Principal Stresses in Two and Three Dimensions

The principal stresses are the components of the stress tensor when the basis is changed in such a way that the shear stress components become zero. To find the principal stresses in two dimensions, we have to find the angle ${\displaystyle \textstyle \theta }$ at which ${\displaystyle \textstyle \sigma _{12}^{'}=0}$. This angle is given by

${\displaystyle \theta ={\cfrac {1}{2}}\tan ^{-1}\left({\frac {2\sigma _{12}}{\sigma _{11}-\sigma _{22}}}\right)}$

Plugging ${\displaystyle \textstyle \theta }$ into the transformation equations for stress we get,

${\displaystyle {\begin{aligned}\sigma _{1}&={\frac {\sigma _{11}+\sigma _{22}}{2}}+{\sqrt {\left({\frac {\sigma _{11}-\sigma _{22}}{2}}\right)^{2}+\sigma _{12}^{2}}}\end{aligned}}}$

Where are the shear tractions usually zero in a body?

The principal stresses in three dimensions are a bit more tedious to calculate. They are given by,

${\displaystyle {\begin{aligned}\sigma _{1}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \phi \\\sigma _{2}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \left(\phi -{\frac {2\pi }{3}}\right)\\\sigma _{3}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \left(\phi -{\frac {4\pi }{3}}\right)\end{aligned}}}$

where,

${\displaystyle {\begin{aligned}\phi &={\cfrac {1}{3}}\cos ^{-1}\left({\frac {2I_{1}^{3}-9I_{1}I_{2}+27I_{3}}{2(I_{1}^{2}-3I_{2})^{3/2}}}\right)\\I_{1}&=\sigma _{11}+\sigma _{22}+\sigma _{33}\\I_{2}&=\sigma _{11}\sigma _{22}+\sigma _{22}\sigma _{33}+\sigma _{33}\sigma _{11}-\sigma _{12}^{2}-\sigma _{23}^{2}-\sigma _{31}^{2}\\I_{3}&=\sigma _{11}\sigma _{22}\sigma _{33}-\sigma _{11}\sigma _{23}^{2}-\sigma _{22}\sigma _{31}^{2}-\sigma _{33}\sigma _{12}^{2}+2\sigma _{12}\sigma _{23}\sigma _{31}\end{aligned}}}$

The quantities ${\displaystyle \textstyle I_{1},I_{2},I_{3}}$ are the stress invariants.

Note: Be careful while implementing above relations in a solver, as the value of:

${\displaystyle {\frac {2I_{1}^{3}-9I_{1}I_{2}+27I_{3}}{2(I_{1}^{2}-3I_{2})^{3/2}}}}$

can be out of range of ${\displaystyle \cos ^{-1}}$, which is (-1, 1).

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