Fractional viscoelastic cohesive zone model with long-range interaction

This paper is devoted to model the time-dependent crack propagation along a nonlocal viscoelastic in- terface. We consider this kind of problem where two main aspects of the mechanical behavior influence the crack propagation, namely, the long-range interaction and the rheological phenomena

Filter equation by fractional calculus

Aim 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

The 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

Analysis of Fractional Viscoelastic Material With Mechanical Parameters Dependent on Random Temperature

It 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

Stochastic analysis of a non-local fractional viscoelastic beam forced by Gaussian white noise

Recently, a displacement-based non-local beam model has been developed and the relative finite element (FE) formulation with closed-form expressions of the elastic and fractional viscoelastic matrices has also been obtained. The static and quasi-static response has been already investigated. This work investigates the stochastic response of the non-local fractional viscoelastic beam, forced by a Gaussian white noise. In this context, by taking into account the mass of the beam, the system of coupled fractional differential equations ruling the beam motion can be decoupled with the method of the fractional order state variable expansion and statistics of the motion of the beam can be readily found

Finite element method for a nonlocal Timoshenko beam model

A 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

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

In this paper a three-dimensional isotropic fractional viscoelastic model is examined. It is shown that if different time scales for the volumetric and deviatoric components are assumed, the Poisson ratio is time varying function; in particular viscoelastic Poisson ratio may be obtained both increasing and decreasing with time. Moreover, it is shown that, from a theoretical point of view, one-dimensional fractional constitutive laws for normal stress and strain components are not correct to fit uniaxial experimental test, unless the time scale of deviatoric and volumetric are equal. Finally, the model is proved to satisfy correspondence principles also for the viscoelastic Poisson’s ratio and some issues about thermodynamic consistency of the model are addressed

On the numerical implementation of a 3D fractional viscoelastic constitutive model

The aim of this paper is the implementation of a 3D fraction al viscoelastic constitutive law in a user material subroutine (UMAT) in the finite element software Abaqus. Essential to the implementation of the model is access to the strain history at each Gauss point of each element in a constructive manner. Details of the UMAT and comparison with some analytical results are presented in order to show that the fractional viscoelastic constitutive law has been successfully implemented

Code for the stochastic analysis of springpot forced by α-stable white noise processes

This is Matlab files to perform stochastic analysis of a springpot forced by α-stable white noise process. The code evaluates the solution with Montecarlo simulation, with the discrete approach (Eq. 34) of the paper "Analytical response and Markovianity of systems governed by fractional differential equations driven by α-stable white noise processes" and with the analytical solution found in the above mentioned paper in Eq. (44). The code also evaluates the Mean Absolute Percentage Error between the discrete approach in Eq. (34) of the above mentioned paper and the analytical solution in Eq. (44).THIS DATASET IS ARCHIVED AT DANS/EASY, BUT NOT ACCESSIBLE HERE. TO VIEW A LIST OF FILES AND ACCESS THE FILES IN THIS DATASET CLICK ON THE DOI-LINK ABOV