• Previous Article
    Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation
  • NHM Home
  • This Issue
  • Next Article
    The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points
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

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.
Citation: Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect. Networks & 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. doi: 10.1137/S0036141003426471.

[2]

G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence,, Applied Mathematics and Optimization, 27 (1993), 105. 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. doi: 10.1051/m2an:2007018.

[4]

A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications, 22 (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. 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. 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. doi: 10.1177/1081286502007001229.

[8]

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

[9]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", $2^{nd}$ edition, 78 (2008).

[10]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Progress in Nonlinear Differential Equations and their Applications, 8 (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. 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, (1992).

[13]

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

[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. doi: 10.1142/S0218202511005210.

[15]

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

[16]

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

[17]

G. Zanzotto, The Cauchy-Born hypothesis, nonlinear elasticity and mechanical twinning in crystals,, Acta Crystallographica Section A, 52 (1996), 839. 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. doi: 10.1137/S0036141003426471.

[2]

G. Anzellotti and S. Baldo, Asymptotic development by $\Gamma$-convergence,, Applied Mathematics and Optimization, 27 (1993), 105. 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. doi: 10.1051/m2an:2007018.

[4]

A. Braides, "$\Gamma$-Convergence for Beginners,", Oxford Lecture Series in Mathematics and its Applications, 22 (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. 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. 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. doi: 10.1177/1081286502007001229.

[8]

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

[9]

B. Dacorogna, "Direct Methods in the Calculus of Variations,", $2^{nd}$ edition, 78 (2008).

[10]

G. Dal Maso, "An Introduction to $\Gamma$-Convergence,", Progress in Nonlinear Differential Equations and their Applications, 8 (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. 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, (1992).

[13]

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

[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. doi: 10.1142/S0218202511005210.

[15]

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

[16]

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

[17]

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

[1]

Marco Cicalese, Antonio DeSimone, Caterina Ida Zeppieri. Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers. Networks & Heterogeneous Media, 2009, 4 (4) : 667-708. doi: 10.3934/nhm.2009.4.667

[2]

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

[3]

Manuel Friedrich, Bernd Schmidt. On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime. Networks & Heterogeneous Media, 2015, 10 (2) : 321-342. doi: 10.3934/nhm.2015.10.321

[4]

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

[5]

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

[6]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060

[7]

Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings. Networks & Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111

[8]

Phoebus Rosakis. Continuum surface energy from a lattice model. Networks & Heterogeneous Media, 2014, 9 (3) : 453-476. doi: 10.3934/nhm.2014.9.453

[9]

Gabriella Bretti, Ciro D’Apice, Rosanna Manzo, Benedetto Piccoli. A continuum-discrete model for supply chains dynamics. Networks & Heterogeneous Media, 2007, 2 (4) : 661-694. doi: 10.3934/nhm.2007.2.661

[10]

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

[11]

Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure & Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963

[12]

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

[13]

Zhiyuan Geng, Wei Wang, Pingwen Zhang, Zhifei Zhang. Stability of half-degree point defect profiles for 2-D nematic liquid crystal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6227-6242. doi: 10.3934/dcds.2017269

[14]

Giuseppe Marino, Hong-Kun Xu. Convergence of generalized proximal point algorithms. Communications on Pure & Applied Analysis, 2004, 3 (4) : 791-808. doi: 10.3934/cpaa.2004.3.791

[15]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

[16]

Yang Xiang, Xiaodong Yan. Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2989-3021. doi: 10.3934/dcdsb.2017183

[17]

Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014

[18]

Robert I. McLachlan, G. R. W. Quispel. Discrete gradient methods have an energy conservation law. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1099-1104. doi: 10.3934/dcds.2014.34.1099

[19]

Dmitry Jakobson and Alexander Strohmaier. High-energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows. Electronic Research Announcements, 2006, 12: 87-94.

[20]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 0 (0) : 1-19. doi: 10.3934/jdg.2019016

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]