Since the principal stresses are orthogonal, we can, once we calculate their directions by the procedure , imagine that now the x, y, z, axes of Figure are oriented in the principal directions.

**Principal Space
and the Octahedral Representation **

Since the principal stresses are
orthogonal, we can, once we calculate their directions by the procedure in
Section, imagine that now the *x*, *y*, *z*, axes of Figure
2.4 are oriented in the principal directions. Thus plane *ABC*, as redrawn
in Figure 2.5a, is normal to a straight line *x*' from the origin with

direction cosines with respect to the 1,
2, 3 axes. For convenience let the *z*'
axis lie in the 1-3 plane. From Equation (2.12) then:

Of particular interest is the special
case in Figure 2.5a where l, *m*, *n*
all equal 1/ 3^{1/3} . This direction is the
diagonal of a unit cube and thus often called the 'space diagonal.' For this
orientation:

The subscript designation 'oct' or o is
used since there are eight such faces normal to the space diagonals forming a
regular octahedron as in Figue 2.5d. The total 'stress resultant' on an
octahedral plane is:

As already mentioned,
the remarkable property of this octahedral orien-tation is, as shown by
Equation (2.28), that the stress or strain tensors uncouple naturally* into
invariant isotropic and deviatoric components when viewed in this perspective
in principal space. Moreover, the angle in Equation (2.30) is related to *I*_{?}_{3}
or more specifically to *I _{s}*

The corresponding octahedral strain
components are, by analogy:

with components corresponding to
Equations (2.29) and (2.30).

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Civil : Principles of Solid Mechanics : Strain and Stress : Principal Space and the Octahedral Representation |

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