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 an octahedron as it is illustrated in Fig. 1.
The direction cosines of the octahedral plane are equal to (since the plane forms equal angles with the coordinate axes and ). The stress tensor acting on the point (origin) has the form:
where , and 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 is given by:
where is the first invariant of the stress tensor and is the mean or hydrostatic stress (see article: Deviatoric stress and invariants). Next, the octahedral shear stress is given by:
where is the second invariant of the stress tensor and is the second invariant of the deviatoric stress tensor.
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:
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