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IssuesArchive of Issues2004-1pp.132-138

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Yu. V. Suvorova, "Yu. N. Rabotnov's nonlinear hereditary-type equation and its applications," Mech. Solids. 39 (1), 132-138 (2004)
Year 2004 Volume 39 Number 1 Pages 132-138
Title Yu. N. Rabotnov's nonlinear hereditary-type equation and its applications
Author(s) Yu. V. Suvorova (Moscow)
Abstract The development of the hereditary mechanics was begun by Boltzmann in the middle of the 19th century and then was rather significantly advanced by Volterra [1, 2] who constructed the class of integral equations suitable for the description of hereditary properties in biological and physical systems. Since then, for a long time, the hereditary mechanics has been primarily the subject matter of mathematical research. Its numerous applications to the description of deformation processes in viscoelastic media are associated with the name of an outstanding scientist Yu. N. Rabotnov. He suggested the nonlinear equation and the kernel that combined the properties of weak singularity and the exponential. These equation and kernel were first published in [3] and subsequently were called Rabotnov's equation and kernel. For twenty years, Rabotnov's equation and kernel had remained the subject matter of mathematical exercises (for details, see [4, 5]). The cycle of experiments with glass-reinforced plastics performed at the Mechanical Engineering Research Institute of the USSR Academy of Sciences in the late 1960s showed that the hereditary concept can be successfully applied to the description of the long-term and short-term creep in such materials [6-8]. The subsequent development of this approach confirmed its advantages. It turns out that different deformation processes (e.g., quasi-static loading, short-tern and long-term creep, stress relaxation, and cyclic loading) can be described by one equation with the same set of parameters. This enables the strength design of structural members and long-term prediction of their behavior in realistic operating conditions not only for composites but also for metals and polymers [9-15].
1.  V. Volterra, A Mathematical Theory of the Struggle for Life [Russian translation], Nauka, Moscow, 1976.
2.  V. Volterra, Theory of Functionals and of Integral and Integro-differential Equations [Russian translation], Nauka, Moscow, 1982.
3.  Yu. N. Rabotnov, "Equilibrium of an elastic medium with aftereffect," PMM [Applied Mathematics and Mechanics], Vol. 12, No. 1, pp. 63-62, 1948.
4.  Yu. N. Rabotnov, Creep in Structural Elements [in Russian], Nauka, Moscow, 1966.
5.  Yu. N. Rabotnov, Fundamentals of Hereditary Solid Mechanics [in Russian], Nauka, Moscow, 1966.
6.  Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanychev, "Applications of the linear theory of heredity to the description of temporal effects in polymer materials," Mekhanika Polymerov, No. 1, pp. 74-87, 1971.
7.  Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanychev, "Nonlinear creep of the glass-reinforced plastic TS8/3-250," Mekhanika Polymerov, No. 3, pp. 391-397, 1971.
8.  Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanychev, "The relation of creep characteristics of glass-reinforced plastics to the instantaneous strain curve," Mekhanika Polymerov, No. 4, pp. 624-628, 1971.
9.  Yu. N. Rabotnov and Yu. V. Suvorova, "The deformation law for metals subjected to a uniaxial loading," Izv. AN SSSR. MTT [Mechanics of Solids], No. 4, pp. 41-54, 1972.
10.  Yu. N. Rabotnov and J. V. Suvorova, "The nonlinear hereditary-type stress-strain relation for metals," Int. J. Solids and Structures, Vol. 14, No. 3, pp. 173-185, 1978.
11.  Yu. V. Suvorova, "The condition of plastic deformation of metals for various loading modes," Izv. AN SSSR. MTT [Mechanics of Solids], No. 1, pp. 73-79, 1974.
12.  Yu. V. Suvorova, "Nonlinear effects of deformation of time-dependent media," Mekhanika Polymerov, No. 6, pp. 976-980, 1976.
13.  J. V. Suvorova, "The influence of time and temperature on the reinforced plastics strength," in Failure Mechanics of Composites, pp. 177-214, North-Holland, Amsterdam, 1985,
14.  J. V. Suvorova, N. G. Ohlson, and S. I. Alexeeva, "An approach to the description of time-dependent materials," Materials and Design, Vol. 24, No. 4, pp. 293-297, 2003.
15.  J. V. Suvorova, N. G. Ohlson, and S. I. Alexeeva, "Temperature influence in the description of time-dependent materials," Materials and Design, Vol. 24, No. 4, pp. 299-304, 2003.
16.  Yu. V. Suvorova and S. I. Alekseeva, "Engineering applications of a hereditary-type model to the description of the behavior of polymers and composites with polymeric matrix," Zavodskaya Laboratoriya. Diagnostika Materialov, Vol. 66, No. 5, pp. 47-51, 2000.
17.  Yu. V. Suvorova and S. I. Alekseeva, "Engineering applications of a nonlinear hereditary-type temperature-dependent model," Zavodskaya Laboratoriya. Diagnostika Materialov, Vol. 66, No. 6, pp. 48-52, 2000.
18.  E. N. Zvonov, N. I. Malinin, L. Kh. Papernik, and B. M. Tseitlin, "Computer-aided identification of creep characteristics for linear elastically hereditary materials," Inzh. Zh. MTT, No. 5, pp. 72-76, 1968.
19.  V. N. Rivkind, "A technique for the determination of the fractional exponential function for the description of the creep curve," Svoistva Sudostroitel'nykh Stekloplastikov i Metody ikh Kontrolya, No. 3, pp. 111-114, 1974.
20.  A. Ya. Gol'dman, V. V. Shcherbak, E. N. Kislov, and E. N. Dvorskii, "A method for determining the parameters for the description of the creep curve for elastically hereditary materials on the basis of tables of Rabotnov's эα-functions," Mashinovedenie, No. 6, pp. 77-82, 1977.
21.  V. S. Ekel'chik and V. M. Ryabov, "The utilization of one class of hereditary-type kernels in the equations of linear elasticity," Mekhanika Kompozitnykh Materialov, No. 3, pp. 393-404, 1981.
22.  Yu. V. Suvorova and A. V. Mosin, "Identification of parameters of Rabotnov's fractional exponential function on the basis of an integral transform and modern software," Problemy Mashinostroeniya i Avtomatizatsii, No. 4, pp. 54-56, 2002.
23.  A. V. Mosin, "The calculation of parameters of a nonlinear hereditary-type constitutive equation," Problemy Mashinostroeniya i Nadezhnosti Mashin, No. 2, pp. 83-88, 2002.
Received 07 November 2003
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