Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium

In the present study, free transverse vibration of a cracked functionally graded (FG) size dependent Timoshenko nanobeam which is resting on polymer elastic foundation is investigated. It is supposed that the material properties of the FG nanobeam are varying continuously across the thickness according to the power-law distribution. To considering the small scale effect, the Eringen’s nonlocal theory is used and for accounting the effect of polymer elastic foundation, the Winkler model is proposed. For this purpose, the equations of motion of the FG Timoshenko nanobeam and boundary conditions are obtained by using Hamilton’s principle. To find the analytical solutions for equations of motion of FG nanobeam, the separation of variable method is employed. Two cases of boundary conditions i.e., simply supported-simply supported (SS) and clamped–clamped (CC) are investigated in the present work. Numerical results are demonstrating the good agreement between the results of present article and some cases available in the literature. The emphasis of the present study is based on investigating the effect of various parameters such as crack severity, crack position, gradient index, mode number, nonlocal parameter, elastic foundation parameter and nanobeam length. It is clearly revealed that the vibrational behavior of a FG nanobeam is significantly depending on these effects. Also, these numerical results can be serving as benchmarks for future studies of FG nanobeams.