Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory

Present investigation deals with the free vibration characteristics of nanoscale-beams resting on elastic Pasternak\'s foundation based on nonlocal strain-gradient theory and a higher order hyperbolic beam model which captures shear deformation effect without using any shear correction factor. The nanobeam is lying on twoparameters elastic foundation consist of lower spring layers as well as a shear layer. Nonlocal strain gradient theory takes into account two scale parameters for modeling the small size effects of nanostructures more accurately. Hamilton\'s principal is utilized to derive the governing equations of embedded strain gradient nanobeam and, after that, analytical solutions are provided for simply supported conditions to solve the governing equations. The obtained results are compared with those predicted by the previous articles available in literature. Finally, the impacts of nonlocal parameter, length scale parameter, slenderness ratio, elastic medium, on vibration frequencies of nanosize beams are all evaluated.