Plastic Deformation Instabilities: Lambert Solutions of Mecking-Lücke Equation with Delay

Hilout, Saïd; Boutat, Mohammed; Grilhé, Jean

doi:https://doi.org/10.1155/2007/45951

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2007 | Article ID 045951 | https://doi.org/10.1155/2007/45951

Plastic Deformation Instabilities: Lambert Solutions of Mecking-Lücke Equation with Delay

Saïd Hilout,¹Mohammed Boutat,²and Jean Grilhé³

Academic Editor: Jan Awrejcewicz

Received24 Feb 2006

Revised02 Sept 2006

Accepted05 Feb 2007

Published22 Mar 2007

Abstract

The aim of this paper is the study of instabilities during plastic deformation at constant cross‐head velocity. The deformation is supposed to be controlled by the emission of dislocation loops. Under some hypothesis analogous to the Mecking‐Lücke relation, we derive a linear delay differential‐difference equation. The “retarded” time term appears as the phase shift between the time of loop nucleation and the time at which the mean strain is recorded. We show the existence of the solution of strain equation. We give an analytic approach of solution using Lambert functions. The stability is also investigated close to the stable solution using a linearization of the number of nucleated loops functions.

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Copyright

Copyright © 2007 Saïd Hilout et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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