Centennial jubilee of Academician Rabotnov and contemporary handling of his fractional operator

Abstract

This note is dedicated to the centennial jubilee of the Russian Academician Yury Rabotnov, who was a pioneer in the application of fractional operators in Mechanics of Solids. In the present Note, the authors wish to present a contemporary handling of Rabotnov operators introduced in 1948 and to show their connectedness with fractional derivatives, what provides the ageless interest to his ideas and results.MSC 2010 : 01A60, 26A33, 33E12, 34A08, 74D05, 74G50 Key Words and Phrases: fractional calculus, Rabotnov fractional exponential function, generalized Rabotnov fractional operator model * On February 24, 2014, Russian science and fractional calculus community all over the world celebrated the centennial anniversary of the birthday of the Russian Academician Yury Nikolaevich Rabotnov who was an outstanding scientist in the field of Solid Mechanics and made a basic contribution to the development of the theory of elasticity and plasticity, the theory of shells and stability of elastic and viscoplastic systems, the creep theory of metals and the hereditary theory of elasticity. He actively worked in the new directions of damage and fracture mechanics and mechanics of composite materials.

Experiment and modelling of the strain-rate-dependent response during in vitro degradation of PLA fibres

“…a, d and b are material parameters, which are used to adjust the shear thinning decay at a higher strain rate. This function was originally proposed by Bird et al [50] and has been modified ever since by other researchers depending upon the requirement of the polymeric behaviour [51][52][53][54].…”

Viscoelasticity of periodontal ligament: an analytical model

“…A general theory of mechanics of hereditary materials was suggested by Rabotnov ( 1980 ) using integral equations; Koeller ( 1984 ) reviewed the application of integral equations to viscoelasticity and introduced fractional calculus into the Rabotnov’s theory employing a structural spring-dashpot model, used to generalize the classical mechanical models. Rossikhin and Shitikova ( 2015 ) summarized the Rabotnov’s theory (see also ( Rossikhin and Shitikova 2014 )). The Rabotnov’s fractional exponential function is related to a well-known Mittag-Leffler function ( Gorenflo et al 2014 ).…”

“…where τ s is the relaxation time; , E 0 and E ∞ are, respectively, the instantaneous (glassy) and relaxed (rubbery) elastic moduli ( Rossikhin and Shitikova 2015 ); and is the Rabotnov’s fractional exponential function, which describes the relaxation of volume and shear stresses. It was introduced by Rabotnov in the following form ( Rabotnov 1948 ; 1980 ): where 0< γ <1 is the fractional parameter.…”

FCAA Related News, Events and Books (FCAA–Volume 20–2–2017)

“…The main achievements were reported in several state-of-the-art papers [10,16,24,25,29]. Utilizing the generalized Hertzian contact law and algebra of dimensionless fractional Rabotnov operators, which was developed recently by Prof. Rossikhin [29,31,32], the theory of impact of viscoelastic bodies has been constructed during last five years by Prof Rossikhin and his research team [27,29,30,34,36,37].…”