Thermodynamical restrictions and wave propagation for a class of fractional order viscoelastic rods. (English) Zbl 1217.74064

Atanacković, Teodor M.; Konjik, Sanja; Oparnica, Ljubica; Zorica, Dušan

Abstr. Appl. Anal. 2011, Article ID 975694, 32 p. (2011).

Summary: We discuss thermodynamical restrictions for a linear constitutive equation containing fractional derivatives of stress and strain of different orders. Such an equation generalizes several known models. The restrictions on coefficients are derived by using entropy inequality for isothermal processes. In addition, we study waves in a rod of finite length modelled by a linear fractional constitutive equation. In particular, we examine stress relaxation and creep and compare results with the quasistatic analysis.

MSC:

74K10	Rods (beams, columns, shafts, arches, rings, etc.)
74F05	Thermal effects in solid mechanics
26A33	Fractional derivatives and integrals
34A08	Fractional ordinary differential equations

References:

[1]	T. M. Atanacković, S. Pilipović, and D. Zorica, “Time distributed-order diffusion-wave equation. I. Volterra-type equation,” Proceedings of The Royal Society of London. Series A, vol. 465, no. 2106, pp. 1869-1891, 2009. · Zbl 1186.35106
[2]	T. M. Atanacković, S. Pilipović, and D. Zorica, “Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations,” Proceedings of The Royal Society of London. Series A, vol. 465, no. 2106, pp. 1893-1917, 2009. · Zbl 1186.35107
[3]	M. Enelund and G. A. Lesieutre, “Time domain modeling of damping using anelastic displacement fields and fractional calculus,” International Journal of Solids and Structures, vol. 36, no. 29, pp. 4447-4472, 1999. · Zbl 0943.74008
[4]	H. H. Sherief, A. M. A. El-Sayed, and A. M. Abd El-Latief, “Fractional order theory of thermoelasticity,” International Journal of Solids and Structures, vol. 47, no. 2, pp. 269-275, 2010. · Zbl 1183.74051
[5]	F. Cortés and M. J. Elejabarrieta, “Structural vibration of flexural beams with thick unconstrained layer damping,” International Journal of Solids and Structures, vol. 45, no. 22-23, pp. 5805-5813, 2008. · Zbl 1177.74186
[6]	F. Barpi and S. Valente, “A fractional order rate approach for modeling concrete structures subjected to creep and fracture,” International Journal of Solids and Structures, vol. 41, no. 9-10, pp. 2607-2621, 2004. · Zbl 1179.74125
[7]	O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368-379, 2002. · Zbl 1070.49013
[8]	T. M. Atanacković, S. Konjik, and S. Pilipović, “Variational problems with fractional derivatives: Euler-Lagrange equations,” Journal of Physics. A, vol. 41, no. 9, pp. 095201-095213, 2008. · Zbl 1175.49020
[9]	T. M. Atanacković, “A modified Zener model of a viscoelastic body,” Continuum Mechanics and Thermodynamics, vol. 14, no. 2, pp. 137-148, 2002. · Zbl 1032.74015
[10]	M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, vol. 12 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1992. · Zbl 0753.73003
[11]	A. D. Drozdov, Viscoelastic Structures: Mechanics of Growth and Aging, Academic Press, San Diego, Calif, USA, 1998. · Zbl 0990.74002
[12]	S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1993. · Zbl 0818.26003
[13]	R. L. Bagley and P. J. Torvik, “On the fractional calculus model of viscoelastic behavior,” Journal of Rheology, vol. 30, no. 1, pp. 133-155, 1986. · Zbl 0613.73034
[14]	H. Schiessel, C. Friedrich, and A. Blumen, “Applications to problems in polymer physics and rheology,” in Applications of Fractional Calculus in Physics, R. Hilfer, Ed., pp. 331-376, World Scientific Publishing, River Edge, NJ, USA, 2000. · Zbl 1046.82039
[15]	T. M. Atanacković, S. Pilipović, and D. Zorica, “Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod,” International Journal of Engineering Science, vol. 49, no. 2, pp. 175-190, 2011. · Zbl 1231.74144
[16]	T. M. Atanacković, S. Pilipović, and D. Zorica, “Distributed-order fractional wave equation on a finite domain. Creep and forced oscillations of a rod,” submitted to Continuum Mechanics and Thermodynamics. · Zbl 1272.74324
[17]	S. Konjik, L. Oparnica, and D. Zorica, “Waves in fractional Zener type viscoelastic media,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 259-268, 2010. · Zbl 1185.35280
[18]	S. Konjik, L. Oparnica, and D. Zorica, “Waves in fractional Zener type viscoelastic media,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 259-268, 2010. · Zbl 1185.35280
[19]	Yu. A. Rossikhin and M. V. Shitikova, “New method for solving dynamic problems of fractional derivative viscoelasticity,” International Journal of Engineering Science, vol. 39, no. 2, pp. 149-176, 2001.
[20]	Yu. A. Rossikhin and M. V. Shitikova, “Analysis of dynamic behaviour of viscoelastic rods whose rheological models contain fractional derivatives of two different orders,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 81, no. 6, pp. 363-376, 2001. · Zbl 1047.74026
[21]	T. M. Atanacković, “On a distributed derivative model of a viscoelastic body,” Comptes Rendus, vol. 331, no. 10, pp. 687-692, 2003. · Zbl 1177.74093
[22]	T. M. Atanacković, L. Oparnica, and S. Pilipović, “Distributional framework for solving fractional differential equations,” Integral Transforms and Special Functions, vol. 20, no. 3-4, pp. 215-222, 2009. · Zbl 1170.26003
[23]	K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011