material time derivative of the water density with respect to the moving solid phase and the relative velocity vws are introduced. Then the derivatives are carried out, the quantity of water lost through evaporation is neglected and the material time derivative of the porosity is expressed through Equation (2.79), yielding
The mass balance equation for air is derived in a similar way
(2.81)
To obtain the equations of Section 2.4.2, further simplifications are needed, which are introduced next.
An updated Lagrangian framework is used where the reference configuration is the last converged configuration of the solid phase. Further, the strain increments within each time step are small. Because of this, we can neglect the convective terms in all the balance equations. Neglecting in the linear momentum balance Equation (2.78) further the relative accelerations of the fluid phases with respect to the solid phase yields the equilibrium Equation (2.34a)
(2.82)
The linear momentum balance equation for fluids (2.76) by omitting all acceleration terms, as in Section 2.2.2, can be written for water
(2.83)
where
and
and for air
(2.85)
where
and
The phase densities appearing in Sections 2.2–2.4 are intrinsic phase averaged densities as indicated above.
The mass balance equation for water is obtained from Equation (2.80), taking into account the reference system chosen, dividing by ρw, developing the divergence term of the relative velocity and neglecting the gradient of water density. This yields
(2.87)
where the first of Equation (2.84) has been taken into account. This coincides with Equation (2.41a) for incompressible grains (α = 1) except for the source term and the second‐order term due to the change in fluid density. This last one could be introduced in the constitutive relationship (2.75).
Similarly, the mass balance equation for air becomes
(2.88)
where again the first of Equation (2.86) has been taken into account and the gradient of water density has been neglected. Similar remarks as for the water mass balance equation
