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Extra resources for Contest Problem Book IV: Annual High School Examinations, 1973-1982
11 writing ~q. where VB is, by definition, the virtual work of the stress per unit mass, show that Hence prove that VB is an exaet differential vB = ~u, if and only if, T ~(J is an exaet differential. When it exists the function u is ealled the strain energy density per unit mass. 13. In the notation of Ex. 12 show that astrain energy density exists if either T = eonstant, isothermal deformation, or ~(J = 0, adiabatie deformation. 14. Prove that the strain energy function is whether the material is isotropie or not.
40 by replacing the arc AB by the c10sed contour C. 40 (1) and (2). 60 hypothesis (ii) the second terms on the right-hand side vanish when ABis replaced by the c10sed contour C. 60 (3). 60 (5), [zz W(z)Ja is imaginary and therefore contributes nothing to the moment. 4o'Qd- B- 1 B= - 1 ':Y oz off Z c M-MB=-Re MB= ~ r/:. ["w(z)-z'w(z)Jc 4o'Q - Re ':Y z ozaff dz. 725. New expressions for the stress resultants 43 Since by definition 4 o2Qfoz oz = V we have, using Stokes's complex theorem, XB-iYB=-i:PVdZ=-2!
Relation of the stress resultants and dislocations to the complex stresses 45 Differentiate (1) und (2) and substitute in (6), (7). 1 i I n(:>e+1) , 2 p, Yl BI = n (:>e + 1) F= :>ep1+2p, OCli I EI = , n(:>e+1)' iM(1)+ N(1) n ) (11) When the plate has n-holes bounded by non-intersecting contours Cs, (s = 1, 2, ... , we 'W( ) ;.. ) (13) + k(z). (14) Now let Po denote the resultant of external forces applied to the outer boundary C and the body force and reversed mass acceleration acting over the whole area enclosed by C.