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

## TL;DR

This paper introduces a generalized logarithmic creep law using Hadamard fractional calculus, extending the Lomnitz law with a new parameter to model ultra slow kinetics in rheology.

## Contribution

It develops a new fractional calculus-based model generalizing the Lomnitz law, deriving its stress-strain relation and relaxation function with memory effects and time-varying viscosity.

## Key findings

- Derived a generalized creep law with a parameter 
- Numerically solved the relaxation function as a Volterra integral equation
- Characterized the model's behavior in creep and relaxation representations

## 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.