By Konstantin Z Markov (ed.)

This article provides in a unified approach sleek geometric equipment in analytical mechanics in response to the applying of fibre bundles, jet manifold formalism and the similar inspiration of connection. Non-relativistic mechanics is noticeable as a specific box thought over a one-dimensional base. in truth, the concept that of connection is the most important hyperlink during the publication. 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 less than various reference frames. distinctive themes on geometric BRST mechanics, relativistic mechanics and others, including many examples, also are handled

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18) for the mean value of the strain field (e(x)) becomes 2 (e{x)) = e (x) -pjK{x- x')tf (X - x')P e^\x') 0 dx'. 40) After excluding the external field eo(x) from Eqs. 41) *(x) = 1 - t f ( x ) . Thus we get an equation of convolution type for e l ^ x ) . The Fourier transform reduces Eq. \k). 42) 0 The same notations for the Fourier transforms of the functions are used here, but w i t h the argument k instead of x. The solution of Eq. \k)=(l-pK*(k)Poy 1 gives (e(k)). 43) rier transform to Eq. 40) and taking takii into account Eq.

103) 0 where the functions /o(7) and / i ( 7 ) (7 = a/a > 1) are the same as i n Eq. 95). If 7 ^> 1 the coefficients dj, t = 1 , 2 , . . , 6 , i n Eq. 103), w i t h i n the accuracy of the terms of order 7 , are transformed to the forms: 3 - 1 7 d\ = ^ o ( l - 4 K ) + - y - ^ ( « o - 2), 47 rf = - ^ ( 3 / 27 s £ - l ) o 2 0 d = - ^ ( l + 2« ), 7 ) 4- -^°-(4 - K ) , 47 d = -2fi 0 5 0 d = - ^ ( l 7 0 + K ). e 0 In the l i m i t 7 —> 0 0 we have D(m) = - 2 ^ [ P i ( m ) - (1 - 2/c )P (m)]. 0 0 2 The other l i m i t (7 —» 1) corresponds to a correlation hole of the shape of a sphere.

Elastic Medium Reinforced with Stiff Flakes In this section we apply the above results to the calculation of elastic moduli of composites reinforced w i t h stiff inclusions having the shape of flattened spheroids. The middle surface of each inclusion is thus a circular axea of random radius a. The material of the inclusions is supposed to be isotopic. I n this case the tensor A ( m ) in Eq. 79) coincides w i t h the tensor Q(m), defined by Eq. 23). Let us consider the tensor A(m), defined by Eq.