Some basic results in the mathematical analysis of dislocation storage and annihilation in stage II and stage III strain hardening

Eighty years since G.I. Taylor presented the first empirical explanation for the yield stress and strain hardening of metals, generally accepted formalisms to explain strain hardening have been established. The most important of these empirical laws have eluded full mathematical analysis until now. The Taylor equation provides a relationship between yield stress and strain. The semi-empirical Kocks–Mecking model provides a description of the physical phenomena of dislocation storage and annihilation, but lacks information on the formation of substructure and its effects. Here, a recently developed mathematical analysis of the corresponding phenomena is presented, which reproduces the essential equations in strain hardening. The first important result is the proof that the Taylor equation is the unique solution of a fundamental evolution equation for dislocation density. The second one demonstrates that dislocation storage and annihilation are exactly additive. The latter conclusion is independent of substructure, explaining the success of the Kocks–Mecking model during stages II and III.