, Volume 30, Issue 5, pp 577–589 | Cite as

Finite-scale microstructures and metastability in one-dimensional elasticity

  • Lev Truskinovsky
  • Giovanni Zanzotto


This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy [1]. Following some previous work in this area, we suitably regularize the problem in order to investigate this degenerancy. Our approach gives an explicit framework for the the study of the rich variety offinite-scale equilibrium microstructures for the bar in a hard loading device, and their stability properties. In this way we clarify the role of interfacial energy in creating finitescale microstructures, by considering the combined effect of the oscillation-inducing and oscillation-inhibiting terms in the energy functional.

Key words

Microstructures Interface energy Non convex variational problems Phase transitions Solid mechanics 


Il lavoro riguarda la non unicità messa in luce da J.L. Ericksen nella sua analisi dell'equilibrio di barre elastiche con energia non convessa. Seguendo le linee di precedenti lavori, per investigare questa degenerazione si ricorre ad una regolarizzazione del problema e si dà un esplicito quadro di riferimento per lo studio della ricca varietà delle microstrutture di scala finita e della loro stabilità. Si chiarisce in particolare il ruolo dell'energia di interfaccia nella creazione di microstrutture di scala finita considerando l'effetto combinato di termini inibitori e favorevoli all'insorgere di oscillazioni nel funzionale energia.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ericksen, J.L. ‘Equilibrium of bars’,J. Elasticity,5 (1975) 191–202.Google Scholar
  2. 2.
    James, R.D. ‘Deformation of shape-memory materials’, inMat. Res. Soc. Symp. Proc., Liu C.Y., Kunsmann, H., Otsuka, K., Wuttig, M. (eds.),246, (1992) 81–90.Google Scholar
  3. 3.
    Truskinovsky, L., and Zanzotto, G. ‘Ericksen's bar revisited’, in preparation.Google Scholar
  4. 4.
    Müller, S. ‘Minimizing sequences for nonconvex functionals, phase transitions and singular perturbations’, in Kirchgässner, K. (ed.),Problems Involving Change of Type, Lecture Notes in Physics, vol.359, Springer, Berlin, (1990), pp. 31–44.Google Scholar
  5. 5.
    Müller, S. ‘Singular perturbation as a selection criterion for periodic minimizing sequences’,Calculus of variations,1 (1993) 169–204.Google Scholar
  6. 7.
    Fu, S., Müller, I., and Xu, H. ‘The interior of the pseudoelastic hysteresis’,Mat. Res. Soc. Symp. Proc. Liu, C.Y., Kunsmann, H., Otsuka, K., Wuttig, M. (eds.)246, (1992) 39–42.Google Scholar
  7. 8.
    Ortín, J. ‘Preisach modeling of hysteresis for a pseudoelastic Cu-Zn-Al single crystal’,J. Appl. Phys.,71 (1992) 1454–1461.Google Scholar
  8. 9.
    Fedelich, B., and Zanzotto, G., ‘Hysteresis in discrete systems of possibly interacting elements with a two-well energy’,J. Nonlin. Sci.,2 (1992) 319–342.Google Scholar
  9. 10.
    Eshelby, J.D. ‘The continuum theory of lattice defects’,Solids State Phys. 3 (1956) 79–144.Google Scholar
  10. 11.
    Abeyaratne, R., Chu, C., and James, R.D. ‘Kinetics and hysteresis in martensitic single crystals’, in Brinson, L.C., Moran, B. (eds),Mechanics of phase transformations and shape-memory alloys, AMD-Vol. 189, ASME 1994.Google Scholar
  11. 12.
    Schryvers, D., Ma, Y., Toth, L., and Tanner, L. ‘Electron microscopy study of the formation of Ni5Al3 in a Ni62.5Al37.5 B2 alloy. II: plate crystallography’,Acta Metall. Mater. (1995), in press.Google Scholar
  12. 13.
    Tan, S., and Xu, H. ‘Observations on a CuAlNi single crystal’,Cont. Mech. Thermodyn.,2 (1990) 241–244.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Lev Truskinovsky
    • 1
  • Giovanni Zanzotto
    • 2
  1. 1.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Dipartimento di Metodi e Modelli Matematici per le Scienze ApplicateUniversità di PadovaPadovaItaly

Personalised recommendations