The constitutive models introduced in the previous sections express the constitutive stress-strain relation of the material. As soon as the water is involved, the material has to be considered as a multi-phase porous media with two phases: the solid matrix (for which we introduced the stress-strain constitutive relations) and the water phase. The two phases are coupled since the pressure acting in the water may affect the mechanical behaviour of the material. Also, the material deformation may modify the pressure in the water. Such hydro-mechanical coupling is well represented by
the Terzaghi effective stress concept for saturated conditions (when water is filling all the pores) (Terzaghi, 1943). It shows the importance of the consideration of the water phase in the analysis of the mechanical behaviour of the material.
In the case of non saturation (water is no longer filling all the pores) the effective stress, expressed as a function of the externally applied stresses and the internal fluid pressures, converts a multi-phase porous media to a mechanically equivalent, single-phase, single-stress state continuum (Khalili et al., 2005). It enters the elastic as well as elasto-plastic constitutive equations of the solid phase, linking a change in stress to strain or any other relevant quantity of the soil skeleton; e. g. see Laloui et al. (2003). As a first approximation, let us consider a force Fn applied on a porous medium (constituted by a solid matrix and pores) through an area A. In this case, we can define a total stress:
If we consider only the part of the load acting on the solid matrix (and deforming it), we may define an effective stress as the part of the load acting on the solid area (XSi) (Fig. 9.9):
The effective stress may be simply defined as that emanating from the elastic (mechanical) straining of the solid skeleton:
ee = Cea’ (9.18)
in which ee is the elastic strain of the solid skeleton, Ce is the drained compliance matrix, and a’ is the effective stress tensor.
In a saturated medium, the effective stress is expressed as the difference between total stress, a, and pore water pressure, u (Terzaghi, 1943):
Fig. 9.9 An illustration of inter-granular stresses
a’ = a — u (9.19)
In an unsaturated granular material with several pressures of different fluid constituents, the effective stress is expressed as:
n
a’= a — ^2 amumI (9.20)
m = 1
in which am is the effective stress parameter, um is the phase pressure, and m = 1, 2,… n represents the number of fluid phases within the system. I is the second order identity tensor. This equation is close to the one of Bishop (1959) for a three — phase material (solid, water and air):
a’ = (a — Ua) + X (Ua — u) (9.21)
where u is the pore water pressure, ua is the pore air pressure, x is an empirical parameter, which has a value of 1 for saturated soils and 0 for dry soils. It represents the proportion of soil suction that contributes to the effective stress. Several attempts have been made to correlate this parameter to the degree of saturation and the suction (Bishop, 1959; Khalili & Khabbaz 1998). As the parameter x seems path-dependent, several authors, starting from Bishop and Blight (1963), proposed the use of two sets of independent “effective” stress fields combining the total stress a, and the pore-air and pore-water pressures, ua and u (Fredlund & Morgenstern, 1977). In the literature the net stress a = a — ua and the suction 5 = ua — u are commonly chosen (Alonso et al, 1990). In general, this net stress concept will be defined in invariant terms using the independent stress variables p(= (ai + a2 + a3)/3), q and 5.
Another way to describe the behaviour is to use the “saturated effective stress” a’ = a — uw and the suction, 5 (Laloui et al., 2001). This combination has the advantage — among others — of permitting a smooth transition from fully saturated to unsaturated condition.
Continuing with the approach having two sets of independent stresses, the strain rate obtained for the elasto-plastic behaviour may be decomposed into elastic and plastic parts:
є ij = єу + єp
each of which results from mechanical and suction variations, as follows. The elastic increment єe. is composed of a mechanical and a hydraulic strain increment:
(9.23)
where ej is the elastic mechanical strain increment induced by the variation of the effective stress a’, 1 eevh is the reversible hydraulic strain increment, Ee is the classical elastic tensor and к is a proportionality coefficient which describes the hydraulic behaviour.
Similarly the plastic strain increment is also deduced from mechanical and hydraulic loads by considering two plastic mechanisms derived from two yield limits:
(9.24)
pm
Where eij is the mechanical plastic strain increment, associated with the mechani-
1 ph
cal yield surface and з eV is the hydraulic plastic strain increment, associated with the hydraulic yield surface.
Using an effective stress approach together with the non-linear models presented in the preceding section allows users to partly take into account the moisture variation effects on mechanical behaviour. However, more fundamental modifications are probably needed. The next section indicates some tools to advance in that direction.
