Metallurgical and Materials Transactions A

, Volume 37, Issue 9, pp 2701–2714 | Cite as

Numerical modeling of diffusion-induced deformation

  • J. A. Dantzig
  • W. J. Boettinger
  • J. A. Warren
  • G. B. McFadden
  • S. R. Coriell
  • R. F. Sekerka


We present a numerical approach to modeling the deformation induced by the Kirkendall effect in binary alloys. The governing equations for isothermal binary diffusion are formulated with respect to inert markers and also with respect to the volume-averaged velocity. Relations necessary to convert between the two formulations are derived. Whereas the marker formulation is the natural one in which to pose constitutive laws, the volume formulation provides certain computational advantages. We therefore compute the diffusion and deformation with respect to the volume-centered velocity and then determine the corresponding fields with respect to the markers. Several problems involving one-dimensional (1-D) diffusion couples are solved for verification, and a problem involving two-dimensional (2-D) diffusion in a lap joint is solved to illustrate the power of the method in a more complex geometry.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W.J. Boettinger, G.B. McFadden, S.R. Coriell, R.F. Sekerka, and J.A. Warren:Acta Mater., 2005, vol. 53, pp. 1995–2008.CrossRefGoogle Scholar
  2. 2.
    G.B. Stephenson:Acta Metall., 1988, vol. 36 (10), pp. 2663–83.CrossRefGoogle Scholar
  3. 3.
    I. Daruka, I.A. Szabó, D.L. Beke, C. Cserháti, A. Kodentsov, and F.J.J. Van Loo:Acta Mater., 1966, vol. 44, pp. 4981–93.CrossRefGoogle Scholar
  4. 4.
    J. Philibert:Atom Movements Diffusion and Mass Transport in Solids, Les Éditions de Physique, Les Ulis, France, 1991 (translated by S.J. Rothman).Google Scholar
  5. 5.
    R.B. Bird, W.E. Stewart, and E.N. Lightfoot:Transport Phenomena, Wiley, New York, NY, 1960.Google Scholar
  6. 6.
    R.F. Sekerka:Progr. Mater. Sci., 2004, vol. 49, pp. 511–36.CrossRefGoogle Scholar
  7. 7.
    Metals Reference Book, 5th ed., C.J. Smithells, ed., Butterworth and Co., London, 1976, p. 975.Google Scholar
  8. 8.
    M.S. Engelman, R. Sani, P.M. Gresho, and M. Bercovier:Int. J. Num. Meth. Eng., 1982, vol. 2, pp. 25–42.Google Scholar
  9. 9.
    P.M. Gresho and R.L. Sani:Incompressible Flow and the Finite Element Method, Wiley, New York, NY, 1998.Google Scholar
  10. 10.
    R. Voigt and V. Ruth:J. Phys. Condens. Matter, 1995, vol. 7, pp. 2655–66.CrossRefGoogle Scholar

Copyright information

© ASM International & TMS-The Minerals, Metals and Materials Society 2006

Authors and Affiliations

  • J. A. Dantzig
    • 1
  • W. J. Boettinger
    • 2
  • J. A. Warren
    • 3
  • G. B. McFadden
    • 4
  • S. R. Coriell
    • 5
  • R. F. Sekerka
    • 6
  1. 1.the Department of Mechanical and Industrial EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Metallurgy DivisionUSA
  3. 3.Thermodynamics and Kinetics GroupMetallurgy DivisionUSA
  4. 4.Mathematical and Computation Sciences Division, Information Technology LaboratoryUSA
  5. 5.the National Institute of Standards and TechnologyGaithersburgUSA
  6. 6.Carnegie Mellon UniversityPittsburghUSA

Personalised recommendations