A small-distortion geometric model for elasto-plasticity in single crystals driven by the motion of dislocations

Abstract

A central problem in nonlinear elasto-plasticity theory is to formulate a geometrically nonlinear, large-strain model of elasto-plasticity in single crystals in which plastic flow is driven directly by the motion of dislocations and which allows for a homogenization procedure from discrete dislocation lines to dislocation densities. In this work, such a model is introduced based on two hypotheses: (1) The Small-Distortion Hypothesis posits that the total plastic distortion may be arbitrarily large but its dominant part admits a representation as a gradient, meaning that the distorting effect of dislocations is small relative to the specimen size. Concretely, this hypothesis is realized here through a"multiplicative Helmholtz-type decomposition", which splits an arbitrary matrix field into the product of a gradient and a matrix field with controlled curl. (2) The Line-Tension Approximation treats all dislocations as infinitesimally thin lines whose local stress fields are individually negligible compared to specimen-scale stresses and only matter in the aggregate. The resulting model, termed the Small-Distortion Geometric Model, furthermore employs the so-called Space-Time Framework, which furnishes a geometric language to precisely describe the advection of dislocation lines. As such, it provides a physically grounded account of crystal plasticity in an idealized mesoscopic setting that bridges dislocation mechanics and continuum elasto-plasticity.