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 β∈[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 β: (i) Elasto-Viscous (EV) materials for 0≤β≤1/2; (ii) Visco-Elastic (VE) materials for 1/2≤β≤1. 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 β∈[0, 1].

Appendix: Fractional calculus

In this appendix we introduce some fundamental concepts on fractional calculus.

The fractional calculus is the natural extension of ordinary differential calculus. In fact, it extends the concepts of derivation and integration to non-integer and complex order.

The fractional calculus was born in the 1695 when G.W. Leibniz introduced the half derivate concept in a note to G. de l’Hôpital [24]. Subsequent studies have focused by different mathematicians [25]: J.B.J. Fourier, P.S. Laplace, L. Euler, S.F. Lacorix, N.H. Abel, etc.

The first definition of fractional operator is probably attributable to J. Liouville, who in 1832 gave the impulse to research by formulating the definition of fractional derivative of exponential function. In 1847, an important contribution was given by G.F.B. Riemann, who introduced their own definition of fractional integral. Following, N.Ya. Sonin unified formulations of Liouville and Riemann from multiple Cauchy integration formula [25], obtaining the following expression of fractional integral:

(A.1)

Equation (A.1) is known in literature as a fractional integral of Riemann-Liouville, since ℜ(β)>0, and it is valid for β∈ℂ.

To obtain the Riemann-Liouville fractional derivative just think that the derivative of order n can be considered as the derivative of order n+m of the mth primitive function, and then generalizing, we have:

$$ \bigl(\mathrm{D}_{a^+}^{\beta}f \bigr) (t)=\frac{1}{\varGamma (n-\beta )} \biggl({\frac{d}{dt}} \biggr)^n\int_a^t{ \frac{f(\tau)}{(t-\tau )^{\,\beta-n+1}}}\,d\tau $$

(A.2)

valid for (n−1)<ℜ(β)<n.

Another definition of fractional integro-differential operator was provided in 1967 by M. Caputo [26]. This definition is easier to handle for the solution of physical problems. The Caputo fractional derivative has the following expression:

$$ \bigl(_{\mathrm{C}}\mathrm{D}_{a^+}^{\beta}f \bigr) (t)= \frac{1}{\varGamma (n-\beta )}\int_a^t {\frac{f^{(n)}(\tau)}{{(t-\tau)^{\,\beta +1-n}}}} \,d\tau $$

(A.3)

Equation (A.3) is valid for (n−1)<β<n. The expression obtained is the result of an interpolation between the integer order derivatives, in fact, for β→n, the expression becomes an nth derivative of f(t).

It can be observed that the expressions (A.3) and (A.2) coincide if we start from initial conditions zero (f(a)=0).

Another definition of fractional operator, which is suitable for the techniques of discretization, it’s known as Grünwald-Letnikov differintegral [25] and is given as

(A.4)

Equation (A.4) defines in the same time two different operators, fractional derivate (for β>0) and fractional integral (for β<0).

Many definitions of fractional operators exist but are not reported for brevity sake’s. For in-depth studies look at previous citied books and [27–30].