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

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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

Abstract
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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 and 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.
[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.

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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.
[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.

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