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December  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 and 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), 1-37. 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), 881-943. 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), 671-709.

[4]

X. Blanc, C. L. and P.-L. Lions, From molecular models to continuum mechanics, Arch. Rational Mech. Anal., 164 (2002), 341-381. 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), 391-426. doi: 10.1051/m2an:2007018.

[6]

A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313-321.

[7]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, New York, 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), 515-539. doi: 10.4171/JEMS/65.

[9]

B. Dacorogna, Direct Methods in the Calculus of Variation, $2^{nd}$ edition, Springer, New York, 2008.

[10]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 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, New York, 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), 445-478. doi: 10.1007/s00332-002-0495-z.

[13]

S. Haussühl, Die Abweichungen von den Cauchy-Relationen, Phys. kondens. Materie, 6 (1967), 181-192.

[14]

N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three-point interactions, Math. Models Methods Appl. Sci., 22 (2012), 21pp. doi: 10.1142/S0218202512500327.

[15]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.

[16]

B. Schmidt, On the derivation of linear elasticity from atomistic models, Networks and Heterogeneous Media, 4 (2009), 789-812. 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), 1-55. 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), 1-37. 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), 881-943. 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), 671-709.

[4]

X. Blanc, C. L. and P.-L. Lions, From molecular models to continuum mechanics, Arch. Rational Mech. Anal., 164 (2002), 341-381. 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), 391-426. doi: 10.1051/m2an:2007018.

[6]

A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313-321.

[7]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, New York, 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), 515-539. doi: 10.4171/JEMS/65.

[9]

B. Dacorogna, Direct Methods in the Calculus of Variation, $2^{nd}$ edition, Springer, New York, 2008.

[10]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 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, New York, 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), 445-478. doi: 10.1007/s00332-002-0495-z.

[13]

S. Haussühl, Die Abweichungen von den Cauchy-Relationen, Phys. kondens. Materie, 6 (1967), 181-192.

[14]

N. Meunier, O. Pantz and A. Raoult, Elastic limit of square lattices with three-point interactions, Math. Models Methods Appl. Sci., 22 (2012), 21pp. doi: 10.1142/S0218202512500327.

[15]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.

[16]

B. Schmidt, On the derivation of linear elasticity from atomistic models, Networks and Heterogeneous Media, 4 (2009), 789-812. 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), 1-55. doi: 10.1007/s00205-008-0138-0.

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