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

Equation 14.6.8

Equation 14.6.9

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We now wish to re-examine the torsion of elastic cylinders for the case where the material is nonhomogeneous. The basic formulation and particular solutions were given in Chapter 9 for the homogeneous case and in Chapter 11 for anisotropic materials. Although a vast amount of work has been devoted to these problems, only a few studies have investigated the corresponding inhomoge- neous case. Early work on the torsion of nonhomogeneous cylinders includes Lekhnitskii (1981); later studies were done by Rooney and Ferrari (1995) and Horgan and Chan (1999c). As expected, most closed-form analytical solutions for the inhomogeneous problem are limited to cylinders of revolution, normally with circular cross-sections. Following the work of Horgan and Chan (1999c), we consider the torsion of a right circular cyl- inder of radius a, as shown in Fig. 14.24. The cylindrical body is assumed to be isotropic, but with graded shear modulus that is a function only of the radial coordinate u = u(r). The usual boundary conditions require zero tractions on the lateral boundary S and a resultant pure torque loading T'over each end section R.