. A treatise on the mathematical theory of elasticity . itions of zero tractionacross a boundary are cos (x, v) .-^ - cos {y, v) 5—^ =0, - cos ix, v) ,-^ -H cos(w, v) ^ = 0,6y^ dxdy cxdy ^ da? and these are the same as OH \oy) ds \dxj where ds denotes an element of the boundary. Hence cj^iox and d^/^y areconstant along the boundary, and we have ds\^ drJ dsV dx by J ds ds dx ds dy * See the theorem (ii) of Article 59. APPLIED TO PLANE STRAIN 215 154, 155] It follows that a boundary free from traction in the (r, 6) system is trans-formed into a boundary subject to normal tension in the (r, 0) sy
. A treatise on the mathematical theory of elasticity . itions of zero tractionacross a boundary are cos (x, v) .-^ - cos {y, v) 5—^ =0, - cos ix, v) ,-^ -H cos(w, v) ^ = 0,6y^ dxdy cxdy ^ da? and these are the same as OH \oy) ds \dxj where ds denotes an element of the boundary. Hence cj^iox and d^/^y areconstant along the boundary, and we have ds\^ drJ dsV dx by J ds ds dx ds dy * See the theorem (ii) of Article 59. APPLIED TO PLANE STRAIN 215 154, 155] It follows that a boundary free from traction in the (r, 6) system is trans-formed into a boundary subject to normal tension in the (r, 0) tension has the same value at all points of the transformed boundary,and its effect is known and can be allowed for. 155. Equilibrium of a circular disk under forces in its plane*. (i) We may now apply the transformation of inversion to the problem of Articles149, 150. Let 0 be a point of a fixed straight line OA (Fig. 19). If ffA were the boundary ofthe section of a body in which there was plane strain produced by a force F directed along. Fig. 19. O&X, the stress-function at P would be -ir^FrO sin 6, where r stands for (yP; and thismay be written — ir~^F6y, where y is the ordinate of P referred to OX. When we invertthe system with respect to 0, taking k=0O, P is transformed to P, and the new stress-function is -ir-^ri^F(ei + e^lc^ylri^, where 5j and ^-6^ are the angles XOP, XffP, andwe have written r^ for OP, and y for the ordinate of P referred to OX. Further the lineOA is transformed into a circle through 0, 0, and the angle 2a which 00 subtends atthe centre is equal to twice the angle AOA. Hence the function -7r~^Fy{6^ + d2) is thestress-function corresponding with equal and opposite isolated forces, each of magnitude F,acting as thrust in the line 00, together with a certain constant normal tension round thebounding circle. To find the magnitude of this tension, we observe that, when P is on the circle, ?•, cosec 02 = ^2 cosec 6^ — k cosec (^i -I- ^2) =
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