Consolidation of a triaxial test specimen

The Cam-clay model assumes that the soil has no stiffness at zero stress, so that some initial (compressive) stress state must be defined for the material. In this case we assume that the soil sample is under an initial hydrostatic pressure of 0.1 MPa (14.5 lb/in²), and this confining pressure remains constant throughout the test. Since the soil may drain through the platen, this pressure is carried as an effective stress in the soil skeleton. This initial stress state is defined using initial conditions. In this particular example it is trivial to see that this initial stress state is in equilibrium with the external distributed pressure of the same magnitude. In more complex cases it may not be so simple to ensure that the discrete, finite element model is in equilibrium with the geostatic loading. Accordingly, the first step of any analysis involving an initial stress state should be a geostatic step. In that step the geostatic external loads (in this case the pressure on the specimen) should be specified. Abaqus will then check whether the initial stress state is in equilibrium with these loads. If it is not, Abaqus will iterate and attempt to establish an equilibrium stress field that balances the prescribed tractions. Such iteration does not occur in this case since the prescribed initial stress is in equilibrium with the applied tractions.

The specimen is compressed to 40% of its initial height over 34.56 × 10⁶ sec (400 days). Although this represents a large strain of the specimen, geometric nonlinearity is ignored in this example because we wish to examine the effects of the material nonlinearity, and we only report the stress-strain response at points, rather than overall load-deflection response that will be predicted quite inaccurately unless geometric nonlinearity is included. The loading is applied in a transient soils consolidation step specifying the time period, with an associated boundary condition prescribing the travel of the platen during that time. The platen is assumed to drain freely throughout the analysis. This is specified by a boundary condition, fixing the pore pressure at zero on the top edge of the mesh. The loading is intended to represent very slow compression, sufficiently slow that the pore pressures never rise to any significant values. We can obtain a rough idea of this time scale by noting that a characteristic time for pore pressure dissipation is $T=H^2{\gamma }_{\omega }/kE$, where H is a typical dimension from the draining surface (60 mm, 2.362 in, in this case); ${\gamma }_{\omega }$ is the specific weight of the pore fluid (1.0 × 10⁴ N/m³, 0.0369 lb/in³); k is the permeability of the soil (0.1728 mm/day, 6.803 × 10⁻³ in/day); and $E$ is a typical soil modulus, which we compute as $E=p/\kappa$, where $\kappa$ is the logarithmic bulk modulus and p is a typical mean normal effective stress. T is, thus, estimated as 0.05 days. This is about the time it takes for pore pressures to drop to 5% of their initial values, following sudden application of a load (see Terzaghi and Peck, 1967). Since the time scale chosen for the loading of the test specimen in this example is very long compared to this value, no significant pore pressures should ever arise in the analysis.

The same analysis could be performed by using a static procedure, in which case the coupled, effective stress formulation element type could be replaced with an element that models soil deformation only. We choose to use the coupled element type and the soils consolidation procedure to exercise these features.

The accuracy of the equilibrium solution within a time increment is controlled by iterating until the out-of-balance forces reduce to a small fraction of an average force magnitude calculated internally by Abaqus. The rough platen causes an nonhomogeneous stress state, which tends to cause an underestimation of this average force magnitude since stresses are locally higher in the region of the mesh near the platen and the reference force magnitude is averaged over the entire mesh. To avoid iterating to excessive accuracy, we have overridden the default calculation of the average force magnitude and have defined that typical actual nodal forces will be of the order 100 N (22.52 lb). This is done using solution controls. The increment size choice is automatic, determined by allowing a maximum pore pressure change ($\mathrm{\Delta }{u}_w^{max}$) of 0.16 KPa (.023 lb/in²) per increment, which should give sufficient definition of the solution.