Quantum Effects on Thermally Induced Vibrations of Functionally Graded Perforated Nanobeams

ABSTRACT

This article presents a quantum‐mechanical analysis of thermally induced free vibrations of functionally graded (FG) perforated nanobeams. A unified analytical framework is developed by combining Euler–Bernoulli beam theory, nonlocal strain gradient theory, and quantum statistical mechanics to simultaneously account for material gradation, perforation, nanoscale effects, and quantum effects. The governing vibration equation is derived using Hamilton's principle and solved analytically for simply‐supported FG perforated nanobeams using Navier's method. Quantum statistical mechanics is employed to evaluate the modal and root mean square (RMS) vibration amplitudes. Quantum statistics are particularly relevant at nanoscale dimensions and low temperatures, where the discrete nature of vibrational energy becomes significant and classical statistical mechanics may become inadequate. The formulation is validated against available literature results for special cases, showing excellent agreement. A detailed parametric study examines the effects of gradient index, number of hole rows, filling ratio, nanoscale parameters, temperature, and nanobeam dimensions. The results show that quantum theory consistently predicts lower RMS amplitudes than classical theory. Moreover, quantum formulation converges with considerably fewer vibration modes, demonstrating its computational efficiency. The proposed framework provides an efficient analytical tool for quantum thermal vibration analysis of FG perforated nanobeams and offers useful insights for nanobeam based nanoelectromechanical systems.