Thermal and magnetic effects on the vibration of a cracked nanobeam embedded in an elastic medium

Abstract

In this study, we develop a model to describe the free vibration behavior of a cracked nanobeam embedded in an elastic medium by considering the effects of longitudinal magnetic field and temperature change. In order to take into account the small-scale and thermal effects, the Euler-Bernoulli beam theory based on the nonlocal elasticity constitutive relation is reformulated for one-dimensional nanoscale systems. In addition, the effect of a longitudinal magnetic field is introduced by considering the Lorenz magnetic force obtained from the classical Maxwell equation. To develop a model of a cracked nanobeam, we suppose that a nanobeam consists of two segments connected by a rotational spring that is located in the position of the cracked section. The surrounding elastic medium is represented by the Winkler-type elastic foundation. Influences of the nonlocal parameter, stiffness of rotational spring, temperature change and magnetic field on the system frequencies are investigated for two types of boundary conditions. Also, the first four mode shape functions for the considered boundary conditions are shown for various values of the crack position.