Skip to main content
SpringerLink
Log in

Quasicontinuum Models of Dynamic Phase Transitions

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

Abstract

We propose a series of quasicontinuum approximations for the simplest lattice model of a fully dynamic martensitic phase transition in one dimension. The approximations are dispersive and include various non-classical corrections to both kinetic and potential energies. We show that the well-posed quasicontinuum theory can be constructed in such a way that the associated closed-form kinetic relation is in an excellent agreement with the predictions of the discrete theory.

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. Abeyaratne R., Knowles J. (1993). A continuum model of a thermoelastic solid capable of undergoing phase transitions. J. Mech. Phys. Solids 41:541–571

    Article  MATH  MathSciNet  Google Scholar 

  2. Abeyaratne R., Vedantam S. (2003). A lattice-based model of the kinetics of twin boundary motion. J. Mech. Phys. Solids 51:1675–1700

    Article  MATH  MathSciNet  Google Scholar 

  3. Benjamin T.B., Bona J.L., Mahony J.J. (1972). Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. Lond. 272:47–78

    MATH  MathSciNet  Google Scholar 

  4. Caflich R.E., Miksis M., Papanicolaou G., Ting L. (1986). Wave propagation in bubbly liquid. Lect. Appl. Math. 23:327–343

    Google Scholar 

  5. Collins M.A. (1981). A quasicontinuum approximation for solitons in an atomic chain. Chem. Phys. Lett. 77:342–347

    Article  MathSciNet  Google Scholar 

  6. Gavrilyuk S., Saurel R. (2002). Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comp. Phys. 175:326–360

    Article  MATH  MathSciNet  Google Scholar 

  7. Gustaffsson B., Kreiss H.O., Oliger J. (1995). Time-dependent Problems and Difference Methods Pure and Applied Mathematics. Wiley, New York

    Google Scholar 

  8. Hochstrasser D., Mertens F.G., Büttner H. (1989). An iterative method for the calculation of narrow solitary excitations on atomic chains. Phys. D 35:259–266

    Article  Google Scholar 

  9. Kevrekidis, P.G., Kevrekidis, I.G., Bishop, A.R., Titi, E.S.: Continuum approach to discreteness. Phys. Rev. E 65:046,613 (2002)

    Google Scholar 

  10. Kevrekidis P.G., Weinstein M.I. (2000). Dynamics of lattice kinks. Phys. D 142:113–152

    Article  MATH  MathSciNet  Google Scholar 

  11. Kreiss H.O., Busenhart H.U. (2001). Time-dependent Partial Differential Equations and Their Numerical Solution. Birkhäuser Verlag, Basel

    Google Scholar 

  12. Kresse O., Truskinovsky L. (2003). Mobility of lattice defects: discrete and continuum approaches. J. Mech. Phys. Solids 51:1305–1332

    Article  MATH  MathSciNet  Google Scholar 

  13. Kresse O., Truskinovsky L. (2004). Lattice friction for crystalline defects: from dislocations to cracks. J. Mech. Phys. Solids 52:2521–2543

    Article  MATH  MathSciNet  Google Scholar 

  14. Kunin I. (1982). Elastic Media with Microstructure I: One-dimensional Models, Solid-State Sciences, vol 26. Springer, Berlin Heidelberg New York

    Google Scholar 

  15. Lax P.D., Levermore C.D., Venakides S. (1994). The generation and propagation of oscillations in dispersive initial value problems and their limiting behavior. In: Fokas A.S., Zakharov V.E. (eds) Important Developments in Soliton Theory, Springer Ser Nonlin Dynam. Springer, Berlin Heidelberg New York, pp. 205–241

    Google Scholar 

  16. Marder M., Gross S. (1995). Origin of crack tip instabilities. J. Mech. Phys. Solids 43:1–48

    Article  MATH  MathSciNet  Google Scholar 

  17. Mindlin R.D. (1965). Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1:417–438

    Article  Google Scholar 

  18. Rayleigh J.W.S. (1945). The theory of sound, vol I:II. Dover Publications, New York

    Google Scholar 

  19. Rogula, D.: Introduction to nonlocal theory of material media. In: Nonlocal Theory of material media, CISM Courses and Lectures, vol. 268, pp. 123–222, Springer, Wien (1982)

