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      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)

      Its success is due to a great part to experimental reasons: in many problems, the air pressure may be considered constant and equal to the atmospheric pressure and in laboratory experiments, it is preferable to vary water pressure. However, it has to be made clear that the choice of the stress variables in constitutive modeling is a different problem from the choice of controlling variables in the experimental investigation (Jommi 2000). The transformation of the experimentally measured quantities is straightforward from the net stress to the form (2.51), see (Bolzon and Schrefler 1995).

      Its drawback lies in the fact that in the presence of saturated and unsaturated states, the stress tensor changes between the two states; when the pore air is absent, the constitutive equations for saturated states cannot be recovered from those for unsaturated states without additional control (Sheng et al. 2004). This precludes practically its application in soil dynamics; capturing liquefaction becomes a problem.

      The compressibility of the solid grains was also considered by Khalili et al. (2000) in their stress tensor

      (2.54)

      where a₁, a₂ are the effective stress parameters defined as , , c_s is the grain compressibility, c the drained compressibility of the soil structure, and c_m the tangent compressibility of the soil structure with respect to a change in capillary pressure.

      In the following, we make use of (2.51) with the assumption that solid grains are incompressible, i.e. α = 1. The governing equations are derived again, using the hybrid mixture theory, as has been done by Schrefler (1995) and Lewis and Schrefler (1998). Isothermal conditions are assumed to hold, as throughout this book. For the full non‐isothermal case, the interested reader is referred to Lewis and Schrefler (1998) and Schrefler (2002).

      We first recall briefly the kinematics of the system.

      2.5.1 Kinematic Equations

      As indicated in Chapter 1, a multiphase medium can be described as the superposition of all π phases, π = 1, 2, …, κ, i.e. in the current configuration, each spatial point x is simultaneously occupied by material points X^π of all phases. The state of motion of each phase is however described independently.

      In a Lagrangian or material description of motion, the position of each material point x^π at time t is a function of its placement in a chosen reference

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