Computational Geomechanics. Manuel Pastor

Computational Geomechanics

already at hand so that only the interaction has to be taken into account. This type of solution procedure is extensively investigated for the dynamic case by Markert et al. (2010) and, as already mentioned, by Park and Felippa (1983), Park (1983), and Zienkiewicz et al. (1988). A splitting procedure will be used in Section 5.5 both for the dynamic case and consolidation allowing for same interpolation for both u and p. Partitioned solutions are quite common in consolidation and have been extensively treated in Lewis and Schrefler (1998). From the investigation of the iteration convergence within a time step, Turska and Schrefler (1993) found the existence of a lower limit for Δt/h² which means that it is not always possible to decrease Δt without also decreasing the mesh size h. Such a limit was also found by Murthy et al. (1989) for Poisson‐type equations and by Rank et al. (1983) for transient finite element analyses by invoking the discrete maximum principle.

3.2.4.3 The Consolidation Equation

In the standard treatment of consolidation equation (see, for instance, Lewis and Schrefler 1998), the acceleration terms are generally omitted a priori. However, as explained above, there is no disadvantage in writing the full dynamic formulation for solving such a problem. The procedure simply reduces the multiplier of the mass matrix M to a negligible value without influencing in any way the numerical stability, provided, of course, that an implicit integration scheme is used.

3.2.4.4 Static Problems – Undrained and Fully Drained Behavior

Steady state (static) conditions will only be reached under the extremes of undrained or fully drained behavior. This can be deduced by rewriting the two, discrete, governing Equations (3.23) and (3.28) omitting terms involving time derivatives. The equations now become:

(3.55) integral Underscript normal upper Omega Endscripts bold upper B Superscript normal upper T Baseline bold sigma Superscript double-prime Baseline d upper Omega minus bold upper Q bold p overbar Superscript normal w Baseline minus bold f Subscript n Superscript left-parenthesis 1 right-parenthesis Baseline equals 0

and

(3.56) bold upper H bold p overbar Superscript normal w Baseline minus bold f Subscript n Superscript left-parenthesis 2 right-parenthesis Baseline equals 0

with the effective stresses given by (3.27) and are defined incrementally as

(3.57) normal d bold sigma Superscript double-prime Baseline equals bold upper D left-parenthesis normal d bold epsilon minus normal d bold epsilon Superscript 0 Baseline right-parenthesis equals bold upper D left-parenthesis bold upper B normal d bold u overbar minus normal d bold epsilon Superscript 0 Baseline right-parenthesis

First, we observe that the equations are uncoupled and that the second of these, i.e. (3.56) can be solved independently of the first for the water pressures. Indeed, in this solution, the negative pressure zones and, hence, the partially saturated regions can be readily determined following the procedures outlined in the previous chapter.

With bold p overbar ^w determined as

(3.58) bold p overbar Superscript w Baseline identical-to bold upper H Superscript negative 1 Baseline bold f Superscript left-parenthesis 2 right-parenthesis

the first Equation (3.55) coupled with the appropriate constitutive law (3.57) can be solved once the history of the load applied has been specified.

The solution so obtained is, of course, the well‐known, drained, behavior.

The case of undrained behavior is somewhat more complex. We note that with k = 0, i.e. with totally impermeable behavior

(3.59) bold upper H equals bold 0 and bold f Superscript left-parenthesis 2 right-parenthesis Baseline equals bold 0

But on re‐examining Equation (3.28), we find that it becomes

(3.60) bold upper Q overTilde Superscript normal upper T Baseline ModifyingAbove Above bold u overbar With ampersand c period dotab semicolon plus bold upper S ModifyingAbove Above bold p overbar With ampersand c period dotab semicolon Superscript w Baseline equals 0

which, on integration, establishes a unique relationship between bold u overbar and bold p overbar _w which is not time‐dependent

(3.61) bold upper Q overTilde Superscript normal upper T Baseline bold u overbar plus bold upper S bold p overbar Superscript w Baseline equals 0

assuming that the initial condition of bold u overbar = 0 and bold p overbar _w = 0 coincides.

Equation (3.61) now has to be solved together with (3.55). If S = 0, i.e. no compressibility is admitted, then we have the problem already discussed in the previous Section 3.2.3 in which only certain bold u overbar − bold p Superscript w Baseline overbar interpolations are permissible (as shown in Figure 3.2). However, if S ≠ 0 _{can be eliminated directly and the solution concerns only the variable .}

Solving (3.61) for bold p Superscript w Baseline overbar which can only be done provided that some fluid compressibility is available giving S ≠ 0, then (3.55) and the constitutive

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