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On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime
1. | Universität Augsburg, Institut für Mathematik, Universitätsstr. 14, 86159 Augsburg, Germany, Germany |
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. |
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. |
[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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