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By Professor Isaak A. Kunin (auth.)
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Additional info for Elastic Media with Microstructure II: Three-Dimensional Models
Consideration of such a system in the scope of the model of simple structure is equivalent to postulating that the angles of rotation of the rods' ends are equal to zero. This means that masses located at the knots are displaced in slots which prevent their rotation. Howevel, such a construction is not, in general invariant with respect to rotation. For an adequate description of the system of rods a model of a medium with complex structure is needed, as we have seen in [Ref. 12, Chap. 3], when 32 2.
1. 2. ;') . ;'. 12) 48 3. 13) where pas'fJ def gafJps'. If we forget about the quasicontinuum, then this Lagrangian describes the most general linear model of a three-dimensional medium of complex structure. The corresponding equations of motion have the form [Ref. 12, Eq. 14) where, in the general case, pas'f3(x, x') is the kernel of a nonlocal moment-ofinertia operator. Let us now consider the physical meaning of the collective cell variables was(x) andfsa(x). 10,12) we find that WOa is the displacement of the mass center of the cell and fOa is the average body force density.
App(x, x') can be found, as in [Ref. 12, Sect. 4]. Thus the equations for the medium with weak dispersion belong to the well studied class of equations with a small coefficient of higher derivatives. It is necessary to take this coefficient into account when constructing solutions as well as when determining the field of applicability. In particular, such equations are also connected with phenomena of the boundary-layer type. In an obvious way, the equations for higher-order approximations can be written down, but they have no significance, since they do not describe qualitatively new phenomena and may provide at most a small correction.