• Previous Article
    Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings
  • NHM Home
  • This Issue
  • Next Article
    Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures
2013, 8(4): 879-912. doi: 10.3934/nhm.2013.8.879

On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth

1. 

Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany, Germany

Received  November 2012 Published  November 2013

We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.
Citation: Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Networks & Heterogeneous Media, 2013, 8 (4) : 879-912. doi: 10.3934/nhm.2013.8.879
References:
[1]

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

[2]

R. Alicandro, M. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity,, Arch. Rational Mech. Anal., 200 (2011), 881. doi: 10.1007/s00205-010-0378-7.

[3]

R. Alicandro, M. Focardi and M. S. Gelli, Finite-difference approximation of energies in fracture mechanics,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 671.

[4]

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

[5]

X. Blanc, C. L. and P.-L. Lions, Atomistic to continuum limits for computational materials science,, Math. Model. Numer. Anal., 41 (2007), 391. doi: 10.1051/m2an:2007018.

[6]

A. Braides, Homogenization of some almost periodic coercive functional,, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313.

[7]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,, Oxford University Press, (1998).

[8]

S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to $SO(n)$,, J. Eur. Math. Soc. (JEMS), 8 (2006), 515. doi: 10.4171/JEMS/65.

[9]

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

[10]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993). doi: 10.1007/978-1-4612-0327-8.

[11]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$-Spaces,, Springer, (2007).

[12]

G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice,, J. Nonlinear Sci., 12 (2002), 445. doi: 10.1007/s00332-002-0495-z.

[13]

S. Haussühl, Die Abweichungen von den Cauchy-Relationen,, Phys. kondens. Materie, 6 (1967), 181.

[14]

N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three-point interactions,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500327.

[15]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials,, Arch. Rational Mech. Anal., 99 (1987), 189. doi: 10.1007/BF00284506.

[16]

B. Schmidt, On the derivation of linear elasticity from atomistic models,, Networks and Heterogeneous Media, 4 (2009), 789. doi: 10.3934/nhm.2009.4.789.

[17]

B. Schmidt, On the passage from atomic to continuum theory for thin films,, Arch. Rational Mech. Anal., 190 (2008), 1. doi: 10.1007/s00205-008-0138-0.

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 J. Math. Anal., 36 (2004), 1. doi: 10.1137/S0036141003426471.

[2]

R. Alicandro, M. Cicalese and A. Gloria, Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity,, Arch. Rational Mech. Anal., 200 (2011), 881. doi: 10.1007/s00205-010-0378-7.

[3]

R. Alicandro, M. Focardi and M. S. Gelli, Finite-difference approximation of energies in fracture mechanics,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 671.

[4]

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

[5]

X. Blanc, C. L. and P.-L. Lions, Atomistic to continuum limits for computational materials science,, Math. Model. Numer. Anal., 41 (2007), 391. doi: 10.1051/m2an:2007018.

[6]

A. Braides, Homogenization of some almost periodic coercive functional,, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313.

[7]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals,, Oxford University Press, (1998).

[8]

S. Conti, G. Dolzmann, B. Kirchheim and S. Müller, Sufficient conditions for the validity of the Cauchy-Born rule close to $SO(n)$,, J. Eur. Math. Soc. (JEMS), 8 (2006), 515. doi: 10.4171/JEMS/65.

[9]

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

[10]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäuser, (1993). doi: 10.1007/978-1-4612-0327-8.

[11]

I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$-Spaces,, Springer, (2007).

[12]

G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice,, J. Nonlinear Sci., 12 (2002), 445. doi: 10.1007/s00332-002-0495-z.

[13]

S. Haussühl, Die Abweichungen von den Cauchy-Relationen,, Phys. kondens. Materie, 6 (1967), 181.

[14]

N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three-point interactions,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512500327.

[15]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials,, Arch. Rational Mech. Anal., 99 (1987), 189. doi: 10.1007/BF00284506.

[16]

B. Schmidt, On the derivation of linear elasticity from atomistic models,, Networks and Heterogeneous Media, 4 (2009), 789. doi: 10.3934/nhm.2009.4.789.

[17]

B. Schmidt, On the passage from atomic to continuum theory for thin films,, Arch. Rational Mech. Anal., 190 (2008), 1. doi: 10.1007/s00205-008-0138-0.

[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]

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

[3]

Bernd Schmidt. On the derivation of linear elasticity from atomistic models. Networks & Heterogeneous Media, 2009, 4 (4) : 789-812. doi: 10.3934/nhm.2009.4.789

[4]

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

[5]

Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks & Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551

[6]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432

[7]

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75

[8]

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

[9]

Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085

[10]

Annie Raoult. Symmetry groups in nonlinear elasticity: an exercise in vintage mathematics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 435-456. doi: 10.3934/cpaa.2009.8.435

[11]

Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237

[12]

Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks & Heterogeneous Media, 2014, 9 (3) : 417-432. doi: 10.3934/nhm.2014.9.417

[13]

Victor Isakov. Carleman estimates for some anisotropic elasticity systems and applications. Evolution Equations & Control Theory, 2012, 1 (1) : 141-154. doi: 10.3934/eect.2012.1.141

[14]

Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51

[15]

Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893

[16]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184

[17]

Brian Straughan. Shocks and acceleration waves in modern continuum mechanics and in social systems. Evolution Equations & Control Theory, 2014, 3 (3) : 541-555. doi: 10.3934/eect.2014.3.541

[18]

Steffen Klassert, Daniel Lenz, Peter Stollmann. Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1553-1571. doi: 10.3934/dcds.2011.29.1553

[19]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[20]

Hilde De Ridder, Hennie De Schepper, Frank Sommen. The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1097-1109. doi: 10.3934/cpaa.2011.10.1097

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]