Thermal vibration analysis of nanoplates based on the higher-order nonlocal strain gradient theory by an analytical approach

In this paper, a new formulation for analyzing free vibration of thin rectangular nanoplates under different thermal conditions is obtained based on the higher-order nonlocal strain gradient theory. Governing equations and non-classical boundary conditions of the nanoplate are derived by using the variational approach. The exact solution is obtained as a function of higher-order and lower-order nonlocal parameters, strain gradient length scale and temperature difference using Navier solution procedure. The influences of small-scale parameters on the vibrational behavior of the nanoplate are investigated for various thermal conditions. High and low temperature conditions are considered to study the effects of changes in temperature and small-scale parameters. It has been shown that increasing the nonlocal parameters decrease the natural frequency of the nanoplate, while increasing the strain gradient length scale will increase it. Also, the natural frequency of the nanoplate will increase by increasing the temperature difference in low temperature conditions, but it will decrease by increasing the temperature difference in high temperature conditions. Non-uniform behaviors are reported for some cases and softening effect and hardening effect are studied. To validate the solutions, the results are compared with previous researches.