On a viscoelastic rod with constitutive equation containing fractional derivatives of two different orders

Mathematics and Mechanics of Solids

Volumn 9, Issue 6, 2004, Pages 629-656

  Stankovic, B ^a   Atanackovic, T M ^a  

^a UNIVERSITY OF NOVI SAD   (Serbia)

Author keywords

Fractional derivatives; Tempered distributions; Viscoelastic rod

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; EQUATIONS OF MOTION; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; VIBRATIONS (MECHANICAL);

FRACTIONAL DERIVATIVES; LATERAL VIBRATION; TEMPERED DISTRIBUTIONS; VISCOELASTIC ROD;

VISCOELASTICITY;

EID: 10944248936     PISSN: 10812865     EISSN: None     Source Type: Journal    
DOI: 10.1177/1081286504036219     Document Type: Article

References (23)

1. Caputo, M. and Mainardi, F. A new dissipation model based on memory mechanism, Pure and Applied Geophysics, 91, 134-147 (1971).
2. Caputo, M. and Mainardi, F. Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento (Ser. II), 1, 161-198 (1971).
3. Oldham, K. B. and Spanier, J. The Fractional Calculus, Academic, New York, 1974.
4. Bagely, R. L. and Torvik, P. J. On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30, 133-155 (1986).
5. Bagely, R. L. and Torvik, P. J. A generalized derivative model for an elastomer damper, The Shock and Vibration Bulletin, 49, 135-143 (1979).
6. Bagely, R. L. and Torvik, P. J. A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27, 201-210 (1983).
7. Gorenflo, R. and Mainardi, F. Fractional Calculus: Integral and Differential Equations of Fractional Order, pp. 223-276, ed. A. Carpinteri & F. Mainardi, Springer, Vienna, 1997.
8. Podlubny, I. Fractional Differential Equations, Academic, San Diego, CA, 1999.
9. Li, G.-G., Zhu, Z.-Y. and Cheng, C.-J. Dynamical stability of viscoelastic column with fractional derivative constitutive relation, Applied Mathematics and Mechanics, 22, 294-303 (2001).
10. Atanackovic, T. M. A modified Zener model of a viscoelastic body, Continuum Mechanics and Thermodynamics, 14, 137-148 (2002).
11. Rossikhin, Yu. A. and Shitikova, M. V. A new method for solving dynamic problems of fractional derivative viscoelasticity, International Journal of Engineering Science, 39, 149-176 (2001).
12. Rossikhin, Yu. A. and Shitikova, M. V. Analysis of dynamic behavior of viscoelastic rods whose rheogical models contain fractional derivatives of two different orders, Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 81, 363-376 (2001).
13. Atanackovic, T. M. and Stankovic, B. Dynamics of a viscoelastic rod of fractional derivative type, Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 82, 377-386 (2002).
14. Stankovic, B. and Atanackovic, T. M. Dynamics of a rod made of generalized Kelvin-Voigt viscoelastic material, Journal of Mathematical Analysis and Applications, 268, 550-562 (2002).
15. Atanackovic, T. M. Stability Theory of Elastic Rods, World Scientific, Singapore, 1997.
16. Komatsu, H. Laplace transforms of hyperfunctions - A new foundation of the Heaviside calculus, Journal of the Faculty of Sciences, University of Tokyo, Sec. IA, 34, 805-820 (1987).
17. Stankovic, B. Solution of mathematical models by localization, Bulletin de l'Académie Serbe des Sciences et Arts, Class des Sciences Mathématiques et Naturalles, CXXIII, 1-17 (2002).
18. Vladimirov, V. S. Generalized Functions in Mathematical Physics, Mir, Moscow, 1979.
19. Vladimirov, V. S. Equations of Mathematical Physics, Nauka, Moscow, 1988 (in Russian).
20. Doetsch, G. Handbuch der Laplace-Transformation, I and II, Birkhäuser, Basel, 1950, 1955.
21. Erdélyi, A. Higher Transcendental Functions, McGraw-Hill, New York, 1955.
22. Marden, M. The Geometry of the Zeros of a Polynomial in a Complex Variable, American Mathematical Society, New York, 1949.
23. Stankovic, B. and Atanackovic, T. M. On a model of a viscoelastic rod, Fractional Calculus and Applied Analysis, 4, 501-522 (2001).