Torsion and plastic deformations

[Dell]

the foollowing hollow shaft has a torsion applied to it so that the plastic area reaches the inner radius, what is the magnitude of the maximum residual shearing stress?

http://lh3.ggpht.com/_H4Iz7SmBrbk/S9XK9ZwAbQI/AAAAAAAACFU/ueYSDTK5qMM/s800/Capture.JPG

i don't see why there should be any residual stress? if the shaft entirely becomes plastic- which as far as i can tell is what happens here- why should i have any residual stress,
i know that when i have an elastic core, the elastic core wants to return to its original placement and the plastic area wants to stay with its residual strain, leaving each of them a stress and the placement being somewhere inbetween. but why is this so in this case too??

 

[KataKoniK]

I can explain the concept of residual stress in this situation. When a torsion is applied to a hollow shaft, the material experiences both elastic and plastic deformation. The elastic deformation causes the material to return to its original shape once the load is removed, while the plastic deformation causes permanent changes in the material's shape.

In the case of a hollow shaft with a torsion applied, the plastic area reaches the inner radius due to the high amount of stress applied. This means that the material in this area has undergone plastic deformation and will not return to its original shape once the load is removed. However, the elastic core of the shaft will still try to return to its original shape, causing a mismatch in the material's shape.

This mismatch results in residual stress, which is the stress that remains in the material even after the load is removed. In this case, the maximum residual shearing stress will occur at the interface between the plastic and elastic regions, where the mismatch in shape is the greatest. The magnitude of this stress will depend on the material properties, the amount of torsion applied, and the geometry of the shaft.

Therefore, even though the entire shaft may become plastic, there will still be residual stress present due to the mismatch in shape between the plastic and elastic regions. This concept is important to consider in engineering and design, as residual stress can affect the overall strength and durability of a material.