September  2014, 9(3): 453-476. doi: 10.3934/nhm.2014.9.453

Continuum surface energy from a lattice model

1. 

Department of Applied Mathematics, University of Crete, Heraklion 70013, Greece

Received  January 2014 Revised  June 2014 Published  October 2014

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.
Citation: Phoebus Rosakis. Continuum surface energy from a lattice model. Networks & Heterogeneous Media, 2014, 9 (3) : 453-476. doi: 10.3934/nhm.2014.9.453
References:
[1]

I. Bárány and D. G. Larman, The convex hull of the integer points in a large ball,, Mathematische Annalen, 312 (1998), 167. doi: 10.1007/s002080050217. Google Scholar

[2]

A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra,, New Perspectives in Algebraic Combinatorics, 38 (1999), 91. Google Scholar

[3]

X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics,, Arch. Rational Mech. Anal., 164 (2002), 341. doi: 10.1007/s00205-002-0218-5. Google Scholar

[4]

M. Beck and S. Robins, Computing the Continuous Discretely: Integer Point Enumeration in Polyhedra,, Undergraduate Texts in Mathematics, (2007). Google Scholar

[5]

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

[6]

J. G. Van der Corput, Over Roosterpunten in het Platte vlak (de Beteekenis van de Methoden van Voronoi en Pfeiffer),, Noordhoff, (1919). Google Scholar

[7]

B. Dacorogna and C.-E Pfister, Wulff theorem and best constant in Sobolev inequality,, Journal de mathématiques pures et appliquées, 71 (1992), 97. Google Scholar

[8]

A. Eichler, J. Hafner, J. Furthmüller and G. Kresse, Structural and electronic properties of rhodium surfaces: an ab initio approach,, Surface Science, 346 (1996), 300. doi: 10.1016/0039-6028(95)00906-X. Google Scholar

[9]

I. Fonseca, The Wulff theorem revisited,, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 432 (1991), 125. doi: 10.1098/rspa.1991.0009. Google Scholar

[10]

C. Herring, Some theorems on the free energies of crystal surfaces,, Physical Review, 82 (1951), 87. doi: 10.1103/PhysRev.82.87. Google Scholar

[11]

M. N. Huxley, Exponential sums and lattice points III,, Proceedings of the London Mathematical Society, 87 (2003), 591. doi: 10.1112/S0024611503014485. Google Scholar

[12]

A. Ivic, E. Krätzel, M. Kühleitner and W. G. Nowak, Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic,, preprint, (). Google Scholar

[13]

C. Mora-Corral, Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity,, Interfaces and Free Boundaries, 11 (2009), 421. doi: 10.4171/IFB/217. Google Scholar

[14]

G. A. Pick, Geometrisches zur Zahlenlehre,, Sitzenber. Lotos Naturwissen Zeitschrift (Prague), 19 (1899), 311. Google Scholar

[15]

J. E. Reeve, On the volume of lattice polyhedra,, Proc. London Math. Soc., 3 (1957), 378. Google Scholar

[16]

P. Rosakis, Surface and Interfacial Energy in Three Dimensional Crystals,, in progress, (2014). Google Scholar

[17]

J. D. Sally and P. Sally, Roots to research: A vertical development of mathematical problems,, American Mathematical Society, (2007). Google Scholar

[18]

A. V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions,, Multiscale Modeling and Simulation, 9 (2011), 905. doi: 10.1137/100792421. Google Scholar

[19]

F. Theil, Surface energies in a two-dimensional mass-spring model for crystals,, ESAIM Math. Model. Numer. Anal., 45 (2011), 873. doi: 10.1051/m2an/2010106. Google Scholar

[20]

K.-M. Tsang, Counting lattice points in the sphere,, Bull. London Math. Soc., 32 (2000), 679. doi: 10.1112/S0024609300007505. Google Scholar

show all references

References:
[1]

I. Bárány and D. G. Larman, The convex hull of the integer points in a large ball,, Mathematische Annalen, 312 (1998), 167. doi: 10.1007/s002080050217. Google Scholar

[2]

A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra,, New Perspectives in Algebraic Combinatorics, 38 (1999), 91. Google Scholar

[3]

X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics,, Arch. Rational Mech. Anal., 164 (2002), 341. doi: 10.1007/s00205-002-0218-5. Google Scholar

[4]

M. Beck and S. Robins, Computing the Continuous Discretely: Integer Point Enumeration in Polyhedra,, Undergraduate Texts in Mathematics, (2007). Google Scholar

[5]

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

[6]

J. G. Van der Corput, Over Roosterpunten in het Platte vlak (de Beteekenis van de Methoden van Voronoi en Pfeiffer),, Noordhoff, (1919). Google Scholar

[7]

B. Dacorogna and C.-E Pfister, Wulff theorem and best constant in Sobolev inequality,, Journal de mathématiques pures et appliquées, 71 (1992), 97. Google Scholar

[8]

A. Eichler, J. Hafner, J. Furthmüller and G. Kresse, Structural and electronic properties of rhodium surfaces: an ab initio approach,, Surface Science, 346 (1996), 300. doi: 10.1016/0039-6028(95)00906-X. Google Scholar

[9]

I. Fonseca, The Wulff theorem revisited,, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 432 (1991), 125. doi: 10.1098/rspa.1991.0009. Google Scholar

[10]

C. Herring, Some theorems on the free energies of crystal surfaces,, Physical Review, 82 (1951), 87. doi: 10.1103/PhysRev.82.87. Google Scholar

[11]

M. N. Huxley, Exponential sums and lattice points III,, Proceedings of the London Mathematical Society, 87 (2003), 591. doi: 10.1112/S0024611503014485. Google Scholar

[12]

A. Ivic, E. Krätzel, M. Kühleitner and W. G. Nowak, Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic,, preprint, (). Google Scholar

[13]

C. Mora-Corral, Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity,, Interfaces and Free Boundaries, 11 (2009), 421. doi: 10.4171/IFB/217. Google Scholar

[14]

G. A. Pick, Geometrisches zur Zahlenlehre,, Sitzenber. Lotos Naturwissen Zeitschrift (Prague), 19 (1899), 311. Google Scholar

[15]

J. E. Reeve, On the volume of lattice polyhedra,, Proc. London Math. Soc., 3 (1957), 378. Google Scholar

[16]

P. Rosakis, Surface and Interfacial Energy in Three Dimensional Crystals,, in progress, (2014). Google Scholar

[17]

J. D. Sally and P. Sally, Roots to research: A vertical development of mathematical problems,, American Mathematical Society, (2007). Google Scholar

[18]

A. V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions,, Multiscale Modeling and Simulation, 9 (2011), 905. doi: 10.1137/100792421. Google Scholar

[19]

F. Theil, Surface energies in a two-dimensional mass-spring model for crystals,, ESAIM Math. Model. Numer. Anal., 45 (2011), 873. doi: 10.1051/m2an/2010106. Google Scholar

[20]

K.-M. Tsang, Counting lattice points in the sphere,, Bull. London Math. Soc., 32 (2000), 679. doi: 10.1112/S0024609300007505. Google Scholar

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