For the torsion problem discussed in Section 14.6, explicitly justify the reductions in polar coordinates summarized by

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For the torsion problem discussed in Section 14.6, explicitly justify the reductions in polar coordinates summarized by relations (14.6.8) and (14.6.9).

Data from section 14.6

We now wish to re-examine the torsion of elastic cylinders for the case where the material is nonhomogeneous.

X S a R T Z

The beginning formulation steps remain the same as presented previously, and thus the displace- ments,

It again becomes useful to introduce the Prandtl stress function, p = (x,y)   Txz  dy' Tyz = so that the

Incorporation of the boundary condition that tractions vanish on the lateral surface S leads to identical

Ur = U=0, ug = arz er = eg=e = erz = ero = 0, with boundary condition o(a) = 0. Toz = d dr e oz or=00=0= T=

Equation 14.6.8

uy = ug = 0, u0 = drz er=el=e==en=ero = 0, et 0 = 60 = 0 = tr = tro = 0,  ar 2 =

Equation 14.6.9

Toz || 1 d r dr d dr r do dr  = -2

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