Thermal vibration and buckling analysis of two-phase nanobeams embedded in size dependent elastic medium

Abstract

In this paper, vibration and buckling of two-phase nanobeams embedded in size dependent elastic medium and under thermal load are analyzed. Due to paradoxes of common differential nonlocal elasticity, such as neglecting the size effect of uniform loads, failure to satisfy the constitutive boundary conditions, associated to transformation of integral nonlocal equation to differential one, and incompatibility between the results of differential nonlocal with those of integral nonlocal, the size dependent effects of nanobeam, elastic medium and thermal load are taken into account simultaneously by using two-phase local/nonlocal Eringen's elasticity, for the first time. Governing equations and corresponding boundary conditions are derived using Hamilton's principle. To obtain natural vibration frequencies as well as critical buckling temperature, three different methods of solution are presented, i.e. exact solution, Generalized Differential Quadrature Method (GDQM) and Finite Element Method (FEM) which is based on the integral form of two-phase elasticity. Several comparison studies are conducted to examine the validity of the present formulation and results. The effects of applying two-phase elasticity on elastic medium and thermal load in vibration and buckling of nanobeams with different boundary conditions are investigated in details. Differences appeared in present results, especially in the cases with higher temperature and nonlocality as well as stiffer elastic medium, reveal that the size dependency of elastic medium and uniform thermal load, which is neglected by differential nonlocal, must be considered by employing two-phase elasticity.