Generalized fractional viscothermoelastic nanobeam under the classical Caputo and the new Caputo-Fabrizio definitions of fractional derivatives

In this paper, we introduce a new unified formula that governs two different definitions of fractional derivative; the classical Caputo definition and Caputo-Fabrizio's new definition. Hence, an analysis has been constructed for fractional viscothermoelastic, isotropic, and homogeneous nanobeams. The governing equations of the viscothermoelastic nanobeam have been constructed in the context of the non-Fourier heat conduction law with one relaxation of Lord and Shulman (L-S). Laplace transform has been applied, and its inversions have been calculated by using the Tzou method of approximation. The numerical results have been validated for a thermoelastic rectangular nanobeam of silicon as a case when it is subjected to ramp-type heating and simply supported. The fractional-order parameter based on the two types of fractional derivatives has significant impacts on all the studied functions except the temperature increment function. The results based on the two types of fractional derivatives are different, although they generate the same behavior of the thermomechanical waves. The ramp-time heat parameter has significant effects on all the studied functions.