A new Timoshenko beam model incorporating microstructure and surface energy effects

Abstract

A new Timoshenko beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and complete boundary conditions for a Timoshenko beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure-and surface energy-dependent size effect. In addition, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, unlike existing Timoshenko beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as limiting cases and recovers the Bernoulli–Euler beam model incorporating the two effects as a special case. Also, the current model reduces to the classical Timoshenko beam model when the microstructure dependence, surface energy and Poisson’s effect are all suppressed. To demonstrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. In addition, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that given by the classical model, with the difference between them being significantly large for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally.