Computational Geomechanics. Manuel Pastor

of the approximation based on the u–p form in which the dependent variables are the displacement of the soil matrix and the pore pressure characterized by a single fluid, i.e. water. However, we shall allow incomplete saturation to exist assuming that the air pressure is zero.

      The formulation thus embraces all the features of the u–p approximation of Sections 2.2.2, 2.3.1, and 2.3.2 and is the basis of a code capable of solving all low‐frequency dynamic problems, consolidation problems, and static drained or undrained problems of soil mechanics. Only two dimensions will be considered here and in the examples which follow, but extension to three dimensions is obvious.

      The code based on the formulation contained in this part of the chapter is named SWANDYNE (indicating its Swansea origin) and its outline was presented in literature by Zienkiewicz et al. (1990a, b). The implicit form used allows long periods to be studied. Indeed, with suitable accuracy control, such codes can be used both for earthquake phenomena limited to hundreds of seconds or consolidation problem with a duration of hundreds of days.

3.2 u‐p Discretization for a General Geomechanics’ Finite Element Code

      3.2.1 Summary of the General Governing Equations

      We will report here the basic governing equation derived in the previous chapter. However, we shall limit ourselves to the 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

(3.8)

      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))

(3.10)

generally taken as constant and σ is the total stress with components

      (3.11)

      The effective stress is defined as in (2.1)

(3.12)

      where α again is a constant usually taken for soils as

(3.13)

      and p the effective pressure defined by (2.24) with p_a = 0.

(3.14)

      The effective stress σ″ is computed from an appropriate constitutive law generally defined as “increments” by (2.2)

(3.15)

      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 p_w. The effective stresses are determined at any stage by a sum of all previous increments and the value of p_w determines the parameters S_w (saturation) and χ_w (effective area). On occasion, the approximation

(3.16)

      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

(3.17)

      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.