By Konstantin Z Markov (ed.)

This article offers in a unified method glossy geometric equipment in analytical mechanics in accordance with the applying of fibre bundles, jet manifold formalism and the similar idea of connection. Non-relativistic mechanics is noticeable as a specific box thought over a one-dimensional base. in reality, the idea that of connection is the most important hyperlink in the course of the booklet. within the gauge scheme of mechanics, connections look as reference frames, dynamic equations, and ion Lagrangian and Hamiltonian formalisms. Non-inertial forces, power conservation legislation and different phenomena regarding reference frames are analyzed; that leads the reader to observable physics. The gauge formula of classical mechanics is prolonged to quantum mechanics lower than diversified reference frames. specified themes on geometric BRST mechanics, relativistic mechanics and others, including many examples, also are handled

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**Example text**

115) i/ (2 - r ) ( 4 - r ) 0 The curves ' 1 ' , '2' and '3', shown i n Fig. 115). They are compared w i t h the experimental data cited in Ref. 20. 5). Experimental data are approximated by the dash line with small circles for (E /Eo) and dash-dot line for (V*/VQ). The statistical analysis shows that the set of cracks investigated i n Ref. 20 is satisfactorily defined by the model w i t h the restriction on crack intersection. The formulas for the effective elastic constants obtained for this model are i n good agreement w i t h experimental data.

88), which enters Eq. 93) for the effective elastic moduli tensor of the composite. The function ^! (x) is assumed to have the symmetry of ellipsoid which is coaxial w i t h the considered inclusion. Let the semiaxes of this ellipsoid be a i = a = a and a . The axis 1 3 is directed along the normal m to the middle surface of the inclusion Q,. I n this case the tensor A(m) has the form (7 = a/a > 1): m 2 3 3 A{m) = ArPiim) + A (P (m) - ^P (m)) + A (P (m) + P (m)) 2 1 2 +A P (m) s A ^ W - l ^ - ^ f o + fi], 5 3 4 + A P (m), 6 6 A = ~\(2-K )f 2 3 0 0 + f ], l A = 3 -~f lt 33 fia (, ^ 1 !

The method of effective field is used again for the description of the interaction between inclusions. As before we assume that every inclusion of the orientation m is located i n a constant local effective stress field o-»(m)(cr»(m) = Co£„(m)). To the accuracy of dominant terms i n the expansion of elastic fields w i t h respect to the small parameters of the problem 61 and 6 , the strain and stress fields i n the medium w i t h such inclusions are represented i n the form which follows from Eq.