A mechanical approach to fractional non-local thermoelasticity

Abstract

In recent years fractional di erential calculus applications have been developed in physics, chemistry as well as in engineering elds. Fractional order integrals and derivatives ex- tend the well-known de nitions of integer-order primitives and derivatives of the ordinary di erential calculus to real-order operators. Engineering applications of these concepts dealt with viscoelastic models, stochastic dy- namics as well as with the, recently developed, fractional-order thermoelasticity [3]. In these elds the main use of fractional operators has been concerned with the interpolation between the heat ux and its time-rate of change, that is related to the well-known second sound e ect. In other recent studies [2] a fractional, non-local thermoelastic model has been proposed as a particular case of the non-local, integral, thermoelasticity introduced at the mid of the seventies [1]. In this study the autors aim to provide a mechanical framework to account for fractional, non-local e ects in thermoelasticity. A mechanical model that corresponds to long-range heat ux is introduced and, on this basis, a modi ed version of the Fourier heat ux equa- tion is obtained. Such an equation involves spatial Marchaud fractional derivatives of the temperature eld as well as Riemann-Liouville fractional derivatives of the heat ux with respect to time variable to account for second sound effects.