The Von Mises stress equation on wikipedia does not balance out

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The discussion centers on the inconsistency found in the Von Mises stress equation presented on Wikipedia. Users analyze the mathematical expansion of the equation, revealing discrepancies between terms that suggest it does not balance correctly. There is confusion regarding the definition of SVM, initially thought to represent Von Mises stress but possibly indicating deviatoric stress instead. Participants seek clarification on the relationship between the stress components and the definitions involved in the equations. The conversation highlights the need for precision in the definitions and formulations related to stress in materials science.
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The Von Mises stress relationship on Wikipedia shows an expanded expression using components is equal to 3/2*s_ij*s_ij but this is not possible.
On Wikipedia for Von Mises stress, it shows the following equation:

1706216828219.png


But this does not work out. If I expand the second term I get:

$$ \sigma_v^2 = 1/2[(\sigma_{11}^2-2\sigma_{11}\sigma_{22}+\sigma_{22}^2+\sigma_{22}^2-2\sigma_{22}\sigma_{33}+\sigma_{33}^2+\sigma_{33}^2-2\sigma_{33}\sigma_{11}+\sigma_{11}^2)+6(\sigma_{12}^2+\sigma_{13}^2+\sigma_{23}^2)] $$

$$ \sigma_v^2 = 1/2(2\sigma_{11}^2 + 2\sigma_{22}^2+2\sigma_{33}^2-2\sigma_{11}\sigma_{22}-2\sigma_{22}\sigma_{33}-2\sigma_{33}\sigma_{11}+6\sigma_{12}^2+6\sigma_{13}^2+6\sigma_{23}^2) $$

$$ \sigma_v^2 = \sigma_{11}^2 + \sigma_{22}^2+\sigma_{33}^2-\sigma_{11}\sigma_{22}-\sigma_{22}\sigma_{33}-\sigma_{33}\sigma_{11}+3\sigma_{12}^2+3\sigma_{13}^2+3\sigma_{23}^2 $$

And I don't see how this can be equal to the third term, when expanded equals to:

$$ 3/2s_{ij}s_{ij} = 3/2(\sigma_{11}^2+\sigma_{22}^2+\sigma_{33}^2+\sigma_{12}^2+\sigma_{13}^2+\sigma_{23}^2) $$
 
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Not in a good place to look at this, but I believe s is deviatoric stress.
 
Frabjous said:
Not in a good place to look at this, but I believe s is deviatoric stress.
You are correct! But then, what is the definition of the following?

$$SVM = sqrt(3/2*\sigma_{ij}*\sigma_{ij})$$

sigma is stress. Previously I thought Svm is Von Mises Stress, but now it might be deviatoric stress, s. Then I am not sure what SVM defined as such is supposed to be.
 
sijij-(σkk/3)δij
Both s and σ appear in the equation
 
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