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ADVANCES IN BONDED BLOCK MODELING

Walton, G., Colorado School of Mines Sinha, S., Colorado School of Mines

ABSTRACT

Stress-induced damage of sparsely fractured rockmasses around excavations has historically been modeled using continuum methods, but these approaches are highly phenomenological in nature and, as a consequence of their formulation, cannot be used to predict the influence of rock reinforcement on ground behaviour. Relatively new discontinuum tools such as bonded block models (BBMs) have the potential to overcome these limitations. This paper documents recent advances in bonded block modeling at both the laboratory and field scales, with an emphasis on research by the authors. Additionally, the following critical questions about BBMs are answered: (1) What physical phenomena can BBMs reproduce, and how does this depend on the choices made during model development? (2) Can BBMs reproduce field-scale behaviour (not just in terms of strength, but also deformation and dilation characteristics)? (3) Can BBMs quantitatively reproduce the influence that rockbolt reinforcement has on ground behaviour?

Keywords: Bonded block modeling (BBM), stress-induced damage; numerical modeling; ground-support interaction; model complexity

INTRODUCTION

With growing global populations and finite resources available for extraction, excavations continue to advance ever-deeper below ground. Figure 1 illustrates this trend in the context of block caving mining, where undercut depths more than 1 km below surface are now the norm. The transition to deep mining leads to a transition in the types of ground control issues that are encountered. Empirical tools developed to aid in support design for blocky ground under low stress cannot be applied in cases where stress-driven fracturing dominates, and spalling and rockbursting are expected (Martin et al., 1999). Accordingly, there is a need to re-think how we should design excavation support (Kaiser, 2019).

In the field of rock mechanics, numerical models have seen increasing application in the past few decades. Such models are now regularly applied to evaluate stress transfer around excavations, to assess the potential for rockmass damage, and to analyze multiple design alternatives. In practical applications, continuum models represent the primary numerical analysis tool because of their relative simplicity (e.g. ease of development, application, and interpretation) and computational efficiency. At larger scales, elastic models provide reasonable approximations of stress transfer around networks of excavations (e.g. mine-

© 2020, Svenska Bergteknikföreningen och författarna/Swedish Rock Engineering Association and authors.

Bergmekanikdagen 2020

wide modeling), whereas to predict damage processes and excavation-scale displacements, inelastic constitutive models may be necessary (Walton et al., 2015).

Figure 1: Block cave mine undercut depth trends over time (from Eberhardt et al., 2015; data originally compiled by Woo et al., 2013).

Despite the unquestionable value of continuum numerical models as an analysis tool, they have two major limitations when it comes to simulation of stress-induced damage at the excavation scale:

1) It can be difficult to directly link field-scale input parameters to laboratory parameters that are available a priori (Walton, 2019); accordingly, it is not possible to make a model with reasonable predictive capability prior to extensive calibration.

2) While they can be used to assess deformation compatibility between the ground and reinforcement elements (Lorig and Varona, 2013), they cannot simulate the influence that reinforcing elements (e.g. rockbolts) have on ground behaviour. This is an inherent limitation of continuum models, rather than an issue associated with a particular constitutive model or support element (Sinha & Walton, 2019).

The first limitation is, at present, effectively true of almost all numerical models in the field of rock mechanics. However, some discontinuum modeling methods that allow for the explicit simulation of fracture initiation, growth, and dilatancy have the potential to overcome this limitation in the near future. Because such models typically represent the macroscopic behaviour of a rock volume as an emergent response that depends on “fundamental” micro-parameters and basic physics, it should be theoretically possible to develop models that can be calibrated to laboratory-scale data and then directly upscaled for application to in-situ problems. Although some studies have shown promise in predicting field-scale rock strength based on laboratory-scale parameter calibration (Insana


et al., 2016; Farahmand et al., 2018; Li et al., 2019; Dadashzadeh & Diederichs, 2019), the transformative potential of such a predictive modeling approach has not yet been fully demonstrated. In fact, this paper does not address the issue of parameter upscaling any further, and laboratory and field scenarios are addressed separately.

