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arXiv · 1701.03068

A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus

Roberto Garra·Francesco Mainardi·Giorgio Spada
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Abstract

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter $\nu \in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $\nu=1$). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.

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Roberto Garra, Francesco Mainardi, Giorgio Spada. 2017-01-11. A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. https://doi.org/10.1016/j.chaos.2017.03.032

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