Hydro-Mechanical Properties of an Unsaturated Frictional Material

214 APPENDIX A. DETAILS ZOU’S MODEL (2003, 2004)
The average pore volume Vv0 per particle in a soil with pore ratio e, in proportion to r 3 s,
can be written as:
Vv
r3 s
= 4π · e
3
(A.2)
When the contact regions in a soil are wetted or dried to a position angle α, according to
the so-called capillary law (Fredlund & Rahardjo 1993b) the dimensionless suction σu(α) in
pore water, in relation to α can be expressed as follows:
σu(α) = (ua − uw) · rs
Ts
=
(2 − sin α − 2 cos α) · cos α
1 − sin α(1 − cos α) − cos α(2 − cos α)
(A.3)
where ua and uw are air and water pressure respectively, Ts is the so-called surface tension of
capillary water (e.g. Ts = 72.75mN/m for the temperature T = 20C ◦ ).
In Zou (2003), Zou (2004) it was proposed that the form of the ideal symmetrical frustum-
shaped pores can be described using a function y relating to the radius r (Fig. 2.22(b)) as
following:
y
rs
= a −
b
r 2 /r 2 s − c
(0 ≤ y ≤ rs) (A.4)
where a, b and c are three constants that can be determined according to geometrical and
physical boundary conditions using following equations:
�
�
a 1 �ρ((αmax) a (a − 1) ln − + ξ)
a − 1 a
2 − ϱ 2 (αmax) �
and
= 4e
−
3nf0
nc · Vw1(αmax)
2π · nf0 · r3 − ρ
s
2 (αmax)
b = a (a − 1) �� ρ(αmax) + ξ) 2� − ρ 2 (αmax) �
c = ρ 2 (αmax) − b
(A.5)
(A.6)
(A.7)
a
where ρ(αmax) = (1−cos αmax)/ cos αmax and ξ is a form parameter which describes the form
of the symmetrical frustum-shaped pores.
When a symmetrical frustum-shaped pore is filled with water up to y, from Eq. A.4 and
by integration the volume Vw1(y) of the pore water in the symmetrical frustum-shaped pore
can be written as:
Vw1(y)
r 3 s
�
= π c y
+ b · ln
rs
a
a − y/rs
�
(0 ≤ y ≤ rs) (A.8)
The total volume Vw(y, α) of pore water per particle during the main imbibition and
drainage processes as well as during the secondary imbibition and drainage processes, de-
pending on y and/or α, can be written as:
Vw(y, α)
r 3 s
= nf (y) Vw1(y)
r 3 s
+ nfi(y, α) Vw1(yi)
r 3 s
Vw1(α)
+ nc
2r3 s
(A.9)