A direct theory of strain-gradient special Cosserat rods via the principle of virtual work
Abstract
Motivated by recent advances in the synthesis of effective one-dimensional mechanical metamaterials, we develop a direct theory of strain-gradient special Cosserat rods based on the principle of virtual work. The internal virtual work is formulated intrinsically in terms of variations of the classical Cosserat strains and their spatial derivatives, without requiring a three-dimensional parent material or being restricted to elastic materials. Objectivity and insensitivity to rigid virtual displacements characterize the admissible internal-work density, while localization yields the dynamic field equations, higher-order contact interactions, natural boundary conditions, and balances of linear and angular momentum. For inextensible and unshearable rods, the constrained principle determines a unique multiplier functional, although its representation by reactive force and double-force fields possesses a gauge freedom absent from classical rod theory. We also formulate objective elastic, hyperelastic, and viscoelastic constitutive laws, extend transverse hemitropy, transverse isotropy, and flip symmetry to strain-gradient rods, and derive invariant representations of symmetric stored energies. As an application, we analyze a planar inextensible cantilever subjected to a compressive terminal force, with stored energy depending on the spatial derivative of curvature. We prove that the eigenvalues of the linearized problem are simple and coincide precisely with the loads at which bifurcation from the straight state occurs. In addition, each bifurcation point generates a connected global branch of nonlinear equilibria that is locally analytic near its origin.
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BibTeX
@misc{hallyburton2026direct,
title = {{A direct theory of strain-gradient special Cosserat rods via the principle of virtual work}},
author = {R. S. Hallyburton and C. Rodriguez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19171},
eprint = {2610.19171},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19171},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A direct theory of strain-gradient special Cosserat rods via the principle of virtual work
AU - R. S. Hallyburton
AU - C. Rodriguez
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.19171
UR - https://arxiv.org/abs/2610.19171
AB - Motivated by recent advances in the synthesis of effective one-dimensional mechanical metamaterials, we develop a direct theory of strain-gradient special Cosserat rods based on the principle of virtual work. The internal virtual work is formulated intrinsically in terms of variations of the classical Cosserat strains and their spatial derivatives, without requiring a three-dimensional parent material or being restricted to elastic materials. Objectivity and insensitivity to rigid virtual displacements characterize the admissible internal-work density, while localization yields the dynamic field equations, higher-order contact interactions, natural boundary conditions, and balances of linear and angular momentum. For inextensible and unshearable rods, the constrained principle determines a unique multiplier functional, although its representation by reactive force and double-force fields possesses a gauge freedom absent from classical rod theory. We also formulate objective elastic, hyperelastic, and viscoelastic constitutive laws, extend transverse hemitropy, transverse isotropy, and flip symmetry to strain-gradient rods, and derive invariant representations of symmetric stored energies. As an application, we analyze a planar inextensible cantilever subjected to a compressive terminal force, with stored energy depending on the spatial derivative of curvature. We prove that the eigenvalues of the linearized problem are simple and coincide precisely with the loads at which bifurcation from the straight state occurs. In addition, each bifurcation point generates a connected global branch of nonlinear equilibria that is locally analytic near its origin.
ER -