Fractional-order theory of thermoelasticicty. I: generalization of the Fourier equation

The paper deals with the generalization of Fourier-type relations in the context of fractional-order calculus. The instantaneous temperature-flux equation of the Fourier-type diffusion is generalized, introducing a self-similar, fractal-type mass clustering at the micro scale. In this setting, the resulting conduction equation at the macro scale yields a Caputo's fractional derivative with order β ∈[0, 1] of temperature gradient that generalizes the Fourier conduction equation. The order of the fractional-derivative has been related to the fractal assembly of the microstructure and some preliminary observations about the thermodynamical restrictions of the coefficients and the state functions related to the fractional-order Fourier equation have been introduced. The distribution and temperature increase in simple rigid conductors have also been reported to investigate the influence of the derivation order in the temperature field.