Stochastic Analysis of a Nonlocal Fractional Viscoelastic Bar Forced by Gaussian White Noise

Abstract

Recently, a displacement-based nonlocal bar model has been developed. The model is based on the assumption that nonlocal forces can be modeled as viscoelastic (VE) long-range interactions mutually exerted by nonadjacent bar segments due to their relative motion; the classical local stress resultants are also present in the model. A finite element (FE) formulation with closed-form expressions of the elastic and viscoelastic matrices has also been obtained. Specifically, Caputo's fractional derivative has been used in order to model viscoelastic long-range interaction. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the nonlocal fractional viscoelastic bar introduced in previous papers, discretized with the finite element method (FEM), forced by a Gaussian white noise. Since the bar is forced by a Gaussian white noise, dynamical effects cannot be neglected. The system of coupled fractional differential equations ruling the bar motion can be decoupled only by means of the fractional order state variable expansion. It is shown that following this approach Monte Carlo simulation can be performed very efficiently. For simplicity, here the work is limited to the axial response, but can be easily extended to transverse motion.

On the nonlocal bending problem with fractional hereditariness

“…In fact, in nonlocal problem the stress (or the strain) at a point of the continuum is function of the stress (or strain) field of the entire domain [28,29]. Size-effects at small scale [30], strain and stress localizations [31,32], anomalous waves dispersions in complex materials with marked microstructures [33,34], structures forced by long-range force field [35][36][37] are some examples where nonlocal phenomena cannot be neglected. Among the various nonlocal models, Eringen's formulation is probably the most famous one [38,39].…”

Dynamics of Nonlocal Rod by Means of Fractional Laplacian

“…More recently, the peridynamic model has been proposed [10], which relates the derivative of stress to an integral involving difference of displacements through a (peridynamic) kernel. The nonlocal elasticity theory allows to study the behaviour of beams at the nano-scale [11,12], also considering the Timoshenko model [13][14][15], the presence of viscoelastic foundation [16], the response to stochastic actions [17][18][19], and allows considering plane elements at the nano-scale [20,21]. A variational approach can also be used [22] and the effect of boundary conditions on the vibration of the beam can be evaluated [23,24].…”

A Fractional Approach to Non-Newtonian Blood Rheology in Capillary Vessels

“…This approach has been recently overcome by some of the authors introducing a mechanically based model of material long-range interactions [18] that generalizes the peridynamics approach presented at the beginning of the century [19], in the sense that it involves both local and non-local interactions. Several studies of the mechanically based non-local mechanics have been presented in recent scientific literature [20][21][22][23][24] also involving fractional-order calculus, that is a generalization of the well-known classical differential calculus in terms of real (or complex) order of differintegration [25,26].…”

“…These forces are transmitted to relatively long distance by relatively large cells, mainly Red Blood Cells (RBC). These long-range interactions are constructed as volume viscous forces scaled by an attenuation function that decreases the forces mutually exerted by two nonadjacent volume elements as the distance between them increases; the approach is analogous of that successfully used in various micro/nanomechanics problems [20][21][22][23][24]. It is shown that if the attenuation function is chosen as a power law of the distance between two volume elements, the integral representing non-local forces reverts to a fractional derivative operator.…”