This paper is more focused on overcoming the second limitation of continuum models. Specifically, unlike continuum models, discontinuum models have been shown to qualitatively reproduce a wide range of ground-support interaction behaviours with respect to rockmass reinforcement (e.g. bolting) in spalling ground (Sinha & Walton, 2019). Figure 2 presents a conceptual model illustrating how reinforcement influences ground behaviour as well as which portions of the ground-support interaction curve are reproduced in continuum and discontinuum models. Since continuum models enforce strain-continuity throughout the simulation domain (i.e. no discrete separation of model elements is allowed), by virtue of their formulation, such models effectively simulate an “already-reinforced” rockmass where separation of discrete elements is not possible. In other words, continuum models implicitly account for the effects of reinforcement, and any influence of support on the model behavior is therefore negligible. Where the real rockmass behaves in a continuum fashion with minimal support, such as in the case of squeezing ground, this limitation is not significant; it is important, however, where discrete fracture separation is relevant (as is the case for stress-induced spalling).

Figure 2: The effect of reinforcement on ground behaviour as a function of the amount of support installed, with ranges of ground-support interaction represented

by continuum and discontinuum models indicated (Sinha & Walton, 2019).

Because of their advantages relative to continuum models and the relative lack of empirical approaches that are appropriate for support design in deep excavations in spalling or


rockbursting ground, discontinuum models have the potential to become a leading tool for modeling of ground-support interaction at the scale of individual excavations. Before such tools can be applied in practice, however, their capabilities must be fully understood and rigorously demonstrated. With that in mind, this paper will answer three major questions:

What physical phenomena can bonded block models (BBMs) reproduce, and howdoes this depend on the choices made during model development?

Can BBMs reproduce field-scale behaviour (not just in terms of strength, but alsodeformation and dilation characteristics)?

Can BBMs quantitatively reproduce the influence that rockbolt reinforcement hason ground behaviour?

It is noted that while there are many different approaches and software packages that can be applied for the purposes described here (e.g. FDEM – EFLEN, Irazu; BPM – PFC; Lattice Spring Method – SRMTools; DDA; etc.), this paper is focused on the BBM approach as implemented in Itasca’s Universal Distinct Element Code (UDEC). Although many of the findings and conclusions presented here may apply equally well to other discontinuum modeling approaches, further study would be necessary to confirm this by reproducing the results shown here (or similar).

THE BONDED-BLOCK MODELING APPROACH

The BBM approach for the simulation of intact rock damage involves three primary components: blocks, contacts, and zones. Blocks represent “unbreakable” (e.g. continuum) elements within the model, and the boundaries between blocks (“contacts”) represent potential fracture pathways. Inside the blocks, a series of finite difference zones are used to discretize the blocks such that block deformation can be calculated using a continuum constitutive model. While the zones are a purely mathematical construct (as in any continuum model), the specific physical meaning of the blocks and contacts depends on the scale of model that is considered. Regardless of scale, what is most important is that the blocks are sufficiently small such that there is a sufficient number of potential fracture pathways that kinematic constraints on fracture propagation do not dominate the model response (Gao & Stead, 2014; Christianson et al., 2016; Mayer & Stead, 2017).

LABORATORY-SCALE BONDED-BLOCK MODELING

The vast majority of BBM studies to date (e.g. Lan et al., 2010, Farahmand & Diederichs, 2015; Li et al., 2017; Mayer & Stead, 2017) have focused on the simulation of laboratory compression tests. This is natural for such a relatively new type of modeling approach, as laboratory-scale models can be constrained by significantly more data than field-scale models. Additionally, these models are small enough that the individual blocks can be defined with sizes and shapes that approximate those of the individual constituent mineral grains within the rock.


