Generalized thermoelastic dissipation in micro/nano-beams with two-dimensional heat conduction

The precise prediction of quality factor based on thermoelastic dissipation (TED) is of great importance for the optimal design of micro/nano-resonators. In this work, employing the nonlocal-dual-phase-lag (NDPL) model and the modified-couple-stress (MCS) model, analytical TED expressions are developed for the first time for micro/nano-beam resonators with two-dimensional (2D) heat conduction along the thickness and length directions. Initially, the governing equation of NDPL coupled thermoelasticity is established, and then the solutions of 2D fluctuation temperature are obtained by utilizing the Galerkin approach. Subsequently, the natural frequency involving the MCS size-dependence length-scale parameter is achieved based on the governing equation of vibration. Finally, analytical generalized TED models are attained according to the energy-definition approach. Three representative types of constraint boundary conditions for the beam resonator, namely, clamped-clamped, clamped-free, and simply supported, are considered in this study. And the present developed TED models are confirmed to possess perfect generality and excellent convergence performance. Moreover, an expression of equivalent thermal relaxation time in the 2D case is introduced and expressed in terms of the length- and thickness-directional thermal relaxation times. Two typical materials, silicon and gold, are selected in the simulation part. The exploration of fluctuation temperature, TED spectra, and thermal relaxation times considering multiple physical effects is conducted as an emphasis. Results reveal that the effect of heatconduction dimension is crucial for micro/nano-beams with a small ratio of length to thickness, while the significant NDPL effect is dependent upon the ultra-high vibration frequency. Additionally, the MCS sizedependence effect can enhance the natural frequency and improve the quality factor of micro/nano-beam resonators.