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Continuum mechanical aspects of phase transformations in solids

Kontinuumsmechanische Aspekte von Phasenübergängen in Festkörpern

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Summary

Some aspects of “classical” thermodynamics of phase transformations are discussed. Then typical solid state transformations as the displacive and diffusional transformation in metals are explained. A general formulation of the Gibbs free energy is presented including all energy terms required. Based on the “classical” nucleation, the “triggering-off” and the “dissipation” condition, various transformation conditions are formulated taking into account the elasto-plastic deformation of both phases. Transformation conditions presented in the literature over the last 40 years are reviewed and compared to the transformation conditions derived here. The transformation conditions for a spherical region growing under a certain volume change in an elasto-plastic matrix are studied as an example. The relevant analytical expressions are presented and discussed.

Übersicht

Einige Aspekte der „klassischen” Thermodynamik von Phasenumwandlungen werden behandelt. In der Folge werden typische Festkörperumwandlungen wie die displazive und die diffusive Umwandlung in Metallen erörtert. In allgemeiner Formulierung wird die Gibbs Energie unter Berücksichtigung aller erforderlichen Energieterme hergeleitet Basierend auf der klassischen Keimentwicklungsbedingung, einer Formulierung über das Wachstum von Keimen und einem auf der Dissipationsleistung beruhenden Konzept werden verschiedene Umwandlungsbedingungen hergeleitet. Dabei wird in beiden Phasen elasto-plastisches Materialverhalten vorausgesetzt. Die hier formulierten Umwandlungsbedingungen werden mit einigen der in den letzten 40 Jahren publizierten Beziehungen verglichen und bewertet. In einem Beispiel werden diese Bedingungen für eine in einer elasto-plastischen Matrix wachsende Kugel bei einer umwandlungsbedingten Volumsdehnung angewandt. Analytische Ausdrücke werden präsentiert und näher erläutert.

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References

  1. Gibbs, J. W.: On the equilibrium of heterogeneous substances. The scientific papers of J. Willard Gibbs, Vol 1, Toronto: Longmans, Greens, 1906 or New York: Dover, 1961

    Google Scholar 

  2. Christian, J. W.: The theory of transformations in metals and alloys. Second edt., Part I, Oxford, New York et al.: Pergamon Press, 1981

    Google Scholar 

  3. Haasen, P., edt.: Phase transformations in materials. Materials Science and Technology (Cahn, R. W., Haasen, P., Kramer, E. J., edts), Vol. 5, Weinheim et al.: VCH Verlagsgesellschaft, 1991

    Google Scholar 

  4. Jena, A. K., Chaturvedi, M. C.: Phase transformations in materials. Englewood Cliffs: Prentice Hall, 1992

    Google Scholar 

  5. Aifantis, E. C.: On the mechanics of phase transformations. Phase Transformations (Aifantis, E. C., Gittus, J., edts) pp. 233–289. London and New York: Elsevier Applied Science Publishers, 1986

    Google Scholar 

  6. Levitas, V.: Post-bifurcation behaviour in finite elastoplasticity. Applications to strain localization and phase transitions. Hannover: Institut f. Baumechanik und numerische Mechanik, IBNM-Bericht P2/5, 1992

    Google Scholar 

  7. Onat, E. T., Leckie, F. A.: Representation of mechanical behavior in the presence of changing internal structure. J. Appl. Mech. 55 (1988) 1–10

    Google Scholar 

  8. McLellan, A. G.: Non-hydrostatic thermodynamics of chemical systems. Proc. Roy. Soc. London A. 314 (1970) 443–455

    Google Scholar 

  9. Paterson, M. S.: Nonhydrostatic thermodynamics and its geologic applications. Reviews of Geophysics and Space Physics. 11 (1973) 355–389

    Google Scholar 

  10. Patel, J. R., Cohen, M.: Criterion for the action of applied stress in the martensitic transformation. Acta metall. 1 (1953) 531–538

    Google Scholar 

  11. Tanaka, K.: Analysis of transformation superplasticity and shape memory effect. In Inoue, T. et al. (eds.): Computational Plasticity, pp. 43–60. London et al.: Elsevier Applied Science, 1991

