We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Skip to main content
SpringerLink
Log in

Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au23Cu30Zn47 demonstrate the quality of the discrete model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev spaces. New York: Academic Press 1975

    Google Scholar 

  2. Alt, H.W., Hoffmann, K.-H., Niezgódka, M., Sprekels, J.: A numerical study of structural phase transitions in shape memory alloys. Preprint No. 90, Institute for Mathematics, University of Augsburg 1985

  3. Delaey, L., Chandrasekaran, M.: Martensitic transformations. Les Editions de Physique, Les Ulis 1982

  4. Falk, F.: Landau theory and martensitic phase transitions. In: Martensitic transformations, pp 3–15. Les Editions de Physique, Les Ulis 1982

    Google Scholar 

  5. Falk, F.: Ginzburg-Landau theory of static domain walls in shape-memory alloys. Z. Physik B — Condensed Matter51, 177–185 (1983)

    Google Scholar 

  6. Müller, I., Wilmanski, K.: A model for phase transitions in pseudoelastic bodies. II Nuovo Cimento57B, 283–318 (1980)

    Google Scholar 

  7. Niezgódka, M., Sprekels, J.: Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys. Math. Methods Appl. Sci.10, 197–223 (1988)

    Google Scholar 

  8. Niezgódka, M., Zheng, S., Sprekels, J.: Global solutions to a model of structural phase transitions in shape memory alloys. J. Math. Anal. Appl.130, 39–54 (1988)

    Google Scholar 

  9. Sprekels, J.: Global existence for thermomechanical processes with nonconvex free energies of Ginzburg-Landau form. J. Math. Anal. Appl.141, 333–348 (1988)

    Google Scholar 

  10. Sprekels, J.: One-dimensional thermomechanical phase transitions with non-convex potentials of Ginzburg-Landau type. Preprint # 505 of the IMA of the University of Minnesota, Minneapolis 1989

  11. Sprekels, J., Zheng, S.: Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys. Physica D39, 59–76 (1989)

    Google Scholar 

  12. Tiihonen, T.: A numerical approach to a shape memory model. Preprint # 98 of the DFGSPP “Anwendungsbezogene Optimierung und Steuerung”, Augsburg 1988

  13. Zheng, S.: Global solutions to thermomechanical equations with nonconvex Landau-Ginzburg free energy. J. Appl. Math. Physics (ZAMP)40, 111–127 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, University of Warsaw, PKiN, IXp., PL-00-901, Warsaw, Poland

    Marek Niezgódka

  2. Fachbereich 10 der Universität-GH Essen, Postfach 103764, W-4300, Essen I, Germany

    Jürgen Sprekels

Authors
  1. Marek Niezgódka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürgen Sprekels
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by Research Program RP.1.02

Supported by DFG, SPP “Anwendungsbezogene Optimierung und Steuerung”

Rights and permissions

Reprints and permissions

About this article

Cite this article

Niezgódka, M., Sprekels, J. Convergent numerical approximations of the thermomechanical phase transitions in shape memory alloys. Numer. Math. 58, 759–778 (1990). https://doi.org/10.1007/BF01385653

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01385653

Subject classification

Search

Navigation