Skip to main content
SpringerLink
Log in

Bifurcations in the Regularized Ericksen Bar Model

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of Müller’s conjecture (Müller, Calc. Var. 1:169–204, 1993) concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of Müller’s conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a “loop” structure that characterizes (k,3k) bifurcations.

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. Armbruster, D., Dangelmayr, G.: Coupled stationary bifurcations in no-flux boundary value problems. Math. Proc. Camb. Phil. Soc. 101, 167–192 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aston, P.: Scaling laws and bifurcations. In: Roberts, M. Stewart, I. (eds.) Singularity Theory and its Applications, Warwick 1989, Part II. Lecture Notes in Mathematics, vol. 1463, pp. 1–21. Springer-Verlag, Berlin (1991)

    Google Scholar 

  3. Ball, J.M.: Dynamics and minimizing sequences. In: Kirchgässner, K. (ed.) Problems Involving Change of Type. Lecture Notes in Physics, vol. 359, pp. 3–16. Springer-Verlag, Berlin (1990)

    Google Scholar 

  4. Ball, J.M., Holmes, P.J., James, R.D., Pego, R.L., Swart, P.J.: On the dynamics of fine structure. J. Nonlin. Science 1, 17–70 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sanstede, B., Wang, X.: AUTO 97: Continuation and Bifurcation Software for ODEs, http://www.maths.surrey.ac.uk/personal/st/B.Sandstede/publications/auto97.pdf

  6. Ericksen, J.: Equilibrium of bars. J. Elasticity 5, 191–202 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  7. Friesecke, G., McLeod, J.B.: Dynamics as a mechanism preventing the formation of finer and finer microstructure. Arch. Rat. Mech. Anal. 133, 199–247 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Friesecke, G., McLeod, J.B.: Dynamic stability of non-minimizing phase mixtures. Proc. Royal Soc. London A 453, 2427–2436 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Springer-Verlag, New York (1985)

    MATH  Google Scholar 

  10. Grinfeld, M., Novick-Cohen, A.: Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments. Proc. Royal Soc. Edinburgh A 125, 351–370 (1995)

    MATH  MathSciNet  Google Scholar 

  11. Grinfeld, M., Novick-Cohen, A.: The viscous Cahn-Hilliard equation: morse decomposition and structure of the global attractor. Trans. AMS 351, 2375–2406 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grinfeld, M., Novick-Cohen, A.: (in preparation)

  13. Healey, T.J., Miller, U.: Two-phase equilibria in the anti-plane shear of an elastic solid with interfacial effect via global bifurcation. Proc. Royal Soc. A 463, 1117–1134 (2007)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Hunt, G.W.: Hidden (a)symmetries of elastic and plastic bifurcation. Appl. Mech. Rev. 36, 1165–1186 (1986)

    Google Scholar 

  15. Huo, Y., Müller, I.: Interfacial and inhomogeneity penalties in phase transitions. Continuum Mech. Thermodyn. 15, 395–407 (2001)

    Article  ADS  Google Scholar 

  16. Kalies, W.D., Holmes, P.J.: On a dynamical model for phase transformation in nonlinear elasticity. Fields Inst. Commun. 5, 255–269 (1996)

    MathSciNet  Google Scholar 

  17. Kamerich, E.: A Guide to Maple. Springer-Verlag, New York (1999)

    Google Scholar 

  18. Müller, S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. 1, 169–204 (1993)

    Article  MATH  Google Scholar 

  19. Swart, P.J., Holmes, P.J.: Energy minimization and the formation of microstructure in dynamic anti-plane shear. Arch. Rat. Mech. Anal. 121, 37–85 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. Truskinovsky, L., Zanzotto, G.: Ericksen’s bar revisited: energy wiggles. J. Mech. Phys. Solids 44, 1371–1408 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  21. Vainchtein, A.: Dynamics of phase transitions and hysteresis in a viscoelastic Ericksen’s bar on an elastic foundation. J. Elasticity 57, 243–280 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vainchtein, A.: Stick-slip interface motion as a singular limit of the viscosity-capillarity model. Math. Mech. Solids 6, 323–341 (2001)

    Article  MATH  Google Scholar 

  23. Vainchtein, A., Healey, T.J., Rosakis, P.: Bifurcation and metastability in a new one-dimensional model for martensitic phase transition. Comput. Methods Appl. Mech. Engrg. 170, 407–421 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Vainchtein, A., Healey, T., Rosakis, P., Truskinovsky, L.: The role of the spinodal region in one-dimensional martensitic phase transitions. Physica 115D, 29–48 (1998)

    ADS  MathSciNet  Google Scholar 

  25. Vainchtein, A., Rosakis, P.: Hysteresis and stick-slip motion of phase boundaries in dynamic models of phase transitions. J. Nonlinear Sci. 9, 697–719 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Yip, N.K.: Structure of stable solutions of a one-dimensional variational problem. Control, Optim. Calc. Variations 12, 721–751 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Strathclyde, Glasgow, G1 1XH, UK

    M. Grinfeld

  2. Department of Mathematics and Maxwell Institute, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    G. J. Lord

Authors
  1. M. Grinfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. J. Lord
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. J. Lord.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grinfeld, M., Lord, G.J. Bifurcations in the Regularized Ericksen Bar Model. J Elasticity 90, 161–173 (2008). https://doi.org/10.1007/s10659-007-9137-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-007-9137-x

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation