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This study proposes a computationally efficient approach about the dynamic response of a homogeneous, transversely isotropic, thermoelastic micro-beam resonator subjected to time-dependent thermal loading. Due to the shortcomings of power law distributions in fractional derivatives, the usage of some other forms of derivatives with few other kernel functions are proposed. With this motivation, the heat transport equation is defined in an integral form of a common derivative on a slipping interval by incorporating the three-phase-lag memory-dependent heat transfer which is modeled based on Euler-Bernoulli beam theory. The beam is assumed to be clamped-clamped conditions on its axial ends. Employing the Laplace transform technique, the analytical solutions have been obtained. Further, incorporating residual calculus, the inversion of the transformed solutions are performed. The analytical expressions for the deflection has been computed numerically for a Silicon micro-beam using Maple software. The new approach is able to portray material heterogeneities and the non-locality of the new kernel allows the full describing of the memory within structure and media with different scales, which cannot be described by classical fractional derivative and also that of the memory-dependent derivative.

Memory response in the vibration of a micro-scale beam due to time-dependent thermal loading