Morphological Equilibrium and Kinetics of Two-Phase Materials

  • D. Gross
Part of the International Centre for Mechanical Sciences book series (CISM, volume 427)


Many engineering and natural crystalline materials are microscopically not homogeneous but consist of two ore more phases with different material properties. The morphology, i.e. the geometric shape of the phases, can be regarded as a result of stresses and strains induced by their different crystalline structure and by the loading history. On account of diffusive mass transport, the morphology may change with time and eventually tend to a final equilibrium state.. It is self-evident that in such multiphase materials the overall properties are found to depend not only on the properties of each phase, but also on the morphology of the microstructure and its evolution. As an example, Fig. la shows the lens-shaped microstructure of Mg-stabilized Zirconia (ZrO2) formed by a tetragonal in a cubic phase. Similarly, a lamellar structure is found in a geomaterial consisting of a pigeonite in an augite phase, see Fig. lb. In high temperature applications, such as turbine blades, Ni-base alloys are becoming more widely used. Microscopically they consist of Ni3X precipitates, the so-called γ′-phase (where X stands for Al, Ti, Si etc.), embedded in a Ni matrix, the γ-phase. In Fig. 2 the microstructure of such materials is displayed for different heat treatment and loading histories. It can be seen that, depending on the specific parameters, the precipitates may be sphere or cube shaped, they may be oblate or prolate. Furthermore they align along the crystallographic axis and show the so-called rafting behaviour under elevated temperature and external loads. The binary system Ni — Ni3Al has been under careful investigation for several years. Materials science has described the material properties very thoroughly. It was observed that the microstructure of those materials is of great importance for their overall properties, see e.g. Hornbogen and Roth (1967), Ardell and Nicholson (1966), Ardell and Meshkinpour (1994). It is characteristic not only for Ni-base alloys that the crystallographic planes of the two phases are continuous. Such phase boundaries are called coherent. Beside these empirical aspects, progress has been made in understanding the mechanics and thermodynamics of two-phase materials, e.g. Leo and Sekerka (1989), Gurtin and Voorhees (1993), Gurtin (1995), Schmidt and Gross (1997). Herein the configurational forces play a central role. These more theoretical works lay the basis for understanding how the microstructures form and how different loading situations influence the morphology on the micro level. However, pure analytical investigations can only treat very simple phenomena or predict qualitative results, see Johnson and Cahn (1984), Kaganova and Roitburd (1988). Numerical investigation, on the other hand, are capable of overcoming these limitations.


