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

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 $α$, 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 $α\le 1$ yields a continuous transition from a Hooke elastic solid with no creep ($α\to -\infty$) to a Maxwell fluid with linear creep ($α=1$) passing through the Lomnitz viscoelastic body with logarithmic creep ($α=0$), 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 $α\le 1$.