By N. I Muskhelishvili
TO the 1st ENGLISH version. In getting ready this translation, i've got taken the freedom of together with footnotes broadly speaking textual content or putting them in small kind on the acceptable locations. i've got additionally corrected minor misprints with no precise point out .. The Chapters and Sections of the unique textual content were referred to as elements and Chapters respectively, the place the latter were numbered consecutively. the topic index used to be now not inside the Russian unique and the authors' index represents an extension of the unique record of references. during this means the reader might be capable of finding fast the pages on which somebody reference is mentioned. The transliteration challenge has been triumph over by way of printing the names of Russian authors and journals additionally in Russian style. whereas getting ready this translation within the first position for my very own informa tion, the information that it can additionally develop into obtainable to a wide circle of readers has made the trouble doubly beneficial. i believe definite that the reader will proportion with me in my admiration for the simplicity and lucidity of presentation.
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Extra resources for Some Basic Problems of the Mathematical Theory of Elasticity: Fundamental Equations Plane Theory of Elasticity Torsion and Bending
5b) The stress surface is again an ellipsoid, but the normal stresses are now given by N = -c 2 I OH 12 ' indicating that, in contrast to the preceding case, the stresses on all planes are compressive. g. Nl > 0, N2 > 0, N3 < 0. 5d) a ... _. _______ ..... hyperboloid of two sheets. ) + N2YJ 2 -I Na ! 6) (see Fig. 5). 5c); hence the normal stress is given by Nl~2 +c 2 N = I OH 12 Y and it will be tensile. Sd), so that the normal stress which is now compressive is given by N c2 == - -I OH - -12.
In the main part of this book the reader will not be assumed to be conversant with tensor calculus. For the understanding of certain remarks it will be sufficient to study Appendix 1 at the end of this book. The following will help to elucidate the final paragraph of this section. Let there be given a quadratic form 2n(~, 1), ~) = 't"xx~2 where + 't"lI11"fJ2 + 't"zz~2 + 2t'lIz1)~ + 2't"zx~~ + 2't"x1l~'1J, are the components of some (arbitrary) vector and the coefficients are quantities independent of ~, "I), ~, but depending on the direction of the axes of the orthogonal rectilinear coordinate system.
Thus, let P = (~, 1], ~) denote a vector, normal to the considered plane and acting in the same direction as the positive normal n. J Then ~, 'Y), ~. 2), N. p2 == 2n(~, 'Y), ~). 4) Now the following will be noted. The quantity N, by definition, has physical meaning and hence cannot depend on the particular choice of coordinate axes. , the square of the length of the vector) does not depend on this choice. , it must be invariant to transformation of (orthogonal, rectilinear) coordinates. , ... ~'2 + 2Y~:tJ'~' + 2Z~,~'~' + 2X~,~'''I)' == === Xx~2 + Yv"l)2 + Zz~2 + 2Yz~~ + 2Zx~~ + 2Xll~"I).
Some Basic Problems of the Mathematical Theory of Elasticity: Fundamental Equations Plane Theory of Elasticity Torsion and Bending by N. I Muskhelishvili