Closed‐Form and Numerical Solutions for Accelerated Thermally Induced Mesoscopic Vibrations of a Silicon Microbeam

ABSTRACT Silicon exhibits a dependence on size and nonlocal behavior in heat conduction at the nano‐/micro‐scale or at specific temperatures, where the main carriers of heat, known as phonons, travel anomalously instead of through the usual diffusive process. This characteristic can result in thermal conductivity that is either greater or less than expected, depending on the material's size and structure. In this study, the coupled thermally induced vibrations of a rectangular microbeam made of silicon are accommodated in the context of crossover thermoelasticity that assumes the temporal crossover of thermal conduction with acceleration, in the sense that the thermal conductivity starts low and becomes relatively high eventually. The Euler–Bernoulli assumption is considered, and the only cause of the vibrations is the given temperature gradient on the upper surface of the beam. Closed‐form solutions for the lateral deflection and the thermal moment, in the case of a quasistatic Euler–Bernoulli beam, are derived in terms of the generalized Mittag–Leffler functions. For dynamic vibration, a numerical technique based on inverting the Laplace transform is employed to bring the solutions in the real domain. The thermally induced mesoscopic vibrations are found to be accelerated following the thermal conduction transition. Furthermore, the presence of low thermal conductivity in the thermoelastic model is found to be activated on the microscopic scale.