Re-analysis of Euler-Bernoulli microbeam bending problems modelled after Eringen's strain-driven non-local theory, strain gradient theory, and the stress-driven theory: A comparative study

Micro/nano dimensioned structures such as carbon nanotubes (CNTs) and graphene have received widespread attention among researchers worldwide. Due to the superior material properties exhibited by such structures of very low dimensions, these structures have been identified as potential building blocks for next-generation advanced sensors, actuators, and transducers. At such a small scale of physical dimensions, a phenomenon called “size effects” starts to prevail and exerts an influence upon the constitutive behaviour of the micro/nanostructure as a consequence of inter-atomic interactions. Classical continuum mechanics fail to incorporate these effects, so non-classical continuum mechanics, essentially “size-dependent,” were developed. A plethora of non-classical continuum theories exist, out of which the present work lays its focus on three distinct theories, namely Eringen’s strain-driven differential non-local model, strain gradient theory, and the recently proposed stress-driven non-local model. Analysis of Euler-Bernoulli microbeam bending behaviour forms the core objective of this study. Here the numerical results of fundamental and derived quantities emanating from the conventional FEM, Strain gradient FEM (SG-FEM), and analytical solutions based on the stress-driven non-local theory are compared against each other, following which some interesting results are discussed.