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Archive for Rational Mechanics and Analysis

, Volume 224, Issue 3, pp 907–953 | Cite as

Connecting Atomistic and Continuous Models of Elastodynamics

  • Julian Braun
Article
  • 126 Downloads

Abstract

We prove the long-time existence of solutions for the equations of atomistic elastodynamics on a bounded domain with time-dependent boundary values as well as their convergence to a solution of continuum nonlinear elastodynamics as the interatomic distances tend to zero. Here, the continuum energy density is given by the Cauchy–Born rule. The models considered allow for general finite range interactions. To control the stability of large deformations we also prove a new atomistic Gårding inequality.

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References

  1. 1.
    Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17, 35–92 (1964). doi: 10.1002/cpa.3160170104 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blanc X., Le Bris C., Lions P.L.: From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381 (2002). doi: 10.1007/s00205-002-0218-5 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Braun, J.: Connecting atomistic and continuum theories of nonlinear elasticity: rigorous existence and convergence results for the boundary value problems. Ph.D. thesis, Universität Augsburg 2016. To appear in Augsburger Schriften zur Mathematik, Physik und Informatik, Logos Verlag, BerlinGoogle Scholar
  4. 4.
    Braun, J., Schmidt, B.: Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory. Calc. Var. Partial Differ. Equ. 55(125) 2016. doi: 10.1007/s00526-016-1048-x
  5. 5.
    Constantine G.M., Savits T.H.: A multivariate faa di bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996). doi: 10.1090/S0002-9947-96-01501-2 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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). doi: 10.1007/BF00250727 MathSciNetMATHGoogle Scholar
  7. 7.
    E, W., Ming P.: Cauchy–Born rule and the stability of crystalline solids: dynamic problems. Acta Math. Appl. Sin. (English Series) 23, 529–550 (2007). doi: 10.1007/s10255-007-0393
  8. 8.
    E, W., Ming P.: Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007). doi: 10.1007/s00205-006-0031-7
  9. 9.
    Ericksen J.L.: On the Cauchy–Born rule. Math. Mech. Solids 13(3–4), 199–220 (2008). doi: 10.1177/1081286507086898 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin 2001. doi: 10.1007/978-3-642-61798-0
  11. 11.
    Hudson T., Ortner C.: On the stability of Bravais lattices and their Cauchy–Born approximations. ESAIM:M 2(N 46), 81–110 (2012). doi: 10.1051/m2an/2011014 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ortner C., Theil F.: Justification of the Cauchy–Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207, 1025–1073 (2013). doi: 10.1007/s00205-012-0592-6 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Simpson H.C., Spector S.J.: Applications of estimates near the boundary to regularity of solutions in linearized elasticity. SIAM J. Math. Anal. 41(3), 923–935 (2009). doi: 10.1137/080722990 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton 1970Google Scholar
  15. 15.
    Strauss W.A.: On continuity of functions with values in various banach spaces. Pac. J. Math. 19(3), 543–551 (1966). doi: 10.2140/pjm.1966.19.543 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Theil F.: Surface energies in a two-dimensional mass-spring model for crystals. ESAIM Math. Model. Numer. Anal. 45, 873–899 (2011). doi: 10.1051/m2an/2010106 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Valent, T.: Boundary Value Problems of Finite Elasticity. Springer, Berlin 1988. doi: 10.1007/978-1-4612-3736-5

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematics Institute, Zeeman BuildingUniversity of WarwickCoventryUK

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