Page 179 - Analysis and Design of Energy Geostructures

152 Analysis and Design of Energy Geostructures

The use of invariants defined in Eqs (4.22) (4.24) is often practical for a number
of considerations related to constitutive modelling of geomaterials and their interaction
with geostructures. In particular, invariants of physical interest especially in the frame-
work of plasticity theory are I 1 , J 2 and J 3 . Physically, I 1 represents the magnitude of
the mean stress, J 2 represents the magnitude of the deviatoric stress and J 3 determines
the direction of the deviatoric stress (Yu, 2006). The aforementioned concepts can be
analysed graphically in the Haigh Westergaard space, that is the three-dimensional
space where the principal directions of stresses are selected as coordinate axes (cf.
Fig. 4.9). The value of I 1 provides a measure of the distance along the space diagonal
(σ 1 5 σ 2 5 σ 3 5 I 1 =3) from the origin to the current spherical plane, also called octa-
hedral plane, defined as

Figure 4.8 A typical Mohr circle.

Figure 4.9 Representations in the Haigh Westergaard space: definition of key variables in (A) the
π-plane and octahedral plane and (B) the octahedral plane only.