Asymptotically exact theory of functionally graded elastic beams

TL;DR

This paper develops an asymptotically exact one-dimensional theory for functionally graded elastic beams, enabling precise stiffness calculations and rigorous error estimation through variational-asymptotic methods.

Contribution

It introduces a novel variational-asymptotic approach that provides accurate 1D equations and error bounds for functionally graded beams with continuously varying properties.

Findings

Accurate effective stiffness calculations for extension, bending, and torsion.

Rigorous error estimates based on the Prager-Synge identity.

Validation of the asymptotic theory's limits of applicability.

Abstract

We construct a one-dimensional first-order theory for functionally graded elastic beams using the variational-asymptotic method. This approach ensures an asymptotically exact one-dimensional equations, allowing for the precise determination of effective stiffnesses in extension, bending, and torsion via numerical solutions of the dual variational problems on the cross-section. Our theory distinguishes itself by offering a rigorous error estimation based on the Prager-Synge identity, which highlights the limits of accuracy and applicability of the derived one-dimensional model for beams with continuously varying elastic moduli across the cross section.