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Andrzej Hanyga

Publications and source records attributed to Andrzej Hanyga.

At least 19 recordsLinked to original sources
arXiv2021

A comment on a controversial issue: a Generalized Fractional Derivative cannot have a regular kernel

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation. It is shown for some selected classes of functions that a necessary condition for a function to be the kernel of a fractional derivative is an integrable singularity at 0. It is shown that locally integrable completely monotone functions satisfy the Sonine equation if and only if they are singular at 0.

math.CA↗
arXiv2021

On solutions of matrix-valued convolution equations, anisotropic fractional derivatives and their applications in linear and non-linear anisotropic viscoelasticity

A relation between matrix-valued complete Bernstein functions and matrix-valued Stieltjes functions is applied to prove that the solutions of matricial convolution equations with extended LICM kernels belong to special classes of functions. In particular the cases of the solutions of the viscoelastic duality relation and the solutions of the matricial Sonine equation are discussed, with applications in anisotropic linear viscoelasticity and a generalization of fractional calculus. In the first case it is in particular shown that duality of completely monotone relaxation functions and Bernstein creep functions in general requires inclusion in the relaxation function of a Newtonian viscosity term in addition to the memory effects represented by the completely monotone kernel. We define anisotropic generalized fractional derivatives (GFD) by replacing the kernel $t^{-α}/Γ(1-α)$ of the Caputo derivatives with completely monotone matrix-valued kernels which are weakly singular at 0.

math.AP↗
arXiv2019

Effects of Newtonian viscosity and relaxation on linear viscoelastic wave propagation

In an important class of linear viscoelastic media the stress is the superposition of a Newtonian term and a stress relaxation term. It is assumed that the creep compliance is a Bernstein class function, which entails that the relaxation function is LICM. In this paper the effect of Newtonian viscosity term on wave propagation is examined. It is shown that Newtonian viscosity dominates over the features resulting from stress relaxation. For comparison the effect of unbounded relaxation function is also examined. In both cases the wave propagation speed is infinite, but the high-frequency asymptotic behavior of attenuation is different. Various combinations of Newtonian viscosity and relaxation functions and the corresponding creep compliances are summarized.

math-ph↗
arXiv2018

On solutions of a class of matrix-valued convolution equations

We apply a relation between matrix-valued complete Bernstein functions and matrix-valued Stieltjes functions to prove that certain convolution equations for matrix-valued functions have unique solutions in a special class of functions. In particular the cases of the viscoelastic duality theorem and the Sonine equation are discussed, with applications in anisotropic linear viscoelasticity and a generalization of fractional calculus.

math-ph↗
arXiv2018

A simple proof of a duality theorem with applications in viscoelasticity

A new concise proof is given of a duality theorem connecting completely monotone relaxation functions with Bernstein class creep functions. The proof makes use of the theory of complete Bernstein functions and Stieltjes functions and is based on a relation between these two function classes.

math.CA↗
arXiv2015

Asymptotic estimates of viscoelastic Green's functions near the wavefront

Asymptotic behavior of viscoelastic Green's functions near the wavefront is expressed in terms of a causal function $g(t)$ defined in \cite{SerHanJMP} in connection with the Kramers-Kronig dispersion relations. Viscoelastic Green's functions exhibit a discontinuity at the wavefront if $g(0) < \infty$. Estimates of continuous and discontinuous viscoelastic Green's functions near the wavefront are obtained.

math-ph↗
arXiv2015

Plane waves in anisotropic viscoelastic media

Two concepts of plane waves in anisotropic viscoelastic media are studied. One of these concepts allows for the use of methods based on the theory of complete Bernstein functions. This allows for a deeper study of frequency-domain asymptotics of the attenuation function and time-domain regularity at the wavefronts. A relation between the direction of the energy flux density and the attenuation vector is examined under much more general assumptions.

math-ph↗
arXiv2015

Wave propagation in anisotropic viscoelasticity

We extend the theory of complete Bernstein functions to matrix-valued functions and apply it to analyze Green's function of an anisotropic multi-dimension\-al linear viscoelastic problem. Green's function is given by the superposition of plane waves. Each plane wave is expressed in terms of matrix-valued attenuation and dispersion functions given in terms of a matrix-valued positive semi-definite Radon measure. More explicit formulae are obtained for 3D isotropic viscoelastic Green's functions. As an example of an anisotropic medium the transversely isotropic medium with a constant symmetry axis is considered.

