- #1
FQVBSina
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- TL;DR Summary
- 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:
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:
$$ 3/2s_{ij}s_{ij} = 3/2(\sigma_{11}^2+\sigma_{22}^2+\sigma_{33}^2+\sigma_{12}^2+\sigma_{13}^2+\sigma_{23}^2) $$
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:
$$ 3/2s_{ij}s_{ij} = 3/2(\sigma_{11}^2+\sigma_{22}^2+\sigma_{33}^2+\sigma_{12}^2+\sigma_{13}^2+\sigma_{23}^2) $$