View source for Elasticity/Principal stresses

==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 <math>\textstyle \theta</math> at which <math>\textstyle \sigma^{'}_{12} = 0</math>.  This angle is given by

<math>  
\theta =      \cfrac{1}{2}\tan^{-1}\left(\frac{2\sigma_{12}}{\sigma_{11}-\sigma_{22}}\right) </math>

Plugging <math>\textstyle \theta</math> into the [[Coordinate transformations|transformation]] equations for stress we get,
:<math>
\begin{align}
  \sigma_1 &= \frac{\sigma_{11}+\sigma_{22}}{2}  +\sqrt{\left(\frac{\sigma_{11}-\sigma_{22}}{2}\right)^{2} +\sigma_{12}^2}  \end{align} </math>

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,

:<math>
\begin{align}
  \sigma_1 & = \frac{I_1}{3} + \frac{2}{3} \left(\sqrt{I^2_1 - 3I_2}\right)\cos\phi \\
  \sigma_2 & = \frac{I_1}{3} + \frac{2}{3} \left(\sqrt{I^2_1 - 3I_2}\right) \cos\left(\phi-\frac{2\pi}{3}\right)  \\
  \sigma_3 & = \frac{I_1}{3} + \frac{2}{3} \left(\sqrt{I^2_1 - 3I_2}\right)   \cos\left(\phi-\frac{4\pi}{3}\right)  
\end{align}
</math>

where,

:<math>
\begin{align}
\phi & = \cfrac{1}{3}\cos^{-1}\left(\frac{2I_1^3 - 9I_1I_2 + 27 I_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{align}
</math>

The quantities <math>\textstyle I_1, I_2, I_3</math> are the '''stress invariants'''.

Note: Be careful while implementing above relations in a solver, as the value of:
:<math>
\frac{2I_1^3 - 9I_1I_2 + 27 I_3}{2(I_1^2 - 3I_2)^{3/2}}
</math>
can be out of range of <math>\cos^{-1}</math>, which is (-1, 1).
==Related Content==
[[Introduction to Elasticity]]

[[Category:Elasticity]]
[[Category:Structural analysis]]
==External Links==
* [https://civilengineeronline.com/mech/principal-stress.php Online Calculator for Principal Stress and Principal Plane]