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Common for the above categories of numerical analysis is that they do not incorporate pre-existing fractures in the material or combinations of failure through intact material and along discontinuities. In a discontinuum computer code, discontinuities are included into the basic model geometry already from the start of calculation. Among the discontinuum codes, one

can distinguish the distinct element programs UDEC and 3DEC (Cundall, 1971; Itasca, 1994a;

1994b), and the discontinuous deformation analysis program, DDA (Goodman and Ke, 1995;

Pei and Shi, 1995). Both types of programs require that the locations of pre-existing discontinuities are known before an analysis is begun. This often, but not always, requires a rough idea of the governing failure mechanisms. By including a large number of

discontinuities it is also possible, to some extent, to simulate the path which failure will take, under the assumption that failure only occurs along discontinuities. Failure through the intact material can be judged in the same manner as for continuum models, but in these programs it is not possible to simulate the formation of a fracture through the intact rock material.

Simulating the failure path and only accounting for pre-existing discontinuities, was addressed by Einstein (1993). Various models for describing a network of fractures, including the persistence and connectivity of discontinuities, were developed. Simple step path failures developing through linking of pre-existing fractures to a continuous failure surface was simulated by checking for kinematic admissibility of the formed rock block. In this approach, the rock material was assumed to be rigid. All displacements therefore are directly associated with the failure surface. Failure occurs when the shear strength of the discontinuities is exceeded. This approach has some similarities with block theory (Goodman and Shi, 1985) since only failure along discontinuities is allowed.

To be able to simulate fracture growth from pre-existing fractures in a rock mass, slightly different methods have been developed. Common for all of these models is that they are based on principles coming from fracture mechanics. Assuming that bridges of intact rock exist between the discontinuities, failure of these bridges can occur as tensile failure (Mode I) at relatively low stress levels, or as shear failure (Mode II) at higher normal stress levels. In the model developed by Einstein et al. (1983), Mode I failure was assumed to generate the primary fracture through the rock bridge, while Mode II failure would generate secondary shear fractures, as shown in Figure 4.6.

Only Mode I fracturing was believed to be important in establishing a failure surface through the rock material. The critical path for a particular joint configuration was defined as that combination of discontinuities and intact rock bridges having the minimum safety margin, where the safety margin was defined as shearing resistance minus the driving force. If the safety margin turns up negative, failure occurs, otherwise not. Failure is assumed to follow the discontinuity until it terminates. For each exit point of a discontinuity, the path of

minimum safety margin is found and this constitutes the failure direction in the intact material.

The calculation process is then repeated to cover the entire slope geometry. The forces acting on a particular discontinuity were assumed to be due solely to the overburden weight.

It appears as though the developed model and corresponding computer program can only handle configurations with one set of parallel joints. Furthermore, deformation of the intact material is not included in this model. It does not formally satisfy the differential equations of equilibrium, the constitutive equations for the material and the strain compatibility equations, as other numerical methods do (BEM, FEM, FDM and DEM). This approach is very similar to the method used by Call et al. (1977) and West et al. (1985) for calculating the possibility of step path failure and can be regarded as a modification and extension of limit equilibrium methods to account for fracturing through intact rock.

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Figure 4.6 Mode I and Mode II fracturing in a rock slope (after Einstein, 1993).

A more rigorous approach to simulating fracture growth was presented by Napier and Hildyard (1992). In this method, a boundary element model was used for modeling the growth of pre-existing fractures in the rock mass. Both extension (Mode I) and shear (Mode II) fracturing were considered. Fracturing was assumed to continue from an existing crack tip.

In the algorithm, the growth direction is first determined as the angle at which the maximum

"excess shear stress" is obtained. The shear strength is defined through the Mohr-Coulomb criterion. Once the optimum growth angle has been selected, fracture growth occurs if a specified failure criterion is met. Both tension and shear failure criteria are used for this.

Although both are based on the Mohr-Coulomb criterion they follow different slopes of the σ1-σ3-curve. These criteria must be calibrated against physical observations of cracking. This numerical model satisfies the conditions of equilibrium, compatibility and constitutive relations in continuum mechanics, and is thus a much more comprehensive model than the simplified approach outlined by Einstein et al. (1983). The pre-existing fractures as well as the newly propagated fractures are represented by displacement discontinuity elements in the boundary element method. Practical applications of this model have been limited to fracturing around underground openings in brittle rock and borehole breakouts (Napier and Hildyard, 1992;

Kuijpers, 1995), in which fracture "seeds" were generated randomly around the opening or

borehole. Results from the numerical simulations have been found to be in good agreement with field observations of fracturing.

Another approach in which the boundary element method is used and where the pre-existing fractures are represented as displacement discontinuity elements is that presented by Shen (1993). Here, fracture mechanics theory was used for determining whether fracture growth from the tip of a pre-existing fracture can occur. In the calculation scheme, the fracture toughness for Mode I fracturing (KIc) is compared to the stress intensity at the fracture tip (KI), which is calculated numerically. This automatically determines both whether fracture propagation will take place, and if so in which direction propagation will occur. Once a propagation direction is determined, a new displacement discontinuity element is added to the fracture tip (the fracture is extended), and the calculations repeated for the new fracture tip.

