Analysis of thermoelastic damping in a microbeam following a modified strain gradient theory and the Moore-Gibson-Thompson heat equation

It has been proven that mechanical elements display size-dependent behavior in structural and thermal fields at microscales. It has also been found that thermoelastic damping (TED) is one of the dominant reasons in confining the quality factor (Q-factor) of such elements. This paper aims to develop a novel formulation for evaluating TED in microbeams by accounting for the size effect on the mechanical and thermal areas via the nonclassical theory of modified strain gradient (MSG) and the non-Fourier heat conduction model of Moore-Gibson-Thompson (MGT). In the first step, the heat equation for beams is derived within the framework of MGT model. Through this equation, the function of temperature fluctuation can be obtained. Then, the constitutive relations of the beam according to MSG theory (MSGT) are extracted. By using the temperature distribution and nonclassical constitutive relations obtained, the maximum amounts of potential and wasted thermal energies during one cycle of beam vibration are calculated. Finally, by placing the value of these energies in the existing relationship for computing the value of TED, an explicit expression for TED is presented. With the aim of clarifying the sensitivity of TED value to the characteristic parameters of MSGT and MGT model, a variety of numerical data are provided. According to the obtained outcomes, the inclusion of size effect in the structural and thermal equations can cause a remarkable difference compared to the classical model. The dependency of TED on some factors like beam thickness and aspect ratio, vibration mode number and material of the beam is also investigated numerically.