Skip to main content
SpringerLink
Log in

Stored Energy Functions for Phase Transitions in Crystals

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract.

A method is presented to construct nonconvex free energies that are invariant under a symmetry group. Algebraic and geometric methods are used to determine invariant functions with the right location of minimizers. The methods are illustrated for symmetry-breaking martensitic phase transformations. Computer algebra is used to compute a basis of the corresponding class of invariant functions. Several phase transitions, such as cubic-to-orthorhombic, are discussed. An explicit example of an energy for the cubic-to-tetragonal phase transition is given.

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. Abud, M., Sartori, G.: The geometry of orbit-space and natural minima of Higgs potentials. Phys. Lett. B 104, 147–152 (1981)

    Article  MathSciNet  Google Scholar 

  2. Abud, M., Sartori, G.: The geometry of spontaneous symmetry breaking. Ann. Physics 150, 307–372 (1983)

    MathSciNet  MATH  Google Scholar 

  3. Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969

  4. Chevalley, C.: Invariants of finite groups generated by reflections. Am. J. Math. 77, 778–782 (1955)

    MathSciNet  MATH  Google Scholar 

  5. Conti, S., Zanzotto, G.: Reconstructive phase transformations, maximal Ericksen-Pitteri neighborhoods, dislocations and plasticity in crystals. Technical Report 42/2002, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig, 2002

  6. Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. second edition, New York: Springer-Verlag, 1997. (An introduction to computational algebraic geometry and commutative algebra)

  7. Dondl, P., Zimmer, J.: Modeling and simulation of martensitic phase transitions with a triple point. Submitted to J. Mech. Phys. Solids

  8. Eisenbud, D.: Commutative algebra. New York: Springer-Verlag, 1995. (With a view toward algebraic geometry)

  9. Falk, F., Konopka, P.: Three-dimensional Landau theory describing the martensic phase transformation of shape-memory alloys. J. Phy. – Condensed Matter 2, 61–77 (1990)

    Article  Google Scholar 

  10. Field, M.: Lectures on bifurcations, dynamics and symmetry. Volume 356 of Pitman Research Notes in Mathematics Series. Longman, Harlow, 1996

  11. Gatermann, K.: Computer algebra methods for equivariant dynamical systems. Berlin: Springer-Verlag, 2000

  12. Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 2.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001. http://www.singular.uni-kl.de

  13. Hochster, M., Eagon, J.A.: Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93, 1020–1058 (1971)

    MATH  Google Scholar 

  14. Jänich, K.: Differenzierbare G-Mannigfaltigkeiten. Berlin: Springer-Verlag, 1968

  15. Jarić, M.V., Michel, L., Sharp, R.T.: Zeros of covariant vector fields for the point groups: invariant formulation. J. Physique 45, 1–27 (1984)

    MathSciNet  Google Scholar 

  16. Landau, L.D.: On the theory of phase transitions. In: Haar, D.T. (ed.), Collected papers of L. D. Landau. New York: Gordon and Breach Science Publishers, 1967

  17. Lyubarskiu i, G.Y.: The application of group theory in physics. New York: Pergamon Press, 1960

  18. Pitteri, M., Zanzotto, G.: Models for Phase Transitions and Twinning in Crystals. CRC Press, Boca Rota, 2002

  19. Procesi, C., Schwarz, G.: Inequalities defining orbit spaces. Invent. Math. 81, 539–554 (1985)

    MathSciNet  MATH  Google Scholar 

  20. Procesi, C., Schwarz, G.W.: The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories. Phys. Lett. B 161, 117–121 (1985)

    Article  MathSciNet  Google Scholar 

  21. Rumberger, M.: Symmetrische dynamische Systeme: Differenzierbarkeit und linearisierte Stabilität. PhD thesis, Universität Hamburg, Fachbereich Mathematik, 1997

  22. Rumberger, M.: Finitely differentiable invariants. Math. Z. 229, 675–694 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Schönert, M. et~al.: GAP – Groups, Algorithms, and Programming. Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, fifth edition, 1995 http://www-gap.dcs.st-and.ac.uk/~gap/

  24. Schwarz, G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975)

    Article  MATH  Google Scholar 

  25. Smith, G.F., Rivlin, R.S.: The strain-energy function for anisotropic elastic materials. Trans. Am. Math. Soc. 88, 175–193 (1958)

    MATH  Google Scholar 

  26. Sturmfels, B.: Algorithms in invariant theory. Vienna: Springer-Verlag, 1993

  27. Weyl, H.: The classical groups: Their invariants and representations. Fifteenth printing, Princeton Paperbacks, Princeton, NJ: Princeton University Press, 1997

  28. Worfolk, P.A.: Zeros of equivariant vector fields: algorithms for an invariant approach. J. Symbolic Comput. 17, 487–511 (1994)

    Google Scholar 

  29. Zimmer, J.: Mathematische Modellierung und Analyse von Formgedächtnislegierungen in mehreren Raumdimensionen (Mathematical Modeling and Analysis of Shape Memory Alloys in Several Space Dimensions). PhD thesis, Technische Universität München, 2000

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstr, 22 04103, Leipzig, Germany

    Johannes Zimmer

Authors
  1. Johannes Zimmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johannes Zimmer.

Additional information

K. Bhattacharya

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zimmer, J. Stored Energy Functions for Phase Transitions in Crystals. Arch. Rational Mech. Anal. 172, 191–212 (2004). https://doi.org/10.1007/s00205-003-0286-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-003-0286-1

Keywords

Search

Navigation