Archive for Rational Mechanics and Analysis

, Volume 183, Issue 2, pp 241–297 | Cite as

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

  • Weinan E
  • Pingbing Ming


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.


Atomistic Model Triangular Lattice Crystalline Solid Complex Lattice Dynamical Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, 2nd ed., 2003Google Scholar
  2. 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, 35–92MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Saunders College Publishing, 1976Google Scholar
  4. 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, 389–450ADSzbMATHGoogle Scholar
  5. 5.
    Blanc X., Le Bris C., Lions P.-L. (2002) From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, 1954Google Scholar
  7. 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, 23–58CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Daw M.S., Baskes M.I. (1983) Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals. Phys. Rev. Lett. 50, 1285–1288CrossRefADSGoogle Scholar
  9. 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, 6443–6453CrossRefADSGoogle Scholar
  10. 10.
    E W., Ming P. (2004) Analysis of multiscale methods, J. Comput. Math. 22, 210–219MathSciNetzbMATHGoogle Scholar
  11. 11.
    E W., Ming, P.: Cauchy–Born rule and the stability of crystalline solids: Dynamical problems. In preparationGoogle Scholar
  12. 12.
    Engel P. (1986) Geometric Crystallography: An Axiomatic Introduction to Crystallography. D. Reidel Publishing Company, Dordrecht, HollandzbMATHGoogle Scholar
  13. 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, 1984Google Scholar
  14. 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, 445–478CrossRefADSMathSciNetzbMATHGoogle Scholar
  15. 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, 637–645CrossRefADSGoogle Scholar
  16. 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, 317–338ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, F., Ming, P.: Crystal stability and instability. In preparationGoogle Scholar
  18. 18.
    Maradudin A.A., Vosko S.H. (1968) Symmetry properties of the normal vibrations of a crystal. Rev. Modern Phys. 40, 1–37CrossRefADSGoogle Scholar
  19. 19.
    Ming, P.: Crystal stability with traction boundary condition. In preparationGoogle Scholar
  20. 20.
    Stakgold I. (1950) The Cauchy relations in a molecular theory of elasticity. Quart. Appl. Math. 8, 169–186MathSciNetzbMATHGoogle Scholar
  21. 21.
    Stillinger F.H., Weber T.A. (1985) Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 5262–5271CrossRefADSGoogle Scholar
  22. 22.
    Strang G. (1964) Accurate partial difference methods. II: Non-linear problems. Numer. Math. 6, 37–46CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Tersoff J. (1988) Empirical interatomistic potential for carbon, with applications to amorphous carbon. Phys. Rev. Lett. 61, 2879–2882CrossRefADSGoogle Scholar
  24. 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, 1996Google Scholar
  25. 25.
    Valent, T.:Boundary Value Problems of Finite Elasticity. Springer-Verlag, 1988Google Scholar
  26. 26.
    Wallace D.C. (1972) Thermodynamics of Crystals. John Wiley & Sons Inc., New YorkGoogle Scholar
  27. 27.
    Weiner J.H. (1983) Statistical Mechanics of Elasticity. John Wiley & Sons Inc., New YorkzbMATHGoogle Scholar
  28. 28.
    Xiang, Y., Ming, P., Wei, H., E, W.: A generalized Peierls-Nabarro model for curved dislocations. In preparationGoogle Scholar
  29. 29.
    Xuan, Y., E, W.: Instability of crystalline solids under stress. In preparationGoogle Scholar
  30. 30.
    Yang, J., E, W.: Generalized Cauchy–Born rules for sheets, plates and rods, submitted for publication, 2005Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mathematics and PACMPrinceton UniversityPrincetonUSA
  2. 2.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSSChinese Academy of SciencesBeijingChina

Personalised recommendations