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Defects, singularities and waves

Bonilla, L.L. and Carpio Rodríguez, Ana María (2007) Defects, singularities and waves. In Recent advances in nonlinear partial differential equations and applications. Proceedings of Symposia in Applied Mathematics (65). American Mathematical Society, Providence, pp. 131-150. ISBN 978-0-8218-4211-9

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Abstract

Crystal defects such as dislocations are the basis of macroscopic properties such as the strength of materials and control their mechanical, optical and electronic properties. In recent times, advances in electronic microscopy have allowed imaging of atoms and therefore to visualize the core of dislocations, cracks, and so on. In continuum mechanics, dislocations are treated as source terms proportional to delta functions supported on the dislocation line. Cores and crystal structure are not properly considered and it is hard to describe the motion of crystal defects. Unlike defects in fluids (such as vortices), dislocations move only within glide planes, not in arbitrary directions, and they move only when the applied stress surpasses the Peierls stress, which is not infinitesimal. We have proposed a discrete model describing defects in crystal lattices with cubic symmetry and having the standard linear anisotropic elasticity (Navier equations) as its continuum limit. Moving dislocations are traveling waves which become stationary solutions if the applied stress falls below the Peierls value. The corresponding transition is a global bifurcation of the model equations similar to that observed in simpler one-dimensional Frenkel-Kontorova models. Discrete models can also be used to study the interaction of dislocations and the creation of dislocations under sufficient applied stress.

Item Type:Book Section
Additional Information:Conference in honor of Peter D. Lax and Louis Nirenberg on their 80th birthdays, Toledo, Spain, June 7–10, 2006
Uncontrolled Keywords:Crystal; dislocations; Peierls stress; Navier equations; anisotropic elasticity; Frenkel-Kontorova model; Peierls-Nabarrow model
Subjects:Sciences > Physics > Materials
ID Code:15251
Deposited On:18 May 2012 10:13
Last Modified:18 May 2012 10:13

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