Dobson, Matthew, Luskin, Mitchell Barry and Ortner, Christoph (2010) Accuracy of quasicontinuum approximations near instabilities. Journal of the Mechanics and Physics of Solids, Volume 58 (Number 10). pp. 1741-1757. doi:10.1016/j.jmps.2010.06.011

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Abstract

The formation and motion of lattice defects such as cracks, dislocations, or grain boundaries, occurs when the lattice configuration loses stability, that is, when an eigenvalue of the Hessian of the lattice energy functional becomes negative. When the atomistic energy is approximated by a hybrid energy that couples atomistic and continuum models, the accuracy of the approximation can only be guaranteed near deformations where both the atomistic energy as well as the hybrid energy are stable. We propose, therefore, that it is essential for the evaluation of the predictive capability of atomistic-to-continuum coupling methods near instabilities that a theoretical analysis be performed, at least for some representative model problems, that determines whether the hybrid energies remain stable up to the onset of instability of the atomistic energy.

We formulate a one-dimensional model problem with nearest and next-nearest neighbour interactions and use rigorous analysis, asymptotic methods, and numerical experiments to obtain such sharp stability estimates for the basic conservative quasicontinuum (QC) approximations. Our results show that the consistent quasi-nonlocal QC approximation correctly reproduces the stability of the atomistic system, whereas the inconsistent energy-based QC approximation incorrectly predicts instability at a significantly reduced applied load that we describe by an analytic criterion in terms of the derivatives of the atomistic potential.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General)
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Multiscale modeling, Crystalline polymers -- Defects -- Mathematical models, Continuum mechanics
Journal or Publication Title:	Journal of the Mechanics and Physics of Solids
Publisher:	Pergamon-Elsevier Science Ltd.
ISSN:	0022-5096
Official Date:	October 2010
Dates:	Date Event October 2010 Published
Volume:	Volume 58
Number:	Number 10
Page Range:	pp. 1741-1757
DOI:	10.1016/j.jmps.2010.06.011
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	National Science Foundation (U.S.) (NSF), United States. Department of Energy, Institute for Mathematics and Its Applications (IMA), University of Minnesota. Supercomputer Institute, University of Minnesota, Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	DMS-0757355 (NSF), DMS-0811039 (NSF), DE-FG02-05ER25706 (DOE)

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