Elastic Media with Microstructure II: Three-Dimensional by Professor Isaak A. Kunin (auth.)

By Professor Isaak A. Kunin (auth.)

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Let us restrict ourselves to the simplest model of the long-wavelength approximation. IThis definition is more general than the one given in Sect. 3, since the invariance of the terms of the sum with respect to rotation is not required. In connection with this, the definition has a formal nature. 22) in powers of k and consider only a finite number of terms. With k = 0, we have the zeroth approximation, which is considered in the next section. When proceeding to successive approximations, it is expedient to distinguish two cases (Sect.

K) and fJ,(k) are even analytic functions of k. 4) which has the same structure as the tensor of elastic moduli in the local theory of elasticity. However, as distinct from the local theory, this representation for clal'fJ(k) is not unique. 4) terms of the form which would only change the connection between the scalar coefficients of the tensors

E. 6 c~al'fJTj ... Tn(ikTl) 00 n=O ... 1) where the cn's are real constant tensors, which can be explicitly expressed in terms of force constants of the micro-model. 2]. 4) with the elastic-constants tensor c8al'fJ. Note that this approximation is universal in the sense that it is physically correct for an infinite medium, as well as for boundary-value problems. As we saw in the examples of one-dimensional models, when constructing successive approximations, it is usually necessary to distinguish the cases of infinite and finite media.

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