A solid circular phosphor-bronze shaft of radius R = 19.0 mm and length L = 920 mm is…

A solid circular phosphor-bronze shaft of radius R = 19.0 mm and length L = 920 mm is homogeneous, isotropic, and linearly elastic with shear modulus G = 41.0 GPa and Poisson’s ratio ν = 0.33. It occupies the cylinder x² + y² ≤ R², 0 ≤ z ≤ L. The end z = 0 is prevented from rotation in the Saint-Venant sense, and the end z = L is loaded by a torque T = 78.0 N·m about e_z, with no net axial force and no bending moment. Body force is absent. The Saint-Venant torsion field is the three-dimensional displacement u = −α y z e_x + α x z e_y, with warp u_z = 0 and with a constant twist rate α to be determined. Compute the infinitesimal strain and, via isotropic Hooke’s law, the Cauchy stress. Verify that the homogeneous Navier equation holds and that the lateral surface r = R is traction-free pointwise. Determine α from the torque identity T = ∫_A (x σ_zy − y σ_zx) dA on a cross-section, and identify the Saint-Venant torsion constant J defined by T = G J α. Report the peak shear stress, the rotation of the end z = L, and the stored energy, computing the energy both as (1/2) T (α L) and as the volume integral of W. Evaluate the traction on the end z = L and compute its force resultant and its moment resultant about the shaft axis.

The circular Saint-Venant field is isochoric and harmonic, so Navier is automatic; Hooke reduces to \(\boldsymbol{\sigma}=2G\varepsilon\).