arXiv subjects

R. S. Hallyburton

Publications and source records attributed to R. S. Hallyburton.

3 recordsLinked to original sources

arXiv2026

A direct theory of strain-gradient special Cosserat rods via the principle of virtual work

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 a stored energy. 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 study a planar inextensible cantilever under a compressive terminal force. 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.

math-ph↗

arXiv2025

Shell energies derived from three-dimensional isotropic strain-gradient elasticity

We derive a class of two-dimensional shell energies for thin elastic bodies exhibiting small-length scale effects modeled via strain-gradient elasticity. Building on the final author's earlier work on plate models, the kinetic and stored surface energies arise as the leading cubic order-in-thickness expressions for three-dimensional kinetic energies with velocity-gradient effects and a broad class of isotropic stored energies, each possessing an intrinsic length scale $\ell$. These include both classical Toupin-Mindlin and more recent dilatational strain-gradient elastic stored energies. A key insight of this work is that consistent asymptotic reductions of strain-gradient theories necessarily begin at cubic order-in-thickness due to the natural scaling assumption $\ell = O(h)$ where $h$ is the thickness of the body. In the limit as the intrinsic length scales vanish, the theory reduces to Koiter's classical shell energy. We illustrate the theory using the shell energy derived from dilatational strain-gradient elasticity, computing the body force, edge tractions and edge double force densities required to support a variety of finite deformations.

math-ph↗

arXiv2021

New Estimates on the bounds of Brunel's operator

We study the coefficients of the Taylor series expansion of powers of the function $ψ(x)=\frac{1-\sqrt{1-x}}{x}$, where the Brunel operator $A\equiv A(T)$ is defined as $ψ(T)$ for any mean-bounded $T$. We prove several new precise estimates regarding the Taylor coefficients of $ψ^n$ for $n\in\mathbb{N}$. We apply these estimates to give an elementary proof that for any mean-bounded, not necessarily positive operator $T$ on a Banach space $X$, the Brunel operator $A(T):X\to X$ is power-bounded and satisfies $\sup_{n\in\mathbb{N}} \|n(A^n-A^{n+1})\| < \infty$ (equivalently, $A(T)$ is a Ritt operator). Along the way we provide specific details of results announced by A. Brunel and R. Emilion in \cite{Brunel}.

math.DS↗