Fractional Calculus and Waves in Linear Viscoelasticity (original) (raw)
A model of diffusive waves in viscoelasticity based on fractional calculus
1997
The partial differential equation of diffusion is generalized by replacing the first time derivative by a fractional derivative of order a . This generalized equation is shown to govern the propagation of stress waves in viscoelastic solids, which exhibit a power law creep of degree p with 0 < p < l , provided that l < cy = 2p < 2 . For the basic Cauchy and Signaling problems the corresponding Green functions are expressed in terms of an entire function for which integral and series representations are provided. Numerical results are presented which show the transition from a pure diffusion process (a = 1) to a pure wave process.
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.
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.
Waves propagation in a fractional viscoelastic continuum
. In this paper the analysis of waves scattering in a fractional-type viscoelastic material is analyzed. Such a material involves, in the constitutive equation, the presence of non-integer order derivatives of the strain filed yielding a memory-type behavior of the material model. The presence of such a term has been also justified experimentally reporting the relaxation modulus of polymeric materials, obtained from experimental test, that are well-fitted by a power-law of fractional order. Some numerical applications reporting the standing-waves condition of an 1D solid varying the fractional differentiation order has also been reported in the paper.
On the thermodynamically consistent fractional wave equation for viscoelastic solids
Acta Mechanica, 2011
Since the well-known methods for the computation of harmonically induced LAMB waves in elastic plates cannot be applied directly to viscoelastic material, an enhanced material model is developed. It is based on fractional time derivatives and incorporates a fractional KELVIN-VOIGT model. A proof of the thermodynamic consistency of this model is given. On the basis of the developed material model, the fractional wave equation is derived which allows for the calculation of LAMB waves in viscoelastic solids.
Meccanica, 2018
We investigate propagation of waves in the Zener-type viscoelastic media through a model which involves fractional derivatives with a regular kernel. The restrictions on the coefficients in the constitutive equation that follow from the weak form of the dissipation principle are obtained. We formulate a problem of motion of a spatially one dimensional continuum in a dimensionless form. Then, it is considered in the frame of distribution theory. The existence and the uniqueness of a distributional solution as well as the analysis of its regularity are presented. Numerical results provide the illustration of our approach.
Short Communication Statistical origins of fractional derivatives in viscoelasticity
Many linear viscoelastic materials show constitutive behavior involving fractional order derivatives. Linear, time invariant systems without memory have exponential decay in time but, contradictorily, not the power law decay associated with fractional derivatives. The physics literature has noted that apparently- non-exponential decays can be observed when several simultaneously decaying processes have closely spaced exponential decay rates. Many engineer-researchers interested in viscoelastic damping, however, seem unaware of these observations. In this letter I give an unoriginal explanation, but with a fresh engineering flavor, for the appearance of these fractional order derivatives. By this explanation, fractional order damping can be expected from many materials with sufficiently disordered dissipation mechanisms.
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.
Statistical origins of fractional derivatives in viscoelasticity
Journal of Sound and Vibration, 2005
Many linear viscoelastic materials show constitutive behavior involving fractional order derivatives. Linear, time invariant systems without memory have exponential decay in time but, contradictorily, not the power law decay associated with fractional derivatives. The physics literature has noted that apparentlynon-exponential decays can be observed when several simultaneously decaying processes have closely spaced exponential decay rates. Many engineer-researchers interested in viscoelastic damping, however, seem unaware of these observations. In this letter I give an unoriginal explanation, but with a fresh engineering flavor, for the appearance of these fractional order derivatives. By this explanation, fractional order damping can be expected from many materials with sufficiently disordered dissipation mechanisms. r