A microstructure- and surface energy-dependent third-order shear deformation beam model

A new non-classical third-order shear deformation model is developed for Reddy-Levinson beams using a variational formulation based on Hamilton's principle. A modified couple stress theory and a surface elasticity theory are employed. The equations of motion and complete boundary conditions for the beam are obtained simultaneously. The new model contains a material length scale parameter to account for the microstructure effect and three surface elastic constants to describe the surface energy effect. Also, Poisson's effect is incorporated in the new beam model. The current non-classical model recovers the classical elasticity-based third-order shear deformation beam model as a special case when the microstructure, surface energy and Poisson's effects are all suppressed. In addition, the newly developed beam model includes the models considering the microstructure dependence or the surface energy effect alone as limiting cases and reduces to two existing models for Bernoulli-Euler and Timoshenko beams incorporating the microstructure and surface energy effects. To illustrate the new model, the static bending and free vibration problems of a simply supported beam loaded by a concentrated force are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that both the deflection and rotation of the simply supported beam predicted by the current model are smaller than those predicted by the classical model. Also, it is observed that the differences in the deflection and rotation predicted by the two beam models are very large when the beam thickness is sufficiently small, but they are diminishing with the increase in the beam thickness. For the free vibration problem, it is found that the natural frequency predicted by the new model is higher than that predicted by the classical beam model, and the difference is significant for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.