On the Paradox of Anomalous Relative Bending Stiffness of Ultrathin Beams in the Gradient Theory of Elasticity

Abstract

The problem of refined modeling of ultrafine rods, which arose in connection with the need to explain the known experimental data on the significant dependence of the bending stiffness of such ultrathin structures on their thickness if the thickness becomes very small, commensurate as some authors believe with the characteristic dimensions of the material microstructure, is considered. To simulate such effects, Kirchhoff’s theory of thin rods uses gradient theories, nonlocal, micropolar elasticity theories, including scaled construction parameters. However, the obtained simulation results are very contradictory; the question of the reliability of the obtained results and the nature of the scale-dependent effect of the effective bending properties of ultrathin rods remains unresolved. It is shown in the paper that these effects for Kirchhoff and Timoshenko rods can be explained by taking into account the surface properties for ultrathin rods (plates).