elastoplasticity
,”
Comput. Methods Appl. Mech. Eng.
,
48
(1)
, pp.
101
118
.
38.
Gupta
,
S. C.
,
1987
, “
Analytical and Numerical Solutions of Radially Symmetric Inward Solidification Problems in Spherical Geometry
,”
Int. J. Heat. Mass. Transfer.
,
30
(
12
), pp.
2611
2616
.
39.
Borja
,
R. I.
,
2013
,
Plasticity –Modeling & Computation
,
Springer
,
New York
.
40.
Glen
,
J. W.
,
1955
, “
The Creep of Polycrystalline Ice
,”
Proc. R. Soc. London A
,
228
(
1175
), pp.
519
538
.
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