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Viscous dissipation and completely monotonic relaxation moduli

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Abstract

The connection between weak dissipativity and positive definiteness of the relaxation function as well as between monotone energy decay and complete monotonicity of the relaxation function of a linear viscoelastic system is discussed. Some theorems about the composition of completely monotonic functions relevant for polymer rheology are presented.

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Abbreviations

R :

the set of all real numbers

C :

the set of complex numbers

R +:=:

\({\left\{{ x \in R \left| {x \geq 0 }\right. }\right\}}\)

C +:=:

\(\left\{p\left|{{\text{Re}}} \right. p > 0\right\}\)

R ++:=:

\({\left\{{x \in R \left| {x > 0 }\right. }\right\}}\)

Z +=:

set of non-negative integers

R d :

the space of d-dimensional real vectors \( {\mathbf{v}} = {\left( {v_{1} \ldots ,v_{d} } \right)} \)

C d :

the space of d-dimensional complex vectors \( {\left( {v_{1} \ldots ,v_{d} } \right)} \)(v 1..., v d )

z* :

complex conjugate

\(v^{\dag} \) :

Hermitean conjugate

S :

the space of symmetric d× d tensors

Z :

either R d or S, considered as a linear space

W :

the linear space of linear mappings L: Z → Z

\( f * g(t): = \) :

\(\int\limits_{- \infty}^t {f(t - s)\,g(s)\;} \text{d}s \equiv \int\limits_0^\infty {f(s)\;g(t - s)\;\text{d}s}\)

\(\dot e =\) :

de/dt

e t (s) :=:

e(t−s), \(\dot e^t (s): = \dot e(t - s),s \geq 0;\)

θ(t):

the Heaviside unit step function.

Dn : =:

∂n /∂t n

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Acknowledgements

The research was carried out in the framework of the project “Mathematical Modeling of Seismic Attenuation”, sponsored by the Norwegian Scientific Council in the framework of the Petroforsk program in 2002–2004. Critical remarks of Dr M. Seredyńska as well as an anonymous reviewer are acknowledged.

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  1. Department of Geosciences, University of Bergen, Allégaten 41, N5007, Bergen, Norway

    Andrzej Hanyga

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  1. Andrzej Hanyga
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Hanyga, A. Viscous dissipation and completely monotonic relaxation moduli. Rheol Acta 44, 614–621 (2005). https://doi.org/10.1007/s00397-005-0443-6

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