19.9 : الاعمدة الدائرية-المواد اللدنة المرنة

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 ρ_Y.

In the plastic region, the stress is uniformly equal to τ_Y, 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 ρ_Y  as it approaches zero, the limiting value of the plastic torque, corresponding to an entirely plastic deformation, can be determined.