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

The probability density function of the response of a nonlinear system under external α-stable Lévy 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 Lévy 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 Lévy white noise

Einstein-Smoluchowsky equation handled by complex fractional moments

In this paper the response of a non linear half oscillator driven by a-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 a and for any nonlinear function f (X; t)

Experimental characterization of the human meniscal tissue

The meniscus plays a critical role in load transmission, stability and energy dissipation in the knee joint. Loss of the meniscus leads to joint degeneration and osteoarthritis. In a number of cases replacement of the resected meniscal tissue by a synthetic implant might avoid the articular cartilage degeneration. None of the available implants presents optimal biomechanics characteristic due to the fact the biomechanics functionality of the meniscus is not yet fully understood. Mimicking the native biomechanical characteristics of the menisci seems to be the key factor in meniscus replacement functioning. This is extremely challenging due to its complex inhomogeneous microstructure, the lack of a full experimental characterization of the material properties and the lack of 3D theoretical, numerical and computational models which can reproduce and validate the experimental results. The objective of this work is to present the experimental characterization of the anisotropic meniscal tissue at the macroscale. Innovative Biaxial tests have been conducted and the results are new to the literature

Elasto-plastic Material Models Introduce Error in Finite Element Polyethylene Wear Predictions

Introduction: Polyethylene wear of joint replacements can cause severe clinical complications, including; osteolysis, implant loosening, inflammation and pain. Wear simulator testing is often used to assess new designs, but it is expensive and time consuming. It is possible to predict the volume of polyethylene implant wear from finite element models using a modification of Archard's classic wear law [1-2]. Typically, linear elastic isotropic, or elasto-plastic material models are used to represent the polyethylene. The purpose of this study was to investigate whether use of a viscoelastic material model would significantly alter the predicted volumetric wear of a mobile-bearing unicompartmental knee replacement. Materials & Methods: Tensile creep-recovery experiments were performed to characterise the creep and relaxation behaviour of the polyethylene (moulded GUR 4150 samples machined to 180x20x1 mm). Samples were loaded to 3 MPa stress in 4 minutes, and then held for 6 hours, the tensile stress was removed and samples were left to relax for 6 hours. The mechanical test data was used fit to a validated three--dimensional fractional Maxwell viscoelastic constitutive material model [3].An explicit finite element model of a mobile--bearing unicompartmental knee replacement was created, which has been described previously [4]. The medial knee replacement was loaded to 1200 N over a period of 0.2 s. The bearing was meshed using quadratic tetrahedral elements (1.5 mm seeding size based on results of a mesh convergence study), and the femoral component was represented as an analytical rigid body. Wear predictions were made from the contact stress and sliding distance using Archard's law, as has been described in the literature [1-2]. A wear factor of 5.24x10-11 was used based upon the work by Netter et al. [2]. All models were created and solved using ABAQUS finite element software (version 6.14, Simulia, Dassault Systemes).Results: The fractional viscoelastic material model predicted almost twice as much wear (0.119 mm3/million cycles) compared to the elasto-plastic model (0.069 mm3/million cycles). The higher wear prediction was due to both an increased sliding distance and higher contact pressures in the viscoelastic model. Discussion: These preliminary findings indicate the simplified elasto-plastic polyethylene material representation can underestimate wear predictions from numerical simulations. Polyethylene is known to be a viscoelastic material which undergoes creep clinically, and it is not surprising that it is necessary to represent that viscoelastic behaviour to accurately predict implant wear. However, it does increase the complexity and run time of such computational studies, which may be prohibitive.References: [1] Maxian, T. A. et al. “A sliding--distance--coupled finite element formulation for polyethylene wear in total hip arthroplasty. ” (1996) J Biomech 29, p687-692. [2] Netter, J. et al. “Prediction of wear in crosslinked polyethylene unicompartmental knee arthroplasty” (2015) Lubricants 3 p381-393. [3] Alotta, G., et al. “On the behaviour of a three-dimensional fractional viscoelastic constitutive model” Manuscript submitted to Meccanica (2016). [4] Pegg, E.C. et al. “Fracture of mobile unicompartmental knee bearings: A parametric finite element study”. (2013) Proc ImechE Part H: Journal of Engineering in Medicine 227 p1213-1223

Viscoelastic material models for more accurate polyethylene wear estimation

Wear debris from ultra-high molecular weight polyethylene (UHMWPE) 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, experiment 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 elasto–plastic material models. Such models cannot represent the creep or viscoelastic material behaviour and may introduce significant error. However, the magnitude of this error and importance of this simplification has never been determined. This study compares the volume of predicted wear from a standard elasto–plastic model, to a fractional viscoelastic material model. Both models have been fitted to experimental data. Standard tensile tests in accordance with ISO 527-3 and tensile creep-recovery tests were performed to experimentally characterise both (a) the elasto–plastic 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 elasto-plastic material representation. This work quantifies, for the first time, the error in troduced 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 Fractional-Order Model of Biopolyester Containing Naturally Occurring Compounds for Soil Stabilization

Currently, the use of polymers and biopolymers as soil-stabilizer additives for control of the soil degradation, deterioration, and desertification and for improving the arid and semiarid soils has been expanded significantly in the agricultural sector. This research was conducted to determine the effect of naturally occurring compounds, such as quercetin (Q) and sodium montmorillonite (NaMMt) at different weight ratios, in biopolyester, such as polylactic acid (PLA), aiming to formulate ecosustainable materials to control the soil degradation and to protect the environment. As known, the use of sophisticated analytical tools to describe the material rheology and melting properties is nowadays very popular among physicists and material scientists. Certainly, several experimental tests conducted on polymeric- and biopolymeric-based materials, such as rubbers, foams, and hydro/aero gels, show that the relaxation time spectra are a continuous function, and as a consequence, multiple relaxation times are involved in the rheological description of the materials, yielding the need for nonconventional relaxation functions. Indeed, in this work, the considered fractional-order model could be considered a powerful tool to describe and to predict the melting properties of the complex polymer-based systems containing different additives

The finite element implementation of 3D fractional viscoelastic constitutive models

The 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

The human meniscus behaves as a functionally graded fractional porous medium under confined compression conditions

In this study, we observe that the poromechanical parameters in human meniscus vary spatially throughout the tissue. The response is anisotropic and the porosity is functionally graded. To draw these conclusions, we measured the anisotropic permeability and the “aggregate modulus” of the tissue, i.e., the stiffness of the material at equilibrium, after the interstitial fluid has ceased flowing. We estimated those parameters within the central portion of the meniscus in three directions (i.e., vertical, radial and circumferential) by fitting an enhanced model on stress relation confined compression tests. We noticed that a classical biphasic model was not sufficient to reproduce the observed experimental behaviour. We propose a poroelastic model based on the assumption that the fluid flow inside the human meniscus is described by a fractional porous medium equation analogous to Darcy’s law, which involves fractional operators. The fluid flux is then time-dependent for a constant applied pressure gradient (in contrast with the classical Darcy’s law, which describes a time independent fluid flux relation). We show that a fractional poroelastic model is well-suited to describe the flow within the meniscus and to identify the associated parameters (i.e., the order of the time derivative and the permeability). The results indicate that mean values of λβ,β in the central body are λβ=5.5443×10−10m4Ns1−β, β=0.0434, while, in the posterior and anterior regions, are λβ=2.851×10−10m4Ns1−β, β=0.0326 and λβ=1.2636×10−10m4Ns1−β, β=0.0232, respectively. Furthermore, numerical simulations show that the fluid flux diffusion is facilitated in the central part of the meniscus and hindered in the posterior and anterior regions