The small length scale effect for a non-local cantilever beam: a paradox solved

Abstract

Non-local continuum mechanics allows one to account for the small length scale effect thatbecomes significant when dealing with microstructures or nanostructures. Thispaper presents some simplified non-local elastic beam models, for the bendinganalyses of small scale rods. Integral-type or gradient non-local models abandonthe classical assumption of locality, and admit that stress depends not only onthe strain value at that point but also on the strain values of all points on thebody. There is a paradox still unresolved at this stage: some bending solutions ofintegral-based non-local elastic beams have been found to be identical to the classical(local) solution, i.e. the small scale effect is not present at all. One example is theEuler–Bernoulli cantilever nanobeam model with a point load which has application inmicroelectromechanical systems and nanoelectromechanical systems as an actuator. In thispaper, it will be shown that this paradox may be overcome with a gradient elastic modelas well as an integral non-local elastic model that is based on combining thelocal and the non-local curvatures in the constitutive elastic relation. The lattermodel comprises the classical gradient model and Eringen’s integral model, and itsapplication produces small length scale terms in the non-local elastic cantilever beamsolution.