# Using 2nd-Piola-Kirchoff stress rather than Cauchy

1. Jul 5, 2011

### hoomanya

Hi,
I need to use the 2nd Piola Kirchoff Stress (S) rather than the usual Cauchy stress i(sigma) in a general transport equation of a material. I was wondering if I need to replace the stress tensor (usually denoted by sigma) with S, or do I just write sigma in terms of S using the relationship that exists between S and sigma.
Thanks!

2. Jul 9, 2011

### afreiden

I see you have no responses, so I'll give you something to think about -- though, I speak from an FEA standpoint:

Why do you NEED to use the PKII stress, S?
Is it because you are using the Lagrangian Strain tensor, E?

Your stress and strain pair (constitutive relationship - e.x. Hooke's Law) must be work- conjugate, meaning that they must behave the same under rigid body rotation. The PKII stress tensor, S, and the Lagrangian strain tensor, E, are indeed both invariant to rigid body rotation, so they are a work-conjugate pair.

S is only meaningful for small (infinitesimal) strains. You could potentially acquire the Cauchy stress, $\sigma$, from S, as you describe -- but you wouldn't want to use $\sigma$ with E in your constitutive relationship. Keep in mind that $\sigma$ is defined in in a spatial reference frame (it is defined in your original coordinate system, regardless of whether there is rigid body rotation).

If your constitutive relationship is in rate-form, then that's a whole 'nother story.

Also, I remind you that I am an FEA guy, so if nothing I said applies to you, I apologize.

3. Jul 9, 2011

### hoomanya

Thanks. What you said made sense. I also had a read of the wiki page on stress and am a lot more clear now.