shear rupture of massive brittle rock under constant ... · conditions the associated normal-shear...

_____________________________________________________________________________________________ Thesis Summary – R. P. Bewick 1

Thesis Summary

October 2015

Shear Rupture of Massive Brittle Rock under Constant Normal Stress and Stiffness Boundary Conditions

Robert Paul Bewick, Ph.D., P.Eng.

Supervisors:

Peter K Kaiser (recently retired from Laurentian University)

William F Bawden (recently retired from University of Toronto)

Abstract

Shear or fault ruptures occur in deep mines and in the Earth’s brittle crust. Shear rupture as a brittle rock mass failure process is the subject of many investigations in rock physics related to underground mining, rock mass failure, and earthquake generation. Typically, shear rupture processes are investigated under constant stress boundary conditions which may not be a representative boundary condition depending on the proximity of a failure process to a stress-free surface or a deformable boundary.

The goal of the research for this thesis was to improve the understanding of shear rupture zone creation in intact low porosity massive brittle rock masses when deformed under constant normal stiffness boundary conditions. This boundary condition occurs when there is little or no influence of a deforming surface on a rock failure process such as in mine abutments, pillar cores and strike-slip oriented shear ruptures away from the Earth’s ground surface. For this purpose, a particle-based Distinct Element Method (DEM) with a grain-based model was used to generate and calibrate a synthetic intact brittle rock with a polygonal grain structure, i.e., a rock that can fracture by failure along both grain boundaries and through mineral grains.

The calibrated synthetic rock was first used to investigate shear rupture zone creation under constant normal stress boundary conditions and then under constant normal stiffness boundary conditions for a selected range of normal stiffness magnitudes representing various potential in situ scenarios. These simulations provided insight into the shear rupture zone creation process and the resulting shear rupture characteristics under both boundary conditions. It was demonstrated that these characteristics (i.e., the fracturing process, the ultimate rupture zone geometry, the load-displacement response, and the shear rupture zone’s peak and ultimate strengths) are not only a function of the rock or rock mass properties, as would be expected, but in a pre-dominant manner, on the boundary conditions under which the rupture zone was initiated, propagated and eventually coalesced to form a continuous zone of fractured rock.

The understanding gained from these simulations was then applied to interpret the shear rupture process and resulting rock mass response in two underground mine pillar scenarios. It was found that while failure occurred in both cases by shear rupture zone creation, the seismic response and the resulting rock mass response could be attributed to differences in stress versus stiffness boundary conditions. The value and practical application of the knowledge gained from the models of synthetic rock was essential in gaining an understanding of the in situ shear rupture process at these mines. The two pillar rupture cases provided field evidence in support of the hypothesis that boundary conditions affect and often dominate the characteristics of shear rupture zones and thus the failure processes in otherwise comparable ground conditions. The findings of the work have direct impact on the interpretation of rock mass response in highly stressed underground mines and the creation and rupture of faults in the Earth’s brittle crust.


1.0 Introduction

Shear rupture occurs in massive rocks under confined rock mass conditions when uncontrolled crack propagation is inhibited. The fracturing processes in brittle rocks leading to shear rupture are dilatant (Brace et al., 1966; Scholz, 1968; Peng and Johnson, 1972; Hallbauer et al., 1973) because of newly created fractures opening and pre-existing and newly created discontinuities shearing on or overriding asperities. When a shear rupture zone is being created and is surrounded by rock that resists dilatant deformation, confining (i.e., normal) stresses will increase during shear rupture zone creation and, to a lesser extent, during subsequent shearing along the newly formed shear rupture surface. The related increases in confining stress and associated stress-path during fracturing and shear differs from that under constant stress boundary conditions (e.g., Indraratna et al., 2005) and, in the extreme, can be represented by constant stiffness boundary conditions (Obert et al., 1976; Goodman, 1976; Johnston and Lam, 1989; Archambault et al., 1992; Indraratna et al., 1997; McKinnon and Garrido, 1998). An example of the stress-path difference under constant stress and stiffness boundary conditions normal to a shear rupture zone during its creation is illustrated schematically in Figure 1a. Also shown are the related schematic load-displacement curves (Fig. 1b), illustrating that the peak strength is typically reached when the shear stress reaches the yield (rupture or strength) envelope under constant stress conditions, with yield occurring long before the peak strength is reached under constant stiffness conditions. While it was understood that the stress path can reach and follow the failure envelope at different places and thus the resulting stress strain curves would differ, what was not understood was how the characteristics in terms of fracture structure and resulting seismic behaviour would differ.

Figure 1 (a) Schematic stress-paths under constant normal stress and normal stiffness boundary conditions: Ia = peak and yield points, Ib = yield point, II = maximum peak strength. (b) Schematic load-displacement curves for the stress-paths under constant normal stress and stiffness

boundary conditions also showing Ia, Ib, and II. From Bewick et al. (2014c).