In this section, we present a summary of the capabilities of laboratory-scale BBMs to illustrate the importance of consideration of different failure mechanisms in BBMs if a wide range of behaviours are to be simulated. Specifically, we have focused on Creighton Granite, a rock from the Sudbury basin in Ontario, Canada. Figure 3 illustrates block and zone geometries, mineralogical assignments, and loading conditions for a series of laboratory-scale simulations conducted by the authors. Note that triaxial simulations were also conducted simply by the addition of stress boundary conditions along the lateral boundaries of the specimen.

Figure 3: Laboratory-scale bonded-block model setup for (a) Brazilian tensile and (b) uniaxial compressive tests; the inset image shows the finite difference discretization of

the individual mineral “blocks” (Sinha & Walton, 2020a).

Although the model shown in Figure 3 uses different properties for each of the mineral constituents and each possible contact type (based on combinations of adjacent minerals), many BBMs in the literature have made the simplifying assumption that a single set of properties can be assigned to all blocks and another single set of properties can be assigned to all contacts. We refer to these model types as “heterogeneous BBMs” and “homogeneous BBMs”, respectively. Additionally, the models may differ with respect to the constitutive model applied to the blocks (i.e. to the zones inside the blocks); although models using rigid blocks (Kazerani & Zhou, 2010) and inelastic blocks (Noorani & Cai, 2010; Wang & Cai, 2019) have been presented in the literature, it is by far most common to use elastic blocks. Considering these options, there are effectively four types of BBM modeling approaches at the laboratory-scale: homogeneous properties and elastic blocks, homogeneous properties and inelastic blocks, heterogeneous properties and elastic blocks, and heterogeneous properties and inelastic blocks.


Sinha & Walton (2020a) compared the capabilities of these four modeling approaches, both by calibrating models to Creighton Granite laboratory data and through a comprehensive literature review. The findings from that study are summarized in Table 1. It was found that incorporation of block (e.g. mineral grain) stiffness heterogeneity was necessary to allow for accurate simulation of the damage initiation and propagation processes under confined conditions. Additionally, it was determined that the use of an inelastic constitutive model for the blocks (to indirectly simulate intragranular fracturing) is necessary to simulate the transition from specimen-scale extensile fracturing to shear fracturing (both at higher confinements as well as post-yield). It was, however, found that there was no additional benefit to assigning separate strength properties to the various mineral grains, as models using a single set of grain strength parameters could reproduce all observed attributes of the rock behaviour.

Table 1: Capabilities of different BBM rock representations (Sinha & Walton, 2020a).

Macroscopic attributes Homogenous,

Elastic Homogenous,

Inelastic Heterogeneous,

Elastic Heterogeneous,

Inelastic Elastic constants ✓ ✓ ✓ ✓

UCS ✓ ✓ ✓ ✓ Tensile strength ✓ ✓ x ✓

Low confinement peak strength (σ3≤10% UCS) ✓ ✓ ✓ ✓

High confinement peak strength (σ3>10% UCS)

x ✓ x ✓

Unconfined CI ✓ ✓ ✓ ✓ Confined CI ✓ a ✓ a ✓ ✓

Unconfined CD ✓ a ✓ a ✓ ✓ Confined CD x x ✓ ✓

Pre-peak behaviour (confined)

x b ✓ x b ✓

Post-peak behaviour (confined)

x c ✓ x c ✓

Confinement dependent dilatancy ✓ d ✓ d ✓ ✓

x - cannot match; a slightly overestimates; b excessive strain-hardening; c rapid drop; d captures qualitative trends only.

With all this in mind, the appropriate level of complexity for a bonded block model depends on the phenomena of interest to be reproduced (this applies equally to field-scale BBMs). For models focused on extensile crack development, stiffness heterogeneity is most important. For models focused on shear behaviour post-yield or at high confinements,


an inelastic constitutive model is critical. If both are considered, the full range of behaviours observed in the laboratory can be reproduced. For example, the results shown in Figure 4 illustrate the capability of the heterogeneous and inelastic BBM developed by Sinha et al. (2020) to reproduce various laboratory data.

