On the viscoelastic characterization of the Jeffreys–Lomnitz law of creep

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.

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

$$ f(t) = {{\displaystyle} \lim\limits_{n \to \infty} \frac{(-1)^n}{n!} \left[ s^{n+1}\, {\widetilde f}^{(n)}(s)\right]_{s={n}/{t}}} \label{eq:29} $$

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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

$$ f(\gamma)= {{\displaystyle} \lim\limits_{n \to \infty} \frac{(-1)^n}{n!} \left[ \xi^{n+1}\, {\widetilde f}^{(n)}(\xi)\right]_{\xi=n/\gamma}}= \frac{1}{\Gamma(1-\alpha)} \frac{{\rm e}^{-\gamma}}{\gamma^\alpha}\,, \label{eq:30} $$

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with ${\widetilde f}(\xi) := (\xi+1)^{\alpha-1}$ and α < 1. Now

$$\begin{array}{rll} {\widetilde f}^{(n)}(\xi) &=& (-1)^n\, \frac{\Gamma(n-\alpha+1)}{\Gamma(1-\alpha)}\, (\xi + 1)^{\alpha -n -1}\,,\\ {\quad} n &=&1,2, \dots \label{eq:31} \end{array} $$

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so we get

$$ \lim\limits_{n \to \infty} \frac{1}{\Gamma(1-\alpha)}\, \frac{\Gamma(n-\alpha +1)}{\Gamma(n+1)}\, \left(\frac{n}{\gamma}\right)^{n+1}\, \left(1+ \frac{n}{\gamma}\right)^{\alpha-n-1}\,. \label{eq:32} $$

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We easily recognize from Stirling asymptotic formula that for n → ∞

$$ \frac{\Gamma(n-\alpha +1)}{\Gamma(n+1)}\sim n^{-\alpha}\,. \label{eq:33} $$

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Because

$$ \left(\frac{n}{\gamma}\right)^{n+1}\,\left(1+ \frac{n}{\gamma}\right)^{\alpha-n-1} = \left(\frac{n}{\gamma}\right)^\alpha \,\left(1 + \frac{\gamma}{n}\right)^{\alpha-n-1} \label{eq:34} $$

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we finally get

$$\begin{array}{rll} f(\gamma) & = & \frac{1}{\Gamma(1-\alpha)\, \gamma^\alpha}\, \lim\limits_{n \to \infty} \left(1+ \frac{\gamma}{n}\right)^{-n}\, \left(1+ \frac{\gamma}{n}\right)^{\alpha-1} \\ & = & \frac{1}{\Gamma(1-\alpha)} \frac{{\rm e}^{-\gamma}}{\gamma^\alpha}\,, \label{eq:35} \end{array} $$

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in view of the Neper limit

$$ \lim\limits_{n \to \infty} \left(1+ \frac{1}{n}\right)^{n} = {\rm{e}}\,. \label{eq:36} $$

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