# Invariants of the stress tensor (von Mises yield criterion)

balasekar1005
TL;DR Summary
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

#### Attachments

Gold Member
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:
$$II_{\sigma}=\frac{1}{2} \left[ tr(\sigma^{2})-(tr \sigma)^2 \right]=- \sigma_{11} \sigma_{22}+ \sigma_{12} \sigma_{21} - \sigma_{11} \sigma_{33} + \sigma_{13} \sigma_{31} - \sigma_{22} \sigma_{33} + \sigma_{23} \sigma_{32}$$