Skip to main content Accessibility help
×
Home

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

  • Thomas Hudson (a1) and Christoph Ortner (a1)

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.

Copyright

References

Hide All
[1] Alicandro, R. and Cicalese, M., A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 137.
[2] Blanc, X., Le Bris, C. and Lions, P.-L., From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164 (2002) 341381.
[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).
[4] Braides, A. and Gelli, M.S., Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 4166.
[5] Dobson, M., Luskin, M. and Ortner, C., Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids 58 (2010) 17411757.
[6] Dobson, M., Luskin, M. and Ortner, C., Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation. Multiscale Model. Simul. 8 (2010) 782802.
[7] W.E and P. Ming, Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183 (2007) 241297.
[8] Friesecke, G. and Theil, F., Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12 (2002) 445478.
[9] Ghutikonda, V.S. and Elliott, R.S., Stability and elastic properties of the stress-free b2 (cscl-type) crystal for the morse pair potential model. J. Elasticity 92 (2008) 151186.
[10] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993).
[11] O. Gonzalez and A.M. Stuart, A first course in continuum mechanics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2008).
[12] C. Kittel, Introduction to Solid State Physics, 7th ed. John Wiley & Sons, New York, Chichester (1996).
[13] R. Kress, Linear integral equations, Applied Mathematical Sciences 82. Springer-Verlag, 2nd edition, New York (1999).
[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).
[15] X.H. Li and M. Luskin, An analysis of the quasi-nonlocal quasicontinuum approximation of the embedded atom model. arXiv:1008.3628v4.
[16] X.H. Li and M. Luskin, A generalized quasi-nonlocal atomistic-to-continuum coupling method with finite range interaction. arXiv:1007.2336.
[17] M.R. Murty, Problems in analytic number theory, Graduate Texts in Mathematics 206. Springer, 2nd edition, New York (2008). Readings in Mathematics.
[18] Ortner, C., A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D. Math. Comput. 80 (2011) 12651285
[19] Ortner, C. and Süli, E., Analysis of a quasicontinuum method in one dimension. ESAIM: M2AN 42 (2008) 5791.
[20] Schmidt, B., A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul. 5 (2006) 664694.
[21] Theil, F., A proof of crystallization in two dimensions. Commun. Math. Phys. 262 (2006) 209236.
[22] D. Wallace, Thermodynamics of Crystals. Dover Publications, New York (1998).
[23] Zhu, T., Li, J., Van Vliet, K.J., Ogata, S., Yip, S. and Suresh, S., Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper. J. Mech. Phys. Solids 52 (2004) 691724.

Keywords

Related content

Powered by UNSILO

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

  • Thomas Hudson (a1) and Christoph Ortner (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.