Defects in unidimensional structures

TL;DR

This work extends generalized continuum mechanics to slender-beam theories by introducing a linearized, bidimensional, non-holonomic three-field framework with $u$, $P$, and $N$. It derives generalized Euler–Bernoulli and Timoshenko beam models, showing the EB case develops curvature linked to the third derivative $w'''$ with no torsion, while the Timoshenko model supports both curvature and torsion via non-holonomic terms. Variational calculus yields static-equilibrium relations and reveals a Helmholtz-type defect equation for the curvature, with asymptotic behavior when $c\ll b$. The results connect micromorphic-type defect measures to classical beam deformations, offering analytical insight into curvature-disclination and torsion-dislocation phenomena in slender structures and guiding defect-aware design.

Abstract

In a previous work of the first authors, a non-holonomic model, generalising the micromorphic models and allowing for curvature (disclinations) to arise from the kinematic values, was presented. In the present paper, a generalisation of the classical models of Euler-Bernoulli and Timoshenko bending beams based on the mentioned work is proposed. The former is still composed of only one unidimensional scalar field, while the later introduces a third unidimensional scalar field, correcting the second order terms. The generalised Euler-Bernoulli beam is then shown to exhibit curvature (i.e. disclinations) linked to a third order derivative of the displacement, but no torsion (dislocations). Parallelly, the generalised Timoshenko beam is shown to exhibit both curvature and torsion, where the former is linked to the non-holonomy introduced in the generalisation. Lastly, using variational calculus, asymptotic values for the value taken by the curvature in static equilibrium are obtained when the second order contribution becomes negligible; along with an equation for the torsion in the generalised Timoshenko beam.