Figure 4: Calibrated BBM results – (a) Peak and residual strengths in σ1-σ3 space, (b) Average stress-strain curves for 0-60 MPa confinement, (c) CI Thresholds, (d) CD

Thresholds, (e) Normalized peak dilation angle, and, (f) simulated BTS stress-strain curve with fracture pattern just beyond peak shown (the average laboratory BTS

strength was 10.5 MPa). The black and blue datapoints in (a, c-e) are laboratory data, and the red circles are from the BBMs (after Sinha et al., 2020).


After calibration, the model was further shown to demonstrate the cohesion-weakening-friction-strengthening (CWFS) strength evolution trend that is now well accepted for hard rock (Martin, 1997) as an emergent behaviour of the model (Figure 5).

Figure 5: Emergent cohesion-weakening-friction-strengthening behaviour of the calibrated BBM at low confinement (σ3 ≤ 20 MPa) (Sinha et al., 2020).

FIELD-SCALE BONDED-BLOCK MODELING

Studies using field-scale BBMs have been conducted in the past decade, although they are much less common than laboratory-scale studies. Such studies have largely consisted of sensitivity analyses, where the relative influences of loading conditions (Damjanac et al., 2007; Garza-Cruz et al., 2014; 2018), support (Gao et al., 2014; Kang et al., 2015; Bai et al., 2016), or material properties (Garza-Cruz and Pierce, 2014) were tested. While these prior studies have all provided qualitatively “realistic” results, there has been a relative lack of detailed model calibration and validation at the field-scale. Additionally, there has been limited investigation of what components of field-scale models (i.e. block stiffness heterogeneity, contact strength heterogeneity, block inelasticity) are most critical in replicating different behaviours; most prior field-scale studies have employed a homogeneous and elastic block representation (Damjanac et al., 2007; Preston et al., 2013; Gao et al., 2014; Bai et al., 2016).

Whereas laboratory-scale BBMs have direct physical analogs for block stiffness heterogeneity (mineral stiffness heterogeneity), block contact strength heterogeneity (different mineral-to-mineral contact types), and block inelasticity (intragranular fracturing), the size of blocks in field-scale models means that such analogs are no longer useful. In particular, field-scale models have historically used block sizes ranging from 10 cm to 50 cm, primarily due to the computational demand of finer-scale block structures (Damjanac et al., 2007; Preston et al., 2013; Bai et al., 2016; Wang et al., 2019).

While we cannot consider field-scale blocks to correspond to individual mineral grains, they are also not a purely mathematical construct either (i.e. they are unlike the finite


difference zones that purely represent the discretization used to solve a differential equation). While there are block sizes above which model results will be unambiguously incorrect (e.g. due to kinematic restrictions imposed by the block arrangement), there is not necessarily a block size below which the results will become independent of block size (unlike in the case of mesh sensitivity). Rather, block size interacts with the input parameters assigned to a field-scale model to influence model behaviour, and the input parameters for a calibrated model should therefore be considered specific to the block size used. Further research is needed to fully understand this interaction between material parameters and block size, as such a relationship is necessary to develop any kind of up-scaling procedure from laboratory-scale models to field-scale models.

With respect to block shape, Sinha & Walton (2019) demonstrated that for two-dimensional BBMs, a randomly generated Voronoi block structure results in more realistic dilation (bulking) behaviour than a trigon (triangular) block structure (where trigon structures underestimate bulking). There is some evidence, however, to suggest that for three-dimensional models, the extra degrees of freedom with respect to block kinematics allow tetrahedral block geometries to replicate dilation in a realistic manner (where three-dimensional Voronoi block structures tend to overestimate bulking).