    Google Scholar 

  12. Mitter, W.: Umwandlungsplastizität und ihre Berücksichtigung bei der Berechnung von Eigenspannungen. Berlin, Stuttgart: Gebrüder Bornträger, 1987

    Google Scholar 

  13. Leblond, J. B., Devaux, J., Devaux, J. C.: Mathematical modelling of transformation plasticity in steels, I: Case of ideal plastic phases. Int. J. Plasticity 5 (1989) 551–572

    Google Scholar 

  14. Fischer, F. D.: A micromechanical model for transformation plasticity in steels. Acta metall mater. 38 (1990) 1535–1546

    Google Scholar 

  15. Patoor, E., Eberhardt, A., Berveiller, M.: Thermomechanical behaviour of shape memory alloys. Arch. Mech. 40 (1988) 775–794

    Google Scholar 

  16. Patoor, E., Eberhardt, A., Berveiller, M.: Potential pseudoelastique et plasticite de transformation martensitique dans les mono-et polycristaux metalliques. Acta metall. 35 (1987) 2779–2789

    Google Scholar 

  17. Johnson, W. C., Müller, W. H.: Characteristics of phase equilibria in coherent solids. Acta metall. mater. 39 (1991) 89–103

    Google Scholar 

  18. Wayman, C. M.: Introduction to the crystallography of martensitic transformation. New York: The Macmillan Company, 1964

    Google Scholar 

  19. Wechsler, M. S., Liebermann, D. S., Read, T. A.: On the theory of the formation of martensite. J. Metals 197 (1953) 1503–1515

    Google Scholar 

  20. Fischer, F. D.: Transformation induced plasticity in triaxially loaded steel specimens subjected to a martensitic transformation. Eur. J. Mech., A/Solids 11 (1992) 233–244

    Google Scholar 

  21. James, R. D.: Microstructure and weak convergence. Materials instabilities in continuum mechanics (Ball, J. M., edt.), pp. 175–196. Oxford: Clarendon Press, 1988

    Google Scholar 

  22. Marder, A. R.: Structure-property relationships in ferrous transformation products. Phase Transformations in Ferrous Alloys (Marder, A. R., Goldstein, J. I., edts.), pp. 11–41. Warrendale: AIME, 1984

    Google Scholar 

  23. Chadwick, P.: Continuum Mechanics. London: George Allen & Unwin Ltd., pp. 114ff, 1976

    Google Scholar 

  24. Maugin, G. A.: The thermomechanics of plasticity and fracture. Cambridge et al.: Cambridge University Press, 1992, chpt. 9.4

    Google Scholar 

  25. Wang, Z. G., Hwang, K. Ch.: A constitutive relation for pseudoelastic behavior in shape memory alloys. Acta Mech. Sinica 7 (1991) 67–75

    Google Scholar 

  26. Sun, Q. P., Hwang, K. Ch., Yu, S. N.: A micromechanics constitutive model of transformation plasticity with shear and dilatation effect. J. Mech. Phys. Solids 39 (1991) 507–524

    Google Scholar 

  27. Sun, Q. P., Hwang, K. Ch.: Micromechanics modelling for the constitutive behavior of polycrystalline shape memory alloys.—I. Derivation of general relations, II. Study of the individual phenomena. J. Mech. Phys. Solids 41 (1993) 1–17, 19–33

    Google Scholar 

  28. Yamauchi, H., De Fontaine, D.: Elastic interaction of defect clusters with arbitrary strain fields in an anisotropic continuum. Acta metall. 27 (1979) 763–776

    Google Scholar 

  29. Rammerstorfer, F. G., Fischer F. D., Böhm, H. J.: Treatment of micromechanical phenomena by finite elements. Discretization Methods in Structural Mechanics (Kuhn, G., Mang, H., edts), pp. 393–404. Berlin, Heidelberg: Springer-Verlag, 1990

    Google Scholar 

  30. Mura, T.: Micromechanics of defects in solids. Second rev. edt. Dordrecht et al.: Markus Nijhoff Publ., 1987

    Google Scholar 

  31. Schmauder, S.: Die Modellierung zähigkeitsbestimmender Prozesse in Mikrogefügen mit Hilfe der Finite-Element-Methode. Fortschr.-Ber. VDI Reihe 5 Nr 146, Düsseldorf: VDI-Verlag, 1988

