Localization, Physical Instability, and Path-Dependence

In many systems that can be modeled with a numerical modeling code such as PFC, 3DEC, or FLAC3D, there may be several paths that the solution may take, depending on rather small changes in initial conditions. This phenomenon is termed bifurcation. For example, a shear test on an elastic/plastic material may either deform uniformly, or it may exhibit shear bands, in which the shear strain is localized rather than being uniformly distributed. It appears that if a numerical model has enough degrees of freedom (i.e., enough elements), then localization is to be expected. Indeed, theoretical work on the bifurcation process (e.g., [Rudnicki1975], and [Vardoulakis1980]) shows that shear bands form even if the material does not strain-soften, provided that the dilation angle is lower than the friction angle. The “simple” Mohr-Coulomb material should always exhibit localization if enough elements exist to resolve one or more localized bands. A strain-softening material is more prone to produce bands.

Some computer programs appear incapable of reproducing band formation, although the phenomenon is to be expected physically. However, Itasca codes are able to allow bands to develop and evolve in the model, partly because they utilize the dynamic equations of motion (i.e., the kinetic energy that accompanies band formation is released and dissipated in a physically realistic way). Several papers document the use of the Itasca code FLAC in modeling shear band formation ([Cundall1989], [Cundall1990], and [Cundall1991]). These should be consulted for details concerning the solution process. One aspect that is not treated well by FLAC is the thickness of a shear band. In reality, the thickness of a band is determined by internal features of the material, such as grain size. These features are not built into the constitutive models used in FLAC. This difficulty is not a major concern in PFC, because the program is used to model behavior at the particle level. For more discussion on modeling the effects of softening and localization within intact material, the FLAC Manual ([Itasca2008]) should be consulted.

So, some MAJOR help reconciling the PFC and FLAC3D versions is needed here.

One topic that involves chaos, physical instability, and bifurcation is path-dependence. In most nonlinear inelastic systems, there are an infinite number of solutions that satisfy equilibrium, compatibility, and the constitutive relations. There is no “correct” solution to the physical problem unless the path is specified. If the path is not specified, all possible solutions are correct. This situation can cause endless debate among modelers and users, particularly if a seemingly irrelevant parameter in the solution process (e.g., damping) is seen to affect the final result. All of the solutions are valid numerically. For example, a simulation of a mining excavation with low damping may show a large overshoot and, hence, large final displacements, while high damping will eliminate the overshoot and give lower final displacements. Which one is more realistic? It depends on the path. If the excavation is done by explosion (i.e., suddenly), then the solution with overshoot may be the appropriate one; if the excavation is done by pick and shovel (i.e., gradually), then the second case may be more appropriate. For cases in which path-dependence is a factor, modeling should be done in a way that mimics the way the system evolves physically.