Thermoelastic Damping In Micro/nano Resonators Based On Nonlocal Elasticity And Nonlcal Heat Conduction

Resonators are important components in Micro/nanoelectromechanical system(MEMS/NEMS).with their small size,they can oscillate at very high resonant frequency,which enables them excellent sensing and detection ability.Quality factor is a key parameter for resonator,which is defined as the ratio of total energy stored by the system to the average energy dissipation per cycle.The improvement of quality factor can make resonators have higher sensitivity and frequency selectivity,and then improve the accuracy and performance of sensors.Thermoelastic damping(TED)is an important energy dissipation mechanism.For resonators operating under vacuum,TED is the main source of dissipation,which determines the upper limit of quality factor.Hence,it is meaningful to study the TED behaviors of resonators systematically.Establishing TED models for classical resonant devices can quickly predict the TED,and guide the design of high-quality resonator.Up to now,researchers have established a relatively mature TED model framework based on classical continuum mechanics and classical Fourier heat conduction law.However,in order to improve the sensitivity of sensors and integrate more functions in the smaller package,the size of the micro/nano resonator is becoming smaller.The classical continuum theory is not capable of explaining the mechanical properties of micro/nano structure,and the classical Fourier’s law cannot describe the internal heat transfer process of micro/nano structure.In order to predict the TED of resonators more accurately,the influence of mechanical and thermal size effects should be considered.The objective of this paper is to develop analytical TED models for typical micro/nano resonators based on the nonlocal strain gradient theory and nonlocal heat conduction model.Firstly,based on the nonlocal strain gradient constitutive relation,the equation of motion of beam/plate along the thickness direction is derived by micro element force analysis or Hamilton principle.According to the classical boundary conditions and the constitutive boundary conditions,the mode shape and natural frequency of the beam/plate are obtained.Secondly,the periodic volume strain functions as the internal heat source and the heat conduction equation of plate/beam is obtained based on the nonlocal heat conduction model,the temperature field of resonator can be obtained by the thermal mode superposition method.Finally,according to the obtained temperature field function,stress and strain components,the expressions of heat dissipation and total strain energy are calculated,and then the analytical solution of TED is obtained through the energy approach.In the analysis of numerical results,the simply supported beam,fully clamped beam,cantilever beam,simply supported plate and fully clamped plate are considered.By comparing with the predictions of the classical TED model,the accuracy of the obtained model is verified.The influences of various size effects on TED under different thickness,frequency and boundary conditions are analyzed.The results show that the strain gradient effect leads to TED reduction for resonators of any size operating at any frequency.While the effect of nonlocal elasticity on TED depends on the relative position between the natural frequency and the critical frequencies of TED peaks and valleys.The single peak in the TED spectrum is split into two due to the spatial and temporal nonlocality of heat conduction.The two peaks show opposite trends with the increase of phonon mean free path;The nonlocal heat conduction effect is more significant in sub-micron scale working around the critical frequency of TED peaks;For micro/nano resonators with different structural boundary conditions and fixed length scale parameters,the maximum of TED changes little.