Skip to main content
Log in

Cauchy-Born Rule and the Stability of Crystalline Solids: Dynamic Problems

Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Cite this article

Abstract

We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from the well-known Cauchy-Born rule.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Adams, R.A., Fournier, J.J.F. Sobolev spaces. 2nd ed. Academic Press, 2003

  2. Ashcroft. N.W., Mermin, N.D. Solid state physics. Saunders College Publishing, 1976

  3. Blanc, X., Le Bris, C., Lions, P.L. From molecular models to continuum mechanics. Arch. Ration. Mech. Anal., 164(4):341–381 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Born, M. Dynamik der krystallgitter. B.G. Teubner, Leipzig–Berlin, 1915

  5. Born, M., Huang, K. Dynamical theory of crystal lattices. Oxford University Press, 1954

  6. Braides, A., Dal Maso, G., Garroni, A. Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Ration. Mech. Anal., 146(1):23–58 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cauchy, A.L. Sur l'equilibre et le mouvement d'un système de points materiels sollicités par forces d'attraction ou de répulsion mutuelle. Ex. de Math., 3:187–213 (1828)

    Google Scholar 

  8. Conti, S., Dolzmann, G., Kirchheim, B., Müller, S. Sufficient conditions for the validity of the cauchy-born rule close to SO(n). J. Eur. Math. Soc., 8(3):515–530 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dafermos, C.M., Hrusa, W.J. Energy methods for quasilinear hyperbolic initial-boundary value problems. applications to elastodynamics. Arch. Ration. Mech. Anal., 87(3):267–292 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. E, W., Liu, J.G. Projection method. I. Convergence and numerical boundary layers. SIAM J. Numer. Anal., 32(4):1017–1057 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. E, W., Ming, P.B. Cauchy-born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal., 183(2):241–297 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ericksen, J.L. The Cauchy and born hypotheses for crystals. in: phase transformations and material instabilities in solids. ed. by M.E. Gurtin, Academic Press, 1984, 61–77

  13. Friesecke, G., Theil, F. Validity and failure of the cauchy-born hypothesis in a two dimensional mass-spring lattice. J. Nonlinear Sci., 12(5):445–478 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hughes, T.J.R., Kato, T., Marsden, J. Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal., 63(3):273–294 (1977)

    Article  MATH  Google Scholar 

  15. Keating, P.N. Effect of invariance requirements on the elastic strain energy of crystals with application to the diamond structure. Phys. Rev., 145(2):637–645 (1965)

    Article  Google Scholar 

  16. Lennard-Jones, J.E., Devonshire, A.F. Critical and cooperative phenomena. III. a theory of melting and the structure of liquids. Proc. Roy. Soc. Lond., A169(938):317–338 (1939)

    Google Scholar 

  17. Liu, F., Ming, P.B. Crystal stability and instability. In preparation

  18. Love, A.E.H. A treatise on the mathematical theory of elasticity. 4th ed. Cambridge University Press, 1927

  19. Majda, A. Compressible fluid flow and systems of conservation laws in several space variables. Springer- Verlag, New York, 1984

  20. Maradudin, A.A., Vosko, S.H. Symmetry properties of the normal vibrations of a crystal. Rev. Mod. Phy., 40(1):1–37 (1968)

    Article  Google Scholar 

  21. Morrey, C.B. Multiple integrals in the calculus of variations. Springer-Verlag, New York, 1966

  22. Musgrave, M.J.P. Crystal acoustics: Introduction to the study of elastic waves and vibrations in crystals. Holden-Day, Inc., San Francisco, 1970

  23. Parry, G.P. On Phase transitions involving internal strain. Int. J. Solids Structures, 17(4):361–378 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pitteri, M. On ν + 1−Lattices. J. of Elasticity, 15(1):3–25 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pitteri, M., Zanzotto, G. Continuum models for phase transitions and twinning in crystals. Chapman & Hall/CRC Press LLC, 2003

  26. Ruelle, D. Statistical mechanics: rigorous results. Imperial College Press, 3rd ed. 1998

  27. Schochet, S. The incompressible limit in nonlinear elasticity. Comm. Math. Phys., 102(2):207–215 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  28. Shatah, J., Struwe, M. Geometric wave equations. American Mathematical Society, Providence, R.I. 2000

  29. Stakgold, I. The Cauchy relations in a molecular theory of elasticity. Quart. Appl. Math., 8(2):169–186 (1950)

    MATH  MathSciNet  Google Scholar 

  30. Strang, G. Accurate partial difference methods. ii: non-linear problems. Numer. Math., 6(1):37–46 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wallace, D.C. Thermodynamics of crystals. John Wiley & Sons Inc., New York, 1972

  32. Weiner, J.H. Statistical mechanics of elasticity. A Wiley-Interscience Publication, 1983

  33. Xuan, Y., E, W. Instability of crystalline solids under stress. In preparation

  34. Zanzotto, G. On the material symmetry group of elastic crystals and the born rule. Arch. Ration. Mech. Anal., 121(1):1–36 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. Zanzotto, G. The cauchy and born hypothesis, nonlinear elasticity and mechanical twinning in crystals. Acta Cryst., A52(6):839–849 (1996)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E Wei-nan*.

Additional information

*Supported in part by an NSF grant DMS 04-07866, with additional support from the project "Research Team on Complex Systems" of the Chinese Academy of Sciences

**Supported by the National Basic Research Program (No.2005CB321704) and the National Natural Science Foundation of China (No.10571172).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Wei-nan*, E., Ping-bing**, M. Cauchy-Born Rule and the Stability of Crystalline Solids: Dynamic Problems. Acta Mathematicae Applicatae Sinica, English Series 23, 529–550 (2007). https://doi.org/10.1007/s10255-007-0393

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-007-0393

Keywords

2000 MR Subject Classification

Navigation