Gioacchino Alotta

Fractional Tajimi–Kanai model for simulating earthquake ground motion

The ground acceleration is usually modeled as a filtered Gaussian process. The most common model is a Tajimi–Kanai (TK) filter that is a viscoelastic Kelvin–Voigt unit (a spring in parallel with a dashpot) carrying a mass excited by a white noise (acceleration at the bedrock). Based upon the observation that every real material exhibits a power law trend in the creep test, in this paper it is proposed the substitution of the purely viscous element in the Kelvin Voigt element with the so called springpot that is an element having an intermediate behavior between purely elastic (spring) and purely viscous (dashpot) behavior ruled by fractional operator. With this choice two main goals are reached: (i) The viscoelastic behavior of the ground may be simply characterized by performing the creep (or the relaxation) test on a specimen of the ground at the given site; (ii) The number of zero crossing of the absolute acceleration at the free field that for the classical TK model is

Viscoelastic material models for more accurate polyethylene wear estimation

\infty

The moment equation closure method revisited through the use of complex fractional moments

∞ for a true white noise acceleration, remains finite for the proposed model.

Einstein-Smoluchowsky equation handled by complex fractional moments

AbstractA mechanically-based nonlocal Timoshenko beam model, recently proposed by the authors, hinges on the assumption that nonlocal effects can be modeled as elastic long-range volume forces and moments mutually exerted by nonadjacent beam segments, which contribute to the equilibrium of any beam segment along with the classical local stress resultants. Long-range volume forces/moments linearly depend on the product of the volumes of the interacting beam segments, and on pure deformation modes of the beam, through attenuation functions governing the space decay of nonlocal effects. This paper investigates the response of this nonlocal beam model when viscoelastic long-range interactions are included, modeled by Caputo’s fractional derivatives. The finite-element method is used to discretize the pertinent fractional-order equations of motion. Closed-form solutions are obtained for creep tests by typical tools of fractional calculus. Numerical results are presented for various nonlocal parameters.

Finite element method for a nonlocal Timoshenko beam model

Wear debris from ultra-high-molecular-weight polyethylene components used for joint replacement prostheses can cause significant clinical complications, and it is essential to be able to predict implant wear accurately in vitro to prevent unsafe implant designs continuing to clinical trials. The established method to predict wear is simulator testing, but the significant equipment costs, experimental time and equipment availability can be prohibitive. It is possible to predict implant wear using finite element methods, though those reported in the literature simplify the material behaviour of polyethylene and typically use linear or elastoplastic material models. Such models cannot represent the creep or viscoelastic material behaviour and may introduce significant error. However, the magnitude of this error and the importance of this simplification have never been determined. This study compares the volume of predicted wear from a standard elastoplastic model, to a fractional viscoelastic material model. Both models have been fitted to the experimental data. Standard tensile tests in accordance with ISO 527-3 and tensile creep recovery tests were performed to experimentally characterise both (a) the elastoplastic parameters and (b) creep and relaxation behaviour of the ultra-high molecular weight polyethylene. Digital image correlation technique was used in order to measure the strain field. The predicted wear with the two material models was compared for a finite element model of a mobile-bearing unicompartmental knee replacement, and wear predictions were made using Archard’s law. The fractional viscoelastic material model predicted almost ten times as much wear compared to the elastoplastic material representation. This work quantifies, for the first time, the error introduced by use of a simplified material model in polyethylene wear predictions, and shows the importance of representing the viscoelastic behaviour of polyethylene for wear predictions.

On the behavior of a three-dimensional fractional viscoelastic constitutive model

In last decades many strategies for seismic vulnerability mitigation of structures have been studied and experimented; in particular energy dissipation by external devices assumes a great importance for the relative simplicity and efficacy. Among all possible approaches the use of fluid viscous dampers are very interesting, because of their velocity-dependent behaviour and relatively low costs. Application on buildings requires a specific study under seismic excitation and a particular attention on structural details. Nevertheless seismic codes give only general information and in most case the design of a such protection systems results difficult; this problem is relevant also in Italy where last seismic code (NTC 2008) on one hand has increased the performance level to be attributed to buildings, especially public and strategic, on the other hand does not give a guide for the design of external dissipation devices as fluid viscous dampers or others. In this study, after a general description of fluid viscous dampers and the criteria of applying on buildings, a simplified procedure for the design of these devices is discussed also by means an application.

