  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
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Yu. V. Suvorova, "Yu. N. Rabotnov's nonlinear hereditarytype equation and its applications," Mech. Solids. 39 (1), 132138 (2004) 
Year 
2004 
Volume 
39 
Number 
1 
Pages 
132138 
Title 
Yu. N. Rabotnov's nonlinear hereditarytype 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 glassreinforced 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 longterm and
shortterm creep in such materials [68].
The subsequent development of this approach
confirmed its advantages. It turns out that different
deformation processes (e.g., quasistatic loading,
shorttern and longterm 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 longterm prediction of their behavior in realistic
operating conditions not only for composites but also
for metals and polymers [915]. 
References 
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 Integrodifferential 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. 6362, 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. 7487, 1971. 
7.  Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanychev,
"Nonlinear creep of the glassreinforced plastic TS8/3250,"
Mekhanika Polymerov, No. 3, pp. 391397, 1971. 
8.  Yu. N. Rabotnov, L. Kh. Papernik, and E. I. Stepanychev,
"The relation of creep characteristics of glassreinforced
plastics to the instantaneous strain curve,"
Mekhanika Polymerov, No. 4, pp. 624628, 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. 4154, 1972. 
10.  Yu. N. Rabotnov and J. V. Suvorova,
"The nonlinear hereditarytype stressstrain relation
for metals,"
Int. J. Solids and Structures, Vol. 14, No. 3, pp. 173185,
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. 7379, 1974. 
12.  Yu. V. Suvorova,
"Nonlinear effects of deformation of timedependent media,"
Mekhanika Polymerov, No. 6, pp. 976980, 1976. 
13.  J. V. Suvorova,
"The influence of time and temperature on the reinforced
plastics strength,"
in Failure Mechanics of Composites, pp. 177214,
NorthHolland, Amsterdam, 1985, 
14.  J. V. Suvorova, N. G. Ohlson, and S. I. Alexeeva,
"An approach to the description of timedependent
materials,"
Materials and Design, Vol. 24, No. 4, pp. 293297, 2003. 
15.  J. V. Suvorova, N. G. Ohlson, and S. I. Alexeeva,
"Temperature influence in the description of timedependent
materials,"
Materials and Design, Vol. 24, No. 4, pp. 299304, 2003. 
16.  Yu. V. Suvorova and S. I. Alekseeva,
"Engineering applications of a hereditarytype model
to the description of the behavior of polymers and
composites with polymeric matrix,"
Zavodskaya Laboratoriya. Diagnostika Materialov,
Vol. 66, No. 5, pp. 4751, 2000. 
17.  Yu. V. Suvorova and S. I. Alekseeva,
"Engineering applications of a nonlinear hereditarytype
temperaturedependent model,"
Zavodskaya Laboratoriya. Diagnostika Materialov,
Vol. 66, No. 6, pp. 4852, 2000. 
18.  E. N. Zvonov, N. I. Malinin, L. Kh. Papernik,
and B. M. Tseitlin,
"Computeraided identification of creep characteristics for linear
elastically hereditary materials,"
Inzh. Zh. MTT, No. 5, pp. 7276, 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. 111114, 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 э_{α}functions,"
Mashinovedenie, No. 6, pp. 7782, 1977. 
21.  V. S. Ekel'chik and V. M. Ryabov,
"The utilization of one class of hereditarytype kernels
in the equations of linear elasticity,"
Mekhanika Kompozitnykh Materialov, No. 3, pp. 393404,
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. 5456, 2002. 
23.  A. V. Mosin, "The calculation of parameters
of a nonlinear hereditarytype constitutive equation,"
Problemy Mashinostroeniya i Nadezhnosti Mashin,
No. 2, pp. 8388, 2002. 

Received 
07 November 2003 
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