Circular shaft experiencing an axial torque

A solid, circular cross-sectioned shaft experiences an axial torque T, as shown above.

Angle of twist
The hypothesis used in developing the stress and strain in the shaft is that all points on a cross-section of the shaft experience the same angle of twist. The angle of twist for a section of length L is given by the equation shown below.

The animation shown below demonstrations the geometry of deformation of the shaft under the load of the torque.

A freeze-frame of the twisting of the shaft is shown below.

 

Stress distribution on the shaft cross-section
For the solid cross-section shaft with material homogeneity on the cross-section, the both the shear strain and shear stress vary linearly with radial position on the cross-section, as shown below.

 

Stress element for points on the cross-section
For point "a" on the cross-section, the shear stress on the x-face points in the positive z-direction. For point "b" on the cross-section, the shear stress on the x-face points in the negative y-direction. The corresponding stress elements for these two points are shown below.