FLAC3D Modeling • Problem Solving with FLAC3D

Interpretation

Since FLAC3D models a nonlinear system as it evolves in time, the interpretation of results may be more difficult than with a conventional finite-element program that produces a “solution” at the end of its calculation phase. There are several indicators that can be used to assess the state of the numerical model (e.g., whether the system is stable, unstable, or in steady-state plastic flow). The various indicators are described below.

Unbalanced Force and Convergence

Each gridpoint is connected to zones that contribute forces to the gridpoint. At equilibrium, the algebraic sum of these forces is almost zero (i.e., the forces acting on one side of the gridpoint nearly balance those acting on the other). If the unbalanced forces approach a constant nonzero value, this indicates that failure and plastic flow are occurring within the model.

As discussed in Reaching Equilibrium, there are five methods of determining model convergence. Each of these is based on the out-of-balance force acting on the gridpoints. Each of these can be saved as a history and viewed as a graph. These convergence criteria are important in assessing the state of the model.

A rule of thumb is that an average force ratio of 1e-5, an local force ratio of 1e-4, or a convergence of 1.0 all indicate overall convergence. However, values 10 or even 100 times larger may be acceptable depending on the specifics of the model and the degree of precision required (e.g., a much larger value larger may be good enough for an intermediate stage in a sequence).

Note that a low convergence value only indicates that forces balance at all gridpoints. However, steady plastic flow may be occurring without acceleration. In order to distinguish between this condition and “true” equilibrium, other indicators (such as those described below) should be examined.

Gridpoint Velocities

The grid velocities may be assessed either by plotting out the whole field of velocities (plot a “zone-vectors” item and set “Value” to “Velocity”), or by selecting certain key points in the grid and tracking their velocities with histories (e.g., zone history velocity-x, or velocity-y, velocity-z, or velocity). Both types of plots are useful. Steady-state conditions are indicated if the velocity histories show horizontal traces in their final stages. If they have all converged to near-zero (in comparison to their starting values), then absolute equilibrium has occurred; if a history has converged to a nonzero value, then steady plastic-flow is occurring at the gridpoint corresponding to that history. If one or more velocity history plots show fluctuating velocities, then the system is likely to be in a transient condition. Note that velocities are expressed in units of displacement divided by number of steps.

The plot of the field of velocity vectors is more difficult to interpret, since both the magnitudes and the nature of the pattern are important. As with gridpoint forces, velocities never decrease precisely to zero. The magnitude of velocity should be viewed in relation to the displacement that would occur if a significant number of steps (e.g., 1000) were to be executed. For example, if current displacements in the system are of the order of 1 cm, and the maximum velocity in the velocity plot is 10-8 m/step, then 1000 steps would produce an additional displacement of 10-5 m, or 10-3 cm, which is 0.1% of the current displacements. In this case, it can be said that the system is in equilibrium even if the velocities all seem to be “flowing” in one direction. More often, the vectors appear to be random (or almost random) in direction and (possibly) in magnitude. A random velocity field of low amplitude is an infallible indicator of equilibrium and no plastic flow.

If the vectors in the velocity field are coherent (i.e., there is some systematic pattern) and their magnitude is quite large (using the criterion described above), then either plastic flow is occurring or the system is still adjusting elastically (e.g., damped elastic oscillation is taking place). To confirm that continuing plastic flow is occurring, a plot of plasticity indicators should be examined, as described below. If, however, the motion involves elastic oscillation, then the magnitude should be observed in order to indicate whether such movement is significant. Seemingly meaningful patterns of oscillation may be seen but, if amplitude is low, then the motion has no physical significance.

Plastic Indicators

For the plasticity models in FLAC3D, stresses that satisfy the yield criterion may be shown by plotting a zone plot item, setting “ColorBy” to “Label” and setting “Label” to “State”. Such an indication usually denotes that plastic flow is occurring, but it is possible for an element simply to “sit” on the yield surface without any significant flow taking place. It is important to look at the whole pattern of plasticity indicators to see whether a mechanism has developed.

Two types of failure mechanisms are indicated by a standard Mohr-Coulomb plasticity state plot: shear failure and tensile failure. Each type is designated by a different color on the plot.[1] The plot also indicates whether stresses within a zone are currently on the yield surface (i.e., the zone is at active failure now, -n), or the zone has failed earlier in the model run, but now the stresses fall below the yield surface (the zone has failed in the past, -p). Initial plastic flow can occur at the beginning of a simulation, but subsequent stress redistribution unloads the yielding elements so that their stresses no longer satisfy the yield criterion, indicated by shear-p or tension-p (on the plasticity state plot).

A failure mechanism is indicated if there is a contiguous line of active plastic zones (indicated by either shear-n or tension-n) that join two surfaces. The diagnosis is confirmed if the velocity plot also indicates motion corresponding to the same mechanism.

If there is no contiguous line or band of active plastic zones between boundaries, two patterns should be compared before and after the execution of, say, 5000 steps. Is the region of active yield increasing or decreasing? If it is decreasing, then the system is probably heading for equilibrium; if it is increasing, then ultimate failure may be possible.

If a condition of continuing plastic flow has been diagnosed, one further question should be asked: Does the active flow band(s) include zones adjacent to artificial boundaries? The term “artificial boundary” refers to a boundary that does not correspond to a physical entity, but one that exists simply to limit the size of the grid that is used (see Artificial Boundaries). If plastic flow occurs along such a boundary, then the solution is not realistic, because the mechanism of failure is influenced by a nonphysical entity. This comment only applies to the final steady-state solution; intermediate stages may exhibit flow along boundaries.

Histories

In any problem, there are certain variables that are of particular interest (e.g., displacements may be of concern in one problem, but stresses may be of concern in another). Liberal use should be made of the zone history command to track these important variables in the regions of interest. After some timestepping has taken place, plots of these histories often provide a way to find out what the system is doing.

Endnotes