Nonlocal geometric rigidity inequalities and the justification of linearized peridynamics
Abstract
We prove geometric rigidity inequalities for vector fields using double-integral Gagliardo-type quantities. We show that the nonlocal-norm distance of the difference quotient of a vector field from the difference quotient of a single rigid motion is controlled by the same nonlocal norm of a difference characterizing the distance to the manifold of all rigid motions. The inequality is then used to show the variational convergence of a sequence of rescaled nonlocal fully nonconvex integral functionals to a linearized energy functional. Related functionals arise in peridynamic formulations of continuum mechanics via the total strain energy; this variational convergence provides a rigorous justification of linearized peridynamics with volume constraint via the convergence of rescaled displacements. We additionally establish analogous rigidity inequalities in the linearized setting, which can be thought of as nonlocal analogues of Korn's inequality for displacements.
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BibTeX
@misc{scott2026nonlocal,
title = {{Nonlocal geometric rigidity inequalities and the justification of linearized peridynamics}},
author = {James M. Scott},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28268},
eprint = {2610.28268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28268},
}
RIS
TY - UNPB
TI - Nonlocal geometric rigidity inequalities and the justification of linearized peridynamics
AU - James M. Scott
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28268
UR - https://arxiv.org/abs/2610.28268
AB - We prove geometric rigidity inequalities for vector fields using double-integral Gagliardo-type quantities. We show that the nonlocal-norm distance of the difference quotient of a vector field from the difference quotient of a single rigid motion is controlled by the same nonlocal norm of a difference characterizing the distance to the manifold of all rigid motions. The inequality is then used to show the variational convergence of a sequence of rescaled nonlocal fully nonconvex integral functionals to a linearized energy functional. Related functionals arise in peridynamic formulations of continuum mechanics via the total strain energy; this variational convergence provides a rigorous justification of linearized peridynamics with volume constraint via the convergence of rescaled displacements. We additionally establish analogous rigidity inequalities in the linearized setting, which can be thought of as nonlocal analogues of Korn's inequality for displacements.
ER -