math-ph↗
arXiv2014

Elimination of memory from the equations of motion of hereditary viscoelasticity for increased efficiency of numerical integration

A method of eliminating the memory from the equations of motion of linear viscoelasticity is presented. Replacing the unbounded memory by a quadrature over a finite or semi-finite interval leads to considerable reduction of computational effort and storage. The method applies to viscoelastic media with separable completely monotonic relaxation moduli with an explicitly known retardation spectrum. In the seismological Strick-Mainardi model the quadrature is a Gauss-Jacobi quaddrature. The relation to fractional-order viscoelasticity is shown

astro-ph.IM↗
arXiv2014

Attenuation and shock waves in linear hereditary viscoelastic media. Strick-Mainardi, Jeffreys-Lomnitz-Strick and Andrade creep compliances

Dispersion, attenuation and wavefronts in a class of linear viscoelastic media proposed by Strick and Mainardi in 1982 and a related class of models due to Lomnitz, Jeffreys and Strick are studied by a new method due to the Author. Unlike the previously studied explicit models of relaxation modulus or creep compliance, these two classes support propagation of discontinuities. Due to an extension made by Strick either of these two classes of models comprise both viscoelastic solids and fluids.

math-ph↗
arXiv2014

An acoustic wave equation based on viscoelasticity

An acoustic wave equation for pressure accounting for viscoelastic attenuation is derived from viscoelastic equations of motion. It is assumed that the relaxation moduli are completely monotonic. The acoustic equation differs significantly from the equations proposed by Szabo (1994) and in several other papers. Integral representations of dispersion and attenuation are derived. General properties and asymptotic behavior of attenuation and dispersion in the low and high frequency range are studied. The results are compatible with experiments. The relation between the asymptotic properties of attenuation and wavefront singularities is examined. The theory is applied to some classes of viscoelastic models and to the quasi-linear attenuation reported in seismology.

math-ph↗
arXiv2014

Dispersion and attenuation for an acoustic wave equation consistent with viscoelasticity

An acoustic wave equation for pressure accounting for viscoelastic attenuation is derived from viscoelastic equations of motion. It differs significantly from the equations proposed by Szabo. Dispersion and attenuation associated with the viscoelastic wave equation is examined. The theory is applied to three classes of viscoelastic models and to the linear attenuation model.

math-ph↗
arXiv2013

Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli

It is shown that viscoelastic wave dispersion and attenuation in a viscoelastic medium with a completely monotonic relaxation modulus is completely characterized by the phase speed and the dispersion-attenuation spectral measure. The dispersion and attenuation functions are expressed in terms of a single dispersion-attenuation spectral measure. An alternative expression of the mutual dependence of the dispersion and attenuation functions, known as the Kramers-Kronig dispersion relation, is also derived from the theory. The minimum phase aspect of the filters involved in the Green's function is another consequence of the theory. Explicit integral expressions for the attenuation and dispersion functions are obtained for a few analytical relaxation models.

math-ph↗
arXiv2011

Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality.

math-ph↗
arXiv2010

Theory of attenuation and finite propagation speed in viscoelastic media

It is shown that the dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum can be expressed in terms of a positive measure. Both functions have a sublinear growth rate at very high frequencies. In the case of power law attenuation positive relaxation spectrum ensures finite propagation speed. For more general attenuation functions the requirement of finite propagation speed imposes a more stringent condition on the high-frequency behavior of attenuation. It is demonstrated that superlinear power law frequency dependence of attenuation is incompatible with finite speed of propagation and with the assumption of positive relaxation spectrum.

cond-mat.mtrl-sci↗
arXiv2010

Attenuation, dispersion and finite propagation speed in viscoelastic media

It is shown that the dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum have a sublinear growth rate at very high frequencies. A local dispersion relation in parametric form is found. The exact limit between attenuation growth rates compatible and incompatible with finite propagation speed is found. Incompatibility of superlinear frequency dependence of attenuation with finite speed of propagation and with the assumption of positive relaxation spectrum is demonstrated.

math-ph↗