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

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.