Towards a statistical theory of texture evolution in polycrystals

SIAM Journal on Scientific Computing

Volumn 30, Issue 6, 2008, Pages 3150-3169

  Barmak, K ^a   Emelianenko, M ^b   Golovaty, D ^c   Kinderlehrer, D ^d   Ta'Asan, S ^d  

^a CARNEGIE MELLON UNIVERSITY   (United States)

^b GEORGE MASON UNIVERSITY   (United States)

^c University of Akron   (United States)

^d CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Anomalous diffusion; Grain growth; Mesoscopic model; Random walk; Renewal processes; Texture

Indexed keywords

COARSENING; GRAIN BOUNDARIES; POLYCRYSTALLINE MATERIALS; PROBABILITY DISTRIBUTIONS; STATISTICAL METHODS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS; SUPERCONDUCTING MATERIALS; TEXTURES;

ANOMALOUS DIFFUSION; GRAIN-BOUNDARIES; MESOSCOPIC MODELING; MONOCRYSTALLINE; POLYCRYSTALLINE MICROSTRUCTURE; RANDOM WALK; RENEWAL PROCESS; STATISTICAL THEORY; TEXTURE EVOLUTIONS; UPSCALING;

GRAIN GROWTH;

EID: 55149093166     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/070692352     Document Type: Article

References (24)

1. S. AGMON, A. DOUGLIS, AND L. NIRENBERG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Comm. Pure Appl. Math., 17 (1964), pp. 35-92.
2. D. APPLEBAUM, Levy Processes and Stochastic Calculus, Cambridge Stud. Adv. Math. 93, Cambridge University Press, Cambridge, UK, 2004.
3. H. V. ATKINSON, Theories on normal grain growth in pure single phase systems, Acta Metall., 36 (1988), pp. 469-491.
4. K. BARMAK, M. EMELIANENKO, D. GOLOVATY, D. KINDERLEHRER, AND S. TA'ASAN, A new perspective on texture evolution, Int. J. Numer. Anal. Model., to appear.
5. L. BRONSARD AND F. REITICH, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), pp. 355-379.
6. E. ÇINLAR, Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs, NJ, 1997.
7. D. R. COX, Renewal Theory, Methuen and Co., London, 1970.
8. R. GORENFLO, A. VLVOLI, AND F. MAINARDI, Discrete and continuous random walk models for space-time fractional derivatives, J. Math. Sci. (N. Y.), 132 (2006), pp. 614-628.
9. M. GRIGORIU, Stochastic Calculus: Applications in Science and Engineering, Birkhäuser, Boston, 2002.
10. E. A. HOLM, G. N. HASSOLD, AND M. A. MIODOWNIK, On misorientation distribution evolution during anisotropic grain growth, Acta Mater., 49 (2001), pp. 2981-2991.
11. B. D. HUGHES, Random Walks and Random Environments, Volume I: Random Walks, Oxford University Press, New York, 1995.
12. D. KINDERLEHRER, I. LIVSHITS, G. S. ROHRER, S. TA'ASAN, S. YU, AND P. YU, Mesoscale evolution of the grain boundary character distribution, recrystallization and grain growth, Mater. Sci. Forum, 467-470 (2004), pp. 1063-1068.
13. D. KINDERLEHRER, J. LEE, I. LIVSHITS, AND S. TA'ASAN, Mesoscale simulation of grain growth, in Continuum Scale Simulation of Engineering Materials, D. Raabe, F. Roters, F. Barlat, and L.-Q. Chen, eds., Wiley-VCH Verlag, Weinheim, 2004, pp. 361-372.
14. D. KINDERLEHRER, I. LIVSHITS, AND S. TA'ASAN, A variational approach to modeling and simulation of grain growth, SIAM J. Sci. Comput., 28 (2006), pp. 1694-1715.
15. D. KINDERLEHRER AND C. LIU, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), pp. 713-729.
16. F. MAINARDI, M. RABERTO, R. GORENFLO, AND E. SCALAS, Fractional calculus and continuoustime finance II: The waiting-time distribution, Phys. A, 287 (2000), pp. 468-481.
17. P. MELLARS, Going East: New genetic and archaeological perspectives on the modern human colonization of Eurasia, Science, 313 (2006), pp. 796-800.
18. R. METZLER AND J. KLAFTER, The random walk's guide to anomalous diffusion; A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1-77.
19. W. W. MULLINS, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), pp. 900-904.
20. W. W. MULLINS, Solid surface morphologies governed by capillarity, in Metal Surfaces: Structure, Energetics, and Kinetics, ASM, Cleveland, OH, 1963, pp. 17-66.
21. I. PODLUBNY, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
22. M. UPMANYU, G. N. HASSOLD, A. KAZARYAN, E. A. HOLM, Y. WANG, B. PATTON, AND D. J. SROLOVITZ, Boundary mobility and energy anisotropy effects on microstructural evolution during grain growth, Interface Sci., 10 (2002), pp. 201-216.
23. G. H. WEISS, Aspects and Applications of the Random Walk, North-Holland, Amsterdam, 1994.
24. S. B. YUSTE AND L. ACEDO, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862-1874.