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An Atomistic-to-Continuum Analysis of Crystal Cleavage in a Two-Dimensional Model Problem

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Abstract

A two-dimensional atomic mass spring system is investigated for critical fracture loads and its crack path geometry. We rigorously prove that, in the discrete-to-continuum limit, the minimal energy of a crystal under uniaxial tension leads to a universal cleavage law and energy minimizers are either homogeneous elastic deformations or configurations that are completely cracked and do not store elastic energy. Beyond critical loading, the specimen generically cleaves along a unique optimal crystallographic hyperplane. For specific symmetric crystal orientations, however, cleavage might fail. In this case a complete characterization of possible limiting crack geometries is obtained.

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References

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

    MATH  MathSciNet  Google Scholar 

  • Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Boll. Unione Mat. Ital, B 7, 105–123 (1992)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Bourdin, B., Francfort, G.A., Marigo, J.J.: Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797–826 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Braides, A.: Non-local variational limits of discrete systems. Commun. Contemp. Math. 2, 285–297 (2000)

    MATH  MathSciNet  Google Scholar 

  • Braides, A., Cicalese, M.: Surface energies in nonconvex discrete systems. Math. Models Methods Appl. Sci. 17, 985–1037 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Braides, A., Gelli, M.S.: Limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7, 41–66 (2002a)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Braides, A., Lew, A., Ortiz, M.: Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180, 151–182 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Braides, A., Solci, M., Vitali, E.: A derivation of linear elastic energies from pair-interaction atomistic systems. Netw. Heterog. Media 2, 551–567 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Buttazzo, G.: Energies on BV and variational models in fracture mechanics. In: Proceedings of Curvature Flows and Related Topics, Levico, 27 June–2 July 1994. Gakkotosho, Tokyo (1995)

    Google Scholar 

  • Conti, S., Dolzmann, G., Müller, S.: Korn’s second inequality and geometric rigidity with mixed growth conditions (2012). arXiv:1203.1138

  • Dal Maso, G., Toader, R.: A model for quasi-static growth of brittle materials: existence and approximation results. Arch. Ration. Mech. Anal. 162, 101–135 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as Γ-limit of finite elasticity. Set-Valued Anal. 10, 165–183 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Dal Maso, G., Francfort, G.A., Toader, R.: Quasistatic crack growth in nonlinear elasticity. Arch. Ration. Mech. Anal. 176, 165–225 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • De Giorgi, E., Ambrosio, L.: Un nuovo funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 82, 199–210 (1988)

    MATH  Google Scholar 

  • Francfort, G.A., Larsen, C.J.: Existence and convergence for quasi-static evolution in brittle fracture. Commun. Pure Appl. Math. 56, 1465–1500 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Francfort, G.A., Marigo, J.J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Friedrich, M., Schmidt, B.: On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime (2013, in progress)

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

    Article  MATH  Google Scholar 

  • Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. 221, 163–198 (1921)

    Article  Google Scholar 

  • Negri, M.: Finite element approximation of the Griffith’s model in fracture mechanics. Numer. Math. 95, 653–687 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Negri, M.: A discontinuous finite element approximation of free discontinuity problems. Adv. Math. Sci. Appl. 15, 283–306 (2005)

    MATH  MathSciNet  Google Scholar 

  • Schmidt, B.: Linear Γ-limits of multiwell energies in nonlinear elasticity theory. Contin. Mech. Thermodyn. 20, 375–396 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Schmidt, B.: On the derivation of linear elasticity from atomistic models. Netw. Heterog. Media 4, 789–812 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Schmidt, B., Fraternali, F., Ortiz, M.: Eigenfracture: an eigendeformation approach to variational fracture. SIAM Multiscale Model. Simul. 7, 1237–1266 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159, Augsburg, Germany

    Manuel Friedrich & Bernd Schmidt

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  1. Manuel Friedrich
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  2. Bernd Schmidt
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Correspondence to Manuel Friedrich.

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Communicated by Antonio DeSimone.

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Friedrich, M., Schmidt, B. An Atomistic-to-Continuum Analysis of Crystal Cleavage in a Two-Dimensional Model Problem. J Nonlinear Sci 24, 145–183 (2014). https://doi.org/10.1007/s00332-013-9187-0

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  • DOI: https://doi.org/10.1007/s00332-013-9187-0

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