Phase Boundaries in Alloys with Elastic Misfit

  • Harald Garcke
Part of the Progress in Mathematics book series (PM, volume 202)


We study a mathematical model describing phase transformations in alloys with kinetics driven by mass transport and stress. To describe the dynamics, a Cahn-Hilliard system taking elastic effects into account is studied. Existence and uniqueness results for the resulting singular elliptic-parabolic system are given.

In the Cahn-Hilliard model, phase boundaries are described by a diffuse interface with small positive thickness. In the stationary case we identify the sharp interface free boundary problem that arises when the interfacial thickness tends to zero. In particular, we obtain a geometric partition problem generalizing variants of isoperimetric problems to situations where elastic interactions cannot be neglected.


Free Boundary Phase Field Model Interfacial Thickness Elastic Effect Chemical Potential Difference 
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. 1.
    G. Caginalp, An analysis of a phase field model of a free boundary, Arch Rat. Mech. Anal. 92 (1986), 205–245.MathSciNetzbMATHCrossRefGoogle Scholar
  2. M. Carrive, A. Miranville and A. Pi¨¦trusThe Cahn-Hilliard equation for deformable elastic mediaAdv. Math. Sci. Appl. (to appear).Google Scholar
  3. 3.
    P. G. CiarletMathematical ElasticityNorth Holland, Amsterdam, New York, Oxford, 1988.Google Scholar
  4. 4.
    C. M. Elliott and S. LuckhausA generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy, SFB256 University Bonn, Preprint195 (1991).Google Scholar
  5. 5.
    H. Garcke, On mathematical models for phase separation in elastically stressed solidshabilitation thesis, University Bonn, 2000.Google Scholar
  6. 6.
    H. Garcke, M. Rumpf andU. Weikard The Cahn-Hilliard equation with elasticity: Finite element approximation and qualitative studies Interfaces and Free Boundaries 3 (2001), 101–118.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, Princeton University Press, 1983. zbMATHGoogle Scholar
  8. 8.
    E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel, Boston, Stuttgart, 1984.zbMATHGoogle Scholar
  9. 9.
    F. C. Larché and J. W. Cahn, The effect of self-stress on diffusion in solids, Acta Metall., 30 (1982), 1835–1845.CrossRefGoogle Scholar
  10. 10.
    P. H. Leo, J. S. Lowengrub and H. J. Jou, A diffuse interface model for microstructural evolution in elastically stressed solids, Acta Mater., 46 (1998), 2113–2130.CrossRefGoogle Scholar
  11. 11.
    A. M. Meirmanov, The Stefan Problem, De Gruyter, Berlin, 1992. Google Scholar
  12. 12.
    A. Onuki, Ginzburg-Landau approach to elastic effects in the phase separation of solids, J. Phys. Soc. Jpn., 58 (1989), 3065–3068.Google Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Harald Garcke
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

Personalised recommendations