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

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 probability density function response of nonlinear systems may be written in discretized form in terms of complex fractional moment not requiring a closure scheme. Excitations such as ground motion, wind turbulence, sea waves, surface roughness, blasts and impacts loads being stochastic processes induce that structural responses are stochastic processes too. Thus, the analyst is concerned with the problem of the response statistical characterization. However, to consider a model closer to reality a nonlinear system has to be considered, then a complete statistical characterization of the response may be performed by solving the Fokker–Planck– Kolmogorov (FPK) equation, a partial differential equation whose solution is the joint probability density function (PDF) of the response variables (Lin and Cai, 1995). Unfortunately, the FPK equation admits analytical solution in very few cases, for this reason we resort to numerical methods. Among the numerical approaches, more attractive, from a computational point of view, is the moment equation (ME) approach, in which the response statistical characterization is given by the response moments or by other quantities related to the former such as cumulants or quasimoments (Stratanovich, 1997; Ibrahim, 1985). This method consists of writing differential equations for the response statistical moments of any order. However, when dealing with nonlinear systems, a serious problem arises in the ME approach, the entire system is hierarchic in the sense that the equations for the moments of a fixed order, say K, contain moments of order higher than K. In this way, the ME form an infinite hierarchy. In order to overcome this difficulty, the so-called closure methods were born. The key idea is to express the response PDF as a Edgeworth or Gram-Charlier series, truncating it at a certain term. The coefficients of the above mentioned series can be written as functions of the response central moments or of the response cumulants or of the response quasimoments. Thus, neglecting the terms beyond a 12 th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015 2 given order is equivalent to make central moments or cumulants or quasi-moments zero, which makes the ME solvable. The moments of order larger than K are expressed in terms of moments of order equal or lower to K by means of the relationships that are obtained by putting the above cited quantities equal to zero. Recently in Di Paola (2014) it has been introduced the complex fractional moments (CFM) through which the FPK equation has been converted, returning a simple method to perform a PDF response function; in Di Matteo et al. (2014) and in Alotta and Di Paola (2015) the method has been applied successfully also for the case of Kolmogorov-Feller equation (Poissonian white noises) and for fractional FPK equation (αstable white noises), respectively. In this paper it will be explored the useful tool of the complex fractional moments to overcome this moment closure procedure. 1. SERIES FORM OF PDF THROUGH COMPLEX FRACTIONAL MOMENTS Starting from the equation of motion of a nonlinear half oscillator in the form, consider the scalar stochastic process ( ) X t satisfying the stochastic differential equation ( ) ( ) ( ) 0 , 0 X f X t W t X X  = +   =  ɺ (1) Where ( ) W t is a Gaussian white noise with zero mean and correlation function ( ) ( ) ( ) ( ) 0 [ ] 2 τ π δ δ + = = E W t W t S t q t with 0 S being the spectral density of ( ) W t . Moreover ( ) , f X t is a nonlinear function of the process ( ) X t and 0 X is the random variable with assigned PDF in zero ( ) ( ) ,0 X p x p x = . The equation ruling the evolution of the PDF of the response process ( ) X t is the so-called Fokker-Planck (FP) equation that may be written as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2

A Mellin transform approach to wavelet analysis

In this paper the response of a non linear half oscillator driven by α-stable white noise in terms of probability density function (PDF) is investigated. The evolution of the PDF of such a system is ruled by the so called Einstein-Smoluchowsky equation involving, in the diffusive term, the Riesz fractional derivative. The solution is obtained by the use of complex fractional moments of the PDF, calculated with the aid of Mellin transform operator. It is shown that solution can be found for various values of stability index α and for any nonlinear function of the drift term in the stochastic differential equation.