    Google Scholar 

  32. Müller, I., Xu, H.: On the pseudo-elastic hysteresis. Acta metall. mater 39 (1991) 263–271

    Google Scholar 

  33. Müller, I.: On the size of the hysteresis in pseudoelasticity. Continuum Mech. Thermodyn. 1 (1989) 125–142

    Google Scholar 

  34. Müller, I.: Equilibrium between coherent phases. Anisotropy and localization of plastic deformations (Boehler, J.-P. Khan, A. S., edts), pp. 573–576. London et al.: Elsevier Applied Science, 1991

    Google Scholar 

  35. Vacher, P., Lexcellent, Chr.: Study of pseudoelasticity behaviour of polycrystalline shape memory alloys by resistivity measurements and acoustic emission. Mechanical Behaviour of Materials-VI (Jono, M., Inoue, T., edts.), pp. 231–236. Oxford et al.: Pergamon Press, 1991

    Google Scholar 

  36. Kaufman, L., Cohen, M.: Thermodynamics and kinetics of martensitic transformations. Progress in metal physics (Chalmers, B., King, R., edts) pp. 165–246. London et al.: Pergamon Press, 1958

    Google Scholar 

  37. Cohen, M.: Operational nucleation in martensitic transformations. Met. Trans. 3 (1972) 1095–1098

    Google Scholar 

  38. Ortin, J., Planes, A.: Thermodynamics of thermoelastic martensitic transformations. Acta metall. 37 (1989) 1433–1441

    Google Scholar 

  39. Roitburd, A. L., Temkin, D. E.: Plastic deformation and thermodynamic hysteresis in phase transformations in solids. Sov. Phys. Solid State 28 (1986) 432–436

    Google Scholar 

  40. Roitburd, A. L., Temkin, D. E.: Hysteresis of a phase transformation in an elastoplastic medium. Sov. Phys. Doklady 31 (1986) 414–416

    Google Scholar 

  41. Tanaka, K., Fischer, F. D., Oberaigner, E. R.: A continuum mechanical approach to the kinetics and deformation of alloys during martensitic transformations. Int. Conf. Martensitic Transformations-ICOMAT-92 (Perkins, J., edt.), in print, 1993

  42. Tanaka, K., Fischer, F. D.: Deformation analysis of shape memory alloys during thermomechanical processes. Mechanical Behavior of Materials-VI, (Jono, M., Inoue, T., edts.), Vol. III, pp. 249–254. Oxford et al.: Pergamon Press 1991

    Google Scholar 

  43. Kestin, J.: Conservative thermodynamics of irreversible processes. Lecture notes for the CISM course: Internal variables in thermodynamics and continuum mechanics. Udine, 1988

  44. Raniecki, B., Lexcellent, Ch., Tanaka, K.: Thermodynamic models of pseudoelastic behavior of shape memory alloys. Arch. Mech. 44 (1992) 261–284

    Google Scholar 

  45. Rice, J. R.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19 (1971) 433–455

    Google Scholar 

  46. Sun, Q. P., Hwang, K. Ch.: Micromechanics constitutive description of thermoelastic martensitic transformation. To appear in Advances in Applied Mechanics, Vol. 31, (Hutchinson, J. W., Wu, T. Y., edts.). New York et al.: Academic Press, 1992

    Google Scholar 

  47. Hill, R.: Energy-momentum tensors in elastostatics: Some reflections on the general theory. J. Mech. Phys. Solids 34 (1986) 305–317

    Google Scholar 

  48. Washizu, K.: Variational methods in elasticity and plasticity. Oxford, New York et al.: Pergamon Press, 2nd edt., 1975

    Google Scholar 

  49. Roitburd, A. L.: Martensitic transformation as a typical phase transformation in solids. Solid State Physics (Ehrenreich, H. et al. edts.), pp. 317–390. New York, San Francisco, London: Academic Press, 1978

