When materials are subjected to forces that surpass their yield strength, they undergo a process known as plastic deformation. This results in a permanent alteration or strain in their structure. This concept can be specifically applied to circular shafts, where the deformation leads to a change in its shape. The precise evaluation of this plastic deformation requires understanding the stress distribution within the circular shaft, which is achieved by calculating the maximum shearing stress in the material. Once identified, a shearing-stress-strain diagram can be plotted to reveal the maximum shearing strain. It is crucial to remember that the shearing strain has a linear relationship with the distance from the axis of the shaft.

The relationship between shearing strain and radial distance can be determined by substituting the maximum shearing strain value into this equation. Similarly, the relationship between shearing stress and radial distance can also be derived. By using the integral relation and replacing the elemental area and polar moment of inertia with the shaft radius, the ultimate torque that leads to the shaft's failure can be calculated by maximizing the material's ultimate shearing stress value. The equivalent stress derived from this calculation is often referred to as the modulus of rupture in the torsion of the given material. This term represents the maximum stress to endure before the material fails under torsion.