Formation of grain boundaries in ductile single crystals under plane-strain simple shear: a block-coordinate finite element method
Abstract
Large plastic deformation can drive an initially uniform single crystal to spontaneously subdivide into misoriented grains separated by thin dislocation walls -- a pattern-forming instability rooted in the loss of convexity of the crystal's elastic energy at large strain. We study this phenomenon for a ductile crystal in plane-strain simple shear within continuum dislocation theory, using a polyconvex (Ciarlet--Geymonat) elastic energy that guarantees existence of minimizers for the coupled deformation--slip problem. Minimizing over the plastic slip yields a condensed energy of double-well form whose non-quasiconvexity favours a lamellar microstructure; the gradient of the geometrically necessary dislocation density regularizes it, giving the grain boundaries a finite thickness and energy as functions of the misorientation angle. A block-coordinate finite element scheme -- alternating a convex non-smooth solve for the slip with a Levenberg-regularized Newton solve for the deformation -- resolves this microstructure numerically and detects its spontaneous onset, reproducing the lamellar grain structure in agreement with the closed-form analysis.
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BibTeX
@misc{le2026formation,
title = {{Formation of grain boundaries in ductile single crystals under plane-strain simple shear: a block-coordinate finite element method}},
author = {Khanh Chau Le and Thanh Danh Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07016},
eprint = {2609.07016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07016},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Formation of grain boundaries in ductile single crystals under plane-strain simple shear: a block-coordinate finite element method
AU - Khanh Chau Le
AU - Thanh Danh Nguyen
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07016
UR - https://arxiv.org/abs/2609.07016
AB - Large plastic deformation can drive an initially uniform single crystal to spontaneously subdivide into misoriented grains separated by thin dislocation walls -- a pattern-forming instability rooted in the loss of convexity of the crystal's elastic energy at large strain. We study this phenomenon for a ductile crystal in plane-strain simple shear within continuum dislocation theory, using a polyconvex (Ciarlet--Geymonat) elastic energy that guarantees existence of minimizers for the coupled deformation--slip problem. Minimizing over the plastic slip yields a condensed energy of double-well form whose non-quasiconvexity favours a lamellar microstructure; the gradient of the geometrically necessary dislocation density regularizes it, giving the grain boundaries a finite thickness and energy as functions of the misorientation angle. A block-coordinate finite element scheme -- alternating a convex non-smooth solve for the slip with a Levenberg-regularized Newton solve for the deformation -- resolves this microstructure numerically and detects its spontaneous onset, reproducing the lamellar grain structure in agreement with the closed-form analysis.
ER -