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June  2013, 8(2): 501-527. doi: 10.3934/nhm.2013.8.501

Gamma-expansion for a 1D confined Lennard-Jones model with point defect

Thomas Hudson 1,

1. 

Mathematical Institute, 24-29 St Giles', Oxford, OX1 3LB, United Kingdom

Received  April 2012 Revised  April 2013 Published  May 2013

We compute a rigorous asymptotic expansion of the energy of a point defect in a 1D chain of atoms with second neighbour interactions. We propose the Confined Lennard-Jones model for interatomic interactions, where it is assumed that nearest neighbour potentials are globally convex and second neighbour potentials are globally concave. We derive the $\Gamma$-limit for the energy functional as the number of atoms per period tends to infinity and derive an explicit form for the first order term in a $\Gamma$-expansion in terms of an infinite cell problem. We prove exponential decay properties for minimisers of the energy in the infinite cell problem, suggesting that the perturbation to the deformation introduced by the defect is confined to a thin boundary layer.
Keywords: Energy asymptotics, discrete to continuum limits., gamma convergence, point defect.
Mathematics Subject Classification: Primary: 49J45; Secondary: 70C20, 74B20, 74G65, 74Q1.
Citation: Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect. Networks and Heterogeneous Media, 2013, 8 (2) : 501-527. doi: 10.3934/nhm.2013.8.501
References:
[1]

R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM Journal of Mathematical Analysis, 36 (2004), 1-37. doi: 10.1137/S0036141003426471.

[2]

G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence, Applied Mathematics and Optimization, 27 (1993), 105-123. doi: 10.1007/BF01195977.

[3]

X. Blanc, C. Le Bris and P.-L. Lions, Atomistic to continuum limits for computational materials science, M2AN. Mathematical Modelling and Numerical Analysis, 41 (2007), 391-426. doi: 10.1051/m2an:2007018.

[4]

A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[5]

A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Mathematical Models and Methods in Applied Science, 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, Archive for Rational Mechanics and Analysis, 146 (1999), 23-58. doi: 10.1007/s002050050135.

[7]

A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses, Mathematics and Mechanics of Solids, 7 (2002), 41-66. doi: 10.1177/1081286502007001229.

[8]

A. Braides and L. Truskinovsky, Asymptotic expansions by $\Gamma$-convergence, Continuum Mechanics and Thermodynamics, 20 (2008), 21-62. doi: 10.1007/s00161-008-0072-2.

[9]

B. Dacorogna, "Direct Methods in the Calculus of Variations," $2^{nd}$ edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.

[10]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[11]

W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Archive for Rational Mechanics and Analysis, 183 (2007), 241-297. doi: 10.1007/s00205-006-0031-7.

[12]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[13]

C. Ortner and F. Theil, Nonlinear elasticity from atomistic mechanics, to appear in Archive for Rational Mechanics and Analysis, arXiv:1202.3858.

[14]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems, Mathematical Models and Methods in Applied Science, 21 (2011), 777-817. doi: 10.1142/S0218202511005210.

[15]

B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Modeling & Simulation, 5 (2006), 664-694. doi: 10.1137/050646251.

[16]

E. Süli and D. F. Mayers, "An Introduction to Numerical Analysis," Cambridge University Press, Cambridge, 2003.

[17]

G. Zanzotto, The Cauchy-Born hypothesis, nonlinear elasticity and mechanical twinning in crystals, Acta Crystallographica Section A, 52 (1996), 839-849. doi: 10.1107/S0108767396006654.

show all references

References:
[1]

R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM Journal of Mathematical Analysis, 36 (2004), 1-37. doi: 10.1137/S0036141003426471.

[2]

G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence, Applied Mathematics and Optimization, 27 (1993), 105-123. doi: 10.1007/BF01195977.

[3]

X. Blanc, C. Le Bris and P.-L. Lions, Atomistic to continuum limits for computational materials science, M2AN. Mathematical Modelling and Numerical Analysis, 41 (2007), 391-426. doi: 10.1051/m2an:2007018.

[4]

A. Braides, "$\Gamma$-Convergence for Beginners," Oxford Lecture Series in Mathematics and its Applications, 22, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[5]

A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems, Mathematical Models and Methods in Applied Science, 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, Archive for Rational Mechanics and Analysis, 146 (1999), 23-58. doi: 10.1007/s002050050135.

[7]

A. Braides and M. S. Gelli, Continuum limits of discrete systems without convexity hypotheses, Mathematics and Mechanics of Solids, 7 (2002), 41-66. doi: 10.1177/1081286502007001229.

[8]

A. Braides and L. Truskinovsky, Asymptotic expansions by $\Gamma$-convergence, Continuum Mechanics and Thermodynamics, 20 (2008), 21-62. doi: 10.1007/s00161-008-0072-2.

[9]

B. Dacorogna, "Direct Methods in the Calculus of Variations," $2^{nd}$ edition, Applied Mathematical Sciences, 78, Springer, New York, 2008.

[10]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[11]

W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Archive for Rational Mechanics and Analysis, 183 (2007), 241-297. doi: 10.1007/s00205-006-0031-7.

[12]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[13]

C. Ortner and F. Theil, Nonlinear elasticity from atomistic mechanics, to appear in Archive for Rational Mechanics and Analysis, arXiv:1202.3858.

[14]

L. Scardia, A. Schlömerkemper and C. Zanini, Boundary layer energies for nonconvex discrete systems, Mathematical Models and Methods in Applied Science, 21 (2011), 777-817. doi: 10.1142/S0218202511005210.

[15]

B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Modeling & Simulation, 5 (2006), 664-694. doi: 10.1137/050646251.

[16]

E. Süli and D. F. Mayers, "An Introduction to Numerical Analysis," Cambridge University Press, Cambridge, 2003.

[17]

G. Zanzotto, The Cauchy-Born hypothesis, nonlinear elasticity and mechanical twinning in crystals, Acta Crystallographica Section A, 52 (1996), 839-849. doi: 10.1107/S0108767396006654.

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