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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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313-321.  Google Scholar [7] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, New York, 1998.  Google Scholar [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.  Google Scholar [9] B. Dacorogna, Direct Methods in the Calculus of Variation, $2^{nd}$ edition, Springer, New York, 2008.  Google Scholar [10] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [11] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$-Spaces, Springer, New York, 2007.  Google Scholar [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.  Google Scholar [13] S. Haussühl, Die Abweichungen von den Cauchy-Relationen, Phys. kondens. Materie, 6 (1967), 181-192. Google Scholar [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.  Google Scholar [15] S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] A. Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL Mem. Mat., 9 (1985), 313-321.  Google Scholar [7] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, New York, 1998.  Google Scholar [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.  Google Scholar [9] B. Dacorogna, Direct Methods in the Calculus of Variation, $2^{nd}$ edition, Springer, New York, 2008.  Google Scholar [10] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [11] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$-Spaces, Springer, New York, 2007.  Google Scholar [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.  Google Scholar [13] S. Haussühl, Die Abweichungen von den Cauchy-Relationen, Phys. kondens. Materie, 6 (1967), 181-192. Google Scholar [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.  Google Scholar [15] S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212. doi: 10.1007/BF00284506.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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