The study of solid circular shafts under stress shows that within the elastic limit, stress increases directly to the distance from the shaft's center. This relationship holds until the shaft reaches a critical point of stress, beyond which it begins to yield, marking the transition from elastic to plastic deformation. At this crucial juncture, the maximum torque the shaft can endure without permanent deformation is determined, signifying the limit of its elastic behavior.

As torque on the shaft increases, the plastic region develops, surrounding the inner elastic core, characterized by a constant stress level in the plastic area and a linear stress distribution within the elastic core. With continuous torque increase, the plastic zone extends, diminishing the elastic core, until the deformation across the shaft becomes entirely plastic.

The total torque exerted on the shaft is a sum of the torques associated with the elastic and plastic deformation regions. By further analysis and simplification, focusing on the expansion of the plastic region, one can calculate the ultimate plastic torque. The ultimate plastic torque is the maximum torque the shaft can handle before it succumbs to complete plastic deformation, losing its original form entirely.