A discrete mechanical model of fractional hereditary materials

Fractional hereditary materials are characterized for the presence, in the stress-strain relations, of fractional-order operators with order beta a[0,1]. In Di Paola and Zingales (J. Rheol. 56(5):983-1004, 2012) exact mechanical models of such materials have been extensively discussed obtaining two intervals for beta: (i) Elasto-Viscous (EV) materials for 0a parts per thousand currency sign beta a parts per thousand currency sign1/2; (ii) Visco-Elastic (VE) materials for 1/2a parts per thousand currency sign beta a parts per thousand currency sign1. These two ranges correspond to different continuous mechanical models. In this paper a discretization scheme based upon the continuous models proposed in Di Paola and Zingales (J. Rheol. 56(5):983-1004, 2012) useful to obtain a mechanical description of fractional derivative is presented. It is shown that the discretized models are ruled by a set of coupled first order differential equations involving symmetric and positive definite matrices. Modal analysis shows that fractional order operators have a mechanical counterpart that is ruled by a set of Kelvin-Voigt units and each of them provides a proper contribution to the overall response. The robustness of the proposed discretization scheme is assessed in the paper for different classes of external loads and for different values of beta a[0, 1].