A multiscale model for dislocations: From mesoscopic elasticity to macroscopic plasticity

Abstract

A multiscale model for dislocations in single crystals is proposed. The aim of this paper is to provide homogenization of dislocations from the meso-to the macro-scale. In particular we prove a new formula relating macroscopic strain incompatibility and dislocation density. Moreover, it is shown that plasticity is recovered from homogenization of purely elastic mesoscopic crystals with dislocations, where the appropriate functional space of Special functions of Bounded Deformation appears as a natural choice. Nonstandard defect and deformation internal variables and their evolution in time appear while novel differential relations are sketched in the non-Riemannian crystal with a thermodynamical model as based on those results.

Fields of bounded deformation for mesoscopic dislocations

“…The following lemma, proved in [2, 8], illustrates the kind of result required if a countable, instead of a finite family, is considered.…”

“…It is shown in [2] that Assumption 5 is in fact a consequence of the set of assumptions on the strain curl, which are required to prove Theorem 4 for a countable family of lines (cf. [13, 8]) .…”

“…For these reasons, the distributional approach is said to describe the geometry (or statics) of dislocations and disclinations in single crystals. Dynamical and energetic aspects at the mesoscale are for the moment beyond reach, although at the macroscale, dynamics is briefly addressed in two publications [7, 8].…”

The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000)

“…Homogenization is obtained from the continuum scale by a limit procedure which will not be detailled here (cf. [35]), but whose effect is to erase the singularities (isolated ones or those resulting from accumulation) and hence to provide a smooth macroscopic crystal. Basically we postulate the following limits:…”

“…Eq (35). indicates that symbol ð in (33) becomes a true derivation operator in the absence of disclinations.…”