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

• balasekar1005
In summary, The two equations for the second stress invariant differ only in the sign and can be defined within a sign convention. The I invariants are the constants of the characteristic polynomial of the stress tensor used to determine the principal stresses. There is no sign convention for the I invariants, as both positive and negative values are valid. It is important to not confuse J2 with I2. One of the equations for the second stress invariant is shown in the book as being negative.
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

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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.

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

## 1. What is the von Mises yield criterion?

The von Mises yield criterion is a mathematical equation used to determine the onset of yielding (permanent deformation) in a material under stress. It is based on the theory that yielding occurs when the maximum shear stress in a material reaches a certain critical value.

## 2. How is the von Mises yield criterion calculated?

The von Mises yield criterion is calculated using the stress tensor, which is a mathematical representation of the stress state at a point in a material. The equation takes into account the normal stresses (tension or compression) and the shear stresses in three dimensions.

## 3. What is the significance of the invariants of the stress tensor in the von Mises yield criterion?

The invariants of the stress tensor are mathematical quantities that are used in the von Mises yield criterion equation. They represent the magnitude of the stress state at a point in a material and are important in determining the onset of yielding.

## 4. What is the difference between the von Mises yield criterion and other yield criteria?

The von Mises yield criterion is a more general yield criterion compared to others, such as the Tresca yield criterion. It takes into account the shear stresses, which are often the primary cause of yielding in materials, while other criteria only consider the normal stresses.

## 5. What are the applications of the von Mises yield criterion in engineering?

The von Mises yield criterion is commonly used in engineering to predict the failure of materials under stress. It is especially useful in the design of structures and components that are subject to complex stress states, such as in aerospace and automotive industries.

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