Page 260 - Mechanical Behavior of Materials

Section 6.5 Stresses on the Octahedral Planes 261

3 3

(a) Normal (b)
τ
σ
γ
α β
2 2

1 1
Octahedral plane:
α = β = γ
σ = σ
h
τ = τ
h

Figure 6.15 Octahedral plane shown relative to the principal normal stress axes (a), and the
octahedron formed by the similar such planes in all octants (b).

For the special case where α = β = γ , the oblique plane intersects the principal axes at equal
distances from the origin and is called the octahedral plane. Based on equilibrium of forces, the
normal stress on this plane can be shown to be the average of the principal normal stresses:

σ 1 + σ 2 + σ 3
σ h = (6.32)
3
The quantity σ h is called the octahedral normal stress or the hydrostatic stress and was considered
in Chapter 5. Equilibrium also permits the shear stress on the same plane, called the octahedral
shear stress, to be evaluated:

1 2 2 2
τ h = (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) (6.33)
3

In each octant of the principal axes coordinate system, there is a similar plane where the normal
makes equal angles with the axes. The stresses on all eight such planes are the same and are σ h and
τ h . These planes can be thought of as forming an octahedron, as shown in Fig. 6.15(b). Noting that
opposite faces of the octahedron correspond to a single plane, the octahedral stresses act on four
planes.
By evaluating the stress invariants, Eq. 6.29, for the special case of the principal normal stresses,
and after some manipulation, σ h and τ h can be written in terms of the invariants:

I 1 1 2
σ h = , τ h = 2 I − 3I 2 (6.34)
1
3 3