19.9:
Circular Shafts – Elastoplastic Materials
Within the elastic limit, the maximum stress in a solid circular shaft varies linearly with a radial distance from its axis.
As the torque increases, the maximum stress reaches a saturation value at the onset of yield. Substituting this saturation value, the corresponding maximum elastic torque, for which the deformation remains fully elastic, can be calculated.
Substituting for J/c, expresses the torque in terms of radial distance and the maximum stress.
As the torque increases further, a plastic region develops in the shaft around an elastic core of radius ρY.
In the plastic region, the stress is uniformly equal to τY, while in the elastic core, the stress varies linearly with ρ.
Increasing the torque further, the plastic region expands till the deformation is entirely plastic.
The total torque in the solid circular shaft can be expressed as a superposition of torques in the elastic and plastic regions.
Simplifying the equation further and substituting for ρY as it approaches zero, the limiting value of the plastic torque, corresponding to an entirely plastic deformation, can be determined.
19.9:
Circular Shafts – Elastoplastic Materials
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.