Skip to main content
SpringerLink
Log in

Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature. We prove, under certain sharp stability conditions, that the correct nonlinear elasticity model is given by the classical Cauchy–Born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the Cauchy–Born rule. The analysis is carried out for both simple and complex lattices, and for this purpose, we develop the necessary tools for performing asymptotic analysis on such lattices. Our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids.

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. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, 2nd ed., 2003

  2. Agmon S., Douglas A., Nirenberg L. (1964) Estimates near the boundary for solutions of elliptic partial differential equations satisfying boundary condition, II. Comm. Pure Appl. Math. 17, 35–92

    MathSciNet  MATH  Google Scholar 

  3. Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Saunders College Publishing, 1976

  4. Ball J.M., James R.D. (1992) Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. Lond. A 338, 389–450

    ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, 1954

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Daw M.S., Baskes M.I. (1983) Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals. Phys. Rev. Lett. 50, 1285–1288

    Article  ADS  Google Scholar 

  9. Daw M.S., Baskes M.I. (1984) Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29, 6443–6453

    Article  ADS  Google Scholar 

  10. E W., Ming P. (2004) Analysis of multiscale methods, J. Comput. Math. 22, 210–219

    MathSciNet  MATH  Google Scholar 

  11. E W., Ming, P.: Cauchy–Born rule and the stability of crystalline solids: Dynamical problems. In preparation

  12. Engel P. (1986) Geometric Crystallography: An Axiomatic Introduction to Crystallography. D. Reidel Publishing Company, Dordrecht, Holland

    MATH  Google Scholar 

  13. Ericksen, J.L.: The Cauchy and Born hypotheses for crystals. Phase Transformations and Material Instabilities in Solids. Gurtin, M.E. (ed.). Academic Press, 61–77, 1984

  14. Friesecke G., Theil F. (2002) Validity and failure of the Cauchy–Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12, 445–478

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  16. Lennard-Jones J.E., Devonshire A.F. (1939) Critical and cooperative phenomena, III. A theory of melting and the structure of liquids. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 169, 317–338

    Article  ADS  MATH  Google Scholar 

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

  18. Maradudin A.A., Vosko S.H. (1968) Symmetry properties of the normal vibrations of a crystal. Rev. Modern Phys. 40, 1–37

    Article  ADS  Google Scholar 

  19. Ming, P.: Crystal stability with traction boundary condition. In preparation

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

    MathSciNet  MATH  Google Scholar 

  21. Stillinger F.H., Weber T.A. (1985) Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 5262–5271

    Article  ADS  Google Scholar 

  22. Strang G. (1964) Accurate partial difference methods. II: Non-linear problems. Numer. Math. 6, 37–46

    Article  MathSciNet  MATH  Google Scholar 

  23. Tersoff J. (1988) Empirical interatomistic potential for carbon, with applications to amorphous carbon. Phys. Rev. Lett. 61, 2879–2882

    Article  ADS  Google Scholar 

  24. Truskinovsky, L.: Fracture as a phase transition. Contemporary Research in the Mechanics and Mathematics of Materials. Batra, R.C., Beatty, M.F. (ed.) © CIMNE, Barcelona, 322–332, 1996

  25. Valent, T.:Boundary Value Problems of Finite Elasticity. Springer-Verlag, 1988

  26. Wallace D.C. (1972) Thermodynamics of Crystals. John Wiley & Sons Inc., New York

    Google Scholar 

  27. Weiner J.H. (1983) Statistical Mechanics of Elasticity. John Wiley & Sons Inc., New York

    MATH  Google Scholar 

  28. Xiang, Y., Ming, P., Wei, H., E, W.: A generalized Peierls-Nabarro model for curved dislocations. In preparation

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

  30. Yang, J., E, W.: Generalized Cauchy–Born rules for sheets, plates and rods, submitted for publication, 2005

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and PACM, Princeton University, Princeton, NJ, 08544-1000, USA

    Weinan E

  2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, No. 55, Zhong-Guan-Cun East Road, Beijing, 100080, China

    Pingbing Ming

Authors
  1. Weinan E
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pingbing Ming
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weinan E.

Additional information

Communicated by the Editors

Rights and permissions

Reprints and permissions

About this article

Cite this article

E, W., Ming, P. Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems. Arch Rational Mech Anal 183, 241–297 (2007). https://doi.org/10.1007/s00205-006-0031-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-006-0031-7

Keywords

Search

Navigation