The Terzaghi consolidation problem

Abaqus contains no one-dimensional elements for effective stress calculations. Therefore, we use a two-dimensional plane strain mesh, with one element only in the x-direction. Element type CPE4P is chosen to perform the finite-strain analysis, and element type CPE8P is chosen for the small-strain analysis. We recommend the use of linear elements for applications involving finite strain, impact, or complex contact conditions and second-order elements for problems where stress concentrations must be captured accurately or where geometric features such as curved surfaces must be modeled. In this particular example the linear and second-order elements yield almost identical results.

The soil is assumed to be linear elastic, with a Young's modulus of 689.5 GPa (10⁸ lb/in²) and Poisson's ratio of 0.3. The specific weight of the pore fluid is assumed to be 276.8 × 10³ N/m³ (1 lb/in³). The permeability is assumed to vary linearly with the void ratio, with a value of 8.47 × 10⁻⁸ m/sec (2.0 × 10⁻⁴ in/min) at a void ratio of 1.5 and a value of 8.47 × 10⁻⁹ m/sec (2.0 × 10⁻⁵ in/min) at a void ratio of 1.0. The void ratio is assumed to be 1.5 initially throughout the sample. Abaqus uses effective permeability, which is permeability divided by the specific weight of the pore fluid. Therefore, the fluid in this problem is assigned the value 276.8 × 10³ N/m³ (1 lb/in³) for the specific weight (water, for example, has a specific weight of 9965 N/m³, 0.036 lb/in³) and the permeability is scaled accordingly.

The boundary conditions are as follows. On the bottom and two vertical sides, the normal component of displacement is fixed ( $u_y=$ 0 on the bottom and $u_x=$ 0 on the sides), and no flow of pore fluid through the walls is permitted. This latter is the natural boundary condition in the fluid mass conservation equation, so no explicit specifications need to be made (as with zero tractions in the equilibrium equation). On the top surface a uniform downward load (an overburden) is applied suddenly. The magnitude of this load is taken to be 689.5 GPa (10⁸ lb/in²). This large load causes considerable deformation, thus illustrating the difference between the small- and large-strain solutions. This surface allows perfect drainage so that the excess pore pressure is always zero on this surface.

The problem is run in two steps. The first step is a single increment of a transient soils consolidation analysis with an arbitrary time step, with no drainage allowed across the top surface (the natural boundary condition in the mass conservation equation governing the pore fluid flow). This establishes the initial solution: uniform pore pressure equal to the load throughout the body, with no stress carried by the soil skeleton (zero effective stress). The actual consolidation is then done with a second soils consolidation step, using automatic time stepping.

The accuracy of the time integration for the second soils consolidation procedure, during which drainage is occurring, is controlled by specifying the maximum allowable pore pressure change per time step, $\mathrm{\Delta }{u}_w^{max}$. Even in a linear problem this value controls the accuracy of the solution, because the time integration operator is not exact (the backward difference rule is used). In this case $\mathrm{\Delta }{u}_w^{max}$ is chosen as 344.8 GPa (5.0 × 10⁷ lb/in²), which is a relatively large value and, so, should only give moderate accuracy: this is considered to be adequate for the purposes of the example.

An important issue in such consolidation problems is the choice of initial time step. As the governing equations are parabolic, the initial solution (immediately after the sudden change in load) is a local, “skin effect,” solution. In this one-dimensional case the form of the initial solution is sketched in Figure 2 for illustration purposes. With a finite element mesh of reasonable size for modeling the solution at a later time (when the changes in pore pressure have diffused into the bulk of the body soil), this initial solution will be modeled poorly. With smaller initial time steps the difficulty becomes more pronounced, as sketched in Figure 2. As in any transient problem, the spatial element size and the time step are related to the extent that time steps smaller than a certain size give no useful information. This coupling of the spatial and temporal approximations is always most obvious at the start of diffusion problems, immediately after prescribed changes in the boundary values. For this particular case the issue has been discussed in detail by Vermeer and Verruijt (1981), who suggest the simple criterion

where $\mathrm{\Delta }h$ is a characteristic element size near the disturbance (that is, near the draining surface in our case), E is the elastic modulus of the soil skeleton, k is the soil permeability, and ${\gamma }_w$ is the specific weight of the permeating fluid. For our model we choose $\mathrm{\Delta }h=$ 254 mm (10 in); and we have $E=$ 689.5 GPa (10⁸ lb/in²), $k=$8.47 × 10⁻⁸ m/s (2.0 × 10⁻⁴ in/min), ${\gamma }_w=$2.768 × 10⁵ N/m³ (1.0 lb/in³), which gives $\mathrm{\Delta }t\ge$.05 s (0.833 × 10⁻³ min). Based on this calculation, an initial time step of .06 sec (0.001 min) is used. This gives an initial solution with no “overshoot” at all, as expected.

In this case we want to continue the analysis to steady-state conditions. This is defined by asking Abaqus to stop when all pore pressure change rates fall below 11.5 KN/m²/s (100 lb/in²/min).