Two-dimensional modeling of thermoelastic damping in small-sized circular plates with size-dependent behavior in both mechanical and thermal areas

Abstract

Within the realm of micro/nanomechanics, thermoelastic damping (TED) is acknowledged as a contributing factor to energy dissipation in mechanical structures. Consequently, the development of an accurate model for this phenomenon is crucial to get the best performance out of extremely small resonators. Considering the superiority of multi-dimensional heat transfer modeling compared to one-dimensional (1D) modeling as well as the definiteness of the size effect on both mechanical and thermal fields, this paper establishes a mathematical framework to appraise TED in circular micro/nanoplates with two-dimensional (2D) heat conduction by leveraging the capabilities of the modified couple stress theory (MCST) and Moore–Gibson–Thompson (MGT) heat equation. To initiate the investigation, the constitutive and heat equations for circular plates are formulated based on the MCST and MGT model. Through the solution of 2D heat equation, the spatial distribution of temperature within the plate is determined. Employing the previously obtained couple stress-based constitutive equations and temperature distribution function, the mathematical expressions of wasted and elastic energies within a single vibration cycle are derived. Ultimately, the substitution of the derived expressions into the formula of energy dissipation (ED) approach yields an infinite series solution for the computation of TED in small-sized circular plates. Following a comparative analysis of the proposed model with prior studies, convergence studies are conducted to identify the optimal number of terms needed for trustworthy outcomes. A range of numerical results with the aid of simulated model are also prepared, with a focus on analyzing the distinctions between 1D and 2D models as well as the implications of using the MCST and MGT model. The findings betoken apparent deviations between the predictions of the new 2D size-dependent model and the 1D traditional formulation, especially for tiny and relatively thick circular plates.