Micromechanical phase-field model and simulation of eutectic growth with misfit stresses [Elektronische Ressource] / Zohreh Ebrahimi

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2010
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

Micromechanical Phase-Field Model and
Simulation of Eutectic Growth with Mis t
Stresses
Von der Fakult at fur Georessourcen und Materialtechnik
der Rheinisch-Westf alischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines
Doktors der Ingenieurwissenschaften
genehmigte Dissertation
vorgelegt von Master of Science
Zohreh Ebrahimi
aus Shiraz (Iran)
Berichter: Univ.-Prof. Dr.-Ing. Andreas Buhrig-P olaczek Dr.-Ing. Heike Emmerich
Tag der mundlic hen Prufung: 23. November 2010
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfugbarContents
Kurzfassung 1
Abstract 2
Introduction 4
1 Phase Growth in Binary Alloys 13
1.1 Eutectic Growth . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Phase-Field Model Description . . . . . . . . . . . . . . . . . . 20
1.3 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.1 The Simulation Algorithm . . . . . . . . . . . . . . . . 26
1.3.2 The Simulation Results . . . . . . . . . . . . . . . . . . 28
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Free-Volume Growth in Eutectic Colonies with Coupled Heat
and Solute Di usion 31
2.1 Free-Boundary Problem . . . . . . . . . . . . . . . . . . . . . 33
2.2 Phase-Field Modeling Approach . . . . . . . . . . . . . . . . . 36
2.2.1 Coupled Heat and Solute Di usion . . . . . . . . . . . 38
i2.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 Simulations of Thermosolutal Free-Volume Solidi cation 42
2.3.2 E ects of the Thermal Di usion on Free-Growth . . . . 48
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Modeling Coherency Strain between Eutectic Lamellae 53
3.1 Linear Elasticity Theory . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 Generalized Hooke’s law . . . . . . . . . . . . . . . . . 61
3.1.2 Elasticity and Interface Structure . . . . . . . . . . . . 64
3.1.3 Mist Stress . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Phase-Field Model . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.1 Free Energy Function . . . . . . . . . . . . . . . . . . . 67
3.2.2 Elastic Free Energy . . . . . . . . . . . . . . . . . . . . 69
3.2.3 Evolutionary Equations . . . . . . . . . . . . . . . . . 71
3.2.4 Mechanical Equilibrium . . . . . . . . . . . . . . . . . 73
3.3 Simulations of Eutectic Growth . . . . . . . . . . . . . . . . . 78
3.3.1 Model Parameters . . . . . . . . . . . . . . . . . . . . 79
3.3.2 Numerical Tests for directional Solidi cation . . . . . . 81
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Eutectic Growth in Alloys with Coherent Elastic Mis t 91
4.1 Anisotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.1 Linear Elastic Stress-Strain Relation for Cubic Materials 97
4.2 Eutectic Growth in Ti-Fe System . . . . . . . . . . . . . . . . 99
4.2.1 Model Parameters . . . . . . . . . . . . . . . . . . . . 100
4.2.2 Directional Solidi cation of Ti-Fe Alloys . . . . . . . . 1034.2.3 Free-Volume Growth under Elastic E ects . . . . . . . 112
4.2.4 Simulations using Cubic Anisotropy . . . . . . . . . . . 116
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 Free Eutectic Growth with Multiple Orientations 131
5.1 Orientation Field and the Associated Free Energy . . . . . . . 132
5.2 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Summary and Outlook 155
Acknowledgements 160
Bibliography 163
Curriculum Vitae 176List of Figures
1.1 A typical phase-diagram of a binary eutectic alloy, the eutec-
tic concentrations of each phase at the eutectic temperature,
C ;C and C are presented at the eutectic temperature T . . 16 E E
1.2 Di erent types of binary eutectic morphologies, brous (upper-
left picture), lamellar (upper-right picture), irregular pattern
(lower-right picture). . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Eutectic growth with four lamellae, initial condition (left pic-
ture), lamellae with curved front (right picture). . . . . . . . . 18
1.4 Schematic representation of the phase- eld variable , which
varies from =1 in solid to zero in the liquid phase. . . . . . . 20
1.5 A schematic illustration of the simulation algorithm for the
phase- eld model. . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 JH investigations for di usion-limited growth with di erent
growth rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 JH investigations for di erent temperature gradients and con-
stant pulling velocities. . . . . . . . . . . . . . . . . . . . . . . 30
v2.1 Free growth of one nucleus in the center of simulation box
after 5000, 10000 and 15000 time steps (upper), and the cor-
responding dimensionless temperature eld after 15000 time
steps (lower). . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Plots of the temperature eld in the middle of simulation box
along x-axis after 5000, 10000 and 150000 time steps. . . . . 44
2.3 Free-volume eutectic colonies formation of one nucleus in the
eutectic alloy CBr -C Cl for coupled heat and solute di u-4 2 6
sion. The two solid phases and are represented as tree-
like structures with di erent colors. The lamellar structure
(a) and bifurcation e ects (b) are shown. . . . . . . . . . . . . 45
2.4 A fully developed eutectic colony, in which several tip-spliting
envelopes are observed (a) and contour plots of microstructure
after each certain number of cycles (some of the tip-splitting
areas are marked) (b). . . . . . . . . . . . . . . . . . . . . . . 46
2.5 A qualitative comparison between simulation results and ex-
periments. Phase- eld simulation of eutectic colonies for a
symmetric alloy of two nuclei (a) and experimental eutectic
colonies of Ti-Fe observed in [14] (b). . . . . . . . . . . . . . . 47
2.6 Isothermal eutectic free growth with constant cooling rate =
0:015. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1 Deformable body under external forces. . . . . . . . . . . . . . 58
3.2 Components of stress acting on the 2-plane. . . . . . . . . . . 59