Fractional calculus and diffusive stress
Abstract
The main ideas of fractional calculus are recalled. A quasi-static uncoupled theory of diffusive stresses based on the anomalous diffusion equation with fractional derivatives is formulated.
References
P.L. Butzer, U. Westphal, An introduction to fractional calculus, in: R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Sci. Publ., River Edge, 2000, 1-85.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophysical Journal of the Royal Astronomical Society, 13, 1967, 529-539.
##plugins.generic.googleScholarLinks.settings.viewInGS##
M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969. (In Italian).
##plugins.generic.googleScholarLinks.settings.viewInGS##
R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures 378, Springer, Vienna, 1997, 223-276,
##plugins.generic.googleScholarLinks.settings.viewInGS##
R. Gorenflo, F. Mainardi, Fractional calculus and stable probability distributions, Fourth Meeting on Current Ideas in Mechanics and Related Fields (Kraków, 1997), Arch. Mech. (Arch. Mech. Stos.) 50 (1998), no. 3, 377-388.
##plugins.generic.googleScholarLinks.settings.viewInGS##
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
##plugins.generic.googleScholarLinks.settings.viewInGS##
F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), CISM Courses and Lectures 378, Springer, Vienna, 1997, 291-348.
##plugins.generic.googleScholarLinks.settings.viewInGS##
F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 4 (2001), no. 2, 153-192.
##plugins.generic.googleScholarLinks.settings.viewInGS##
W. Nowacki, Thermoelasticity, Translated from the Polish by Henryk Zorski, Second edition, Pergamon Press, Oxford and PWN—Polish Scientific Publishers, Warsaw, 1986.
##plugins.generic.googleScholarLinks.settings.viewInGS##
K.B. Oldham, J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, Vol. 111, Academic Press, New York-London, 1974.
##plugins.generic.googleScholarLinks.settings.viewInGS##
I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of Their Applications, Mathematics in Science and Engineering 198, Academic Press, Inc., San Diego, 1999.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Y.Z. Povstenko, Stresses exerted by a source of diffusion in a case of a nonparabolic diffusion equation, Internat. J. Engrg. Sci. 43 (2005), no. 11-12, 977-991.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stress, J. Thermal Stresses 28 (2005), no. 1, 83-102.
##plugins.generic.googleScholarLinks.settings.viewInGS##
Y.A. Rossikhin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews, 50 (1997), 15-67.
##plugins.generic.googleScholarLinks.settings.viewInGS##
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
##plugins.generic.googleScholarLinks.settings.viewInGS##