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.
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BibTeX
@misc{rindler2026smalldistortion,
title = {{A small-distortion geometric model for elasto-plasticity in single crystals driven by the motion of dislocations}},
author = {Filip Rindler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23728},
eprint = {2609.23728},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23728},
}
RIS
TY - UNPB
TI - A small-distortion geometric model for elasto-plasticity in single crystals driven by the motion of dislocations
AU - Filip Rindler
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23728
UR - https://arxiv.org/abs/2609.23728
AB - 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.
ER -