By Mauro Ferrari, Vladimir T. Granik, Ali Imam, Joseph C. Nadeau

The lately proposed, totally multi-scale concept of doublet mechanics, provided the following in a self-contained shape, deals unprecented possibilities to reconcile the discrete and continuum representations of solids whereas conserving an easy analytical structure and whole compatibility with lattice dynamics and continuum mechanics. Its purposes comprise micro-electro-mechanical structures (MEMS), granular and particulate media, nanotubes and peptide arrays. Novel effects are mentioned, together with the identity of a brand new type of dispersive floor waves, and the presentation of equipment for the experimental selection of the fundamental microstructural parameters. The relationships among doublet mechanics, lattice dynamics, and continuum theories are tested.

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31) except JJo to be zero yields H2 o == O. This means rn o = pat/J/aJ-lo and rn o is not a function of V'T. 31) except 10i to be zero yields H3 0 i == o. This means toi = pat/J/a/oi and toi is not a function of V'T. Up to this point we have not applied the Second Law of Thermodynamics; nothing has been said of which processes are possible and which are not. In this chapter, we retain the viewpoint that different statements of the Second Law are possible, each embodying some aspects of it. 32) t which states that the sum of the entropy from heat conduction q through the surface of the body oBt and from radiation r in the body B t , considered over a closed cycle 1, must be greater than or equal to zero.

35) can be rewritten as K. ~ aT < 0 I) aXi ox; - · (2 37) · Given that K # 0, if eqn. 37) is to hold for arbitrary (near zero) values of aT/ ax; it must be that a()/ ax i = o. Thus, (J is not an explicit function of x. This result, added to the results of parts d) to f) above, show that () is an explicit function of T only. Given that (J ~ 0 and a() / aT ~ 0, O(T) is an invertible function and can replace T as a constitutive variable in all of the foregoing equations and relations. 38) and 8 is not a function of g = grad(()) because t/J is not from part c) above.

Green and Naghdi, 1979). Finally, the results of the analyses are applied to a study of homogeneous linear elastic doublet mechanics. 1) where the superscripted dot ( · ) denotes the material time derivative. In eqn. 1), r is the volume rate of heat supplied per unit mass, q is the heat flux vector, P is the mechanical power per unit volume, E is the internal energy per unit mass, and p is the mass density. 2) e where is the internal rate of entropy production per unit mass, s is the entropy per unit mass, and (J is a function of empirical temperature, T, and other constitutive variables such that (J ~ 0 and a(J/ aT ~ O.

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