Modelling the flexural waves in a nanoplate based on the fractional order nonlocal strain gradient elasticity and thermoelasticity

Abstract

The traveling and the standing flexural waves in an infinite homogeneous elastic nanoplate are studied based on the nonlocal strain gradient elasticity and the thermoelasticity in present work. The spatial and temporal fractional order differential are introduced to reflect the spatial nonlocal and the history-dependent feature of the thermoelastic behavior of material in the small scale. It is a natural extension of the nonlocal strain gradient elasticity and thermoelasticity with integer order. The mathematical model of generalized thermoelasticity with three-phase lag and the fraction order derivatives are used. A new deflection-dependent approximation solution is first proposed to deal with the thermo-elastic coupling in the present work which leads to the frequency-dependent temperature distribution along thickness. The frequency equations of the traveling and standing flexural waves are obtained by two approaches and compared. The dispersive and the attenuation feature of the traveling and standing flexural waves are analyzed based on the numerical results. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, strain gradient parameter, fractional order and thermal parameters on the propagation characteristics of flexural waves. It is found that the spatial and temporal fraction order have different influences on the dispersion and attenuation of flexural waves and the appropriate combination of the spatial and temporal fraction orders can model more complex thermoelastic behavior of nanoplate. Besides, the deflection-dependent approximation solution of temperature distribution reported in the present work is applicable when the wavelength is much greater than the thickness of nanoplate.