By A.Cemal Eringen
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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).
Continuum Physics by A.Cemal Eringen