City Research Online

Abstract

This paper introduces a novel approach to model length-scale effects in micro and nano beams with square and rectangular cross-sections by using the dimensional reduction technique and variational asymptotic method (VAM). Traditional continuum approaches have been advanced by accounting for the length-scale effects, which have gained significant importance in micro- and nano-scale applications like MEMS and NEMS devices, in recent years. In this context, an asymptotically-correct modified strain-gradient beam theory has been developed by incorporating higher-order length-scale parameters through VAM, integrating the strain-gradient theory within the VAM framework for the first time. The formulation is focused on rectangular and square cross-section beams, whereby an originally three-dimensional structure is reduced to an energy-equivalent one-dimensional beam using the modified strain-gradient theory as the underlying constitutive behaviour. A systematic investigation has been conducted to assess the effect of the choice of the order of the material length-scale parameter, providing considerable insights into the appropriateness of the choice regarding the beam behaviour. It is shown that choosing the order of the length-scale parameter as O (
l
) = O (
b
) is the most appropriate, where
l
is the material length-scale parameter, and
b
is the width of the beam. Other two possible choices, O (
l
) = O (
L
),
L
being the beam length and O (
l
) << O (
b
) are shown to be inconsistent because the former ignores the classical strain energy contributions whereas the latter ignores the strain gradient terms. Hence, using O (
l
) = O (
b
), the zeroth- and first-order strain-gradient beam theories have been derived based on the VAM-based dimensional reduction procedure. The results are compared with those available in the literature. The developed theory captures the stiffening effect of the length-scale parameters and establishes an asymptotically-accurate beam model that includes higher strain-gradient effects. The proposed beam theory is applied to carbon nanotube (CNT)-reinforced composite micro beams, to investigate the length-scale-dependent behaviour of the micro beams when subjected to bending loads.

Publication Type:	Article
Additional Information:	Copyright The Author(s) 2026. This the author accepted manuscript of an article published by Sage in Mathematics and Mechanics of Solids. The final version is available online at: https://doi.org/10.1177/10812865261417688
Publisher Keywords:	Strain-gradient theory, variational asymptotic method, micro- beam, analytical model, length-scale parameter, carbon nano tube
Subjects:	Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General) T Technology > TH Building construction
Departments:	School of Science & Technology School of Science & Technology > Department of Engineering
SWORD Depositor:	Symplectic Administrator

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