By Mauro Ferrari, Vladimir T. Granik, Ali Imam, Joseph C. Nadeau
The lately proposed, totally multi-scale conception of doublet mechanics, awarded the following in a self-contained shape, bargains unprecented possibilities to reconcile the discrete and continuum representations of solids whereas conserving an easy analytical layout and entire compatibility with lattice dynamics and continuum mechanics. Its functions 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 tools 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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Extra resources for Advances in Doublet Mechanics
Mon and M. 4 Energy Balance The differential formulation of energy balance for the material is r - div( q) + P + C - e = o. 12) In eqn. 12) r is the volume rate of heat supplied per unit volume, q is the heat flux vector, P and C are the mechanical and chemical power per unit volume, respectively, and e is the internal energy per unit volume. Of course, one could define r as a summation of heat supplies associated with each component, however, in this work, we elect to use the simpler definition given above.
Under some conditions (see, below, Sect. 2 the couple macrostresses Mij vanish. 47) to the equations of motion of the "symmetric" classical continuum. 49)2 become identities. ) eqns. 48) include different force macrostresses: u~f) in eqn. 47) and uW in eqn. 48). 48) at M > 1 become quite different from the equations of motion of both classical and Cosserat continua. 4 Alternative Formulation For later convenience, we express some of the equations developed in the previous sections in a more compact form.
27) 22 1. Introduction (M. Ferrari, V. T. Granik and A. 28) a=l On the other hand, the time rate of change of W is W· ~ (8W · 8W. 29) Comparing eqns. Ja tai 8W = --. 31) ds dt for the granular medium, where T and M are the force and couple vectors per unit area of the surface S, respectively, F is the force per unit volume and p is the density. 32) where v and 8 are arbitrary smooth functions with v(R, 0) = v(R, T) = 0, and setting it equal to zero and using eqn. 30), along with the definitions of the microstrains, yield the following results in terms of the microstresses: 1) Conservation of linear momentum: ~ ~ (_I)X- 1 (TJa)X- 1 LJ LJ a=l X=l = ,• X X 0 0 T ak 1 • • • T ak x 82u i P 8t 2 ' 8 (t ai + Pai) + 8Xkl ••• 8Xk ~.