Fractional differential equations of multiphase hereditary materials and exact mechanical models

Creep and relaxation tests, performed on various materials like polymers, rubbers and so on are well-tted by power-laws with exponent 2 [0; 1] (Nutting (1921), Di Paola et al. (2011)). The consequence of this observation is that the stress-strain relation of hereditary materials is ruled by fractional operators (Scott Blair (1947), Slonimsky (1961)). A large amount of researches have been performed in the second part of the last century with the aim to connect constitutive fractional relations with some mechanical models by means of fractance trees and ladders (see Podlubny (1999)). Recently, Di Paola and Zingales (2012) proposed a mechanical model that corresponds to fractional stress-strain relation with any real exponent and they have proposed a description of above model (Di Paola et al. (2012)). In this study the authors aim to extend the study to cases with more fractional phases and to fractional Kelvin-Voigt model of hereditariness