Longitudinal Vibration of Defected Single-Layer Nanoplate Using Nonlocal Elasticity Theory (Contemporary Mathematics, 2025)
This article deals with the longitudinal vibrations of a defective single-layer nanoplate. Based on the nonlocal elasticity theory, Kirchhoff plate theory, and von Kármán theory, the longitudinal vibration is investigated. The coupled governing equations for the nonlocal elastic displacement and defect concentration fields are derived by means of Eringen's principle. The analytical presentation of the natural frequency of the defective single-layer nanoplate for longitudinal vibration modes is presented. Moreover, the natural frequencies were determined for various values of the defection and nonlocal elasticity factors. The results reported that an increase in the nonlocal parameter increases the frequencies, while the increase in the defect concentration decreases the natural frequencies of the nanoplate for different values of the half wave number throughout the computation frequencies of the nanoplate. The present work is an attempt to learn more about the effects of the small-scale length and defection factors on the longitudinal vibration behavior of single-layer nanoplates. To the authors' knowledge, these effects have not yet been studied deeply. However, longitudinal vibrations are critical for nanoplate applications because they directly influence the mechanical stability, dynamic response, and functional performance of nanoscale devices. Finally, the longitudinal vibrations determine the natural frequencies and resonant behaviors of nanoplates, affecting their ability to serve as key components in applications such as sensors, actuators, and nano-oscillators.
Keywords: nonlocal elasticity theory; Kirchhoff plate theory; von Kármán theory; Eringen's principle; longitudinal vibrations; defected single-layer nanoplate