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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.
KeywordsRiccati Equation Fractional Calculus Path Integral Maxwell Model Relaxation Modulus
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