By Professor Isaak A. Kunin (auth.)
ISBN-10: 3642819605
ISBN-13: 9783642819605
ISBN-10: 3642819621
ISBN-13: 9783642819629
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Extra info for Elastic Media with Microstructure II: Three-Dimensional Models
Example text
K,ucAa,utS(k)utS(k) = -qa(k). 36) For an infinite medium, free vibrations have a special significance. 6 Approximate Models 27 correspond to propagating waves. 38) give the dependence of the frequencies on the wave vector for the corresponding mode of free vibrations. 2) all the modes pass through the points w = 0, k = O. 39) which propagate with the group velocity V,~(k) -def 'Owi(k) 'Ok), . e. spatial dispersion takes place. The absence of nonzero real roots of the equation wi(k) = 0 is a necessary condition of stability of the medium.
E. such media are not gyrotropic. a(x) = (C$a pf3 + c~apf3TJ<2aTJa<2)Cpf3(x) . 10) where C2 has the dimension of ordinary elastic moduli and / is the characteristic scale parameter. The latter can also be considered as a small parameter with respect to which the expansion is carried out. 11) For an inhomogeneous medium, the elastic moduli Co, Ch C2 will be functions of the point x, and it is natural to regard them as changing slowly over distances 30 2. Medium of Simple Structure of the order of I.
8) allows us to construct an explicit expression for the static Green's tensor. 13) 40 2. 3, 4). 5) we have P j( - fl) = 1. An important property of the constructed Green's tensors is the absence of a singularity at r = 0, since Ga~(r) is an even analytic function. This corresponds to a physical meaning of the non local theory for a medium with an elementary unit of length, which models the discrete medium and formally is an effect of truncating the Fourier spectrum. As a result, all the divergencies connected with point sources of force, which are inherent to the model of an elastic continuum are eliminated.
Elastic Media with Microstructure II: Three-Dimensional Models by Professor Isaak A. Kunin (auth.)
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