Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2007 / Article

Research Article | Open Access

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

Saïd Hilout, Mohammed Boutat, Jean Grilhé, "Plastic Deformation Instabilities: Lambert Solutions of Mecking-Lücke Equation with Delay", Mathematical Problems in Engineering, vol. 2007, Article ID 045951, 13 pages, 2007. https://doi.org/10.1155/2007/45951

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

Academic Editor: Jan Awrejcewicz
Received24 Feb 2006
Revised02 Sep 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 © 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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