Because of the sizes of the blocks used in field-scale models, one would not expect block stiffness to be nearly as relevant to model behaviour as it is at the laboratory-scale, and indeed that hypothesis has been confirmed by the authors’ experience. Similarly, although many studies have utilized contact strength heterogeneity in field-scale models (Garza-Cruz et al., 2014; 2018), we have found it to be possible to obtain satisfactory results without adding in this type of model complexity. Inelasticity within the blocks, however, perhaps becomes more important in field-scale models than in laboratory models. This is because the geometries and stress conditions in many field-scale scenarios often lead to the potential for multiple yield mechanisms to occur (e.g. extensional fracturing in the “inner shell” near the excavation and shear damage in the “outer shell”; Valley et al., 2011). Whereas block inelasticity at the laboratory-scale represents intragranular fracturing, block inelasticity at the field-scale simply represents damage at scales finer than can be explicitly represented by the block structure itself; such finer-scale, more distributed damage is often more relevant under confined conditions (e.g. brittle shear damage), similar to intra-granular fracturing at the laboratory-scale (Hofmann et al., 2015; Abdelaziz et al., 2018). Although it may be possible to simulate different failure mechanisms using different scales of bonded-blocks within one another (as has been implemented at the laboratory-scale; Gao et al., 2016; Wang & Cai, 2018), at present this approach remains relatively untested at the field-scale.

Sinha & Walton (2018) demonstrated the limitations of field-scale BBMs with elastic blocks using a hypothetical granite pillar model. In particular, they calibrated pillar models with varying width to height ratios (W/H) to empirical pillar strength data from the literature and expected stress-strain behaviour. They, however, found that it was not


possible to identify a single set of elastic BBM input parameters that could be used to match the expected behaviour for all W/H cases; in particular, lower contact friction properties needed to be used for larger W/H cases, or else the confinement generated by the pillar geometry would effectively lock the individual blocks into place. Subsequent modeling efforts, however, have shown that the use of inelastic block properties can remedy this issue by allowing lower confinement extensional fracturing behaviour to be replicated by the explicit damage and opening of the block contacts while indirectly simulating distributed damage under confinement using inelastic blocks. Figure 6 illustrates such a case, where a fully homogeneous BBM with inelastic blocks could be calibrated to match the expected strength (validated by empirical data) and stress-strain behaviour of hypothetical granite pillars of varying W/H using a single set of input parameters.

Figure 6: Calibrated pillar BBM results (UDEC) as compared to expected stress-strain curves derived by calibration of FLAC3D models to empirical pillar strength

data (see Sinha & Walton, 2018 for more information on the FLAC3D model development and validation).

With the aforementioned aspects of field-scale BBM development in mind, we can now consider specific field-scale case studies that illustrate the capabilities of BBMs in terms of replicating both rockmass behaviour under stress and ground-support interaction.

Creighton Mine Case Study

Creighton Mine in Sudbury, Canada is one of the 10 deepest mines in the world with active mining depths extending below 2.5 km. The main orebody consists of nickel-copper mineralization along the contact between a felsic norite unit and the Huronian footwall rocks. Details on the geological environment and predominant structures at Creighton mine are provided in detail in the literature (Malek et al., 2008; Snelling et al., 2013).


Walton et al. (2015; 2016) developed a large-scale continuum (FLAC3D) model that was calibrated to extensometer measurements made during mining on the 7910 level (Figure 7, left). More recently, Sinha & Walton (2020b) integrated a pillar-scale two-dimensional BBM with this calibrated continuum model by applying the pillar-boundary displacements over the course of mining to the top and bottom of the BBM (Figure 7, right). Although the diagram in Figure 7 shows perfectly uniform and vertical displacement boundaries, the actual displacements applied were non-uniform and included horizontal (shear) components due to the position of the cross-section of interest (A-A’) relative to the active stoping area during the mining period considered. The BBM included the changes in stresses at the lateral edges of the pillar prior to mining of the adjacent drifts using stress boundaries as well as the explicit installation of support elements as used in the mine.

Figure 7: Schematic of the integrated continuum-discontinuum approach that shows the mine-scale FLAC3D model (after Walton, 2014) and the corresponding BBM

pillar model with critical model solution points indicated below (Sinha and Walton, 2020b).

