Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method

This study investigates the free vibration behavior of a double cracked nanobeam composed of bi-directional functionally graded material. The analysis incorporates Eringen's nonlocal elasticity theory and the Euler-Bernoulli theory. The material properties are considered to vary in both the thickness and length directions. The cracked nanobeam is modeled as a series of interconnected sub-beams, with rotational springs placed at the cracked sections. This modeling approach accounts for the discontinuities in rotational displacement resulting from bending, which is directly related to the bending moment transmitted by the cracked section. The problem is solved using the differential quadrature method, which approximates the derivatives of the field quantities by employing a weighted linear sum of the nodal values. By doing so, the problem is transformed into a linear algebraic system. Various supporting cases are examined, and a parametric study is conducted to analyze the impact of the axial and transverse gradient indices, nonlocal parameter, and crack severity on the obtained results.