Consolidation and Swelling Processes in Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 827

Water flow Air flow

v wy + ∂

v wy
∂ y

J ay + ∂

J ay
∂ y

dy dy

Water flow

v wx
∂ y

v wx + ∂

v wx

dy

Soil element
of thickness dz

dy

Air flow

J ax + ∂ J ax
∂

dy

J ax

y

dx

Y

v wy J ay

X

Figure 16.20 Two-dimensional unsteady-state air and water flow.

ε z = normal strain in the z- direction, where:

σ x = total normal stress in the x- direction,

σ y = total normal stress in the y- direction,

dε v = volumetric strain change (i.e., dV v /V 0 ),

σ z = total normal stress in the z- direction,

dV v = total volume change of a soil element, and

σ x − u a = net normal stress in the x- direction,

V 0 = initial total volume of a soil element.

σ y − u a = net normal stress in the y- direction,

σ z − u a =

net normal stress in the z- direction,

Rearranging Eqs. 16.47, 16.48, and 16.49, the change in
net normal stresses can be expressed as a function of normal
strain and matric suction changes:

μ =

Poisson's ratio,

=

E modulus of elasticity or Young's modulus for

the soil structure with respect to a change in
σ

d σ x − u a = βd u a − u w

− u a ,

2 G(dε x + αdε v ) − (16.54)

H = modulus of elasticity for the soil structure with

respect to a change in u a −

d σ y − u a = βd u a − u w

2 G(dε y +

αdε v ) −

(16.55)

u w ,

d σ z − u a = βd u a − u w

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

2 G(dε z + αdε v ) − (16.56)

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

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

where

τ xy = shear stress on the x- plane in the y- direction

(i.e., τ xy = τ yx ),

E

G =

τ yz = shear stress on the y- plane in the z- direction

(i.e., τ yz =

2 ( 1 + μ)

τ zy ),

μ

τ zx = shear stress on the z- plane in the x- direction

(i.e., τ zx =

α =

1 − 2 μ

τ xz ), and

E 2 G

H

1 + u

G = shear modulus.

β = or β =

H( 1 − 2 μ) 1 − 2 u

The change in volumetric strain, dε v , can be written as
the summation of the normal strain changes:

16.6.4 Water Phase Constitutive Relations
The constitutive equation for the water phase defines the
change in volume of water in an unsaturated soil element

dε v = dε x + dε y + dε z (16.53)