A discrete mechanical model of fractional hereditary materials

Abstract

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. \ud \ud 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]

Modeling and stability analysis of a fractional-order Francis hydro-turbine governing system

“…In recent years, owning to the advantages of fractional order calculus in the strong dependence, memory and sufficient preponderance in modeling viscoelastic substance [20][21][22][23][24], many researchers have introduced it to model the mechanical behaviors of polymers, gels foams and the rheology of soft matter and biological tissues [25][26][27][28][29][30][31]. In 2008, Machado et al [32] stated that, while individual dynamics of each element has an integer-order nature, the global dynamics reveal the existence of both integer and fractional dynamics.…”

Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness

“…that has the same mathematical structures as the governing equation of the linear hereditary hierarchy described in [9,10]. Laplace transformation L[u] = u(s) of Equation (29) yields, after some straightforward manipulations:…”

“…The use of power-laws for creep and relaxation allow introducing constitutive stress-strain relations in terms of time non-local convolution integrals with power-law kernels yielding constitutive equations in terms of fractional differential calculus [5][6][7][8]. Among them, mechanical hierarchy has been introduced to capture the power-law evolution [9][10][11] and provide a mechanical description of real-order integro-differential operators [8]. Fractional-order operators are nowadays recognized as an interesting mathematical tool to model non-local problems in spatial coordinates [12][13][14][15][16][17] as well as in time parameter [18][19][20][21].…”

A geometrical multi-scale numerical method for coupled hygro-thermo-mechanical problems in photovoltaic laminates

“…Instead of using a Prony series representation, a fractional calculus approach [8, 9] is herein adopted to synthetically characterize those dependencies. This approach has been proved in [11] to be very effective for parameters identification.…”