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 an octahedron as it is illustrated in Fig. 1.

Octahedral stress planes

Figure 1: Octahedral stress planes.

The direction cosines of the octahedral plane are equal to n_{1}=n_{2}=n_{3}=\frac{1}{\sqrt{3}} (since the plane forms equal angles with the coordinate axes and  n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1). The stress tensor acting on the point O (origin) has the form:

\sigma_{ij}=\left[\begin{array}{ccc}\sigma_{1} & 0 & 0\\0 & \sigma_{2} & 0\\0 & 0 & \sigma_{3}\end{array}\right] (1)

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):

T_{i(oct)}^{(n)}=\sigma_{ij}n_{j} (2)

Then, the octahedral normal stress \sigma_{oct} is given by:

\begin{array}{rl}\sigma_{oct}=&T_{i(oct)}^{(n)}n_{i}=\sigma_{ij}n_{i}n_{j}\\= & \sigma_{1}n_{1}n_{1}+\sigma_{2}n_{2}n_{2}+\sigma_{3}n_{3}n_{3}\\=&\frac{1}{3}\left(\sigma_{1}+\sigma_{2}+\sigma_{3}\right)=\frac{1}{3}I_{1}=p\end{array} (3)

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:

\begin{array}{rl}\tau_{oct}=&\sqrt{T_{i(oct)}^{(n)}T_{i(oct)}^{(n)}-\sigma_{oct}^{2}}\\=&\frac{1}{3}\left[\left(\sigma_{11}-\sigma_{22}\right)^2+\left(\sigma_{22}-\sigma_{33}\right)^2+\left(\sigma_{33}-\sigma_{11}\right)^2\right]^\frac{1}{2}\\=&\frac{1}{3}\sqrt{2I_{1}^{2}-6I_{2}}=\sqrt{\frac{2}{3}J_{2}}\end{array} (4)

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:

\sigma_{ij}=\left[\begin{array}{ccc}4&-2&-1\\-2&3&4\\-1&4&-2\end{array}\right] (5)

Solution

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:

\begin{array}{rl}I_{1}= & \sigma_{11}+\sigma_{22}+\sigma_{33}\\=&5\\I_{2}=&\sigma_{11}\sigma_{22}+\sigma_{22}\sigma_{33}+\sigma_{11}\sigma_{33}-\sigma_{12}^2-\sigma_{23}^2-\sigma_{13}^2\\=&-23\end{array} (6)

Thus the octahedral stresses are:

\sigma_{oct}=\frac{1}{3}I_{1}=\frac{5}{3} (7)

and

\tau_{oct}=\frac{1}{3}\sqrt{2I_{1}^{2}-6I_{2}}=\frac{\sqrt{188}}{3} (8)

Suggested Bibliography

W.F. Chen and D.J. Han. Plasticity for Structural Engineers. Springer-Verlag,  New York, 1988.

Y.C. Fung and P. Tong. Classical and Computational Solid Mechanics. World Scientific, Singapore, 2001.

J.C. Jaeger, N.G.W. Cook and R.W. Zimmerman. Fundamentals of Rock Mechanics. Blackwell Publishing, Malden MA, 4th edition, 2007.

Share
This entry was posted in Stress, Stress & Strain. Bookmark the permalink.

5 Responses to Octahedral stresses

  1. Hemant Deshpande says:

    will you please quote the practical application or case study where octahedral stress equations are used…

    • rockmechs says:

      Octahedral stresses are practically invariants, meaning that they do not depend on the orientation of the coordinate system. Thus, they can be used for failure criteria on Haigh-Westergaard stress space. I am planning to write some article on this subject soon.

  2. mehdi deihim says:

    what is its application in theory of elasticity and to wanna visualize this conception in a sample or …

  3. veeresh says:

    what are the direction cosines of oct. shear stress??

    • rockmechs says:

      I don’t understand your question… The octahedral shear stress is parallel to the octahedral plane. A plane (like the octahedral plane) is defined by its normal unit vector. This normal vector has three components in 3-dimensional space. These components are the projections of the unit vector on the three coordinate axes. The magnitude (length) of each component equals to the cosine of the angle formed between the unit vector and the corresponding coordinate axis. I hope I have helped you.