172 research outputs found
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.Comment: arXiv admin note: substantial text overlap with arXiv:1307.737
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
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
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 show
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
A simple proof of a duality theorem with applications in scalar and anisotropic viscoelasticity
A new concise proof is given of a duality theorem connecting completely
monotone relaxation functions with Bernstein class creep functions in
one-dimensional and anisotropic 3D viscoelasticity. 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.Comment: arXiv admin note: substantial text overlap with arXiv:1804.0369
Positive solutions of viscoelastic problems
In 1,2 or 3 dimensions a scalar wave excited by a non-negative source in a
viscoelastic medium with a non-negative relaxation spectrum or a Newtonian
response or both combined inherits the sign of the source. The key assumption
is a constitutive relation which involves the sum of a Newtonian viscosity term
and a memory term with a completely monotone relaxation kernel. In
higher-dimensional spaces this result holds for sufficiently regular sources.
Two positivity results for vector-valued wave fields including isotropic
viscoelasticity are also obtained. Positivity is also shown to hold under
weakened hypotheses.Comment: 23 pp, 2 figure
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
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
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
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