101,902 research outputs found
Analysis of Fractional Viscoelastic Material With Mechanical Parameters Dependent on Random Temperature
No full textIt is well known that mechanical parameters of polymeric materials, e.g., epoxy resin, are strongly influenced by the temperature. On the other hand, in many applications, the temperature is not known exactly during the design process and this introduces uncertainties in the prevision of the behavior also when the stresses are deterministic. For this reason, in this paper, the mechanical behavior of an epoxy resin is characterized by means of a fractional viscoelastic model at different temperatures; then, a simple method to characterize the response of the fractional viscoelastic material at different temperatures modeled as a random variable with assigned probability density function (PDF) subjected to deterministic stresses is presented. It is found that the first-and second-order statistical moments of the response can be easily evaluated only by the knowledge of the PDF of the temperature and the behavior of the parameters with the temperature. Comparison with Monte Carlo simulations is also performed in order to assess the accuracy and the reliability of the method
Filter equation by fractional calculus
No full textAim of this paper is to represent a causal filter equation for any kind of linear system in the general form L=f(t), where f(t) is the forcing function, x(t) is the output and L is a summation of fractional operators. The exact form of the operator L is obtained by using Mellin transform in complex plane
Exact frequency response of bars with multiple dampers
No full textThe paper addresses the frequency analysis of bars with an arbitrary number of dampers, subjected to harmonically varying loads. Multiple external/internal dampers occurring at the same position along the bar, modelling external damping devices and internal damping due to damage or imperfect connections, are considered. In this context, the challenge is to handle simultaneous discontinuities of the response variables, i.e. axial force/displacement discontinuities at the location of external/internal dampers. Based on the theory of generalized functions, the paper will present exact closed-form expressions of the frequency response under point/polynomial loads, which hold regardless of the number of dampers. In addition, closed-form expressions will be derived for the exact dynamic stiffness matrix and load vector of the bar, to be used in a standard assemblage procedure for an exact frequency response analysis of 2D truss structures. Changes to consider a single damper at a given position are straightforward. Numerical applications show the advantages of the proposed method
The finite element implementation of 3D fractional viscoelastic constitutive models
No full textThe aim of this paper is to present the implementation of 3D fractional viscoelastic constitutive theory presented in Alotta et al., 2016 [1]. Fractional viscoelastic models exactly reproduce the time dependent behaviour of real viscoelastic materials which exhibit a long “fading memory”. From an implementation point of view, this feature implies storing the stress/strain history throughout the simulations which may require a large amount of memory. We propose here a number of strategies to effectively limit the memory required. The form of the constitutive equations are summarized and the finite element implementation in a Newton-Raphson integration scheme is described in detail. The expressions that are needed to be coded in user-defined material subroutines for quasi static and dynamic implicit and explicit analysis (UMAT and VUMAT) in the commercial finite element software ABAQUS are readily provided. In order to demonstrate the accuracy of the numerical implementation we report a number of benchmark problems validated against analytical results. We have also analysed the behaviour of a viscoelastic plate with a hole in order to show the efficiency of these types of models. The source codes for the UMAT and VUMAT are provided as online supplements to this paper
Stochastic analysis of a non-local fractional viscoelastic bar forced by Gaussian white noise
No full textRecently, 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) longrange 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 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
Finite element method for a nonlocal Timoshenko beam model
No full textA finite element method is presented for a nonlocal Timoshenko beam model recently proposed by the authors. The model relies on the key idea that nonlocal effects consist of long-range volume forces and moments exchanged by non-adjacent beam segments, which contribute to the equilibrium of a beam segment along with the classical local stress resultants. The long-range volume forces/moments are linearly depending on the product of the volumes of the interacting beam segments, and their relative motion measured in terms of the pure beam deformation modes, through appropriate attenuation functions governing the spatial decay of nonlocal effects. In this paper, the beam model is reformulated within a variational framework involving a consistent total elastic potential energy functional. The latter serves as a basis to derive a suitable finite element formulation of the equilibrium equations. A local stiffness matrix and a nonlocal stiffness matrix contribute to the global stiffness matrix. While the local stiffness matrix is obtained by a standard assemblage of the classical element stiffness matrices, the nonlocal stiffness matrix is built as the sum of component matrices, each involving the stiffness of the long-range interactions between a couple of finite elements. A remarkable result is that, for most common attenuation functions of nonlocal effects, exact closed-form solutions can be found for every element of the nonlocal stiffness matrix. Numerical applications are presented for a variety of nonlocal parameters, including a comparison with experimental data
Einstein-Smoluchowsky equation handled by complex fractional moments
No full textIn 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
Viscoelastic material models for more accurate polyethylene wear estimation
No full textWear 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
A Mellin transform approach to wavelet analysis
No full textThe paper proposes a fractional calculus approach to continuous wavelet analysis. Upon introducing a Mellin transform expression of the mother wavelet, it is shown that the wavelet transform of an arbitrary function f(t) can be given a fractional representation involving a suitable number of Riesz integrals of f(t), and corresponding fractional moments of the mother wavelet. This result serves as a basis for an original approach to wavelet analysis of linear systems under arbitrary excitations. In particular, using the proposed fractional representation for the wavelet transform of the excitation, it is found that the wavelet transform of the response can readily be computed by a Mellin transform expression, with fractional moments obtained from a set of algebraic equations whose coefficient matrix applies for any scale a of the wavelet transform. Robustness and computationally efficiency of the proposed approach are shown in the paper
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