# American Institute of Mathematical Sciences

March 2018, 13(1): 95-118. doi: 10.3934/nhm.2018005

## On Lennard-Jones systems with finite range interactions and their asymptotic analysis

 1 TU Dresden, Department of Mathematics, Zellescher Weg 12-14, 01069 Dresden, Germany 2 University of Würzburg, Institute of Mathematics, Emil-Fischer-Straẞe 40, 97074 Würzburg, Germany

Received  June 2017 Revised  October 2017 Published  March 2018

The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of $Γ$-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density.

Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions.

A crucial ingredient to our proofs is a novel decomposition of the energy that allows for refined estimates.

Citation: Mathias Schäffner, Anja Schlömerkemper. On Lennard-Jones systems with finite range interactions and their asymptotic analysis. Networks & Heterogeneous Media, 2018, 13 (1) : 95-118. doi: 10.3934/nhm.2018005
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Univ. Press, 2000. [2] J. M. Ball, Some open problems in elasticity, In Geometry, Mechanics and Dynamics, Springer, New York, 2002, 3-59 [3] X. Blanc, C. LeBris and P. L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), 341-381. doi: 10.1007/s00205-002-0218-5. [4] A. Braides, $Γ$-Convergence for Beginners, Oxford Univ. Press, 2002. [5] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Math. Models Methods Appl. Sci., 17 (2007), 985-1037. doi: 10.1142/S0218202507002182. [6] 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), 23-58. doi: 10.1007/s002050050135. [7] A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids, 7 (2002), 41-66. doi: 10.1177/1081286502007001229. [8] A. Braides and M. S. Gelli, The passage from discrete to continuous variational problems: A nonlinear homogenization process, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, 170 (2004), 45-63. [9] A. Braides and M. S. Gelli, From discrete systems to continuous variational problems: An introduction, Lect. Notes Unione Mat. Ital., 2 (2006), 3-77. [10] A. Braides and M. S. Gelli, Asymptotic analysis of microscopic impenetrability constraints for atomistic systems, J. Mech. Phys. Solids, 96 (2016), 235-251. doi: 10.1016/j.jmps.2016.07.016. [11] A. Braides, M. S. Gelli and M. Sigalotti, The passage from nonconvex discrete systems to variational problems in Sobolev spaces: The one-dimensional case, Proc. Steklov Inst. Math., 236 (2002), 395-414. [12] A. Braides, A. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes, Arch. Rational Mech. Anal., 180 (2006), 151-182. doi: 10.1007/s00205-005-0399-9. [13] A. Braides and M. Solci, Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case, Math. Mech. Solids, 21 (2016), 915-930. doi: 10.1177/1081286514544780. [14] A. Braides and L. Truskinovsky, Asymptotic expansions by $Γ$-convergence, Cont. Mech. Thermodyn., 20 (2008), 21-62. doi: 10.1007/s00161-008-0072-2. [15] M. Friedrich and B. Schmidt, An atomistic-to-continuum analysis of crystal cleavage in a two-dimensional model problem, J. Nonlinear Sci., 24 (2014), 145-183. doi: 10.1007/s00332-013-9187-0. [16] M. Friedrich and B. Schmidt, An analysis of crystal cleavage in the passage from atomistic models to continuum theory, Arch. Ration. Mech. Anal., 217 (2015), 263-308. doi: 10.1007/s00205-014-0833-y. [17] M. G. D. Geers, R. H. J. Peerlings, M. A. Peletier and L. Scardia, Asymptotic behaviour of a pile-up of infinite walls of edge dislocations, Arch. Ration. Mech. Anal., 209 (2013), 495-539. doi: 10.1007/s00205-013-0635-7. [18] M. Luskin and C. Ortner, Atomistic-to-continuum coupling, Acta Numer., 22 (2013), 397-508. doi: 10.1017/S0962492913000068. [19] L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems, Math. Models Methods Appl. Sci., 21 (2011), 777-817. doi: 10.1142/S0218202511005210. [20] L. Scardia, A. Schlömerkemper and C. Zanini, Towards uniformly $Γ$-equivalent theories for nonconvex discrete systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 661-686. [21] M. Schäffner and A. Schlömerkemper, On a $Γ$-convergence analysis of a quasicontinuum method, Multiscale Model. Simul., 13 (2015), 132-172. doi: 10.1137/140971439. [22] M. Schäffner, Multiscale analysis of non-convex discrete systems via $Γ$-convergence, Ph. D thesis, University of Würzburg, 2015. [23] E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Phil. Mag. A, 73 (2006), 1529-1563. doi: 10.1080/01418619608243000. [24] L. Truskinovsky, Fracture as a phase transition, Contemporary Research in the Mechanics and Mathematics of Marterials, (1996), 322-332.

