By A.M. Linkov
by the writer to the English version The e-book goals to offer a strong new device of computational mechanics, complicated variable boundary vital equations (CV-BIE). The publication is conceived as a continuation of the classical monograph by means of N. I. Muskhelishvili into the pc period. years have handed because the Russian version of the current ebook. we've seen turning out to be curiosity in numerical simulation of media with inner constitution, and feature proof of the possibility of the recent equipment. The proof used to be in particular transparent in difficulties on the subject of a number of grains, blocks, cracks, inclusions and voids. This triggered me, while getting ready the English version, to put extra emphasis on such issues. the opposite swap used to be encouraged via Professor Graham Gladwell. It used to be he who steered me to abridge the chain of formulae and to extend the variety of examples. Now the reader will locate extra examples exhibiting the aptitude and benefits of the research. the 1st bankruptcy of the e-book encompasses a basic exposition of the speculation of actual variable potentials, together with the hypersingular strength and the hypersingular equations. This makes up for the absence of such exposition in present textbooks, and divulges vital hyperlinks among the genuine variable BIE and the complicated variable opposite numbers. The bankruptcy may help readers who're studying or lecturing at the boundary point method.
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Extra info for Boundary Integral Equations in Elasticity Theory
2) where L is the total boundary of the system of blocks (the contact between surfaces of adjacent blocks is treated as a single line on which mechanical values may experience discontinuities); the direction of traveling along the contact is arbitrary, the normal is directed to the right of this direction; At = tn+ - tn-is the discontinuity of traction on L. As usual, the index "plus" ("minus") refers to a value from the left (right) of the traveling path. On the external boundaries of the blocks we consider t n- = 0, U- = 0, f-L- = O.
4) do not exist in a usual sense. It is understood as a finite-part (Hadamard) integral. It means that to evaluate it, one should exclude a small s-vicinity of the point Xo, and delete all the terms in the integral tending to infritity when 6~O, i. e. keep only the finite part of the integral, and after that come to the limit for s~O. Practical integration based on this defInition is quite simple; this becomes especially evident when integration is performed in complex variables (see part III).
From this, it follows that the potential of the simple layer is continuous through the contour L: The same holds for the hypersingular potential if the normal Ox is taken in the direction of the normal to L at the point Xo to which a field point x tends: The potentials with the index S, that are ts and Us, in general, have different limits as x tends to L from different sides. 1) Au(x o) =u~(xo)-us(xo) = -w s(x o). 1) we conclude that the density WI presents the traction discontinuity At = t/ - ts- on the contour L.