Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 672

Figure 13.6 Translation and deformation of two-dimensional element of unsaturated soil.

between the positive directions of the two axes decreases.
The shear strain components of a three-dimensional element
are formulated as

where:

ε v = volumetric strain.

∂u
∂y +

∂ v
∂x

γ xy = (13.6)

Volumetric strain is equal to the difference between the
volumes of the voids in the element before and after defor-
mation, V v , referenced to the initial volume of the ele-
ment, V 0 :

∂ v
∂z +

∂ w
∂y

γ yz = (13.7)

∂ w
∂x +

∂u
∂z

V v
V 0

γ zx = (13.8)

ε v = (13.11)

where:

The volumetric strain ε v can be used as a deformation state
variable for the overall referential element or the soil struc-
ture. Soil structure volume change is the result of normal
components of deformation.

γ xy = shear strain on the z- plane (i.e., γ xy = γ yx ),

γ yz = shear strain on the x- plane (i.e., γ yz = γ zy ), and

γ zx = shear strain on the y- plane (i.e., γ zx = γ xz ).

The normal and shear strains of the soil structure can be
written as a deformation tensor:
⎡

13.2.3 Water and Air Volume Changes
The unsaturated soil element shown in Fig. 13.6 can be used
to describe changes in the volume of the voids filled with air
and water. The element of unsaturated soil is considered as a
spatial element with respect to the water and air phases. The
change in the volume of each fluid in the voids is defined as
the difference between the fluid volumes in the voids prior
to a change in the stress state and the fluid volumes after
a change in the stress state (Fig. 13.6). The fluid change
(i.e., air or water) per unit initial volume of the soil element

⎤

1
2 γ yx

1
2 γ zx

ε x

⎣ ⎦

1 1

2 γ xy

ε y

2 γ zy

(13.9)

1
2 γ xz

1
2 γ yz

ε z

The sum of the normal strain components along the trace
of the matrix is called volumetric strain:

ε v = ε x + ε y + ε z (13.10)