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© 2012 Achar and Hanneken, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Microscopic Formulation of** 

B.N. Narahari Achar and John W. Hanneken

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51493

**1. Introduction** 

a limit.

**Fractional Theory of Viscoelasticity** 

Viscoelasiticity refers to the phenomenon in which a material body, when deformed exhibits both elastic (akin to solids)and viscous (akin to liquids) behavior. The body stores mechanical energy (elastic behavior) and dissipates it simultaneously(viscous behavior). Linear theory of viscoelasticity treats the body as a linear system which when subjected to an excitation responds with a response function. If the excitation is a stress, the response is a strain and if the excitation is a strain, the response is a stress. Mechanical models involving a spring-mass connected to a dashpot have been used to explain the elastic and viscous behavior.The mathematical structure of the theory and the spring-dash-pot type of mechanical models used and the so called Standard Linear Solid have all been only too well known [1-4]. In recent years methods of fractional calculus have been applied to develop viscoelastic models especially by Caputo and Mainardi [5,6] , Glockle and Nonenmacher [7], and Gorenflo and Mainardi [8]. A recent monograph by Mainardi[9] gives extensive list of references to the literature connecting fractional calculus, linear viscoelasticity and wave motion.All these works treat the phenomenon of viscoelasticity as a macroscopic phenomenon exhibited by matter treated as an elastic continuum albeit including a viscous aspect as well. It should be recognized that matter has an atomic structure and is fundamentally discrete in nature. A microscopic approach would recognize this aspect and a theoretical model would yield the results for the continuum as

It is well established that lattice dynamics provides a microscopic basis for understanding a host of phenomena in condensed matter physics, including mechanical, thermal, dielectric and optical phenomena, which are macroscopic, generally described from the continuum point of view. Since the pioneering work of Born and Von Karman [10], lattice dynamics has developed into a veritable branch of condensed matter physics. Lucid treatment of various

and reproduction in any medium, provided the original work is properly cited.
