# Poisson's ratio for a rigid rod

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1. Dec 3, 2014

### westmckay99

I have a conceptual misunderstanding it seems. Poisson's ratio is the ratio of elastic strain deformation of the transverse and longitudinal components. That being said, if I were to induce thermal stress (heating up) to a rod by keeping its ends (longitudinal component) rigid, would there be a mechanical contribution to the transverse strain on top of the thermal one? My textbook solutions manual seems to think so however I don't understand how you can have a transverse strain when you have no longitudinal one (no change in length since the rod is maintained rigid throughout the thermal stress exposure).

Any insight on this would be greatly appreciated.

2. Dec 3, 2014

### OldEngr63

Imagine a process wherein you heat the rod with the ends free, and then later, while it is hot, compress it back to its original cool free length. I think you will see where the Poisson expansion comes from in this.

3. Dec 3, 2014

### westmckay99

Hi, thanks for the reply! I completely understand that thought experiment, however I see a flaw: the imaginary step of letting it expand longitudinally offsets the compression afterwards, ie it may expand in the transverse direction when you compress it back down but it had already contracted by the same amount originally when you let it expand i the longitudinal direction (net effect of mechanical strain in the transverse direction is null).

4. Dec 4, 2014

### Staff: Mentor

When you heat it with no constraint, it expands with equal strains in all directions to a new stress-free configuration.

Chet

Last edited: Dec 4, 2014
5. Dec 4, 2014

### westmckay99

Hi, Thanks for the reply! However, I'm unsure as to how your comment applies to my question. The scenario I refer to has longitudinal constraints. The thermal expansion in the radial direction I understand but how is it there's also a mechanical contribution (they use the poisson ratio times the thermal stress/modulus of elasticity) when there's is no longitudinal strain to begin with. Refer to my previous reply to OldEngr63 to see my counter argument to the hypothetical thought experiment of assuming it had been free to expand then applying a compression strain to return the expansion to the original length.

Cheers