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