Computational Geomechanics. Manuel Pastor
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to obtain the unique undrained condition.
The existence of the two steady states is well known and what we have indicated here is a process by which various matrices given in the original computer program can be used to obtain either of the steady state solutions. However, this does require an alternative to the original computer program. Though, it is possible to obtain such steady states by the code, using the previous time‐stepping procedure. Two types of undrained conditions exist: (a) when k = 0 throughout; (b) k ≠ 0 but the complete boundary is impermeable. Both cases can be computed with no difficulties.
Provided that the boundary conditions are consistent with the existence of drained and undrained steady state conditions, the time‐stepping process will, in due course, converge with
However, this process may be time‐consuming even if large time steps, Δt are used. A simpler procedure is to use the GN00 scheme with
Equations (3.41) and (3.61) now become, for the undrained problem,
If the material behavior is linearly elastic, then the equation can be solved directly yielding the two unknowns
3.2.5 The Structure of the Numerical Equations Illustrated by their Linear Equivalent
If complete saturation is assumed together with a linear form of the constitutive law, we can write the effective stress simply as
(3.62)
We can now reduce the governing u–p Equations (3.23) and (3.28) to the form given below
(3.63)
and
(3.64)
where
(3.65)
is the well‐known elastic stiffness matrix which is always symmetric in form. S and H are again symmetric matrices defined in (3.31) and (3.30) and
The overall system can be written in the terms of the variable set [
(3.66)
Once again the uncoupled nature of the problem under drained condition is evident (by dropping the time derivatives) giving
(3.67)