Principal stresses

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Principal Stresses in Two and Three Dimensions[edit]

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

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

Plugging \textstyle \theta into the transformation equations for stress we get,


\begin{align}
  \sigma_1 &= \frac{\sigma_{11}+\sigma_{22}}{2}  +\sqrt{\left(\frac{\sigma_{11}-\sigma_{22}}{2}\right)^2 + \sigma_{12}^2} \\
  \sigma_2 &= \frac{\sigma_{11}+\sigma_{22}}{2} -\sqrt{\left(\frac{\sigma_{11}-\sigma_{22}}{2}\right)^2 + \sigma_{12}^2}  \end{align}

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,


\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}

where,


\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}

The quantities \textstyle I_1, I_2, I_3 are the stress invariants.

Related Content[edit]

Introduction to Elasticity