# Rod stretching physics problem

1. Feb 15, 2005

### TSN79

A small test rod is stretched beyond yielding strength with a force F, and then relieved of tension. If it is now heated and cooled, will a force greater or smaller than F be required to stretch the rod further? Personally I think a smaller force will be required since the rod already has began to give way, but I'm not sure how to explain the physics involved. Can someone help me?

2. Feb 15, 2005

### Astronuc

Staff Emeritus
The force F' required for additional strain depends a lot on the original yield point, beyond which plastic or permanent deformation occurs. The yield point depends on the amount of cold-work (dislocation density) in the material or the degree of recrystallization.

Fully-annealed material (no cold-work) has the lowest yield point. The yield point increases with the amount of cold work introduced.

The lower the yield point, the less force required to exceed yield. Below the yield point, the stress ($\sigma$)-strain($\epsilon$)- relationship is essentially linear (Hooke's law), $\sigma$ = E $\epsilon$, where E is the elastic (Young's) modulus.

Beyond the yield point, the stress is given by a more complicated stress-strain law, e.g. $\sigma$ = K E$\epsilon^n$.

If cold working increases the yield point, then more force (greater stress) will be needed to further deform the bar. If the heating is sufficient to anneal the material (i.e. lower the yield point), then less force is necessary. That is one reason that thermo-mechanical treatments do repetitive cold-work/anneal cycles.

3. Feb 18, 2005

### PerennialII

I'd say it can go both ways, along the lines of Astronuc. If we consider the material to have strain hardened during the streching beyond yield and the effects of the thermal cycle have not produced recrystallization or any other microstructural recovery the stress will be greater to re-initiate yielding. On the other hand, if recrystallization / annealing occur the stress for further plasticity can be lower than the final stress of the first cycle, and even lower than the initial yield. Since the point is to introduce the thermal cycle I'd go with the latter option, giving an exact answer would require more details.