 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| T. A. Surguladze, "An application of the fractional Green's function," Mech. Solids. 36 (1), 42-48 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
1 |
Pages |
42-48 |
| Title |
An application of the fractional Green's function |
| Author(s) |
T. A. Surguladze (Moscow) |
| Abstract |
By means of the fractional Green's function, direct and inverse
relations are obtained for some models of viscoelastic bodies.
An advantage of the method is that its application
requires the following standard procedures:
to find the one-sided Green's function for ordinary differential operators;
to find a certain polynomial on the basis of some simple rule; to find fractional
derivatives of elementary functions.
The theory of viscoelasticity is concerned with processes
in which the state of a mechanical system depends on the
entire history of all actions experienced by the system.
In recent years, the intensive development of this theory
has been determined by
numerous technical applications connected with studying the phenomena of
creep in metals, plastics, concrete, rocks, and other bodies.
This theory has acquired great importance for the strength analysis
of new materials, especially, polymers, because of their industrial applications.
A detailed exposition of linear and nonlinear theories of
viscoelasticity can be found in [1-4].
Among the methods to obtain constitutive relations for viscoelastic
materials, we mention those of fractional calculus, which have been
frequently used during the last eighty years. Basic principles of
fractional calculus are given in [5, 6]. [See also T. A.
Surguladze, On Some Applications of Fractional Calculus in Viscoelasticity
[in Russian], Manuscript registered at VINITI RAN, 07.06.2000, No. 1827-B99.] |
| References |
| 1. | A. A. Il'yushin and B. E. Pobedrya, Basic Principles of
Mathematical Viscoelasticity [in Russian], Nauka, Moscow, 1970. |
| 2. | B. E. Pobedrya, "Mathematical theory of nonlinear
viscoelasticity," in Elasticity and Inelasticity [in Russian], No. 3,
pp. 95-173, Izd-vo MGU, Moscow, 1973. |
| 3. | Yu. N. Rabotnov, Elements of Hereditary Mechanics of Solids
[in Russian], Nauka, Moscow, 1977. |
| 4. | D. Blend, Theory of Linear Viscoelasticity [Russian translation],
Mir, Moscow, 1965. |
| 5. | K. B. Oldham and J. Spanier, The Fractional Calculus, Academic
Press, New York, London, 1974. |
| 6. | S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals
and Derivatives and their Applications [in Russian], Nauka i Tekhnika, Minsk, 1987. |
| 7. | T. D. Shermergor, "Application of fractional differential
operators for the description of hereditary properties of materials,"
Zh. Prikl. Mekh. i Tekhn. Fiziki, No. 6, pp. 118-121, 1966. |
| 8. | R. C. Koeller, "Applications of fractional calculus to the theory
of viscoelasticity," J. Appl. Mech., Vol. 51, pp. 299-307, 1984. |
| 9. | R. L. Bagley and P. J. Torvik, "A theoretical basis for the
application of fractional calculus to viscoelasticity," J. Rheology,
Vol. 27, No. 3, pp. 201-210, 1983. |
| 10. | R. L. Bagley and P. J. Torvik, "Fractional calculus- a
different approach to the analysis of viscoelastically damped structures,"
AIAA J., Vol. 21, No. 5, pp. 741-748, 1983. |
| 11. | R. L. Bagley and P. J. Torvik, "On the fractional calculus model
of viscoelastic behavior," J. Rheology, Vol. 30, No. 1, pp. 133-155, 1986. |
| 12. | T. F. Nonnenmacher, "Fractional relaxation equations for
viscoelasticity and related phenomena," in Lect. Notes in Phys.,
Vol. 381, pp. 307-321, Springer-Verlag, Berlin, 1991. |
| 13. | K. S. Miller and B. Ross, "Fractional Green's functions,"
Indian J. Appl. Math, Vol. 22, No. 9, pp. 763-767, 1991. |
| 14. | K. S. Miller, Linear Differential Equations in the Real Domains,
W. W. Norton and Co., New York, 1963. |
|
| Received |
05 June 1999 |
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