2017/01/31 by Roberto Garra, Francesco Mainardi, Giorgio Spada · 1 citation
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Constitutive equation #Creep #Fractional Differential Equations Solutions #Fractional calculus #Generalization #Hadamard transform #Kernel (algebra) #Logarithm #Mathematical analysis #Mathematics #Numerical methods in engineering #Physics #Probabilistic and Robust Engineering Design #Pure mathematics #Relaxation (psychology) #Rheology #Viscoelasticity #cond-mat.mtrl-sci #math-ph #math.CV #math.MP #msc:26A33 #msc:45D05 #msc:74D05 #msc:74L10 #msc:76A10 #physics.geo-ph
paper · pdf · doi:10.1016/j.chaos.2017.03.032
15 pages, 2 figures, to appear in Chaos, Solitons and Fractals (2017)
openalex publication_date 2017/03/31 · arxiv created 2017/04/09 · arxiv updated 2017/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.