Search results for "fractional calculu"

Waves propagation in a fractional viscoelastic continuum

2009

In this paper the analysis of waves scattering in a fractional-type viscoelastic material is analyzed. Such a material involves, in the constitutive equation, the presence of noninteger order derivatives of the strain filed yielding a memory-type behavior of the material model. The presence of such a term has been also justified experimentally reporting the relaxation modulus of polymeric materials, obtained from experimental test, that are well-fitted by a powerlaw of fractional order. Some numerical applications reporting the standing-waves condition of an 1D solid varying the fractional differentiation order has also been reported in the paper.

Vibrations of elastic structures with external nonlinear visco-elastic damping devices

2011

Fractional calculus application to visco-elastic solid

2009

It is widely known that fractional derivative is the best mathematical tool to describe visco-elastic constitutive law. In this paper it is shown that as soon as we assume the creep compliance function as power law type, as in the linearized version of the Nutting equation, then the fractional constitutive law appears in a natural way. Moreover, using Nutting equation for the creep function, the relaxation modulus is also of power law type whose coefficients (intensity and exponent) are strictly related to those of the creep compliance. It follows that by a simple creep test (or relaxation test) by means of a best fitting procedure we may easily evaluate the parameters of Nutting equation a…

Prestress and experimental tests on fractional viscoelastic materials

2013

Creep and/or Relaxation tests on viscoelastic materials show a power-law trend. Based upon Boltzmann superposition principle the constitutive law with a power-law kernel is ruled by the Caputo’s fractional derivative. Fractional constitutive law posses a long memory and then the parameters obtained by best fitting procedures on experimental data are strongly influenced by the prestress on the specimen. As in fact during the relaxation test the imposed history of deformation is not instantaneously applied, since a unit step function may not be realized by the test machine. Aim of this paper, it is shown that, the experimental procedure, and in particular the initial ramp to reach the constan…

Analytical Solutions of Viscoelastic Nonlocal Timoshenko Beams

2022

A consistent nonlocal viscoelastic beam model is proposed in this paper. Specifically, a Timoshenko bending problem, where size- and time-dependent effects cannot be neglected, is investigated. In order to inspect scale phenomena, a stress-driven nonlocal formulation is used, whereas to simulate time-dependent effects, fractional linear viscoelasticity is considered. These two approaches are adopted to develop a new Timoshenko bending model. Analytical solutions and application samples of the proposed formulation are presented. Moreover, in order to show influences of viscoelastic and size effects on mechanical response, parametric analyses are provided. The contributed results can be usefu…

Fractional Calculus as a New Perspective in the Viscoelastic Behaviour of the Intervertebral Disc

2022

The spinal column is the load-bearing structure of the human being along with its components, which together build a strong, resistant, and stable structure, but there are a few different pathologies from which it can suffer, such as herniated discs. The intervertebral disc acts as a shock absorber and ensures the spine’s great capacity to support high loads and different states of stress, thanks to its viscoelastic properties. Some studies have attempted to describe the viscoelastic behaviour of the intervertebral disc using classical rheological models, such as the Kelvin-Voigt, or multi-parameter models. Even if these models partially describe the viscoelastic response of disc, all visco…

Analysis of multi degree of freedom systems with fractional derivative elements of rational order

2014

In this paper a novel method based on complex eigenanalysis in the state variables domain is proposed to uncouple the set of rational order fractional differential equations governing the dynamics of multi-degree-of-freedom system. The traditional complex eigenanalysis is appropriately modified to be applicable to the coupled fractional differential equations. This is done by expanding the dimension of the problem and solving the system in the state variable domain. Examples of applications are given pertaining to multi-degree-of-freedom systems under both deterministic and stochastic loads.

Statistics of nonlinear stochastic dynamical systems under Lévy noises by a convolution quadrature approach

2010

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by L\'evy white noises. The proposed numerical procedure relies on the introduction of an integral transform of Wiener-Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: 1) Statistics of systems with several different drift terms can be handled in an efficie…

On fractional diffusion and continuous time random walks

2003

Abstract A continuous time random walk model is presented with long-tailed waiting time density that approaches a Gaussian distribution in the continuum limit. This example shows that continuous time random walks with long time tails and diffusion equations with a fractional time derivative are in general not asymptotically equivalent.

Probabilistic characterization of nonlinear systems under α-stable white noise via complex fractional moments

2015

Abstract The probability density function of the response of a nonlinear system under external α -stable Levy white noise is ruled by the so called Fractional Fokker–Planck equation. In such equation the diffusive term is the Riesz fractional derivative of the probability density function of the response. The paper deals with the solution of such equation by using the complex fractional moments. The analysis is performed in terms of probability density for a linear and a non-linear half oscillator forced by Levy white noise with different stability indexes α . Numerical results are reported for a wide range of non-linearity of the mechanical system and stability index of the Levy white nois…