External Load Boundary Element Method Interfacial Energy Boundary Integral Equation Elastic Strain Energy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ardell, A., and Maheshwari, A. (1992). Anomalous coarsening behaviour of small volume fractions of Ni3A1 precipitates in binary Ni-Al alloys. Acta metall. mater. 40 (10): 2661–2667.CrossRefGoogle Scholar
  2. Ardell, A., and Meshkinpour, M. (1994). Role of volume fraction in the coarsening of Ni3Si precipitates in binary Ni-Si alloys. Material Science and Engineering A 185: 153–163.Google Scholar
  3. Ardell, A., and Nicholson, R. (1966). On the modulated structure of aged Ni-Al alloys. Acta metall. 14: 1205–1309.Google Scholar
  4. Ardell, A., and Rastogi, P. (1971). The coarsening behaviour of the ry’ precipitate in Nickel-Silicon alloys. Acta metall. 19: 321–330.CrossRefGoogle Scholar
  5. Binder, K., ed. (1986). Monte Carlo methods in statistical physics. Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Singapore, Tokyo: Springer.MATHGoogle Scholar
  6. Binder, K., ed. (1987). Applications of the Monte Carlo method in statistical physics. Berlin, Heidelberg, New York, Barcelona, Hong Kong, London, Milan, Paris, Singapore, Tokyo: Springer.Google Scholar
  7. Brebbia, C., and Telles, J. (1985). Boundary Element Techniques. Springer Verlag.Google Scholar
  8. Eshelby, J. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A241: 376–396.MathSciNetCrossRefMATHGoogle Scholar
  9. Eshelby, J. (1970). Energy relations and the energy-momentum tensor in continuum mechanics, In Kanninen (1970). 77–115.Google Scholar
  10. Fried, E., and Gurtin, M. (1993). Continnum theory of thermally induced phase transitions based on an order parameter. Physica D 68: 326–343.MathSciNetCrossRefMATHGoogle Scholar
  11. Göken, M., and Kempf, M. (1999). Microstructural properties of superalloys investigated by nanoindentations in an atomic force microscope..Acta Mater. (47):1043–1052.Google Scholar
  12. Gurtin, M., and Voorhees, P. (1993). The continuum mechanics of coherent two-phase elastic solids with mass transport. Proc. R. Soc. Lond. A 440: 323–343.MathSciNetCrossRefMATHGoogle Scholar
  13. Gurtin, M. (1995). The nature of configurational forces. Arch. Rational Mech. Anal. 131: 67–100.MathSciNetCrossRefMATHGoogle Scholar
  14. Hoover, W. G., Ashurst, W. T., and Olness, R. J. (1974). Two-dimensional computer studies of crystal stability and fluid viscosity. J. Chem. Phys. 60 (10): 4043–4047.CrossRefGoogle Scholar
  15. Hornbogen, E., and Roth, M. (1967). Die Verteilung hohärenter Teilchen in Nickellegierungen. Z Metallkde 58: 842–855.Google Scholar
  16. Johnson, W., and Cahn, J. (1984). Elastically induced shape bifurcations of inclusions. Acta metal!. 32 (11): 1925–1933.CrossRefGoogle Scholar
  17. Johnson, W., Berkenpas, M., and Laughlin, D. (1988). Precipitate shape transitions during coarsening under uniaxial stress. Acta metall. 36 (2): 3149–3162.CrossRefGoogle Scholar
  18. Kaganova, I., and Roitburd, R. (1988). Equilibrium between elastically-interacting phases. Soy. Phys. JETP 67 (4): 1173–1183.Google Scholar
  19. Kanninen, M., ed. (1970). Inelastic Behaviour of Solids. New York: McGraw Hill.Google Scholar
  20. Kolling, S., and Gross, D. (2000). Description of two-phase materials using discrete atom method. ZAMM 80: S385 - S386.CrossRefMATHGoogle Scholar
  21. Kolling, S., and Gross, D. (accepted for publication in 2001 ). Simulation of microstructural evolution in materials with misfitting precipitates. Journal of Probabilistic Engineering Mechanics.Google Scholar
  22. Lee, J.K. (1995). Coherency strain analysis via discrete atom method. Scr. Met. Mat. 32 (4): 559–564.CrossRefGoogle Scholar
  23. Lee, J.K. (1996a). Effects of applied stress on coherent precipitates via a disctrete atom method. Metals and Materials 2 (3): 183–193.Google Scholar
  24. Lee, J.K. (1996b). A study on coherency strain precipitate morphology via a discrete atom method. Met. Mat. Trans. 27A: 1449–1459.CrossRefGoogle Scholar
  25. Leo, P., and Sekerka, R. (1989). The effect of surface stress on crystal-melt and crystal-crystal equilibrium. Acta metall. 37 (12): 3119–3138.CrossRefGoogle Scholar
  26. Leo, P., Lowengrub, J., and Jou, H. (1998). A diffuse interface model for microstructural evolution in elastically stressed solids. Acta mater. 46 (6): 2113–2130.CrossRefGoogle Scholar
  27. Lubliner, J. (1990). Plasticity Theory. New York: Macmillan Publishing Company.MATHGoogle Scholar
  28. Luenberger, D. (1984). Linear and nonlinear programming. Addison-Wesley, 2 edition.Google Scholar
  29. Mueller, R., and Gross, D. (1998a). 3D equilibrium shapes in two-phase materials. ZAMM 78(2):635–636.Google Scholar
  30. Mueller, R., and Gross, D. (1998b). 3D simulation of equilibrium morphologies of precipitates. Comp. Mat. Sci. 11: 35–44.Google Scholar
  31. Mura, T. (1987). Micromechanics of Defects in Solids. Martinus Nijhoff Publishers.Google Scholar
  32. Nemat-Nasser, S., and Hori, M. (1993). Micromechanics: Overall properties of heterogeneous materials. Amsterdam, London, New York, Tokyo: North Holland.Google Scholar
  33. Schclar, N. (1994). Anisotropic Analysis using Boundary Elements, volume 20 of Topics in Engineering. Southampton UK and Boston USA: Computational Mechanics Publications.MATHGoogle Scholar
  34. Schmidt, I., and Gross, D. (1995). A strategy for determinig the equilibrium shape of an inclusion. Arch. Mech. 47 (2): 379–390.MATHGoogle Scholar
  35. Schmidt, I., and Gross, D. (1997). The equilibrium shape of an elastically inhomogeneous particle. J. Mech. Phys. Solids 45 (9): 1521–1549.MathSciNetCrossRefMATHGoogle Scholar
  36. Schmidt, I., and Gross, D. (1999). Directional coarsening in Ni-base superalloys: analytical results for an elasticity based model. Proc. R. Soc. Lond. 455: 3085–3106.CrossRefGoogle Scholar
  37. Schmidt, I., Mueller, R., and Gross, D. (1998). The effect of elastic inhomogeneity on equilibrium and stability of a two particle morphology. Mechanics of Materials 30: 181–196.CrossRefGoogle Scholar
  38. Schmidt, I. (1997). Gleichgewichtsmorphologien elastischer Einschlüsse. Ph.D. Dissertation, Technische Hochschule Darmstadt, D-64289 Darmstadt.Google Scholar
  39. Su, C., and Voorhees, P. (1996). The dynamics of precipitate evolution in elastically stressed solids–I, inverse coarsening. Acta mater. 44 (5): 1987–1999.CrossRefGoogle Scholar
  40. Thompson, M., Su, C., and Voorhees, P. (1993). The equilibrium shape of a misfitting precipitate. Acta metall. mater. 42 (6): 2107–2122.CrossRefGoogle Scholar
  41. Voorhees, P., McFadden, G., and Johnson, W. (1992). On the morphological development of second-phase particles in elastically-stressed solids. Acta metall. mater. 40 (11): 2979–2992.CrossRefGoogle Scholar
  42. Schoenlein, L.H., Rühle, M., and Heuer, A.H. (1984). In Situ Straining Experoments of Mg-PSZ Single Crystals Adv. in Ceramics 12, Science and Technology of Zirkonia II, Eds: Claussen, N., Rühle, M. and Heuer, A.H., The American Ceramic Society: 275–282.Google Scholar
  43. Ruble, M., and Heuer, A.H. (1984). Phase Transformations in ZrO2-Containing Ceramivs: II The Martensitic Reaction in t-ZrO2 Adv. in Ceramics 12, Science and Technology of Zirkonia II, Eds: Claussen, N., Rühle, M. and Heuer, A.H., The American Ceramic Society: 1–32.Google Scholar
  44. Wahi, R.P. (1997). Nickel base superalloys: Deformation characteristics at elevated temperatures Adv. in Comp. Eng. Sci., Tech. Sci. Press, Eds. Atluri, S.N, Yagawa, G.:85–90Google Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • D. Gross
    • 1
  1. 1.Institute of MechanicsDarmstadtDeutschland

Personalised recommendations