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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Univ. Press, 2000. [2] J. M. Ball, Some open problems in elasticity, In Geometry, Mechanics and Dynamics, Springer, New York, 2002, 3-59 [3] X. Blanc, C. LeBris and P. L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), 341-381. doi: 10.1007/s00205-002-0218-5. [4] A. Braides, $Γ$-Convergence for Beginners, Oxford Univ. Press, 2002. [5] A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Math. Models Methods Appl. Sci., 17 (2007), 985-1037. doi: 10.1142/S0218202507002182. [6] 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), 23-58. doi: 10.1007/s002050050135. [7] A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses, Math. Mech. Solids, 7 (2002), 41-66. doi: 10.1177/1081286502007001229. [8] A. Braides and M. S. Gelli, The passage from discrete to continuous variational problems: A nonlinear homogenization process, in Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publ., Dordrecht, 170 (2004), 45-63. [9] A. Braides and M. S. Gelli, From discrete systems to continuous variational problems: An introduction, Lect. Notes Unione Mat. Ital., 2 (2006), 3-77. [10] A. Braides and M. S. Gelli, Asymptotic analysis of microscopic impenetrability constraints for atomistic systems, J. Mech. Phys. Solids, 96 (2016), 235-251. doi: 10.1016/j.jmps.2016.07.016. [11] A. Braides, M. S. Gelli and M. Sigalotti, The passage from nonconvex discrete systems to variational problems in Sobolev spaces: The one-dimensional case, Proc. Steklov Inst. Math., 236 (2002), 395-414. [12] A. Braides, A. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes, Arch. Rational Mech. Anal., 180 (2006), 151-182. doi: 10.1007/s00205-005-0399-9. [13] A. Braides and M. Solci, Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case, Math. Mech. Solids, 21 (2016), 915-930. doi: 10.1177/1081286514544780. [14] A. Braides and L. Truskinovsky, Asymptotic expansions by $Γ$-convergence, Cont. Mech. Thermodyn., 20 (2008), 21-62. doi: 10.1007/s00161-008-0072-2. [15] M. Friedrich and B. Schmidt, An atomistic-to-continuum analysis of crystal cleavage in a two-dimensional model problem, J. Nonlinear Sci., 24 (2014), 145-183. doi: 10.1007/s00332-013-9187-0. [16] M. Friedrich and B. Schmidt, An analysis of crystal cleavage in the passage from atomistic models to continuum theory, Arch. Ration. Mech. Anal., 217 (2015), 263-308. doi: 10.1007/s00205-014-0833-y. [17] M. G. D. Geers, R. H. J. Peerlings, M. A. Peletier and L. Scardia, Asymptotic behaviour of a pile-up of infinite walls of edge dislocations, Arch. Ration. Mech. Anal., 209 (2013), 495-539. doi: 10.1007/s00205-013-0635-7. [18] M. Luskin and C. Ortner, Atomistic-to-continuum coupling, Acta Numer., 22 (2013), 397-508. doi: 10.1017/S0962492913000068. [19] L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems, Math. Models Methods Appl. Sci., 21 (2011), 777-817. doi: 10.1142/S0218202511005210. [20] L. Scardia, A. Schlömerkemper and C. Zanini, Towards uniformly $Γ$-equivalent theories for nonconvex discrete systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 661-686. [21] M. Schäffner and A. Schlömerkemper, On a $Γ$-convergence analysis of a quasicontinuum method, Multiscale Model. Simul., 13 (2015), 132-172. doi: 10.1137/140971439. [22] M. Schäffner, Multiscale analysis of non-convex discrete systems via $Γ$-convergence, Ph. D thesis, University of Würzburg, 2015. [23] E. Tadmor, M. Ortiz and R. Phillips, Quasicontinuum analysis of defects in solids, Phil. Mag. A, 73 (2006), 1529-1563. doi: 10.1080/01418619608243000. [24] L. Truskinovsky, Fracture as a phase transition, Contemporary Research in the Mechanics and Mathematics of Marterials, (1996), 322-332.
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