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

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

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

$\begin{array}{cc}{\sigma }_1& =\frac{{\sigma }_{11}+{\sigma }_{22}}{2}+\sqrt{{\left(\frac{{\sigma }_{11}-{\sigma }_{22}}{2}\right)}^2+{\sigma }_{12}^2}\end{array}$

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,

$\begin{array}{cc}{\sigma }_1& =\frac{I_1}{3}+\frac{2}{3}\left(\sqrt{I_1^2-3I_2}\right)\cos \varphi \\ {\sigma }_2& =\frac{I_1}{3}+\frac{2}{3}\left(\sqrt{I_1^2-3I_2}\right)\cos (\varphi -\frac{2\pi }{3})\\ {\sigma }_3& =\frac{I_1}{3}+\frac{2}{3}\left(\sqrt{I_1^2-3I_2}\right)\cos (\varphi -\frac{4\pi }{3})\end{array}$

where,

$\begin{array}{cc}\varphi & =\frac{1}{3}{\cos }^{-1}\left(\frac{2I_1^3-9I_1{I}_2+27I_3}{2(I_1^2-3I_2{)}^{3/2}}\right)\\ I_1& ={\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}\\ I_2& ={\sigma }_{11}{\sigma }_{22}+{\sigma }_{22}{\sigma }_{33}+{\sigma }_{33}{\sigma }_{11}-{\sigma }_{12}^2-{\sigma }_{23}^2-{\sigma }_{31}^2\\ I_3& ={\sigma }_{11}{\sigma }_{22}{\sigma }_{33}-{\sigma }_{11}{\sigma }_{23}^2-{\sigma }_{22}{\sigma }_{31}^2-{\sigma }_{33}{\sigma }_{12}^2+2{\sigma }_{12}{\sigma }_{23}{\sigma }_{31}\end{array}$

The quantities $I_1,I_2,I_3$ are the stress invariants.

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

$\frac{2I_1^3-9I_1{I}_2+27I_3}{2(I_1^2-3I_2{)}^{3/2}}$

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

Related Content

Introduction to Elasticity

External Links

• Online Calculator for Principal Stress and Principal Plane