    Google Scholar 

  50. Roitburd, A. L.: Phase equilibrium in solids. Sov. Phys. Solid State 28 (1986) 1716–1718

    Google Scholar 

  51. Kaganova, I. M., Roitburd, A. L.: Effect of plastic deformation on the equilibrium shape of a new-phase inclusion and thermodynamic hysteresis. Sov. Phys. Solid State 31 (1989) 545–550

    Google Scholar 

  52. Berveiller, M., Patoor, E., Buisson, M.: Thermomechanical constitutive equations for shape memory alloys. J. de Physique IV, Colloque C4, Suppl. J. de Physique III, 1 (1991) C4-387–C4-396

    Google Scholar 

  53. Abeyaratne, R., Knowles, J. K.: On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38 (1990) 345–360

    Google Scholar 

  54. Grienfel'd, M.: Continuum methods in the theory of phase transitions in solids. Physics Eart & Planetary Int. 50 (1988) 99–109

    Google Scholar 

  55. Raniecki, B., Tanaka, K.: On the thermodynamic driving force for martensitic phase transformations. Residual Stresses-III ICRS3 (Fujiwara, H. et al., edts.), pp. 196–201. London, New York: Elsevier Appl. Sci., 1992

    Google Scholar 

  56. Ganghoffer, J. F., Denis, S., Gautier, E., Simon, A. Simonsson, K., Sjöström, S.: Mechanical and thermodynamical study of a macroscopically coherent phase transition case of the martensitic transformation. J. de Physique IV, Colloque C4, Suppl. J. de Physique III, 1 (1991) C4-89-C4-94

    Google Scholar 

  57. Liu, I-Sh.: On interface equilibrium and inclusion problems. Continuum Mech. Thermodyn. 4 (1992) 177–186

    Google Scholar 

  58. Lehner, F. K.: Thermodynamics of rock deformation by pressure solution. Deformation processes in minerals, ceramics and rocks (Barber, D. J., Meredith, P. G., edts.) pp. 296–333. London: Unwin Hyman, 1990

    Google Scholar 

  59. Heidug, W.: A thermodynamic theory of fluid infiltrated porous media undergoing large deformations and change of phase. PhD-thesis, Brown Univ., USA, 1985

    Google Scholar 

  60. Gurtin, M. E., Struthers, A.: Multiphase thermomechanics with interfacial structure. 3. Evolving phase boundaries in the presence of bulk deformation. Arch. Rational Mech. Anal. 112 (1990) 97–160

    Google Scholar 

  61. Gurtin, M. E.: The dynamics of solid-solid phase transitions. 1. Coherent interfaces. Research Report No. 92-NA-041, Dept. Mathematics, Carnegie Mellon Univ., 1992

  62. Nishiyama, Z.: Martensitic transformation. New York et al.: Academic Press, 1978

    Google Scholar 

  63. Reckling, K.: Plastizitätstheorie und ihre Anwendungen auf Festigkeitsprobleme. Berlin, Heidelberg, New York: Springer Verlag, 1967

    Google Scholar 

  64. Lee, J. K., Earmme, Y. Y., Aaronson, H. J., Russel, K. C.: Plastic relaxation of the transformation strain energy of a misfitting spherical precipitate: ideal plastic behavior. Met. Trans. 11A (1980) 1837–1847

    Google Scholar 

  65. Lee, E. U.: Thermal stress and strain in a metal matrix composite with a spherical reinforcement particle. Met. Trans. 23A (1992) 2205–2210

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mechanics, University for Mining and Metallurgy, Franz-Josef-Straße 18, A-8700, Leoben, Austria

    F. D. Fischer & E. R. Oberaigner

  2. Laboratoire de Physique et Mecanique de Materiaux, Institut Superieur de Genie Mechanique et Productique, Ile du Saulcy, F-57045, Metz Cedex 01, France

    M. Berveiller

  3. Department of Aerospace Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, 191, Hino/Tokyo, Japan

    K. Tanaka

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Fischer, F.D., Berveiller, M., Tanaka, K. et al. Continuum mechanical aspects of phase transformations in solids. Arch. Appl. Mech. 64, 54–85 (1994). https://doi.org/10.1007/BF00789099

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