Elastic localization in finite structures: assessment of the infinite-domain approximation and validity of weakly nonlinear analysis
Abstract
A common approach to studying elastic localization is to pose the problem on an infinite domain, avoiding explicit treatment of the end conditions. For a finite structure, this approach raises a natural question: how long must the structure be for the infinite-domain approximation to become accurate, and how large is the resulting error? To address this question, we develop a constructive method that provides an explicit length threshold beyond which the approximation error is small and explicitly bounded. Applied to a prototypical model arising from localized buckling of a twisted rod, the method yields the threshold and error bounds analytically. The threshold is found to vary with loading, showing that whether the infinite-domain approximation is appropriate depends on both structural length and loading state. The load-dependent length threshold also clarifies the validity of weakly nonlinear analysis: the finite-domain expansion is appropriate only over a limited range near bifurcation, beyond which the infinite-domain expansion is more appropriate. The method is then applied to the fully nonlinear problem of localized bulging in cylindrical membrane tubes. Numerical evaluation of the model-dependent estimates yields a length threshold and an upper bound on the approximation error, while direct comparisons of the finite- and infinite-domain solutions support these predictions. Together, these results characterize the applicability of the infinite-domain approximation and weakly nonlinear description to elastic localization in finite structures.
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BibTeX
@misc{yu2026elastic,
title = {{Elastic localization in finite structures: assessment of the infinite-domain approximation and validity of weakly nonlinear analysis}},
author = {Xiang Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.15312},
eprint = {2606.15312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.15312},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Elastic localization in finite structures: assessment of the infinite-domain approximation and validity of weakly nonlinear analysis
AU - Xiang Yu
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2606.15312
UR - https://arxiv.org/abs/2606.15312
AB - A common approach to studying elastic localization is to pose the problem on an infinite domain, avoiding explicit treatment of the end conditions. For a finite structure, this approach raises a natural question: how long must the structure be for the infinite-domain approximation to become accurate, and how large is the resulting error? To address this question, we develop a constructive method that provides an explicit length threshold beyond which the approximation error is small and explicitly bounded. Applied to a prototypical model arising from localized buckling of a twisted rod, the method yields the threshold and error bounds analytically. The threshold is found to vary with loading, showing that whether the infinite-domain approximation is appropriate depends on both structural length and loading state. The load-dependent length threshold also clarifies the validity of weakly nonlinear analysis: the finite-domain expansion is appropriate only over a limited range near bifurcation, beyond which the infinite-domain expansion is more appropriate. The method is then applied to the fully nonlinear problem of localized bulging in cylindrical membrane tubes. Numerical evaluation of the model-dependent estimates yields a length threshold and an upper bound on the approximation error, while direct comparisons of the finite- and infinite-domain solutions support these predictions. Together, these results characterize the applicability of the infinite-domain approximation and weakly nonlinear description to elastic localization in finite structures.
ER -