Asymptotically-correct modified strain gradient modelling of micro and nano beams

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.