After iterative parameter adjustment, it was possible to replicate the observed displacements over the full monitoring period at all anchor locations of the multi-point borehole extensometer (MPBX) installed along the mid-height of the pillar at the location of the BBM (Figure 8). This confirms that with a homogeneous, inelastic BBM, it is


possible to quantitatively replicate the field-scale deformation behaviour of intact rock under high stress.

Additionally, although the peak strength of the pillar was not considered as a target for model calibration, following completion of the calibration process, it was determined that the pillar model’s peak strength to UCS ratio (0.45) is nearly identical to that predicted by Hedley & Grant’s (1972) empirical equation for the strength of hard rock pillars (0.47).

Figure 8: (a) Comparison of calibrated BBM displacements and measured displacements for multiple MPBX anchors, and, (b) anchor locations within the pillar

shown in meters (Sinha and Walton, 2020b).

An examination of the model results at the end of monitoring period illustrates how the different model components simulate the different yield mechanisms present in the pillar (Figure 9). Specifically, the block contacts near the pillar peripheries have been damaged, simulating surficial spalling, whereas zone yield deeper in the pillar simulates the development of a confined shear surface. Note that the relative lack of opening of the spalling fractures shown in Figure 9a is a function of the rockbolt reinforcement and 75 mm shotcrete liner in the model (and in the mine).

Figure 9: Calibrated pillar BBM results, showing (a) stress transfer through the pillar, and, (b) inelastic block yield at the final calibrated model state; note the use of UDEC’s compression negative convention in the representation of the vertical stresses

(Sinha and Walton, 2020b).


West Cliff Mine Case Study

The West Cliff Mine provides an ideal case study to illustrate the relative value of local (entry-scale) discontinuum models for the assessment of ground-support interaction. The West Cliff Mine is a longwall coal mine is Australia that was studied extensively using field instrumentation by Colwell (2006) and more recently using continuum numerical models by Mohamed et al. (2016) and Sinha & Walton (2020c). Here, the continuum model results of Sinha & Walton (2020c) are presented, followed by BBM model results for comparison.

Figure 10a shows the continuum model set-up used by Sinha & Walton (2020c), where loading of the pillar of interest from mining of the adjacent headgate panel is simulated through the application of a uniformly increasing stress at the top of the model. The model displacements obtained using the continuum model match the extensometer data from Site A (lightly supported with two bolts) quite well. When the calibrated model parameters from Site A were used to model Site B (identical to Site A in every way, but with two additional bolts and mesh straps), the impact of the additional support in significantly reducing the ground displacements relative to Site A could not be replicated (see Figure 10). This directly illustrates the inability of the continuum model to fully represent the influence of reinforcement on the ground.

Figure 10: (a) Continuum model setup and (b) calibrated model results (Sinha & Walton, 2019).

To overcome this limitation, a discontinuum model was developed with an identical set-up to the continuum model shown in Figure 10, but with a Voronoi BBM representation of the coal pillar. After an iterative trial-and-error calibration process, the model was able to


replicate the observed field displacements at Site A both under development and headgate loading conditions (Figure 11a) as well as the trend in displacement and stress changes at the location of a stress cell 4 m from the edge of the pillar (Figure 11b).

Figure 11: Calibrated BBM results showing agreement with (a) extensometer data, and, (b) stress-displacement evolution at a point 4 m into the pillar.

The calibrated BBM was then used to evaluate the influence of the rockbolts installed in the mine on the observed ground displacements. As shown in Figure 12, the rockbolts influence the ground displacements much more in the BBM than in the continuum model (compare to Figure 10b), and the predicted mid-height pillar displacement for the “4 Bolts” case (29 mm) is much closer to the displacement observed at Site B in-situ (22 mm) than that in the corresponding continuum model (71 mm). This promising result suggests that in future, similar models could be used as part of the design process for rock reinforcement.

Figure 12: Calibrated BBM results with displacements for varying degrees of support.


