Circular Shafts - Elastoplastic Materials

Within the elastic limit, maximum stress in a solid circular shaft varies linearly with radial distance from its axis. This linear relationship holds until the shaft reaches a critical stress point. Understanding circular shaft stresses in linear range is fundamental to predicting when elastic behavior transitions to plastic deformation.

The maximum elastic torque is the torque value at which the shaft reaches its yield point, marking the onset of plastic deformation. Calculating this torque involves substituting the saturation stress value and the polar moment of inertia. This threshold determines the limit of elastic behavior before permanent deformation occurs.

When torque increases beyond the elastic limit, a plastic region develops around an elastic core of radius ρY. The plastic region experiences uniform stress equal to the yield stress, while the elastic core maintains linear stress variation. This mixed deformation state continues until the plastic region expands completely.

In the plastic region, stress remains uniformly constant at the yield stress value. In contrast, the elastic core exhibits linear stress variation with radial distance. This distinct stress behavior in each region allows engineers to calculate total torque by superposing contributions from both deformation zones.

The ultimate plastic torque is the maximum torque a shaft can withstand before complete plastic deformation occurs across its entire cross-section. This limiting value is determined when the elastic core radius approaches zero, meaning the entire shaft has yielded. At this point, the shaft loses its original form entirely.

Total torque is expressed as the superposition of torques in the elastic and plastic regions. By analyzing each region's stress distribution and integrating across the cross-section, engineers sum the individual torque contributions. This method enables prediction of shaft behavior during the transition from elastic to fully plastic states.

The elastic core radius ρY shrinks as applied torque increases beyond the yield point. The plastic region expands inward from the shaft's outer surface, progressively reducing the elastic core. Eventually, when ρY approaches zero, the entire shaft becomes plastic, and no elastic core remains.