Mathematical Problems in Engineering 
Volume 2007 (2007), Article ID 45951, 13 pages
doi:10.1155/2007/45951
Research Article

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

Saïd Hilout,1 Mohammed Boutat,2 and Jean Grilhé3

 1Département de Mathématiques Appliquées et Informatique, Faculté des Sciences et Techniques, Cadi Ayyad University, BP 523, Béni–Mellal 23000, Marrakech, Morocco
2Laboratoire de Mathématiques, CNRS/UMR 6086, Université de Poitiers, Boulevard Marie et Pierre Curie, Téléport 2, BP 30179, Futuroscope Chasseneuil Cedex 86962, France
3Laboratoire de Métallurgie Physique, CNRS/UMR 6630, Université de Poitiers, Bâtiment SP2MI-Téléport 2, Boulevard 3, BP 30179, Futuroscope Chasseneuil Cedex 86962, France
Received 24 February 2006; Revised 2 September 2006; Accepted 5 February 2007

Recommended by Jan Awrejcewicz

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.