By Mauro Ferrari, Vladimir T. Granik, Ali Imam, Joseph C. Nadeau
The lately proposed, totally multi-scale idea of doublet mechanics, awarded the following in a self-contained shape, deals unprecented possibilities to reconcile the discrete and continuum representations of solids whereas keeping an easy analytical structure and whole compatibility with lattice dynamics and continuum mechanics. Its purposes contain micro-electro-mechanical structures (MEMS), granular and particulate media, nanotubes and peptide arrays. Novel effects are said, together with the id of a brand new classification of dispersive floor waves, and the presentation of equipment for the experimental choice of the basic microstructural parameters. The relationships among doublet mechanics, lattice dynamics, and continuum theories are tested.
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Additional info for Advances in Doublet Mechanics
14 1. Introduction (M. Ferrari, V. T. Granik and A. Imam) As discussed in Sect. 3, doublet mechanical analysis has been applied to crystalline solids, as well as granular, particulate, and composite media in the past. An exciting novel vista on the applications of the method is offered in Chap. 10 where Chopra and Zettl review the field of nanotubes. Nanotubes are technological objects of extraordinary interest. They are particularly suitable for doublet mechanical analysis because their discrete nature is evident at the structural scale and, as a result, they are not amenable to being modeled as continuous media.
3) where g denotes grad(O). Rewriting eqn. 3) yields . 2 Balance Laws 29 or, in terms of the Helmholtz free energy ('¢ = E - f) s) . - p(,¢ . + f)s) - q. g -f)- - pf)e +p = ( o. 5) will be referred to as energy/entropy balance equations and must hold for all thermomechanical processes. Definition of a thermomechanical process will be delayed until the mechanical power is discussed further. The only difference in these equations, between continuum mechanics (eM) and doublet mechanics (DM), lies in how the mechanical power per unit volume P is written.
43) leads to the conclusion that the DM microstrain measures are properly invariant. 51) where T/3 is the magnitude of ,/3. -t/3,Tf3,S,c a ,g"/3· g: X),a +... 55) where 9 is the magnitude of grad(O). Again, as in Chap. 2, the dependence of q on can be removed in the linear theory. 59) L (GOta€Ot + HOtaPOt + ROta10t ) + LaA + L M3 abCb, Ot=1 b=1 A = s - SO is an increment of the entropy per unit volume and SO is the entropy per unit volume of the solid in an initial state. 61) G Ota = M1 Ota , H Ota = M2 Ota , B Ot ,8 = D,8Ot, M3 ab = M3ba, COt ,8 = FOt ,8 = ROta = o.
Advances in Doublet Mechanics by Mauro Ferrari, Vladimir T. Granik, Ali Imam, Joseph C. Nadeau