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

TL;DR

This paper extends the Jeffreys-Lomnitz law of creep by allowing its exponent to be negative, revealing a continuous spectrum from elastic solids to Maxwell fluids and providing analytical and numerical tools for characterization.

Contribution

It generalizes the Jeffreys-Lomnitz law to all negative exponents, establishing a unified framework for viscoelastic behavior analysis.

Findings

The extended law remains a Bernstein function for all negative exponents.

A continuous transition from elastic to fluid-like behavior is demonstrated.

Analytical expressions for the spectrum of retardation times are derived.

Abstract

In 1958 Jeffreys 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 $\alpha$, usually limited to the range [0,1] 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 monotonic derivative, with a related spectrum of retardation times. The complete range $\alpha \le 1$ yields a continuous transition from a Hooke elastic solid with no creep ($\alpha \to -\infty$) to a Maxwell fluid with linear creep ($\alpha=1$) passing through the Lomnitz viscoelastic body with logarithmic creep ($\alpha=0$), 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 $\alpha \le 1$.