Analyzing the free vibration behavior of nano-scaled beams subjected to axial loading based on the nonlocal integral elasticity theory

Excellent mechanical, chemical, electrical and thermal properties of nano-materials have made them one of the most interesting fields in the research and industrial centers. The mechanical behavior of these materials can be studied using experiments, atomistic models and continuum mechanics models. Eringen’s nonlocal elasticity theory is the most popular size-dependent continuum theory which has been used by several researchers to study the mechanical behavior of nano-scaled structures. According to nonlocal theory, stress at one point is a function of all strains in the vicinity. In the current article, the free vibration of Euler-Bernoulli nano-scaled beams under an axial compressive load is studied. For this purpose, a finite element method is developed using the principle of total potential energy in conjunction with the nonlocal integral elasticity theory. In addition, the governing equations have been derived and by solving the corresponding eigenvalue problem, the natural frequencies have been obtained. Different geometries and various boundary conditions (including the essential and natural boundary conditions) can be considered in the currently proposed method, properly. The effects of nonlocal parameter, axial compressive loads and various boundary conditions on the vibration behavior of the nano-scaled Euler-Bernoulli beams are studied.

Keywords:

nonlocal elasticity theory , finite element method , Euler-Bernoulli beam , free vibration , axial compressive loading