Hydro-Mechanical Properties of an Unsaturated Frictional Material

Table A.1: Overview of equations for determination of several parameters of Zou’s model
(2004)
processes yi nf (y) nfi(y, α) α
main imbibition
contact regions - 0 0 0 ∼ αmax
frustum-shaped pores y0 = rs nf0 0 αmax
main drainage
frustum-shaped pores y0 = rs nf0(1 − µ)(1 − y/rs) nf0[µ + (1 − µ)y/rs] αmax
contact regions y0 = rs 0 µ · nf0 · α/αmax 0 ∼ αmax
secondary drainage
frustum-shaped pores y1 nf0(1 − µ)(1 − y/y1) nf0[µ + (1 − µ)y/y1] αmax
contact regions y1 0 µ · nf0 · α/αmax 0 ∼ αmax
secondary imbibition
frustum-shaped pores y2 nf0(1 − µ)(1 − y2/rs) nf0[µ + (1 − µ)y2/rs] αmax
(0 ≤ y ≤ rs; 0 ≤ α ≤ αmax, i = 0, 1, 2)
where nf (y) is the average number of the frustum-shaped pores that are wetting or draining
per particle, nfi(y, α) is the average number of the frustum-shaped pores that are filled with
water to y = yi (where yi is the maximal filling height). nf (y), nfi(y), yi and the position
angle α in Eq. A.9 can be determined corresponding to Table A.1 for different imbibition and
drainage processes.
The degree of saturation Sr(y, α), depending on y and/or α , can estimated using the
following equation:
Vw(y, α)
Sr(y, α) = Sr0
Vv
where Sr0 is the degree of saturation for suction (ua − uw) ≈ 0.
215
(A.10)
To determine the suction in pore water during wetting and draining the frustum-shaped
pores, in Zou (Zou 2003, 2004) it is assumed that the contact angle β(y) (Fig. 2.22(b)) as
function of y can be described using the following equation:
�
B
β(y) =
A − y + C (A.11)
rs
The three constants A, B and C can be determined according to physical boundary conditions
using following equations:
A = ζ,
�
π2 B = ζ (ζ − 1)
4 − β2 �
0 , C = ζ · β 2 0 − (ζ − 1) π2
4
where
�
�
1 − sin αmax/2 − cos αmax(1 − cos αmax)
β0 = arccos
1 − sin αmax(1 − cos αmax) − cos αmax(2 − cos αmax)
(A.12)
(A.13)