Page 180 - Analysis and Design of Energy Geostructures

Deformation in the context of energy geostructures 153

ð4:32Þ
σ 1 1 σ 2 1 σ 3 5 I 1 5 3p
This distance reads
p ffiffiffi
ð σ 1 1 σ 2 1 σ 3 Þ 3 p ffiffiffi
AA 5 p ffiffiffi 5 I 1 5 3p ð4:33Þ
0
3 3
The special plane for which the mean stress p is zero is called the π-plane and reads

σ 1 1 σ 2 1 σ 3 5 0 ð4:34Þ

The second invariant of the deviatoric stress tensor J 2 is a measure of the distance
from the space diagonal to the current stress state in the spherical plane. The combina-
tion of J 2 and J 3 through the formulation of the Lode’s angle defines the orientation
of the stress state within this plane and reads
p
1 3 3 J 3
ffiffiffi
θ l 52 sin 21 ð4:35Þ
3 2 J 2 3
The second invariant of the deviatoric stress tensor J 2 is also related to the deviato-
ric stress as (Roscoe and Burland, 1968; Wood, 1990)
p ffiffiffiffiffiffi
q 5 3J 2 ð4:36Þ

that in terms of principal stresses can also be written as

2
2
q 5 1 ð σ 1 2σ 2 Þ 1 σ 2 2σ 3 Þ 1 σ 1 2σ 3 Þ 2 ð4:37Þ
ð
ð
6
In Fig. 4.9, the principal stresses ~σ 1 , ~σ 2 and ~σ 3 are the projections in space of the
principal stresses σ 1 , σ 2 and σ 3 .
4.6 Momentum conservation equation
4.6.1 General

In all cases where inertial forces acting on a material can be neglected or are absent
(i.e. the so-called quasi-static conditions hold), the so-called Cauchy’s first law of
motion reduces to the indefinite equilibrium equations.

4.6.2 Indefinite equilibrium equations
The indefinite equilibrium equations can be derived considering that the stress compo-
nents are continuous functions of the point coordinates and analysing the stress com-
ponents acting on the faces of an element opposite to those highlighted in Fig. 4.7.