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The thermoelastic damping in a micro-beam of rectangular cross-section is studied using the Euler–Bernoulli beam theory incorporating Lord and Shulman's theory of generalized thermoelasticity with memory-dependent heat conduction. By employing the mode analysis method, explicit formulae for the thermoelastic damping, frequency shift and attenuation are derived. The kernel function, the relaxation time, and the memory time are selected to analyze  their effects on thermoelastic damping. The dependence of the inverse quality factoron the boundary conditions, vibration modes, time delay, aspect ratios, and relaxation time of small-scale mechanical resonators is shown for the memory-dependent heat conduction with ideal, linear, and nonlinear kernels. Numerical results of thefactor are calculated and compared with those based on the Lord–Shulman generalized thermoelasticity and the Lifshitz–Roukes model. The obtained numerical results show that the memory-dependent Lord–Shulman model is more accurate in describing thermoelastic damping than Lifshitz–Roukes' model and the classical Lord–Shulman model.

Thermoelastic damping in a micro-beam based on the memory-dependent generalized thermoelasticity