Fractional-order constitutive equations in mechanics and thermodynamics

This chapter is devoted to the application of fractional calculus in mechanics of materials and thermodynamics. The use of fractional calculus in mechanics is related to the definition of fractional-order constitutive equations leading to the class of fractional hereditariness. In this regard, a brief description of the classical rheological models of material hereditariness and a comparison with the fractional elements are reported. It is shown that a rheological hierarchy corresponding to the fractional order stress-strain relation may be defined. Such a model provides a multi-scale mechanical picture of the power-law hereditariness and it leads toward an unique definition of material free energy. The chapter is also devoted to the investigation of the fractional-order Fourier equation. The analysis of the anomalous heat transfer has been conducted with the a multi-scale approach similar to that used in material hereditariness. Thermodynamic consistency of the model has been reported in terms of the irreversible entropy production.