\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

Abstract Related Papers Cited by
  • We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of $\Gamma$-convergence. We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line. As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets.
    Mathematics Subject Classification: Primary: 74R10, 49J45, 70G75.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    G. Alberti and C. Mantegazza, A note on the theory of $SBV$ functions, Boll. Un. Mat. Ital. B (7), 11 (1989), 375-382.

    [2]

    R. Alicandro, M. Focardi and M. S. Gelli, Finite-difference approximation of energies in fracture mechanics, Ann. Scuola Norm. Sup., 29 (2000), 671-709.

    [3]

    L. Ambrosio, A compactness theorem for a special class of functions of bounded variation, Boll. Un. Mat. Ital. B (7), 3 (1989), 857-881.

    [4]

    L. Ambrosio, Existence theory for a new class of variational problems, Arch. Ration. Mech. Anal., 111 (1990), 291-322.doi: 10.1007/BF00376024.

    [5]

    L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford 2000.

    [6]

    A. Braides, $\Gamma$-convergence for Beginners, Oxford University Press, Oxford 2002.doi: 10.1093/acprof:oso/9780198507840.001.0001.

    [7]

    A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Math. Models Methods Appl. Sci., 17 (2007), 985-1037.doi: 10.1142/S0218202507002182.

    [8]

    A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case, Arch. Ration. Mech. Anal., 146 (1999), 23-58.doi: 10.1007/s002050050135.

    [9]

    A. Braides and M. S. Gelli, Limits of discrete systems without convexity hypotheses, Math. Mech. Solids, 7 (2002), 41-66.doi: 10.1177/1081286502007001229.

    [10]

    A. Braides and M. S. Gelli, Limits of discrete systems with long-range interactions, J. Convex Anal., 9 (2002), 363-399.

    [11]

    A. Braides, A. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes, Arch. Ration. Mech. Anal., 180 (2006), 151-182.doi: 10.1007/s00205-005-0399-9.

    [12]

    A. Braides, M. Solci and E. Vitali, A derivation of linear elastic energies from pair-interaction atomistic systems, Netw. Heterog. Media, 2 (2007), 551-567.doi: 10.3934/nhm.2007.2.551.

    [13]

    A. Chambolle, A. Giacomini and M. Ponsiglione, Piecewise rigidity, J. Funct. Anal. Solids, 244 (2007), 134-153.doi: 10.1016/j.jfa.2006.11.006.

    [14]

    S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to $SO(n)$, J. Eur Math. Soc., (JEMS) 8 (2006), 515-530.doi: 10.4171/JEMS/65.

    [15]

    G. Cortesani and R. Toader, A density result in SBV with respect to non-isotropic energies, Nonlinear Analysis, 38 (1999), 585-604.doi: 10.1016/S0362-546X(98)00132-1.

    [16]

    G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston $\cdot$ Basel $\cdot$ Berlin 1993.doi: 10.1007/978-1-4612-0327-8.

    [17]

    E. De Giorgi and L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni, Acc. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur., 82 (1988), 199-210.

    [18]

    H. Federer, Geometric Measure Theory, Springer, New York, 1969.

    [19]

    M. Focardi and M. S. Gelli, Approximation results by difference schemes of fracture energies: The vectorial case, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 469-495.doi: 10.1007/s00030-003-1002-4.

    [20]

    G. A. Francfort and J. J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342.doi: 10.1016/S0022-5096(98)00034-9.

    [21]

    M. Friedrich and B. Schmidt, An atomistic-to-continuum analysis of crystal cleavage in a two-dimensional model problem, J. Nonlin. Sci., 24 (2014), 145-183.doi: 10.1007/s00332-013-9187-0.

    [22]

    G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.doi: 10.1002/cpa.10048.

    [23]

    G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002), 445-478.doi: 10.1007/s00332-002-0495-z.

    [24]

    A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures, Calc. Var. Partial Differential Equations, 22 (2005), 129-172.doi: 10.1007/s00526-004-0269-6.

    [25]

    C. Mora-Corral, Explicit energy-minimizers of incompressible elastic brittle bars under uniaxial extension, C. R. Acad. Sci. Paris, 348 (2010), 1045-1048.doi: 10.1016/j.crma.2010.09.005.

    [26]

    M. Negri, Finite element approximation of the Griffith's model in fracture mechanics, Numer. Math., 95 (2003), 653-687.doi: 10.1007/s00211-003-0456-y.

    [27]

    B. Schmidt, On the derivation of linear elasticity from atomistic models, Netw. Heterog. Media, 4 (2009), 789-812.doi: 10.3934/nhm.2009.4.789.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(76) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return