Stress (mechanics) - Octahedral Stresses

Octahedral Stresses

Considering the principal directions as the coordinate axes, a plane whose normal vector makes equal angles with each of the principal axes (i.e. having direction cosines equal to ) is called an octahedral plane. There are a total of eight octahedral planes (Figure 6). The normal and shear components of the stress tensor on these planes are called octahedral normal stress and octahedral shear stress, respectively.

Knowing that the stress tensor of point O (Figure 6) in the principal axes is

$\sigma_{ij}= \begin{bmatrix} \sigma_1 & 0 & 0\\ 0 & \sigma_2 & 0\\ 0 & 0 & \sigma_3 \end{bmatrix} \,\!$

the stress vector on an octahedral plane is then given by:

$\begin{align} \mathbf{T}_\mathrm{oct}^{(\mathbf{n})}&= \sigma_{ij}n_i\mathbf{e}_j \\ &=\sigma_1n_1\mathbf{e}_1+\sigma_2n_2\mathbf{e}_2+\sigma_3n_3\mathbf{e}_3\\ &=\tfrac{1}{\sqrt{3}}(\sigma_1\mathbf{e}_1+\sigma_2\mathbf{e}_2+\sigma_3\mathbf{e}_3) \end{align} \,\!$

The normal component of the stress vector at point O associated with the octahedral plane is

$\begin{align} \sigma_\mathrm{oct} &= T^{(n)}_in_i \\ &=\sigma_{ij}n_in_j \\ &=\sigma_1n_1n_1+\sigma_2n_2n_2+\sigma_3n_3n_3 \\ &=\tfrac{1}{3}(\sigma_1+\sigma_2+\sigma_3)=\tfrac{1}{3}I_1 \end{align} \,\!$

which is the mean normal stress or hydrostatic stress. This value is the same in all eight octahedral planes. The shear stress on the octahedral plane is then

$\begin{align} \tau_\mathrm{oct} &=\sqrt{T_i^{(n)}T_i^{(n)}-\sigma_\mathrm{n}^2} \\ &=\left^{1/2} \\ &=\tfrac{1}{3}\left^{1/2} = \tfrac{1}{3}\sqrt{2I_1^2-6I_2} = \sqrt{\tfrac{2}{3}J_2} \end{align} \,\!$

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