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

Previous Article
Homogenization of high-contrast and non symmetric conductivities for non periodic columnar structures
NHM Home
This Issue
Next Article
Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings

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

Julian Braun ^1, and Bernd Schmidt ^1,

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

Received November 2012 Published November 2013

Abstract
Related Papers

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.

Keywords: Nonlinear elasticity theory, atomistic systems, discrete-to-continuum limits, Cauchy-Born rule..

Mathematics Subject Classification: 49J45, 74B20, 74Qx.

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.

[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 and Heterogeneous Media, 2009, 4 (4) : 667-708. doi: 10.3934/nhm.2009.4.667
[2]	Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. On the Cauchy-Born approximation at finite temperature for alloys. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3131-3153. doi: 10.3934/dcdsb.2021176
[3]	Manuel Friedrich, Bernd Schmidt. On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime. Networks and Heterogeneous Media, 2015, 10 (2) : 321-342. doi: 10.3934/nhm.2015.10.321
[4]	Bernd Schmidt. On the derivation of linear elasticity from atomistic models. Networks and Heterogeneous Media, 2009, 4 (4) : 789-812. doi: 10.3934/nhm.2009.4.789
[5]	Uri M. Ascher. Discrete processes and their continuous limits. Journal of Dynamics and Games, 2020, 7 (2) : 123-140. doi: 10.3934/jdg.2020008
[6]	Julian Braun, Bernd Schmidt. An atomistic derivation of von-Kármán plate theory. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022019
[7]	Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings. Networks and Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111
[8]	Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432
[9]	Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure and Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75
[10]	Gabriella Bretti, Ciro D’Apice, Rosanna Manzo, Benedetto Piccoli. A continuum-discrete model for supply chains dynamics. Networks and Heterogeneous Media, 2007, 2 (4) : 661-694. doi: 10.3934/nhm.2007.2.661
[11]	Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks and Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551
[12]	Weinan E, Jianfeng Lu. Mathematical theory of solids: From quantum mechanics to continuum models. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5085-5097. doi: 10.3934/dcds.2014.34.5085
[13]	Franco Cardin, Alberto Lovison. Finite mechanical proxies for a class of reducible continuum systems. Networks and Heterogeneous Media, 2014, 9 (3) : 417-432. doi: 10.3934/nhm.2014.9.417
[14]	Annie Raoult. Symmetry groups in nonlinear elasticity: an exercise in vintage mathematics. Communications on Pure and Applied Analysis, 2009, 8 (1) : 435-456. doi: 10.3934/cpaa.2009.8.435
[15]	Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237
[16]	Ezio Di Costanzo, Marta Menci, Eleonora Messina, Roberto Natalini, Antonia Vecchio. A hybrid model of collective motion of discrete particles under alignment and continuum chemotaxis. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 443-472. doi: 10.3934/dcdsb.2019189
[17]	Victor Isakov. Carleman estimates for some anisotropic elasticity systems and applications. Evolution Equations and Control Theory, 2012, 1 (1) : 141-154. doi: 10.3934/eect.2012.1.141
[18]	Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51
[19]	Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic and Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893
[20]	Brian Straughan. Shocks and acceleration waves in modern continuum mechanics and in social systems. Evolution Equations and Control Theory, 2014, 3 (3) : 541-555. doi: 10.3934/eect.2014.3.541

2021 Impact Factor: 1.41

Tools

Metrics

Other articles
by authors

on AIMS
Julian Braun Bernd Schmidt
on Google Scholar
Julian Braun Bernd Schmidt

[Back to Top]