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.
Mots-clés : Bravais lattice, Cauchy-Born model, stability
@article{M2AN_2012__46_1_81_0,
author = {Hudson, Thomas and Ortner, Christoph},
title = {On the stability of {Bravais} lattices and their {Cauchy-Born} approximations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {81--110},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/m2an/2011014},
mrnumber = {2846368},
zbl = {1291.35388},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2011014/}
}
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 PY - 2012 SP - 81 EP - 110 VL - 46 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011014/ DO - 10.1051/m2an/2011014 LA - en ID - M2AN_2012__46_1_81_0 ER -
%0 Journal Article %A Hudson, Thomas %A Ortner, Christoph %T On the stability of Bravais lattices and their Cauchy-Born approximations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 81-110 %V 46 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011014/ %R 10.1051/m2an/2011014 %G en %F M2AN_2012__46_1_81_0
Hudson, Thomas; Ortner, Christoph. On the stability of Bravais lattices and their Cauchy-Born approximations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 1, pp. 81-110. doi : 10.1051/m2an/2011014. http://archive.numdam.org/articles/10.1051/m2an/2011014/
[1] and , A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1-37. | MR | Zbl
[2] , and , From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164 (2002) 341-381. | MR | Zbl
[3] and , 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). | MR | Zbl
[4] and , Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66. | MR | Zbl
[5] , and , Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids 58 (2010) 1741-1757. | MR | Zbl
[6] , and , Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul. 8 (2010) 782-802. | MR | Zbl
[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. | MR | Zbl
[8] and , Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12 (2002) 445-478. | MR | Zbl
[9] and , Stability and elastic properties of the stress-free b2 (cscl-type) crystal for the morse pair potential model. J. Elasticity 92 (2008) 151-186. | MR | Zbl
[10] , Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993). | MR | Zbl
[11] and , A first course in continuum mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2008). | MR | Zbl
[12] , Introduction to Solid State Physics, 7th ed. John Wiley & Sons, New York, Chichester (1996). | Zbl
[13] , Linear integral equations, Applied Mathematical Sciences 82. Springer-Verlag, 2nd edition, New York (1999). | MR | Zbl
[14] and , Theory of elasticity, Course of Theoretical Physics 7. Translated by J.B. Sykes and W.H. Reid. Pergamon Press, London (1959). | MR | Zbl
[15] and , An analysis of the quasi-nonlocal quasicontinuum approximation of the embedded atom model. arXiv:1008.3628v4.
[16] and , A generalized quasi-nonlocal atomistic-to-continuum coupling method with finite range interaction. arXiv:1007.2336. | MR | Zbl
[17] , Problems in analytic number theory, Graduate Texts in Mathematics 206. Springer, 2nd edition, New York (2008). Readings in Mathematics. | MR | Zbl
[18] , A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D. Math. Comput. 80 (2011) 1265-1285 | MR
[19] and , Analysis of a quasicontinuum method in one dimension. ESAIM: M2AN 42 (2008) 57-91. | Numdam | MR | Zbl
[20] , A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul. 5 (2006) 664-694. | MR | Zbl
[21] , A proof of crystallization in two dimensions. Commun. Math. Phys. 262 (2006) 209-236. | MR | Zbl
[22] , Thermodynamics of Crystals. Dover Publications, New York (1998).
[23] , , , , and , Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper. J. Mech. Phys. Solids 52 (2004) 691-724. | Zbl
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