March  2012, 17(2): 661-686. doi: 10.3934/dcdsb.2012.17.661

Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems

1. 

Eindhoven University of Technology, PO Box 513, WH 4.10, 5600 MB Eindhoven, Netherlands

2. 

Institute for Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany

3. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received  August 2010 Revised  August 2011 Published  December 2011

In this paper we consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We prescribe the positions in the deformed configuration of the first two and the last two atoms of the chain.
    We are interested in a good approximation of the discrete energy of this system for a large number of atoms, i.e., in the continuum limit.
    We show that the canonical expansion by $\Gamma$-convergence does not provide an accurate approximation of the discrete energy if the boundary conditions for the deformation are close to the threshold between elastic and fracture regimes. This suggests that a uniformly $\Gamma$-equivalent approximation of the energy should be made, as introduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard $\Gamma$-expansion.
    In this spirit we provide a uniformly $\Gamma$-equivalent approximation of the discrete energy at first order, which arises as the $\Gamma$-limit of a suitably scaled functional.
Citation: Lucia Scardia, Anja Schlömerkemper, Chiara Zanini. Towards uniformly $\Gamma$-equivalent theories for nonconvex discrete systems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 661-686. doi: 10.3934/dcdsb.2012.17.661
References:
[1]

G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence,, Appl. Math. Optim., 27 (1993), 105.  doi: 10.1007/BF01195977.  Google Scholar

[2]

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

[3]

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

[4]

A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses,, Math. Mech. Solids, 7 (2002), 41.  doi: 10.1177/1081286502007001229.  Google Scholar

[5]

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

[6]

A. Braides, "$\Gamma$-Convergence for Beginners,'', Oxford Lecture Series in Mathematics and its Applications, 22 (2002).   Google Scholar

[7]

A. Braides and L. Truskinovsky, Asymptotic expansions by $\Gamma$-convergence,, Cont. Mech. Thermodyn., 20 (2008), 21.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[8]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993).   Google Scholar

[9]

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.  doi: 10.1007/s00030-003-1002-4.  Google Scholar

[10]

O. Nguyen and M. Ortiz, Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior,, J. Mech. Phys. Solids, 50 (2002), 1727.  doi: 10.1016/S0022-5096(01)00133-8.  Google Scholar

[11]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems,, Math. Models Methods Appl. Sci., 21 (2011), 777.  doi: 10.1142/S0218202511005210.  Google Scholar

[12]

V. B. Shenoy, R. Miller, E. B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum models of interfacial structure and deformation,, Phys. Rev. Lett., 80 (1998), 742.  doi: 10.1103/PhysRevLett.80.742.  Google Scholar

[13]

L. Truskinovsky, Fracture as a phase transition,, in, (1996), 322.   Google Scholar

show all references

References:
[1]

G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence,, Appl. Math. Optim., 27 (1993), 105.  doi: 10.1007/BF01195977.  Google Scholar

[2]

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

[3]

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

[4]

A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses,, Math. Mech. Solids, 7 (2002), 41.  doi: 10.1177/1081286502007001229.  Google Scholar

[5]

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

[6]

A. Braides, "$\Gamma$-Convergence for Beginners,'', Oxford Lecture Series in Mathematics and its Applications, 22 (2002).   Google Scholar

[7]

A. Braides and L. Truskinovsky, Asymptotic expansions by $\Gamma$-convergence,, Cont. Mech. Thermodyn., 20 (2008), 21.  doi: 10.1007/s00161-008-0072-2.  Google Scholar

[8]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,'', Progress in Nonlinear Differential Equations and their Applications, 8 (1993).   Google Scholar

[9]

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.  doi: 10.1007/s00030-003-1002-4.  Google Scholar

[10]

O. Nguyen and M. Ortiz, Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior,, J. Mech. Phys. Solids, 50 (2002), 1727.  doi: 10.1016/S0022-5096(01)00133-8.  Google Scholar

[11]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems,, Math. Models Methods Appl. Sci., 21 (2011), 777.  doi: 10.1142/S0218202511005210.  Google Scholar

[12]

V. B. Shenoy, R. Miller, E. B. Tadmor, R. Phillips and M. Ortiz, Quasicontinuum models of interfacial structure and deformation,, Phys. Rev. Lett., 80 (1998), 742.  doi: 10.1103/PhysRevLett.80.742.  Google Scholar

[13]

L. Truskinovsky, Fracture as a phase transition,, in, (1996), 322.   Google Scholar

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