CONCLUSIONS

This paper has documented recent advances in bonded block modeling, both at the laboratory-scale and field-scale. Laboratory-scale modeling has confirmed that certain types of BBMs can replicate all commonly observed small-scale behaviours of brittle low porosity rocks. Block inelasticity is crucial to the simulation of small-scale shear damage that is relevant under confined conditions, and this conclusion was found to apply equally well to field-scale models. Two case studies have been used to demonstrate that BBMs are indeed capable of replicating field-scale behaviour of sparsely fractured brittle rockmasses, including strength and deformation characteristics. Perhaps most significantly, it has also been established that such models may represent a promising approach for simulation of ground-support interaction as could be used in rock reinforcement design. While it may not be practical to calibrate model results to specific displacements in most practical use cases, less strictly calibrated BBMs can still be used as design tools to draw general conclusions about brittle rock damage and ground-support interaction.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the use of UDEC licenses provided as part of the Itasca Educational Partnership program in the development of some of the models presented here. The authors would like to thank Dr. Ehsan Ghazvinian for reviewing a draft of this paper. Dr. Walton would also like to acknowledge his Computational Geomechanics Research group at the Colorado School of Mines (www.mines.edu/geomech) for their ongoing contributions to the group’s research efforts.

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remote seismicity in Creighton Mine, Sudbury, Canada. International Journal of Rock Mechanics and Mining Sciences. 2013; 58:166-79.

Valley B, Kim BH, Suorineni FT, Bahrani N, Bewick RP and Kaiser PK. Influence of confinement dependent failure processes on rock mass strength at depth. Harmonising Rock Engineering and the Environment. 2011: 376.

Walton G. Improving continuum models for excavations in rock masses under high stress through an enhanced understanding of post-yield dilatancy. Ph.D. Thesis. Queen’s University, Kingston, Canada. 2014.

Walton G, Diederichs M and Punkkinen A. The influence of constitutive model selection on predicted stresses and yield in deep mine pillars–A case study at the Creighton mine, Sudbury, Canada. Geomechanics and Tunnelling. 2015; 8(5):441-9.

Walton G, Diederichs M, Punkkinen A and Whitmore J. Back analysis of a pillar monitoring experiment at 2.4 km depth in the Sudbury Basin, Canada. International Journal of Rock Mechanics and Mining Sciences. 2016; 85:33-51.

Walton G. Initial guidelines for the selection of input parameters for cohesion-weakening-friction-strengthening (CWFS) analysis of excavations in brittle rock. Tunnelling and Underground Space Technology. 2019; 84:189-200.

Wang X and Cai M. Modeling of brittle rock failure considering inter-and intra-grain contact failures. Computers and Geotechnics. 2018; 101:224-44.

Wang X and Cai M. A comprehensive parametric study of grain-based models for rock failure process simulation. International Journal of Rock Mechanics and Mining Sciences. 2019; 115:60-76.

Wang X, Wu Y, Li X and Liang S. Numerical investigation into evolution of crack and stress in residual coal pillars under the influence of longwall mining of the adjacent underlying coal seam. Shock and Vibration. 2019.

Woo KS, Eberhardt E, Elmo D and Stead D. Empirical investigation and characterization of surface subsidence related to block cave mining. International Journal of Rock Mechanics and Mining Sciences. 2013; 61:31-42.

Wu W, Bai JB, Wang XY, Yan S and Wu SX. Numerical Study of Failure Mechanisms and Control Techniques for a Gob-Side Yield Pillar in the Sijiazhuang Coal Mine, China. Rock Mechanics and Rock Engineering. 2019; 52(4):1231-45.

Sinha S, Shirole D and Walton G. Investigation of the micromechanical damage process in a granitic rock using an inelastic Bonded Block Model (BBM). Journal of Geophysical Research: Solid Earth. 2020 (under review).


Walton Paper Final Draft (1) · 2020. 3. 9. · $'9$1&(6 ,1 %21'(' %/2&. 02'(/,1* :dowrq * &rorudgr...

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Transcript of Walton Paper Final Draft (1) · 2020. 3. 9. · $'9$1&(6 ,1 %21'(' %/2&. 02'(/,1* :dowrq * &rorudgr...