Formation of shear bands through discrete localized plastic increments
Abstract
We study rate-independent systems in which the evolution is obtained by incremental minimization of elastic energy and one-homogeneous dissipation under constraints on the admissible inelastic increments. Such constraints arise naturally in several models of inelastic materials, including shear-transformation-zone models, intermittent plasticity, and quantized plasticity, where evolution proceeds through prescribed elementary activation events rather than arbitrary plastic increments. We establish the existence of a continuous-time limit for the corresponding constrained incremental scheme under suitable assumptions. As an application, we formulate a simplified shear-transformation-zone model consisting of localized activation events on a lattice embedded in a two-dimensional elastic medium. We prove that this model generates shear-band formation through the successive activation of neighboring transformation zones. More precisely, we show that every admissible evolution remains localized on a growing line segment in the infinite-domain setting and that the same mechanism persists on sufficiently large bounded domains. These results provide a rigorous mathematical framework for constrained activation models and show that restrictions on admissible elementary events alone constitute a mechanism for localization in rate-independent systems.
Related
- Fully Discrete Active Flux Method based on Transported Acoustic Increments for the Compressible Euler Equations
- Parameter Estimation for Unnormalized Discrete Models via Empirically Localized Deformed Bregman Divergence
- ϕ-Null controllability of a semi-discrete linear KdV equation with localized interior control
- On the coincidence of discrete and continuous definitions of Entanglement of Formation
- Energy-Dissipative, High-Order BDF Schemes for Simulating Geometric Flows via Energy Increments
BibTeX
@misc{olbermann2026formation,
title = {{Formation of shear bands through discrete localized plastic increments}},
author = {Heiner Olbermann and Hidde Schönberger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37584},
eprint = {2609.37584},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37584},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Formation of shear bands through discrete localized plastic increments
AU - Heiner Olbermann
AU - Hidde Schönberger
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37584
UR - https://arxiv.org/abs/2609.37584
AB - We study rate-independent systems in which the evolution is obtained by incremental minimization of elastic energy and one-homogeneous dissipation under constraints on the admissible inelastic increments. Such constraints arise naturally in several models of inelastic materials, including shear-transformation-zone models, intermittent plasticity, and quantized plasticity, where evolution proceeds through prescribed elementary activation events rather than arbitrary plastic increments. We establish the existence of a continuous-time limit for the corresponding constrained incremental scheme under suitable assumptions. As an application, we formulate a simplified shear-transformation-zone model consisting of localized activation events on a lattice embedded in a two-dimensional elastic medium. We prove that this model generates shear-band formation through the successive activation of neighboring transformation zones. More precisely, we show that every admissible evolution remains localized on a growing line segment in the infinite-domain setting and that the same mechanism persists on sufficiently large bounded domains. These results provide a rigorous mathematical framework for constrained activation models and show that restrictions on admissible elementary events alone constitute a mechanism for localization in rate-independent systems.
ER -