# American Institute of Mathematical Sciences

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
 [1] Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427 [2] Grigor Nika, Bogdan Vernescu. Rate of convergence for a multi-scale model of dilute emulsions with non-uniform surface tension. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1553-1564. doi: 10.3934/dcdss.2016062 [3] Jean-Philippe Bernard, Emmanuel Frénod, Antoine Rousseau. Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects. Conference Publications, 2013, 2013 (special) : 69-76. doi: 10.3934/proc.2013.2013.69 [4] Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200 [5] Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Multi-scale model of bladder cancer development. Conference Publications, 2011, 2011 (Special) : 803-812. doi: 10.3934/proc.2011.2011.803 [6] Markus Gahn. Multi-scale modeling of processes in porous media - coupling reaction-diffusion processes in the solid and the fluid phase and on the separating interfaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6511-6531. doi: 10.3934/dcdsb.2019151 [7] Thomas Blanc, Mihaï Bostan. Multi-scale analysis for highly anisotropic parabolic problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 335-399. doi: 10.3934/dcdsb.2019186 [8] Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013 [9] Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449 [10] Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135 [11] Gianni Dal Maso, Flaviana Iurlano. Fracture models as $\Gamma$-limits of damage models. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1657-1686. doi: 10.3934/cpaa.2013.12.1657 [12] Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451 [13] Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501 [14] Roberto Alicandro, Giuliano Lazzaroni, Mariapia Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Networks & Heterogeneous Media, 2018, 13 (1) : 1-26. doi: 10.3934/nhm.2018001 [15] Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259 [16] Gianni Dal Maso. Ennio De Giorgi and $\mathbf\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1017-1021. doi: 10.3934/dcds.2011.31.1017 [17] Alexander Mielke. Deriving amplitude equations via evolutionary $\Gamma$-convergence. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2679-2700. doi: 10.3934/dcds.2015.35.2679 [18] Fujun Zhou, Junde Wu, Shangbin Cui. Existence and asymptotic behavior of solutions to a moving boundary problem modeling the growth of multi-layer tumors. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1669-1688. doi: 10.3934/cpaa.2009.8.1669 [19] Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 [20] Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1

2019 Impact Factor: 1.27