Search results for "fractional calculu"
showing 10 items of 145 documents
On the influence of the initial ramp for a correct definition of the parameters of fractional viscoelastic materials
2014
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. Usually an initial ramp is present in the deformation history and the time at which the deformation attains the maximum …
Approximate survival probability determination of hysteretic systems with fractional derivative elements
2018
Abstract A Galerkin scheme-based approach is developed for determining the survival probability and first-passage probability of a randomly excited hysteretic systems endowed with fractional derivative elements. Specifically, by employing a combination of statistical linearization and of stochastic averaging, the amplitude of the system response is modeled as one-dimensional Markovian Process. In this manner the corresponding backward Kolmogorov equation which governs the evolution of the survival probability of the system is determined. An approximate solution of this equation is sought by employing a Galerkin scheme in which a convenient set of confluent hypergeometric functions is used a…
A Non-Local Two Dimensional Foundation Model
2012
Classical foundation models such as the Pasternak and the Reissner models have been recently reformulated within the framework of non-local mechanics, by using the gradient theory of elasticity. To contribute to the research effort in this field, this paper presents a two-dimensional foundation model built by using a mechanically based non-local elasticity theory, recently proposed by the authors. The foundation is thought of as an ensemble of soil column elements resting on an elastic base. It is assumed that each column element is acted upon by a local Winkler-like reaction force exerted by the elastic base, by contact shear forces and volume forces due, respectively, to adjacent and non-…
Solution strategies for 1D elastic continuum with long-range interactions: Smooth and fractional decay
2010
Abstract An elastic continuum model with long-range forces is addressed in this study within the context of approximate analytical methods. Such a model stems from a mechanically-based approach to non-local theory where long-range central forces are introduced between non-adjacent volume elements. Specifically, long-range forces depend on the relative displacement, on the volume product between interacting elements and they are proportional to a proper, material-dependent, distance-decaying function. Smooth-decay functions lead to integro-differential governing equations whereas hypersingular, fractional-decay functions lead to a fractional differential governing equation of Marchaud type. …
Stochastic response of a fractional vibroimpact system
2017
Abstract The paper proposes a method to investigate the stochastic dynamics of a vibroimpact single-degree-of-freedom fractional system under a Gaussian white noise input. It is assumed that the system has a hard type impact against a one-sided motionless barrier, which is located at the system’s equilibrium position; furthermore, the system under study is endowed with an element modeled with fractional derivative. The proposed method is based on stochastic averaging technique and overcome the particular difficulty due to the presence of fractional derivative of an absolute value function; particularly an analytical expression for the system’s mean squared response amplitude is presented an…
Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables
2012
Abstract The aim of this paper is the probabilistic representation of the probability density function (PDF) or the characteristic function (CF) in terms of fractional moments of complex order. It is shown that such complex moments are related to Riesz and complementary Riesz integrals at the origin. By invoking the inverse Mellin transform theorem, the PDF or the CF is exactly evaluated in integral form in terms of complex fractional moments. Discretization leads to the conclusion that with few fractional moments the whole PDF or CF may be restored. Application to the pathological case of an α -stable random variable is discussed in detail, showing the impressive capability to characterize…
Fokker Planck equation solved in terms of complex fractional moments
2014
Abstract In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones, the solution of the FPK equation in terms of a finite number of complex moments may be easily found. It is shown that the probability density function (PDF) solution of the FPK equation is restored in the whole domain, including the trend at infinity with the exception of the value of the PDF in zero.
The Multiscale Stochastic Model of Fractional Hereditary Materials (FHM)
2013
Abstract In a recent paper the authors proposed a mechanical model corresponding, exactly, to fractional hereditary materials (FHM). Fractional derivation index 13 E [0,1/2] corresponds to a mechanical model composed by a column of massless newtonian fluid resting on a bed of independent linear springs. Fractional derivation index 13 E [1/2, 1], corresponds, instead, to a mechanical model constituted by massless, shear-type elastic column resting on a bed of linear independent dashpots. The real-order of derivation is related to the exponent of the power-law decay of mechanical characteristics. In this paper the authors aim to introduce a multiscale fractance description of FHM in presence …
A mechanical approach to fractional non-local thermoelasticity
2010
In recent years fractional di erential calculus applications have been developed in physics, chemistry as well as in engineering elds. Fractional order integrals and derivatives ex- tend the well-known de nitions of integer-order primitives and derivatives of the ordinary di erential calculus to real-order operators. Engineering applications of these concepts dealt with viscoelastic models, stochastic dy- namics as well as with the, recently developed, fractional-order thermoelasticity [3]. In these elds the main use of fractional operators has been concerned with the interpolation between the heat ux and its time-rate of change, that is related to the well-known second sound e ect. In othe…
Fractional differential calculus for 3D mechanically based non-local elasticity
2011
This paper aims to formulate the three-dimensional (3D) problem of non-local elasticity in terms of fractional differential operators. The non-local continuum is framed in the context of the mechanically based non-local elasticity established by the authors in a previous study; Non-local interactions are expressed in terms of central body forces depending on the relative displacement between non-adjacent volume elements as well as on the product of interacting volumes. The non-local, long-range interactions are assumed to be proportional to a power-law decaying function of the interaction distance. It is shown that, as far as an unbounded domain is considered, the elastic equilibrium proble…