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

Author

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  • Garra, Roberto
  • Mainardi, Francesco
  • Spada, Giorgio

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 ν ∈ (0, 1], the logarithmic creep law known in rheology as Lomnitz law (obtained for ν=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.

Suggested Citation

  • Garra, Roberto & Mainardi, Francesco & Spada, Giorgio, 2017. "A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 333-338.
    • Handle: RePEc:eee:chsofr:v:102:y:2017:i:c:p:333-338
    DOI: 10.1016/j.chaos.2017.03.032
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    Cited by:

    1. Garra, R. & Consiglio, A. & Mainardi, F., 2022. "A note on a modified fractional Maxwell model," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    2. Li, Jing & Ma, Li, 2023. "A unified Maxwell model with time-varying viscosity via ψ-Caputo fractional derivative coined," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).
    3. Zhao, Zhengang & Zheng, Yunying, 2023. "A Galerkin finite element method for the space Hadamard fractional partial differential equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 272-289.
    4. Charles Wing Ho Green & Yanzhi Liu & Yubin Yan, 2021. "Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes," Mathematics, MDPI, vol. 9(21), pages 1-25, October.
    5. Ivano Colombaro & Andrea Giusti & Silvia Vitali, 2018. "Storage and Dissipation of Energy in Prabhakar Viscoelasticity," Mathematics, MDPI, vol. 6(2), pages 1-9, January.

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