Modeling the Size-Dependent Nanostructures: Incorporating the Bulk and Surface Effects

Abstract

To precisely model the size dependencies in nanostructures, the surface effect and bulk effect are incorporated. From the physical point of view, size dependencies stem from not only the surface, but also the bulk. The surface energy theory and strain gradient elasticity theory are introduced to characterize the surface effect and bulk effect, respectively. The new models for Bernoulli-Euler and Timoshenko beams are developed. Governing equations, initial conditions, and boundary conditions are derived simultaneously by using Hamilton’s principle. The new models, incorporating the Poisson effect, contain three material length scale parameters and three surface elasticity constants to capture the size effect in the bulk and surface layer of the beam, respectively. The models recover the models, where either the bulk effect or the surface effect is considered, and also can degenerate into the corresponding modified couple stress models or the classical models when some constants are ignored. In addition, the new Timoshenko beam model recovers the new Bernoulli-Euler beam when shear deformation is ignored. To illustrate the new models, the static bending and free vibration problems of the simply supported nanoscale Bernoulli-Euler and Timoshenko beams are solved, respectively. Numerical results reveal that the differences in the deflection, rotation, and natural frequency predicted by the present model and the other models are large when the beam thickness is small. These differences, however, are decreasing or even diminishing with the increase in the size of the beams. The models may guide the precise design of nanobeam-based devices for a wide range of potential applications.