Elasticity/Stress example 4

Example 4

Given:

The octahedral plane is the plane that is equally inclined to the directions of the three principal stresses. For any given stress of state there are eight such planes.

Show:

1. The normal traction on an octahedral plane is given by

${𝐭}_n^{\text{oct}}=\frac{1}{3}({\sigma }_1+{\sigma }_2+{\sigma }_3)=\frac{1}{3}{I}_{\mathit{\sigma }}$.

1. The projected shear traction on an octahedral plane is given by

${𝐭}_s^{\text{oct}}=\frac{1}{3}\sqrt{{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_2)}^2}=\frac{1}{3}\sqrt{2I_{\mathit{\sigma }}^2-6I{I}_{\mathit{\sigma }}}.$

Here $({\sigma }_1,{\sigma }_2,{\sigma }_3)$ are the principal stresses and $(I_{\mathit{\sigma }},I{I}_{\mathit{\sigma }})$ are the first two invariants of the stress tensor ($\mathit{\sigma }$).

Solution

Let us take the basis as the directions of the principal stresses $\widehat{𝐧}1$, $\widehat{𝐧}2$, $\widehat{𝐧}3$. Then the stress tensor is given by

$\left[\mathit{\sigma }\right]=\left[\begin{array}{ccc}{\sigma }_1& 0& 0\\ 0& {\sigma }_2& 0\\ 0& 0& {\sigma }_3\end{array}\right]$

If ${\widehat{𝐧}}_o$ is the direction of the normal to an octahedral plane, then the components of this normal with respect to the principal basis are $n_{o1}$, $n_{o2}$, and $n_{o3}$. The normal is oriented in such a manner that it makes equal angles with the principal directions. Therefore, $n_{o1}=n_{o2}=n_{o3}=n_o$. Since $n_{o1}^2+n_{o2}^2+n_{o3}^2=1$, we have $n_o=1/\sqrt{(}3)$.

The traction vector on an octahedral plane is given by

${𝐭}_o={\widehat{𝐧}}_o\bullet \left[\mathit{\sigma }\right]=n_o{\sigma }_1,n_o{\sigma }_2,n_o{\sigma }_3$

The normal traction is,

$N={𝐭}_o\bullet {\widehat{𝐧}}_o=n_o^2{\sigma }_1+n_o^2{\sigma }_2+n_o^2{\sigma }_3$

Now, $I_{\sigma }=({\sigma }_1+{\sigma }_2+{\sigma }_3)$. Therefore,

$N=(1/3)({\sigma }_1+{\sigma }_2+{\sigma }_3)=(1/3)I_{\sigma }$

The projected shear traction is given by

$S=\sqrt{{𝐭}_o\bullet {𝐭}_o-N^2}$

Therefore,

$S=\sqrt{n_o^2{\sigma }_1^2+n_o^2{\sigma }_2^2+n_o^2{\sigma }_3^2-(1/9)({\sigma }_1+{\sigma }_2+{\sigma }_3{)}^2}$

Also,

$I{I}_{\sigma }={\sigma }_1{\sigma }_2+{\sigma }_2{\sigma }_3+{\sigma }_3{\sigma }_1$

If you do the algebra for S, you will get the required relations.