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June  2015, 10(2): 321-342. doi: 10.3934/nhm.2015.10.321

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

Manuel Friedrich 1, and  Bernd Schmidt 1,

1. 

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

Received  March 2014 Revised  October 2014 Published  April 2015

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.
Keywords: Brittle materials, free discontinuity problems., atomistic models, variational fracture, discrete-to-continuum limits.
Mathematics Subject Classification: Primary: 74R10, 49J45, 70G7.
Citation: Manuel Friedrich, Bernd Schmidt. On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime. Networks & Heterogeneous Media, 2015, 10 (2) : 321-342. doi: 10.3934/nhm.2015.10.321
References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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