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Variational Concepts with Applications to Microstructural Evolution

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Abstract

In systems at elevated temperature the development of the microstructure of a material is controlled by diffusional and interface migration processes. As first step the description of the microstructure is reduced to a finite number of time-dependent characteristic parameters (CPs). Then the Thermodynamic Extremal Principle (TEP) is engaged to develop the evolution equations for these characteristic parameters. This treatment is demonstrated on a bamboo-structured material system predicting the spatial and time distribution of chemical composition as well as the deformation state.

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References

  1. Andersson, J.O. and Agren, J.: Models for numerical treatment of multicomponent diffusion in simple phases, J. Appl. Phys. 72, 1992, 1350–1355.

    Article  Google Scholar 

  2. Belova, I.V. and Murch, G.E.: Analysis of interdiffusion data in multicomponent alloys to extract fundamental diffusion information, J. Phase Equil. Diff. 27, 2006, 629–637.

    Google Scholar 

  3. Chen, Q., Jeppsson, J. and Agren, J.: Analytical treatment of diffusion during precipitate growth in multicomponent systems, Acta Mater. 56, 2008, 1890–1896.

    Article  Google Scholar 

  4. Cocks, A.C.F., Gill, S.P.A. and Pan, J.-Z.: Modeling microstructure evolution in engineering materials, in E. van der Giessen and Th. Y. Wu (Eds.), Advances in Applied Mechanics, Vol. 36, Academic Press, San Diego, 1999, pp. 81–162.

    Google Scholar 

  5. Cornet, J.-F. and Calais, D.: Etude de l’effet Kirkendall d’apres les equations de Darken. J. Phys. Chem. Solids 33, 1972, 1675–1684.

    Article  Google Scholar 

  6. Danielewski, M., Wierzba, B., Bachorczyk-Nagy, R. and Pietrzyk M.: Three-dimensional interdiffusion under stress field in Fe-Ni-Cu alloys, J. Phase Equilibria Diff. 27, 2006, 691–698.

    Google Scholar 

  7. Darken, L.S.: Diffusion, mobility and their interrelation through free energy in binary metallic systems, Trans. Amer. Inst. Min. Metall. Engrg. 175, 1948, 184–201.

    Google Scholar 

  8. Fratzl, P., Fischer, F.D. and Svoboda, J.: Energy dissipation and stability of propagating interfaces. Phys. Rev. Lett. 95, 2005, 195702-1–195702-4.

    Article  Google Scholar 

  9. Gamsjäger, E., Svoboda, J., Fischer, F.D. and Rettenmayr, M.: Kinetics of solute driven melting and solidification, Acta Mater. 55, 2007, 2599–2607.

    Article  Google Scholar 

  10. Gamsjäger, E., Svoboda, J. and Fischer, F.D.: Solute drag or diffusion processes in a migrating thick interface, Philos. Mag. Lett. 88, 2008, 415–420.

    Article  Google Scholar 

  11. Hackl, K. and Fischer, F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials, Proc. Royal Soc. A 464, 2008, 117–132.

    Article  MATH  MathSciNet  Google Scholar 

  12. Helander, T. and Agren, J.: Diffusion in the B2-B.C.C. Phase of the Al-Fe-Ni system – Application of a phenomenological model, Acta Mater. 47, 1999, 3291–3300.

    Article  Google Scholar 

  13. Hillert, M.: Phase Equilibria, Phase Diagrams and Phase Transformations, 2nd edn., Cambridge University Press, Cambridge, 2008.

    Google Scholar 

  14. Kattner, U.R. and Campbell, C.E.: Modelling of thermodynamics and diffusion in multicomponent systems, Mat. Sci. Technol. 25, 2009, 443–459.

    Article  Google Scholar 

  15. Manning, J.R.: Diffusion Kinetics for Atoms in Crystals, Van Nostrand, Princeton, NJ, 1968.

    Google Scholar 

  16. Moleko, L.K. and Allnatt, A.R.: Exact linear relations between the phenomenological coefficients for matter transport in a random alloy, Philos. Mag. A 58, 1988, 677–681.

    Article  Google Scholar 

  17. Moroz, A.: A variational framework for nonlinear chemical thermodynamics employing the maximum energy dissipation principle. J. Phys. Chem. B 113, 2009, 8086–8090.

    Article  Google Scholar 

  18. Nakajima, H.: The discovery and acceptance of the Kirkendall effect: The result of a short research career. JOM 49, 1997, 15–19.

    Article  Google Scholar 

  19. Onsager, L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 1931, 405–426.

    Article  MATH  Google Scholar 

  20. Onsager, L.: Theories and problems of liquid diffusion. Ann. N.Y. Acad. Sci. 46, 1945, 241–265.

    Article  Google Scholar 

  21. Sonderegger, B., Kozeschnik, E., Leitner, H., Clemens, H., Svoboda, J. and Fischer, FD.: Computational analysis of the precipitation kinetics in a complex tool steel. Int. J. Mat. Res. 99, 2008, 410–415.

    Google Scholar 

  22. Svoboda, J. and Riedel, H.: Quasi-equilibrium sintering for coupled grain-boundary and surface diffusion, Acta Mater. 43, 1995, 499–506.

    Article  Google Scholar 

  23. Svoboda, J., Fischer, F.D., Fratzl, P. and Kozeschnik, E.: Modelling of kinetics in multicomponent multi-phase systems with spherical precipitates I. – Theory, Mat. Sci. Engrg. A 385, 2004, 166–174.

    Google Scholar 

  24. Svoboda, J., Fischer, F.D. and Fratzl, P.: Diffusion and creep in multi-component alloys with non-ideal sources and sinks for vacancies, Acta Mater. 54, 2006, 3043–3053.

    Article  Google Scholar 

  25. Svoboda, J., Vala, J., Gamsjäger, E. and Fischer, F.D.: A thick-interface model for diffusive and massive phase transformation in substitutional alloys, Acta Mater. 54, 2006, 3953–3960.

    Article  Google Scholar 

  26. Svoboda, J. and Fischer, F.D.: A new approach to modelling of non-steady grain growth, Acta Mater. 55, 2007, 4467–4474.

    Article  Google Scholar 

  27. Svoboda, J., Fischer, F.D. and Mayrhofer, P.H.: A model for evolution of shape changing precipitates in multicomponent systems, Acta Mater. 56, 2008, 4896–4904.

    Article  Google Scholar 

  28. Svoboda, J., Fischer, F.D. and Gamsjäger, E.: Simulation of chemically driven inelastic strain in multi-component systems with non-ideal sources and sinks for vacancies, Acta Mater. 56, 2008, 351–357.

    Article  Google Scholar 

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Correspondence to F. D. Fischer .

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Fischer, F.D., Svoboda, J., Hackl, K. (2010). Variational Concepts with Applications to Microstructural Evolution. In: Hackl, K. (eds) IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials. IUTAM Bookseries, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9195-6_6

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  • DOI: https://doi.org/10.1007/978-90-481-9195-6_6

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  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-9194-9

  • Online ISBN: 978-90-481-9195-6

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