Equilibrium of an elastic medium with after-effect

Abstract

Abstract Editorial Note: The original version of the paper is usually cited as Yu.N. Rabotnov, Equilibrium of an elastic medium with after-effect (in Russian). Prikladnaya Matematika i Mekhanika (J. Appl. Math. Mech.) 12, No 1 (1948), 53–62. This is a re-printed version of the paper, translated from Russian into English by Marina Shitikova, under the kind permission by the Editorial Board of the journal “Prikladnaya Matematika i Mekhanika”, Institute of Mechanics — Russian Academy of Sciences, http://pmm.ipmnet.ru/ru/, a journal published with English translation since 1958, as “Journal of Applied Mathematics and Mechanics” by Elsevier http://www.journals.elsevier.com/journal-of-applied-mathematics-and-mechanics/. Note that nowadays, the fractional exponential function ∋α (β, t) introduced by Rabotnov as (2.5), is known also as the Rabotnov function and as a special case of the Mittag-Leffler function widely used in fractional calculus. Rabotnov spoke also about differential equations with fractional derivatives (Sect. 3) but preferred to work with integral equations methods. Yury Nikolaevich Rabotnov (24 February 1914–15 May 1985) was a great Russian scientist in the field of Mechanics. More about his life and contributions, the readers can find in this journal (Fract. Calc. Appl. Anal.), in the articles: D. Valério, J. Tenreiro Machado, V. Kiryakova, 17, No 2 (2014), 552–578; DOI: 10.2478/s13540-014-0185-1; http://link.springer.com/article/10.2478/s13540-014-0185-1 Yu. A. Rossikhin and M. V. Shitikova, 10, No 2 (2007), 111–122; http://www.math.bas.bf/~fcaa Yu.A. Rossikhin, M.V. Shitikova, 17, No 3 (2014) — this same issue, 674–683; DOI: 10.2478/s13540-014-0192-2.