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

2017; Elsevier BV; Volume: 102; Linguagem: Inglês

10.1016/j.chaos.2017.03.032

1873-2887

Roberto Garra, Francesco Mainardi, Giorgio Spada,

Numerical methods in engineering

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.