sigma plus bold m p Superscript a"/>
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
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 Xπ 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 xπ at time t is a function of its placement in a chosen reference configuration, Xπ and of the current time t
To keep this mapping continuous and bijective at all times, the determinant of the Jacobian of this transformation must not equal zero and must be strictly positive, since it is equal to the determinant of the deformation gradient tensor Fπ
(2.56)
where Uπ is the right stretch tensor, Vπ the left stretch tensor, and the skew‐symmetric tensor Rπ gives the rigid body rotation. Differentiation with respect to the appropriate coordinates of the reference or actual configuration is respectively denoted by comma or slash, i.e.
(2.57)
Because of the non‐singularity of the Lagrangian relationship (2.55), its inverse can be written and the Eulerian or spatial description of motion follows
(2.58)
The material time derivative of any differentiable function fπ(X, t) given in its spatial description and referred to a moving particle of the π phase is
If superscript α is used for
2.5.2 Microscopic Balance Equations
In the hybrid mixture theories, the microscopic situation of any π phase is first described by the classical equations of continuum mechanics. At the interfaces to other constituents, the material properties and thermodynamic quantities may present step discontinuities. As throughout the book, the effects of the interfaces are here not taken into account explicitly. These are introduced, e.g. in Schrefler (2002) and Gray and Schrefler (2001, 2007).
For a thermodynamic property ψ, the conservation equation within the π phase may be written as
