Numerical Modeling: An Efficient Tool for Analyzing the Behavior of Constructions - Multiscale Geomechanics: From Soil to Engineering Projects - page 264

of density and of particle-size on the capillary pressure is taken into account in the
parameter π max
c

which is written as follows:

2πT
K(e)R

π max
c

= [9.64]

where K(e) is a function of void ratio e given by: K(e)=0.32e 2 +4.06e+0.11 . Taibi
shows that this relationship can be generalized to soils by replacing parameter R by
d 10 (maximum diameter of the particles with 10% of the total weight) which appears to
be the most representative parameter. Recent results on Perafita sand [HAR 02] show
that, in the presence of a large quantity of fines, parameter R may be less than d 10
(ratio of 1/20). We also impose a continuity between the pore pressure and capillary
stress at the limit of saturated and unsaturated domains.

This condition deduced from the results with a high degree of saturation allows us
to use a single framework to model saturated and unsaturated geomaterials. Indeed,
the same modeling tools can be used for analyzing actual cases where the state of the
in situ material varies with hydrological and hydroclimatical conditions. For example,
we give an expression of function π c (P c ) , which ensures that the latter can be retained
for a first approximation:

tanh( P c − p c d
π max
c

π c = π max ) [9.65]

c

This simple mathematical relationship ensures the continuity of pore pressure and
capillary stress in saturated and unsaturated cases, but may prove to be too stiff or
not valid over the entire range of pressures. It must be adapted to suit the material
and the range of pressures considered. Coussy and Dangla [COU 02a] extending the
energy approach of non-linear poroelasticity in Biot's saturated media [BIO 41] to the
case of unsaturated soils give the expression of the capillary stress which reduces to a
pressure ( π ):

π = S w p w + S a p a − U(S w ,T) [9.66]

where U is the free energy stored in interfaces and T is the superficial tension.

In some studies, the existence of an effective stress is rejected and the behavior is
defined by either using a state surface which connects the total stress, void ratio and
capillary pressure (Bishop and Blight [BIS 63], Matyas and Radhakrishna [MAT 68],
Fredlund and Morgenstern [FRE 77], Lloret and Alonso [LLO 85]) or elastoplastic
models written in terms of total stress (Alonso et al. [ALO 90] and Toll [TOL 90]).