The Fractional Derivative and its Application to Mechanics. Analysis of Viscoelastic Structures Described with Generalized Fractional Derivative Models. -.Lecture presented at the Johannes Kepler University, Linz, Austria, March 4, 2014 (original) (raw)

Generalized fractional derivatives and their applications to mechanical systems

Archive of Applied Mechanics, 2014

New fractional derivatives, termed henceforth generalized fractional derivatives (GFDs), are introduced. Their definition is based on the concept that fractional derivatives (FDs) interpolate the integer-order derivatives. This idea generates infinite classes of FDs. The new FDs provide, beside the fractional order, any number of free parameters to better calibrate the response of a physical system or procedure. Their usefulness and consequences are subject of further investigation. Like the Caputo FD, the GFDs allow the application of initial conditions having direct physical significance. A numerical method is also developed for the solution of differential equations involving GFDs. Mechanical systems including fractional oscillators, viscoelastic plane bodies and plates described by such equations are analyzed.

On fractional modelling of viscoelastic mechanical systems

Mechanics Research Communications, 2016

Since Leibniz's fractional derivative, introduced by Lazopoulos [1], has physical meaning contrary to other fractional derivatives, the viscoelastic mechanical systems are modelled with the help of Leibniz fractional derivative. The compliance and relaxation behaviour of the viscoelastic systems is revisited and comparison with the conventional systems and the existing fractional viscoelastic systems is presented.

Notes on computational aspects of the fractional-order viscoelastic model

Journal of Engineering Mathematics

This paper deals with the computational aspect of the investigation of the relaxation properties of viscoelastic materials. The constitutive fractional Zener model is considered under continuous deformation with a jump at the origin. The analytical solution of this equation is obtained by the Laplace transform method. It is derived in a closed form in the terms of the Mittag-Leffler function. The method of numerical evaluation of the Mittag-Leffler function for arbitrary negative arguments which corresponds to physically meaningful interpretation is demonstrated. A numerical example is given to illustrate the effectiveness of this result.

On a fractional derivative type of a viscoelastic body

Theoretical and Applied Mechanics, 2002

We study a viscoelastic body, in a linear stress state with fractional derivative type of dissipation. The model was formulated in [1]. Here we derive restrictions on the model that follow from Clausius-Duhem inequality. Several known constitutive equations are derived as special cases of our model. Two examples are discussed.

Fractional calculus in viscoelasticity: An experimental study

Communications in Nonlinear Science and Numerical Simulation, 2010

Viscoelastic properties of soft biological tissues provide information that may be useful in medical diagnosis. Noninvasive elasticity imaging techniques, such as Magnetic Resonance Elastography (MRE), reconstruct viscoelastic material properties from dynamic displacement images. The reconstruction algorithms employed in these techniques assume a certain viscoelastic material model and the results are sensitive to the model chosen. Developing a better model for the viscoelasticity of soft tissue-like materials could improve the diagnostic capability of MRE. The well known ''integer derivative" viscoelastic models of Voigt and Kelvin, and variations of them, cannot represent the more complicated rate dependency of material behavior of biological tissues over a broad spectral range. Recently the ''fractional derivative" models have been investigated by a number of researchers. Fractional order models approximate the viscoelastic material behavior of materials through the corresponding fractional differential equations. This paper focuses on the tissue mimicking materials CF-11 and gelatin, and compares fractional and integer order models to describe their behavior under harmonic mechanical loading. Specifically, Rayleigh (surface) waves on CF-11 and gelatin phantoms are studied, experimentally and theoretically, in order to develop an independent test bed for assessing viscoelastic material models that will ultimately be used in MRE reconstruction algorithms.

Fractional Calculus and Waves in Linear Viscoelasticity

2010

Readership: Graduate and PhD students in applied mathematics, classical physics, mechanical engineering and chemical physics; academic institutions; research centers. Key Features • Contains accessible mathematical language for easy understanding • Features ample examples to reiterate concepts in the book • Makes extensive use of graphical images • Includes a large and informative general bibliography for further research

Fractional Calculus with Applications in Mechanics

Fractional Calculus with Applications in Mechanics, 2014

In this section, we review some basic properties of fractional integrals and derivatives, which we will need later in the analysis of concrete problems. This section contains results from various books and papers [

A generalised fractional derivative approach to viscoelastic material properties measurement

Applied Mathematics and Computation, 2005

Viscoelastic materials are largely used as a means to provide damping to structures, thus mitigating resonant vibration responses. Devices made with viscoelastic materials such as isolators, dynamic vibration neutralisers (also called dynamic vibration absorbers), sandwich panels and structural links can be designed for highly efficient vibration control. To properly devise a vibration control strategy with viscoelastic materials, two basic dynamic

The types of derivatives and bifurcation in fractional mechanics

AIP Conference Proceedings, 2020

Fractional calculus appears to be a powerful tool of solid mechanics. In recent years several papers have been published including various forms of non-localities. Two basic fields can be distinguished, there are non-locality in space and in time. When term non-locality is used in its original meaning the value of some quantity in an internal point of the body is determined by the values of other quantities in a whole region around that location. The second one means to include hereditary effects like non-classical viscosity and study fractional visco-elasticity or visco-plasticity. In non-linear stability investigation the way of the loss of stability is studied and classified as a generic static or dynamic bifurcation. To do that step some kind of regular condition is necessary. This condition is connected with non-locality. Generally such behaviour is a result of viscous (time derivative) and gradient dependent terms in the constitutive equations. Such derivatives are not always of first order. There are materials where tests justify models with fractional order derivatives. Moreover, there are (for example Riemann-Liouville) type of fractional derivatives which are non-local itself. Thus by defining strain by fractional derivation of the displacement field a non-local quantity appears instead of the conventional (local) strain. In such a way various versions of non-localities are obtained by using various types of fractional derivatives. The paper studies how the selection of that fractional derivative effects the way of the loss of stability. Basically the study aims constitutive modelling via instability phenomena, that is, by observing the way of loss of stability of some material we can be informed about the form of fractional derivative in its mathematical model. ELEMENTS OF FRACTIONAL CALCULUS Fractional order (say α th) generalization of an integral is a key element. When such operator is combined with a first order derivation the result is an (1 − α) th order fractional derivative. In this section that way is followed to construct fractional derivatives.