use of the condensed, vectorial form of these which is convenient for finite element discretization. The tensorial form of the equations can be found in Section 3.3.
The overall equilibrium or momentum balance equation is given by (2.11) and is copied here for completeness as
In the above and in all the following equations, the relative fluid acceleration terms are omitted as only the u–p form is being considered.
The strain matrix S is defined in two dimensions as (see (2.10))
(3.9)
Here u is the displacement vector and ρ the total density of the mixture (see (2.19))
generally taken as constant and σ is the total stress with components
(3.11)
The effective stress is defined as in (2.1)
where α again is a constant usually taken for soils as
and p the effective pressure defined by (2.24) with pa = 0.
The effective stress σ″ is computed from an appropriate constitutive law generally defined as “increments” by (2.2)
where D is the tangent matrix dependent on the state variables and history and ε 0 corresponds to thermal and creep strains.
The main variables of the problem are thus u and pw. The effective stresses are determined at any stage by a sum of all previous increments and the value of pw determines the parameters Sw (saturation) and χw (effective area). On occasion, the approximation
can be used.
An additional equation is supplied by the mass conservation coupled with fluid momentum balance. This is conveniently given by (2.33b) which can be written as
with k = k (S w).
The contribution of the solid acceleration is neglected in this equation. Its inclusion in the equation will render the final equation system nonsymmetric (see Leung 1984) and the effect of this omission has been investigated in Chan (1988) who found it to be insignificant. However, it has been included in the force term of the computer code SWANDYNE‐II (Chan 1995) although it is neglected in the left‐hand side of the final algebraic equation when the symmetric solution procedure is used.
The above set defines the complete equation system for solution of the problem defined providing the necessary boundary conditions have been specified as in (2.18) and (2.19), i.e.
and