  20. Rosenau P. (1986). Dynamics of nonlinear mass-spring chains near the continuum limit. Phys. Lett. A 118(5):222–227

    Article  MathSciNet  Google Scholar 

  21. Rosenau P. (1988). Dynamics of dense discrete systems. Prog. Theor. Phys. 79:1028–1042

    Article  Google Scholar 

  22. Rosenau P. (2003). Hamiltonian dynamics of dense chains and lattices: or how to correct the continuum. Phys. Lett. A 311(1):39–52

    Article  MATH  MathSciNet  Google Scholar 

  23. Sharma, B.L.: The kinetic relation of a Peierls dislocation in a higher-gradient dispersive continuum. Ph.D. Thesis, Cornell University (2004).

  24. Slepyan L.I. (2002). Models and phenomena in fracture mechanics. Springer, Berlin Heidelberg New York

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Theil F., Levitas V. (2000). A study of a hamiltonian model for martensitic phase transformations including microkinetic energy. Math. Mech. Solids 5(3):337–368

    MATH  MathSciNet  Google Scholar 

  27. Toupin R.A. (1964). Theories of elasticity with couple-stresses. Arch. Ration Appl. Mech. Anal. 17:85–112

    MATH  MathSciNet  Google Scholar 

  28. Triantafyllidis N., Bardenhagen S. (1993). On higher order gradient continuum theories in 1-D nonlinear elasticity Derivation from and comparison to the corresponding discrete models. J. Elast. 33(3):259–293

    Article  MATH  MathSciNet  Google Scholar 

  29. Truskinovsky L. (1987). Dynamics of nonequilibrium phase boundaries in a heat conducting elastic medium. J. Appl. Math. Mech. 51:777–784

    Article  MathSciNet  Google Scholar 

  30. Truskinovsky L., Vainchtein A. (2003). Peierls–Nabarro landscape for martensitic phase transitions. Phys. Rev. B. 67, 172,103

    Google Scholar 

  31. Truskinovsky L., Vainchtein A. (2004). The origin of nucleation peak in transformational plasticity. J. Mech. Phys. Solids 52:1421–1446

    Article  MATH  MathSciNet  Google Scholar 

  32. Truskinovsky L., Vainchtein A. (2005). Explicit kinetic relation from “first principles”. In: Steinmann P., Maugin G. (eds) Mech Material Forces. Springer, Berlin Heidelberg New York, pp. 43–50

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Truskinovsky L., Vainchtein A. (2005). Quasicontinuum modeling of short-wave instabilities in crystal lattices. Philos. Mag. 85:4055–4065

    Article  Google Scholar 

  35. Wattis J.A.D. (1993). Approximations to solitary waves on lattices, II: quasicontinuum methods for fast and slow waves. J. Phys. A 26:1193–1209

    Article  MATH  MathSciNet  Google Scholar 

  36. Wattis J.A.D. (1996). Approximations to solitary waves on lattices, III: the monatomic lattice with second-neighbour interactions. J. Phys. A 29:8139–8157

    Article  MATH  MathSciNet  Google Scholar 

  37. Wattis J.A.D. (1998). Stationary breather modes of generalized nonlinear Klein-Gordon lattices. J. Phys. A 31:3301–3323

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mechanique des Solides, CNRS-UMR 7649, Ecole Polytechnique, 91128, Palaiseau, France

    Lev Truskinovsky & Anna Vainchtein

Authors
  1. Lev Truskinovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Vainchtein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Vainchtein.

Additional information

Communicated by A. DeSimone

Rights and permissions

Reprints and permissions

About this article

Cite this article

Truskinovsky, L., Vainchtein, A. Quasicontinuum Models of Dynamic Phase Transitions. Continuum Mech. Thermodyn. 18, 1–21 (2006). https://doi.org/10.1007/s00161-006-0018-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-006-0018-5

Keywords

PACS

Search

Navigation