Skip to main content
SpringerLink
Log in

Still states of bistable lattices, compatibility, and phase transition

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

We study a two-dimensional triangular lattice made of bistable rods. Each rod has two equilibrium lengths, and thus its energy has two equal minima. A rod undergoes a phase transition when its elongation exceeds a critical value. The lattice is subject to a homogeneous strain and is periodic with a sufficiently large period. The effective strain of a periodic element is defined. After phase transitions, the lattice rods are in two different states and lattice strain is inhomogeneous, the Cauchy–Born rule is not applicable. We show that the lattice has a number of deformed still states that carry no stresses. These states densely cover a neutral region in the space of entries of effective strains. In this region, the minimal energy of the periodic lattice is asymptotically close to zero. When the period goes to infinity, the effective energy of such lattices has the “flat bottom” which we explicitly describe. The compatibility of the partially transited lattice is studied. We derive compatibility conditions for lattices and demonstrate a family of compatible lattices (strips) that densely covers the flat bottom region. Under an additional assumption of the small difference of two equilibrium lengths, we demonstrate that the still structures continuously vary with the effective strain and prove a linear dependence of the average strain on the concentration of transited rods.

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. Ansini N., Braides A., Piat V.C.: Interactions between homogenization and phase-transition processes. Tr. Mat. Inst. Steklova 236, 373–385 (2002)

    Google Scholar 

  2. Ansini N., Braides A., Piat V.C.: Gradient theory of phase transitions in inhomogeneous media. Proc. R. Soc. Edinb. A 133, 265–296 (2003)

    Article  MATH  Google Scholar 

  3. Ariyawansa K.A., Berlyand L., Panchenko A.: A network model of geometrically constrained deformations of granular materials. Netw. Heterog. Media 3(1), 125–148 (2008)

    MATH  MathSciNet  Google Scholar 

  4. Ayzenberg-Stepanenko M.V., Slepyan L.I.: Resonant-frequency primitive waveforms and star waves in lattices. J. Sound Vib. 313(3–5), 812–821 (2008)

    Article  ADS  Google Scholar 

  5. Bagi K.: Stress and strain in granular asseblies. Mech. Mater. 22, 165–177 (1996)

    Article  Google Scholar 

  6. Braides A., Defrancheschi A.: Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  7. Cherkaev A., Vinogradov V., Leealavanichkul S.: The waves of damage in elastic lattices with waiting links. Des. Simul. Mech. Mater. 38, 748–756 (2006)

    Article  Google Scholar 

  8. Cherkaev A., Zhornitskaya L.: Dynamics of damage in two-dimensional structures with waiting links. In: Movchan, A.B. (ed.) Asymptotics, Singularities and Homogenisation in Problems of Mechanics, pp. 273–284. Kluwer, Dordrecht (2004)

    Chapter  Google Scholar 

  9. Cherkaev A., Zhornitskaya L.: Protective structures with waiting links and their damage elovulion. Multibody Syst. Dyn. 13, 53–67 (2005)

    Article  MATH  Google Scholar 

  10. Cherkaev A., Cherkaev E., Slepyan L.: Transition waves in bistable structures I: delocalization of damage ii: Analytical solution, wave speed, and energy dissipation. J. Mech. Phys. Solids 53(2), 383–436 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Love A.E.H.: A treatise on the mathematical theory of elasticity. Dover Publications, New York (1977)

    Google Scholar 

  12. Maso G.D.: Introduction to Γ-convergence. Burkauser, Basel (1993)

    Google Scholar 

  13. Ming W.E.P.: Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mishuris G.S., Slepyan L.I., Movchan A.B.: Dynamics of a bridged crack in a discrete lattice. Quart. J. Mech. Appl. Math. 61(2), 151–160 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mishuris G.S., Slepyan L.I., Movchan A.B.: Localised knife waves in a structured interface. J. Mech. Phys. Solids 57(12), 1958–1979 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  16. Sataki M.: Tensorial form definitions of discrete-mechanical quantities for granular assemblies. Int. J. Solids Struct. 41, 5775–5791 (2004)

    Article  Google Scholar 

  17. Slepyan L.I.: Feeding and dissipative waves in fracture and phase transition, III. Triangular-cell lattice. J. Mech. Phys. Solids 49(12), 2839–2875 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Splepyan L.I.: Models and Phenomena in Fracture Mechanics. Springer, Berlin (2002)

    Google Scholar 

  19. Slepyan L.I., Ayzenberg-Stepanenko M.V.: Some surprising phenomena in weak-bond fracture of a triangular lattice. J. Mech. Phys. Solids 50(8), 1591–1625 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Slepyan L.I., Ayzenberg-Stepanenko M.V.: Localized transition waves in bistable-bond lattices. J. Mech. Phys. Solids 52, 1447–1479 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Slepyan L., Cherkaev A., Cherkaev E.: Transition waves in bistable structures. I. Delocalization of damage. J. Mech. Phys. Solids 53, 383–405 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Slepyan L., Cherkaev A., Cherkaev E.: Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation. J. Mech. Phys. Solids 53, 407–436 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Truskinovsky L., Vainchtein A.: Kinetics of martensitic phase trasformations: lattice model. SIAM J. Appl. Math. 66(2), 533–553 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  24. Truskinovsky L., Vainchtein A.: Quasicontinuum models of dynamic phase transitions. Contin. Mech. Thermodyn. 18, 1–21 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Andrej Cherkaev

  2. Department of Mathematics, Washington State University, Pullman, WA, USA

    Andrei Kouznetsov & Alexander Panchenko

Authors
  1. Andrej Cherkaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrei Kouznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Panchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Panchenko.

Additional information

Communicated by Dr. Lev Truskinovsky.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cherkaev, A., Kouznetsov, A. & Panchenko, A. Still states of bistable lattices, compatibility, and phase transition. Continuum Mech. Thermodyn. 22, 421–444 (2010). https://doi.org/10.1007/s00161-010-0161-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-010-0161-x

Keywords

Search

Navigation