A New Framework for Unidimensional Structures Based on Generalised Continua

TL;DR

This paper develops a hierarchical family of beam models from higher-order elasticity, unifying classical and defect-inclusive models through different kinematic regimes, and demonstrates their applicability to material defects in beam structures.

Contribution

It introduces a comprehensive framework unifying various beam models based on generalized continua, including new non-holonomic models that incorporate defects like dislocations and disclinations.

Findings

Holonomic model reduces to higher-order Euler--Bernoulli beam.

Semi-holonomic model generalizes Timoshenko beam.

Non-holonomic model captures dislocations and disclinations.

Abstract

The present work introduces a family of beam models derived from a three-dimensional higher-order elasticity framework. By incorporating three kinematic fields - the macroscopic displacement u, the micro-distortion tensor P, and the third-order tensor N - the study systematically explores three regimes: holonomic, semi-holonomic, and non-holonomic. These regimes correspond to varying levels of kinematic constraints, ranging from classical elasticity to a fully relaxed model. The holonomic case reduces to a higher-order Euler--Bernoulli beam model, while the semi-holonomic case generalises the Timoshenko beam model. The non-holonomic case provides a unified framework that naturally incorporates both dislocations and disclinations. Furthermore, the holonomic and semi-holonomic models are shown to emerge as singular limits of the non-holonomic model by increasing specific penalty coefficients. Simplified ordinary differential equation systems are derived for specific cases, such as pure traction and bending, illustrating the practical applicability of the models. The results highlight the hierarchical structure of the proposed framework and its ability to capture material defects in beam-like structures.