Computational Geomechanics. Manuel Pastor

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of the solid pressure p_s, i.e. the pressure acting on the solid grains is introduced (viz. Hassanizadeh and Gray 1990). From the application of the second principle of thermodynamics, it appears that under conditions close to thermodynamic equilibrium

      (2.46)

      where is a coefficient and θ_s and θ_w the absolute temperatures of solid and water, respectively. Under the assumption of local thermodynamic equilibrium, where temperatures are equal and rate terms vanish, it follows that

(2.47)

and the effective stress principle in the form of (2.45b) can be derived following Hassanizadeh and Gray (1990). Equation (2.47) holds also under nonequilibrium conditions; it has however to be assumed that the solid grains remain incompressible, the same assumption as with the mixture theory.

Let us consider now the partially saturated case. There are again several conflicting expressions in literature. The first expression for partially saturated soils was developed by Bishop (1959) and may be written as follows, see Equations (2.36), (2.24), and (1.19)

(2.48)

      where χ_w is the Bishop parameter, usually a function of the degree of saturation, see (1.22a) and (1.22b). The same expression, but with Bishop’s parameter equal to the water degree of saturation was derived by Lewis and Schrefler (1982, 1987) using volume averaging. Hassanizadeh and Gray (1990) find for the partially saturated case under the assumption of the following form of the Helmholtz free energy for the solid A^s = A^s(ρ^s, ε_ij, θ^s, S_w) that

(2.49)

      which considering thermodynamic equilibrium conditions or nonequilibrium conditions but incompressible solid grains reduces to

(2.50)

Taking into account that S_a + S_w = 1, the solid pressure (2.50) coincides with that of Equation (2.48) if χ_w = S_w, which is often the case in soil mechanics as shown by Nuth and Laloui (2008). The effective stress can then simply be written as

(2.51)

      Coussy (1995) obtains under the assumption of a simpler form of the functional dependence of the Helmholtz free energy of the solid phase A^s = A^s(ε_ij, θ^s, S_w) as above:

(2.52)

where dp_c = dp_a − dp_w is the capillary pressure increment. This equation has an incremental form and differs substantially from the previous ones, i.e. it is not an exact differential and its use in soil dynamics is not straightforward because the solid pressure‐like term has to be integrated in each time step even with a linear elastic effective stress‐strain relationship, being the capillary pressure‐saturation relationship in general nonlinear, see Figure 1.6. The practical implication of these different formulations for slow phenomena has been investigated in detail by Schrefler and Gawin (1996). It was concluded that in many soil mechanics situations, the resulting differences are small and appear usually after long‐lasting variations of the moisture content. Only several cycles of drying and wetting would produce significant differences. The new stress tensor of Coussy (2004) in finite form coincides with (2.52) when passing to the differential form.

The thermodynamic consistency of the different formulations has been investigated in Gray and Schrefler (2001, 2007) and Borja (2004): (2.45b), (2.51), and (2.52) are thermodynamically consistent while the conditions under which (2.45a) and (2.49) are consistent are given in Gray and Schrefler (2007).

For the sake of completeness, we recall two other formulations of the stress tensor in partially saturated soils which are currently used.

      A previously commonly used form of a stress tensor in partially saturated soil mechanics is the net stress, introduced by Fredlund and Morgenstern (1977). The net stress is defined as the difference between the total stress and the air pressure (no assumption is needed for the grain compressibility)

      (2.53) Скачать книгу