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 )