ehsan allahyari - PhD candidate, Faculty of Mechanical Engineering, K. N. Toosi University of Technology
masoud asgari - assistam‘ professor; Faculty of Mechanical Engineering, K N. Toosi university of Technology,
mahshid mahbod - Msc student, Faculty of Mechanical Engineering, K. N. Toast University of Technology

In this article, it is focused on the purely analytical technique called multiple scales, which, when combined with a nonlocal strain gradient theory, yield very powerfuland versatile technique. The multiple-scale analysis is one of the well-known methods for solving various nonlinear problems analytically and nonlocal strain gradient theoryis proposed to introduce both of the length scales into a single theory. Surface elasticity theory of Gurtin and Murdoch is also employed to consider small scale effectby presenting the nonlocal parameter and the surface effects at the top and bottom of the bulk part of the nanoplate as a membrane with different mechanicalproperties. The equations of motion can be obtained by Von Karnian nonlinear strain—displacement relationship and introducing the airy stress function. The presentedmethod is verified by comparing the results with their counterparts reported in open literature and a good agreement is observed for two different boundaryconditions. Effects of various parameters such as nonlocal parameters, surface effects, mode number, temperature change, elastic medium coefficients for two kinds ofboundary conditions are also discussed.

Analytical technique, Multiple scales, Nonlocal strain gradient theory, Surface effects