Continuum Models

In continuum models, the displacement field will always be continuous. The location of the failure surface can only be judged by the concentration of shear strain in the model. No actual failure surface discontinuity is formed and it can thus be difficult to continue to analyze the behavior of the slope after the first failure surface has formed. Simulation of rotational shear failure in a continuum slope (no discontinuities) using the continuum code FLAC (Cundall, 1976; Itasca, 1995a) is shown in Figure 4.4.

FLAC (Version 3.30)

Figure 4.4 Calculated displacement vectors at failure in a plastic material, using the continuum finite difference program FLAC. The slope height is 300 meters and the slope angle 40°.

The location of the failure surface, judged from the onset of displacement or shear strain (not shown), is in general very similar to that predicted from limit equilibrium methods. Still, the

"failure surface" is not very distinct and for large slope deformations, the assumption of a continuum may not be realistic.

To overcome some of these obstacles, several new approaches have been proposed among which two major categories can be identified. The first category is conventional continuum numerical analysis but with new constitutive models which better replicate the actual rock material behavior, while the second category is concerned with simulating localization of shear bands in the material.

In the first category, one finds the Cosserat plasticity models. Classical continuum models, including those where the rock mass (intact material and discontinuities) is modeled as an equivalent continuum, are not accurate in regions of high stress gradients. They cannot account for the bending stiffness of layers, or blocks, of intact rock. The Cosserat continuum model account for both rotational and translational degrees of freedom in a material, unlike classical plasticity models in which only translational degrees of freedom are included. Hence, moments and internal spins are taken into account in the formulation, and a non-symmetric stress tensor is allowed (Dawson and Cundall, 1995). Cosserat models are thus better adapted to modeling discontinuous media. Dai et al. (1994) presented an application of Cosserat models for the simulation of slope stability. A reasonable failure development was

reproduced, while still using a very crude model, but an actual failure surface discontinuity is not formed in such a model.

In the second category of numerical models, an attempt is made to simulate the localization of shear bands in the intact material. Most geologic materials exhibit softening behavior as the strength of the material is exceeded, and a residual strength is achieved. Typical for these materials is that the plastic shear strain will localize in thin bands in the material, rather than being uniformly distributed. There is also evidence that such localization can occur in non-softening, plastic materials provided that the material is allowed to dilate (Itasca, 1995a;

Zienkiewicz et al., 1995). The simplest approach to model localization involves the use of strain-softening material models in which the rock mass is given different peak and residual strengths (see Figure 3.17). Simulation of a progressive failure behavior is thus made easier and more realistic. There are several commercially available codes which have built-in strain-softening material models, for example, FLAC (Itasca, 1995a).

However, the simulation of shear band localization using numerical models presents some particular difficulties. One problem is that the localization tends to be mesh-dependent. This

means that, as shown by Zienkiewicz et al. (1995) and Schlangen (1995), the shear bands have a tendency to follow the patterns in the discretized mesh of the model rather than develop freely without interference of the mesh. Furthermore, large numbers of elements are required to produce localization in a numerical model, and two models of the same problem geometry but with slightly different discretizations may also produce different results. In many

conventional models, the thickness of the shear bands is not simulated accurately. This is the case for the finite difference code FLAC in which the shear band collapses down to the smallest width that can be resolved by the mesh, normally one to three elements wide (Itasca, 1995a). In conventional finite element methods a more diffuse failure surface is formed which does not represent the actual failure mechanism very well (Larsson et al., 1992a). The

inherent mesh-dependency is tackled by adjusting the mesh to follow the expected shear bands. This satisfies continuity of displacements over the element boundaries, but requires complete re-meshing (smaller elements, more elements and better orientation of the elements) of the model during the calculations, something which is very time-consuming.

New developments in this area are based on bifurcation theory and new numerical schemes to simulate localization (Chambon et al., 1994). Bifurcation in a certain structural system means that the solution can take different paths, depending upon small changes in the initial

conditions. In the inter-element approach, Larsson (1990) and Larsson et al. (1992a, 1992b, 1994) developed a finite element model which could simulate localization in soil materials with the help of bifurcation analysis. In the bifurcation analysis, criteria for the formation of shear bands were formulated in terms of a critical value for the hardening modulus or a critical stress level, and the associated critical bifurcation directions. The bifurcation directions could be evaluated using analytical methods. In Larsson's approach, an incremental plasticity

formulation was used along with constitutive relations which were adapted for soil materials (Mohr-Coulomb, double-cap and Cam-Clay models). Furthermore, strains were discontinuous over the element boundaries (as opposed to classical finite element model, see above). The finite element mesh was also adapted to the critical bifurcation directions. During the

calculations, some re-meshing is still necessary, and this is carried out by aligning the element sides with the bifurcation directions, thus obtaining the most finely discretized mesh in the region of interest. Larsson et al. (1992a, 1992b) presented an example where the new model was applied to a slope stability problem yielding a good representation of localized

deformations at failure (Figure 4.5).

Figure 4.5 Slip line developed at the final load step in a finite element model of a slope loaded by an elastic footing (from Larsson et al., 1994).

Klisinski et al. (1995) used a different approach termed "inner softening band" in which

softening occurs through displacement within an element. Cracks can be introduced anywhere and in any orientation within the element mesh, and at any time during the calculations. The results are less sensitive to the size and geometry of the finite element mesh. This method is, however, still in its early development and has to date only been applied to the cracking of concrete. Other attempts include those in which various new constitutive relations have been combined with mesh refinement techniques to obtain a more precise representation of the shear band. Hicks and Mar (1994) have, in this regard, also included groundwater pressure in their model. In a recent paper, Zienkiewicz et al. (1995) discussed these and other efforts at modeling localization and concluded that more robust formulations are necessary to avoid some of the numerical difficulties commonly encountered. The procedure proposed by Zienkiewicz et al. (1995) is more general and less dependent on the initial mesh. Since re-meshing was used, this procedure is also very time-consuming, and thus none of the developed model techniques can as yet be considered to be practically useful tools for slope stability analysis. It must also be questioned whether a numerical simulation of a very distinct slip surface is warranted. In reality, the failure surface can be relatively thick (see Section 3.3.3) and perhaps a less rigorous but much less time-consuming model is satisfactory for our purposes.