Shen (1993) also presented a more comprehensive model in which both Mode I and Mode II fracturing, so called mixed mode fracturing, was incorporated. Fracture toughness for shear fracturing is difficult to determine and hence, a different criterion for fracture growth was adopted. The existing G-criterion states that fracture propagation will occur when the strain energy release rate (G) in an elastic medium is larger than the surface energy required to separate the material (Gc). This criterion was modified to ensure that, as had been observed in laboratory tests, Mode I fracturing would occur prior to Mode II fracturing. Shen (1993) successfully simulated fracture propagation and the coalescence of fractures observed in experimental tests on small scale specimens of gypsum (Figure 4.7).

The last two models are definitely the most comprehensive of the fracture growth models presented here. However, the model by Shen (1993) has not yet been applied to any practical problem of larger scale. Currently, there are limitations to the number of pre-existing

fractures, or seeds, that can be modeled. Also, for these models to become practically useful, different sizes of these seeds must be possible to include. The approach by Napier and Hildyard (1992) has only been tested for underground problems, but it is likely that it can handle surface problems as well. For both models, it is essential that calibrations against observed failures are conducted to obtain correct values for the parameters in the growth criterion.

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Figure 4.7 Two stages of observed and simulated fracture propagation and coalescence (after Shen, 1993).

The simulation of shear band localization in continuum models has in fact many similarities with the fracture mechanics approach described above. A fracture mechanics criterion for fracture growth could probably also be used as a complement to the standard plasticity formulation. Nonetheless, it is probably easier to use the localization approach for a continuum material. The real benefit of using a fracture mechanics approach comes when applying the model to a discontinuous material in which failure along discontinuities is mixed with fracturing through the intact rock material.

A very recent addition to the family of numerical programs which perhaps can be used for slope analysis is the two-dimensional version of the Particle Flow Code (PFC^2D) developed by Itasca (1995b), see also Cundall and Strack (1979). PFC is a distinct element program in which the individual blocks are modeled as circular particles. The particles may represent grains in a granular material like sand, or they may be bonded together to represent a solid material. By bonding particles together as blocks, this provides a unique possibility for simulating both fracturing through intact material, as well as failure along pre-existing discontinuities. Whether the program can be used for modeling rock slopes and progressive failure development still remains to be seen.

The above models for simulating progressive failure are all two-dimensional, although FLAC, UDEC and PFC are also available in three-dimensional versions. For a two-dimensional analysis, an assumption must be made regarding the third dimension. For oval open pits, the assumption of plane strain or plane stress is valid at the long sides of the pit. Two-dimensional models could thus be used for the design of large portions of the pit, with a few important exceptions such as the corners. Extension of the above models for localization and fracture propagation to three dimensions does not come easily. Models for simulation of fracture propagation in rock have not yet been developed for application to three-dimensional slope geometries. It is also questionable whether it is feasible to do so, without making substantial simplifications of the fracture geometry in three dimensions.

An important factor to consider is the ability to model groundwater flow. In commercially available programs like FLAC and UDEC (Itasca 1993, 1994), groundwater flow can be modeled explicitly, both as flow in the rock matrix and flow in the rock joints. Consequently, the effect of groundwater pressure on the slope can be quantified directly. It must be

remembered, however, that coupled discontinuum analysis can be fairly complex.

4.4.4 Summary

1. Numerical analysis can be used to calculate both stress and deformation in a slope, and different materials and constitutive relations can relatively easily be incorporated.

Numerical models can, with the correct input data, be used also for making predictions of slope behavior. Furthermore, sensitivity analyses are readily carried out using numerical modeling.

2. To simulate the actual failure mechanism, an assumption of the failure surface is generally not necessary. Existing continuum codes can, to some extent, simulate the location and shape of the failure surface developing in a slope, although an actual discontinuity is not developed. Discontinuum codes can simulate failure along pre-existing discontinuities. It is, however, much more difficult to simulate failure both along discontinuities and through intact rock in the same model.

3. More development is necessary before a general method is available which can

simulate both fracturing through intact material and slip and separation of pre-existing discontinuities. Most of the methods described above have not advanced to the stage where they can be used routinely as a tool for predicting slope failures. Furthermore, simulating crack propagation in a slope may require an explicit description of the

fracture geometry (pre-existing discontinuities and intact rock bridges), and this is almost impossible to accomplish for a large scale slope.

4. Rather than attempting to simulate the exact mechanism in a single numerical model, several different models with different fracture patterns and different material models and properties, can be analyzed. Using this approach, the important factors governing a certain slope behavior can be identified. This approach is perhaps most useful when comparing results with field observations and measurements, see e.g., Board et al.

(1996). Simpler models are used in favor of more complex models, which also facilitates collection and choice of input data.