State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 104

The second stress invariants of the first and second stress
tensors, respectively, are

3.5.2.1 Stress Points (Lambe, 1967)
Geotechnical engineering analyses often require an under-
standing of the development or change in the stress state
resulting from following various loading patterns. These
changes can be visualized by drawing a series of Mohr circles
which follow the loading process. However, the pattern of
the stress state change may become confusing when the
loading pattern is complex. There are benefits associated
with using an approximate single-stress-point representation
from a Mohr circle to indicate the stress path followed during
a process (Lambe, 1967). A selected stress point at the top
of the Mohr circle (i.e., point of maximum deviator stress)
can be used to represent the stress path being followed. The
stress point suggested by Lambe (1967) is representative of
stress changes in a two-dimensional plane strain problem.
Figure 3.24 shows a Mohr circle for a two-dimensional
case where the vertical and horizontal planes are principal
planes. The stress point selected to represent the Mohr circle
has coordinates of ( p, q, r ), where

I 21 = (σ 1 − u a )(σ 2 − u a ) + (σ 2 − u a )(σ 3 − u a )

+ (σ 3 − u a )(σ 1 − u a ) (3.52)

and

u w ) 2

I 22 = 3 (u a − (3.53)

where:
I 21 =

second stress invariant of the first tensor and

second stress invariant of the second tensor.
The third stress invariants of the first and second stress
tensors, respectively, are

I 22 =

I 31 = (σ 1 − u a )(σ 2 − u a )(σ 3 − u a ) (3.54)

u w ) 3

I 32 = (u a − (3.55)

where:
I 31 =

third stress invariant of the first tensor and

σ v +

u a

σ 1 +

u a
(3.57)

σ h σ 3

I 32 = third stress invariant of the second tensor.

p = − or p = −

2 2

The stress invariants of the second tensor, I 12 , I 22 , and
I 32 , are related as follows:
I 32 = 3 I 12 3

σ v − σ 1 −

σ h σ 3

q = q =

or (3.58)

1
9 I 12 I 22

2 2

= (3.56)

r = u a − u w (3.59)

Therefore, only one stress invariant is required to repre-
sent the second tensor. In other words, a total of four stress
invariants are required to characterize the stress state of an
unsaturated soil as opposed to three stress invariants for a
saturated soil.

where:
(σ v − u a ) =

vertical net normal stress and

(σ h − u a ) =

horizontal net normal stress.

Figure 3.24 Designation of stress point for unsaturated soil using extended Mohr circle.