Invariants of the stress tensor (von Mises yield criterion)

[balasekar1005]

TL;DR

I see different versions of the second invariant of the cauchy stress tensor.

Hello all,

I am trying to understand the von Mises yield criterion and stumbled across two equations for the second stress invariant. Although the only difference is a difference in signs (negative and positive), it has been bothering me. Attached are the two versions. Which one is correct and if both are correct, why is there a change in sign?

Thank you,
Bala

 

[Frabjous]

I have not done theoretical stuff in a long time, so take what I say with a grain of salt ...

The I invariants are the constants of the characteristic polynomial of the stress tensor used to determine the principal stresses so that you can define them to within a sign depending on how you choose to write the equation. What I cannot remember, is if there is a sign convention. Since you are finding both, my guess is that there is not one.

BTW, make sure that you do not confuse J₂ with I₂.

 

[FEAnalyst]

Here's what I've found in one of the books:
$$II_{\sigma}=\frac{1}{2} \left[ tr(\sigma^{2})-(tr \sigma)^2 \right]=- \sigma_{11} \sigma_{22}+ \sigma_{12} \sigma_{21} - \sigma_{11} \sigma_{33} + \sigma_{13} \sigma_{31} - \sigma_{22} \sigma_{33} + \sigma_{23} \sigma_{32}$$