On the stability of Bravais lattices and their Cauchy–Born approximations

Thomas Hudson; Christoph Ortner

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 1, page 81-110
  • ISSN: 0764-583X

Abstract

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We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze the relationship between atomistic and Cauchy–Born stability regions, and the convergence of atomistic stability regions as the cell size tends to infinity.

How to cite

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Hudson, Thomas, and Ortner, Christoph. "On the stability of Bravais lattices and their Cauchy–Born approximations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.1 (2012): 81-110. <http://eudml.org/doc/273191>.

@article{Hudson2012,
abstract = {We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze the relationship between atomistic and Cauchy–Born stability regions, and the convergence of atomistic stability regions as the cell size tends to infinity.},
author = {Hudson, Thomas, Ortner, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Bravais lattice; Cauchy–Born model; stability; Cauchy-Born model},
language = {eng},
number = {1},
pages = {81-110},
publisher = {EDP-Sciences},
title = {On the stability of Bravais lattices and their Cauchy–Born approximations},
url = {http://eudml.org/doc/273191},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Hudson, Thomas
AU - Ortner, Christoph
TI - On the stability of Bravais lattices and their Cauchy–Born approximations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 1
SP - 81
EP - 110
AB - We investigate the stability of Bravais lattices and their Cauchy–Born approximations under periodic perturbations. We formulate a general interaction law and derive its Cauchy–Born continuum limit. We then analyze the atomistic and Cauchy–Born stability regions, that is, the sets of all matrices that describe a stable Bravais lattice in the atomistic and Cauchy–Born models respectively. Motivated by recent results in one dimension on the stability of atomistic/continuum coupling methods, we analyze the relationship between atomistic and Cauchy–Born stability regions, and the convergence of atomistic stability regions as the cell size tends to infinity.
LA - eng
KW - Bravais lattice; Cauchy–Born model; stability; Cauchy-Born model
UR - http://eudml.org/doc/273191
ER -

References

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  1. [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. Zbl1070.49009MR2083851
  2. [2] X. Blanc, C. Le Bris and P.-L. Lions, From molecular models to continuum mechanics. Arch. Ration. Mech. Anal.164 (2002) 341–381. Zbl1028.74005MR1933632
  3. [3] M. Born and K. Huang, Dynamical theory of crystal lattices. Oxford Classic Texts in the Physical Sciences. The Clarendon Press Oxford University Press, New York, Reprint of the 1954 original (1988). Zbl0908.01039MR1654161
  4. [4] A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41–66. Zbl1024.74004MR1900933
  5. [5] M. Dobson, M. Luskin and C. Ortner, Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids58 (2010) 1741–1757. Zbl1200.74005MR2742030
  6. [6] M. Dobson, M. Luskin and C. Ortner, Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul.8 (2010) 782–802. Zbl1225.82009MR2609639
  7. [7] W.E and P. Ming, Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183 (2007) 241–297. Zbl1106.74019MR2278407
  8. [8] 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. Zbl1084.74501MR1923388
  9. [9] V.S. Ghutikonda and R.S. Elliott, Stability and elastic properties of the stress-free b2 (cscl-type) crystal for the morse pair potential model. J. Elasticity92 (2008) 151–186. Zbl1147.74010MR2417286
  10. [10] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993). Zbl0786.35001MR1239172
  11. [11] O. Gonzalez and A.M. Stuart, A first course in continuum mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2008). Zbl1143.74001MR2378978
  12. [12] C. Kittel, Introduction to Solid State Physics, 7th ed. John Wiley & Sons, New York, Chichester (1996). Zbl0052.45506
  13. [13] R. Kress, Linear integral equations, Applied Mathematical Sciences 82. Springer-Verlag, 2nd edition, New York (1999). Zbl0920.45001MR1723850
  14. [14] L.D. Landau and E.M. Lifshitz, Theory of elasticity, Course of Theoretical Physics 7. Translated by J.B. Sykes and W.H. Reid. Pergamon Press, London (1959). Zbl0146.22405MR106584
  15. [15] X.H. Li and M. Luskin, An analysis of the quasi-nonlocal quasicontinuum approximation of the embedded atom model. arXiv:1008.3628v4. 
  16. [16] X.H. Li and M. Luskin, A generalized quasi-nonlocal atomistic-to-continuum coupling method with finite range interaction. arXiv:1007.2336. Zbl1241.82078MR2911393
  17. [17] M.R. Murty, Problems in analytic number theory, Graduate Texts in Mathematics 206. Springer, 2nd edition, New York (2008). Readings in Mathematics. Zbl1190.11001MR2376618
  18. [18] C. Ortner, A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D. Math. Comput.80 (2011) 1265–1285 Zbl05918690MR2785458
  19. [19] C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension. ESAIM: M2AN 42 (2008) 57–91. Zbl1139.74004MR2387422
  20. [20] B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul.5 (2006) 664–694. Zbl1117.49018MR2247767
  21. [21] F. Theil, A proof of crystallization in two dimensions. Commun. Math. Phys.262 (2006) 209–236. Zbl1113.82016MR2200888
  22. [22] D. Wallace, Thermodynamics of Crystals. Dover Publications, New York (1998). 
  23. [23] T. Zhu, J. Li, K.J. Van Vliet, S. Ogata, S. Yip and S. Suresh, Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper. J. Mech. Phys. Solids52 (2004) 691–724. Zbl1106.74316

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