zbMATH — the first resource for mathematics

Cauchy-Born rule and the stability of crystalline solids: Static problems. (English) Zbl 1106.74019
Summary: We study the connection between atomistic and continuum models for elastic deformation of crystalline solids at zero temperature. We prove, under certain sharp stability conditions, that the correct nonlinear elasticity model can be given by classical Cauchy-Born rule in the sense that elastically deformed states of atomistic model are closely approximated by solutions of continuum model with stored energy functionals obtained from 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 in crystalline solids.

74E15 Crystalline structure
74A25 Molecular, statistical, and kinetic theories in solid mechanics
82D25 Statistical mechanical studies of crystals
Full Text: DOI
[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 · Zbl 0123.28706 · doi:10.1002/cpa.3160170104
[3] Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Saunders College Publishing, 1976 · Zbl 1107.82300
[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 · Zbl 0758.73009 · doi:10.1098/rsta.1992.0013
[5] Blanc X., Le Bris C., Lions P.-L. (2002) From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381 · Zbl 1028.74005 · doi:10.1007/s00205-002-0218-5
[6] Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, 1954 · Zbl 0057.44601
[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 · Zbl 0945.74006 · doi:10.1007/s002050050135
[8] Daw M.S., Baskes M.I. (1983) Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals. Phys. Rev. Lett. 50, 1285–1288 · doi:10.1103/PhysRevLett.50.1285
[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 · doi:10.1103/PhysRevB.29.6443
[10] E W., Ming P. (2004) Analysis of multiscale methods, J. Comput. Math. 22, 210–219 · Zbl 1046.65108
[11] E W., Ming, P.: Cauchy–Born rule and the stability of crystalline solids: Dynamical problems. In preparation · Zbl 1106.74019
[12] Engel P. (1986) Geometric Crystallography: An Axiomatic Introduction to Crystallography. D. Reidel Publishing Company, Dordrecht, Holland · Zbl 0659.51001
[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 · Zbl 0567.73112
[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 · Zbl 1084.74501 · doi:10.1007/s00332-002-0495-z
[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 · doi:10.1103/PhysRev.145.637
[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 · Zbl 0020.32703 · doi:10.1098/rspa.1939.0002
[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 · doi:10.1103/RevModPhys.40.1
[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 · Zbl 0037.42901
[21] Stillinger F.H., Weber T.A. (1985) Computer simulation of local order in condensed phases of silicon. Phys. Rev. B 31, 5262–5271 · doi:10.1103/PhysRevB.31.5262
[22] Strang G. (1964) Accurate partial difference methods. II: Non-linear problems. Numer. Math. 6, 37–46 · Zbl 0143.38204 · doi:10.1007/BF01386051
[23] Tersoff J. (1988) Empirical interatomistic potential for carbon, with applications to amorphous carbon. Phys. Rev. Lett. 61, 2879–2882 · doi:10.1103/PhysRevLett.61.2879
[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 · Zbl 0648.73019
[26] Wallace D.C. (1972) Thermodynamics of Crystals. John Wiley & Sons Inc., New York · Zbl 0242.15005
[27] Weiner J.H. (1983) Statistical Mechanics of Elasticity. John Wiley & Sons Inc., New York · Zbl 0616.73034
[28] Xiang, Y., Ming, P., Wei, H., E, W.: A generalized Peierls-Nabarro model for curved dislocations. In preparation · Zbl 1364.74027
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.