Linearization in nonlocal elasticity via Gamma-convergence
Abstract
Linearization is a classical tool in continuum mechanics that connects nonlinear models of solids to their linear counterparts through the small-strain limit, as in the passage from finite to linear elasticity. Here, we study a nonlocal analogue of this limit process in the context of state-based peridynamics, starting from a class of models in nonlocal hyperelasticity subject to nonlocal Dirichlet boundary conditions. Recent formal calculations in the literature indicate that the linearized versions of these models are closely related to the widely used Eringen models. The main purpose of this note is to make this connection rigorous by establishing Γ-convergence of the associated energy functionals and a corresponding compactness result that together imply strong convergence of minimizers. Our proof relies on a local-to-nonlocal transfer argument that reduces the problem to a well-known local linearization result.
Related
- Sharp thresholds and convergence for a nonlocal bistable equation
- Full-Range Γ-Convergence of Weak-Type Nonlocal Functionals
- Endogenies and linearization in the non-virtually connected case
- Bi-parameter local linearization for the stochastic wave equation with rough noise
- Discrete-time feedback linearization control for nonsmooth constrained optimization
BibTeX
@misc{kreisbeck2026linearization,
title = {{Linearization in nonlocal elasticity via Gamma-convergence}},
author = {Carolin Kreisbeck and Carlos Mora-Corral and Felix Seifert},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23114},
eprint = {2610.23114},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23114},
}
RIS
TY - UNPB
TI - Linearization in nonlocal elasticity via Gamma-convergence
AU - Carolin Kreisbeck
AU - Carlos Mora-Corral
AU - Felix Seifert
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23114
UR - https://arxiv.org/abs/2610.23114
AB - Linearization is a classical tool in continuum mechanics that connects nonlinear models of solids to their linear counterparts through the small-strain limit, as in the passage from finite to linear elasticity. Here, we study a nonlocal analogue of this limit process in the context of state-based peridynamics, starting from a class of models in nonlocal hyperelasticity subject to nonlocal Dirichlet boundary conditions. Recent formal calculations in the literature indicate that the linearized versions of these models are closely related to the widely used Eringen models. The main purpose of this note is to make this connection rigorous by establishing Γ-convergence of the associated energy functionals and a corresponding compactness result that together imply strong convergence of minimizers. Our proof relies on a local-to-nonlocal transfer argument that reduces the problem to a well-known local linearization result.
ER -