Abstract
In 1958, Jeffreys (Geophys J R Astron Soc 1:92–95) proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys–Lomnitz law of creep by allowing its power law exponent α, usually limited to the range 0 ≤ α ≤ 1 to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotone derivative, with a related spectrum of retardation times. The complete range α ≤ 1 yields a continuous transition from a Hooke elastic solid with no creep \(\left(\alpha \,\to\, -\infty\right)\) to a Maxwell fluid with linear creep \(\left(\alpha \,=\,1\right)\) passing through the Lomnitz viscoelastic body with logarithmic creep \(\left(\alpha\, =0\right)\), which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys–Lomnitz creep law extended to all α ≤ 1.
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Notes
For the most relevant properties of the Laplace transform, we refer, e.g., to the Appendix (Eq. 32) of the well-known treatise by Tschoegl (1989, pp. 560–570). In the following, we use the notation \(f(t)\,\div\, \widetilde f(s)\) to denote the juxtaposition of an original function f(t) with its Laplace transform \(\widetilde f(s)\).
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Acknowledgements
FM would like to remind his personal contacts with Sir Harold Jeffreys (in 1973–1974) and with Prof. Ellis Strick (in 1980–1984) who, in some way, even if after so many years, have inspired this research work. GS acknowledges COST Action ES0701 “Improved Constraints on Models of Glacial Isostatic Adjustment”. We are grateful to two anonymous reviewers for helpful suggestions. The figures have been drawn using the generic mapping tools public domain software of Wessel and Smith (1998).
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Appendix: The Post–Widder formula for the retardation spectrum
Appendix: The Post–Widder formula for the retardation spectrum
The Post–Widder formula provides the original function f(t) from its Laplace transform \(\widetilde f(s)\), known with all its derivatives on the real semiaxis of the complex Laplace plane, via a limit of infinite sequence as
where \({\widetilde f}^{(n)}\) is the n–th derivative of \(\widetilde{f}\) with respect to the Laplace variable s. This is the case when the Laplace transform is proved to be an analytic function on the right-half s-plane. For other details, we refer, e.g., to Tschoegl (1989).
The purpose of this Appendix is to prove the validity of Eq. 27 by using (in a suitable way) the Post–Widder formula. In our case, denoting in Eq. 31 t by γ and s by ξ, we must verify that
with \({\widetilde f}(\xi) := (\xi+1)^{\alpha-1}\) and α < 1. Now
so we get
We easily recognize from Stirling asymptotic formula that for n → ∞
Because
we finally get
in view of the Neper limit
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Mainardi, F., Spada, G. On the viscoelastic characterization of the Jeffreys–Lomnitz law of creep. Rheol Acta 51, 783–791 (2012). https://doi.org/10.1007/s00397-012-0634-x
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DOI: https://doi.org/10.1007/s00397-012-0634-x