Abstract
Rheology is concerned with the flow and deformation of material Traditionally it is the study of the behavior of material bodies treated as continuous media rather than as aggregates of interacting particles. The macroscopic equations of motion can, in principle, be obtained by coarse-graining the microscopic force laws, much as the Navier-Stokes equations of classical hydrodynamics are obtained by averaging the microscopic momentum equations of the individual particles in a fluid. The macroscopic equations are not as simple for a solid as they are for a liquid in that the symmetry, compressibility, and temperature properties are quite different in the two cases. These differences and others are due to the fact that the interactions among the particles are strong and long-range in a solid and the interactions among the particles are weaker and shorter-range in a liquid. The theoretical difficulties in constructing the averages necessary to go from the microscopic to the macroscopic domains are quite interesting, but their pursuit would lead us too far afield. Therefore we restrict our discussion to the classical models of the 19th century and use phenomenological arguments to generalize the traditional rheological equations to the fractional calculus.
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Bibliography
H. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York (1962).
W. N. Findley, J. S. Lai and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover, New York (1976).
P. Garbaczewski, in Chaos-The Interplay between Stochastic and Deterministic Behavior, P. Garbaczewski, M. Wolf and A. Weron, eds., Springer-Verlag, Berlin (1995).
W. G. Glöekle and T. F. Nonnenmacher, A fractional Calculus approach to self-similar protein dynamics, Biophys. J. 68, 46–53 (1995).
W. G. Glöekle and T. F. Nonnenmacher, Fox function representation of non-Debye relaxation processes, J. Stat. Phys. 71 (1993) 741.
W. G. Glöckle and T. F. Nonnenmacher, Fractional relaxation and the time-temperature superposition principle, Rheol. Acta 33 (1994) 337.
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition, Academic, New York (1980).
M. Kac, Probability and Related Topics in Physical Sciences, Interscience, New York (1959).
M. Köpf, R. Metzler, O. Haferkamp and T. F. Nonnenmacher, NMR Studier of Anomalous Diffusion in Biological Tissue: Experimental Observations of Lévy Stable Processes, in Fractals in Biology and Medicine, vol. II, G. A. Losa, D. Merlini, T. R. Nonnenmacher and E. R. Weibel, eds., Birkhäuser, Basel (1998).
D. Kusnezov, A. Bulgac and G. D. Dang, Phys. Rev. Lett 82, 1136 (1999).
N. Laskin, Fractional Quantum Mechanics and Lévy Path Integrals, preprint.
R. Metzler, W. G. Glöekle, T. F. Nonnenmacher and B. J. West, Fractional tuning of the Riccati equation, Fractals 5, 597 (1997).
E. Montroll, On the quantum analogue of the Lévy distribution, in Physical Reality and Mathematical Description, C. Mehra, M. Reidel, eds., Dordrecht, The Netherlands, 501–508 (1974).
T. F. Nonnenmacher and R. Metzler, Fractals 3, 557 (1995).
P. G. Nutting, J. Franklin Inst. 191, 679 (1921).
P. G. Nutting, Proc. Am. Soc. Test. Mater. 21, 1162 (1921).
Yu. N. Rabotnov, Elements of Hereditary Solid Mechanics, MIR, Moscow (1980).
H. Schiessel, R. Metzler, A. Blumen and T. F. Nonnenmacher, Generalized viscoelastic models: their fractional equations with solutions, J. Phys.: Math. Gen. 28, 6567–84 (1995).
G. W. Scott Blair, B. C. Veinoglou and J. E. Caffyn, Limitations of the Newtonian time scale in relation to non-equilibrium rheological states and a theory of quasi-properties, Proc. Roy. Soc. Ser. A 187, 69 (1947).
G. W. Scott Blair, A Survey of General and Applied Rheology, Pitman, London (1949).
R. K. Schofield and W. G. Blair, Proc. R. Soc. A 138, 707 (1932).
B. J. West, Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails, Studies of Nonlinear Phenomena in the Life Sciences vol. 7, World Scientific, Singapore (1999).
B. J. West, Quantum Lévy Propagators, J. Phys. Chem. B 104, 3830 (2000).
F. W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science, World Scientific, Singapore (1986).
N. Wiener, Nonlinear Problems in Random Theory, MIT Press, Cambridge, MA (1958).
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West, B.J., Bologna, M., Grigolini, P. (2003). Fractional Rheology. In: Physics of Fractal Operators. Institute for Nonlinear Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21746-8_7
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DOI: https://doi.org/10.1007/978-0-387-21746-8_7
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