Octahedral stresses
Octahedral stresses we call the normal and shear stresses that are acting on some specific planes inside the stressed body, the octahedral planes. If we consider the principal directions as the coordinate axes (see also the article: Principal stresses and stress invariants), then the plane whose normal vector forms equal angles with the coordinate system is called octahedral plane. There are eight such planes forming a regular octahedron as it is illustrated in Fig. 1.
The direction cosines of the octahedral plane are equal to \( n_{1}=n_{2}=n_{3}=1/\sqrt{3} \) (since the plane forms equal angles with the coordinate axes andÂ \( n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1 \) ). The stress tensor acting on the point \( O \) (origin) has the form:
where \( \sigma_{1} \), \( \sigma_{2} \) and \( \sigma_{3} \) are the principal stresses. The components of the stress vector acting on the plane are given by (for more details see: Calculation of normal and shear stress on a plane):
Then, the octahedral normal stress \( \sigma_{oct} \) is given by:
where \( I_{1} \) is the first invariant of the stress tensor and \( p \) is the mean or hydrostatic stress (see article: Deviatoric stress and invariants). Next, the octahedral shear stress \( \tau_{oct} \) is given by:
where \( I_{2} \) is the second invariant of the stress tensor and \( J_{2}\) is the second invariant of the deviatoric stress tensor.
Example
Calculate the octahedral stresses for the following stress tensor:
According to equations (3) and (4) we only have to calculate the first and second invariant of the stress tensor. From article Principal stresses and stress invariants we get:
Thus the octahedral stresses are:
and
Related articles

Principal stresses and stress invariants
Deviatoric stress and invariants
Transformation of a tensor to a new coordinate system
Calculation of normal and shear stress on a plane
Mohr's circle in 3 dimensions
3 comments

20220603T10:53:55.999467+00:00
Add more diagram

20211010T08:56:46.929901+00:00
Thank you for useful article. 1. equation in line below fig 1. : should be: n1= n2= n3 = 1/sqrt(3)

20211011T12:17:11+00:00
Thank you Ahmet for your comment. You are correct, it is 1/sqrt(3).