Shear rupture processes are typically studied under constant stress boundary conditions (e.g., Lajtai, 1969; Petit, 1988; Lockner et al., 1991; Sonnenberg et al., 2003; Wong et al., 2005). In direct shear, under constant normal stress boundary conditions, as demonstrated in the thesis (Bewick 2013), the rupture mechanism, rupture zone geometry, and shear stress versus horizontal displacement response of an intact brittle specimen are dependent on the normal stress to uniaxial compressive strength ratio (σn/UCS) (Bewick et al., 2014a and b) as illustrated by Figure 2 and Figure 3:

At low ratios (σn/UCS <0.17) (Fig. 2a and Fig. 3), rupture is dominated by tensile splitting fracture modes; a process that occurs at or just after the yield point when the peak shear strength is reached. The load-displacement response is brittle with a large post-peak strength drop and the rupture zone is relatively thin and fairly planar; and

At higher ratios (σn/UCS 0.17 to <1.0), rupture progressively involves more shear. First, an array of en échelon fractures develop (consisting of either tensile or shear mechanism at time of creation depending on the σn/UCS ratio) followed by linkages of the fracture array across the specimen leading to a shear rupture surface with a relatively wide damage zone. The load-displacement response is strain-weakening at the lower limit of the range of σn/UCS (Fig. 2b and Fig. 3) to one with no to little post-peak strength drop at high σn/UCS (Fig. 2c and Fig. 3). The rupture zone is relatively wide, discontinuous, and irregular.

Figure 2 Rupture in direct shear under constant normal stress boundary conditions showing change in rupture zone geometry, mechanism and crack patterns, and load-displacement response with increasing applied normal stress in (a) to (c). DEM simulation rupture zone images (orange –

grain boundary, black – intra-grain tensile fractures) showing (a) change in rupture mode from tensile splitting, (b) shear rupture via tensile en échelon fracturing, and (c) shear rupture via shear en échelon fracture arrays. Also shown (i to iii) are particle velocity vectors to illustrate explicit

displacement patterns and mechanisms at the time of rupture. From Bewick et al. (2014c).


Previous investigations on the effect of constant normal stiffness boundary conditions in direct shear on intact specimens (Obert et al., 1976; Archambault et al., 1992) focused on the strength characteristics (strength envelope shape) of the materials tested. The previous investigations showed that under constant normal stiffness boundary conditions the associated normal-shear stress-path generally follows the strength envelope generated from tests under constant normal stress (as in the schematic stress-path for normal stiffness boundary conditions in Fig. 1a). Only a few investigations have been completed on intact brittle rocks deformed under non-constant stress boundary conditions. Hallbauer et al. (1973), using copper jacketed cylindrical specimens of quartzite deformed in triaxial compression, stopped tests at predetermined locations along the loading path and removed the specimens for sectioning and microscope observations. In these experiments, the lateral stress magnitudes were not kept constant during loading (a result of the copper jacket) and increased during deformation. They found that shear rupture in the specimens initiated and began to propagate pre-peak strength (Jaeger and Cook, 1976). Their test results provide some insight into brittle rock specimen rupture under non-constant stress boundary conditions; shear ruptures were generated before peak as opposed to post-peak strength as determined from constant stress boundary conditions.

The majority of the experiments used to investigate shear rupture of massive rock in the brittle field have been conducted using constant stress boundary conditions. This boundary condition may not prevail in nature (during earthquakes) or in mining (during rockbursts) when fracturing processes leading to shear rupture zone creation are constrained (i.e., away from free surfaces). A limited number of experiments using intact brittle rock have been completed on specimens under constant stiffness boundary conditions. Thus, the understanding of shear rupture zone creation under different boundary conditions is incomplete and formed the main target for the research reported in this thesis.


Figure 3 Division of rupture in the synthetic rock based on the σn/UCS ratio considering rupture mechanisms, rupture zone geometries, and idealized shear stress versus horizontal displacement responses. Fracture in the rupture zone images, orange grain boundary and black mineral

grain tensile fractures. Simulations were completed using the calibrated DEM and scaled to represent different strength categories of sandstone based on the classification proposed by Trollop in Deere (1968). From Bewick et al. (2014b).


2.0 Approach and Methodology

The creation of rupture zones in a calibrated numerically generated synthetic rock specimen under constant normal stress and stiffness boundary conditions was investigated using the commercially available particle based Distinct Element Method (DEM), Particle Flow Code in Two Dimensions (PFC2D v4.00-190) (Itasca, 2011) and its embedded Grain Based Method (GBM) (Potyondy, 2010; Itasca, 2011). The understanding gained from the numerical simulations was then applied to re-interpret two pillar case histories which were found to have failed by shear rupture zone creation (Coulson, 2009). In this way, the value and practical application of the gained understanding was shown. Coulson (2009) reported that the failure processes in, and the micro-seismic behaviour of the two pillar cases differed and it is demonstrated in the thesis (Bewick 2013) that this can be related to differing boundary conditions. These two cases provided field evidence in support of the thesis hypothesis that boundary conditions affect shear rupture zone characteristics and thus mining-induced failure processes in otherwise comparable ground conditions.

2.1 Simulation procedure

PFC2D models the movement and interaction of particles which are represented as rigid circular disks that can overlap at contacts. The particles abide by Newton’s laws of motion, and a force-displacement law is applied to each particle-particle contact. Boundary conditions (such as constant stress or velocity) are applied along walls which are rigid borders. Particles are not bonded to walls and shear can occur at particle-wall interfaces. PFC2D uses an explicit finite difference method where a calculation cycle is carried out using a time-stepping routine (Potyondy and Cundall, 2004).

The Grain Based Method (GBM) (Potyondy, 2010) in PFC2D v4.00-190 is used to generate a realistic synthetic rock specimen (Fig. 4a) with a grain structure that mimics deformable, breakable, polygonal grains cemented along their adjoining sides. Each grain (an analogue for a rock block) (Fig. 4b) is composed of a cemented circular particle (disk) assembly and the grains are cemented along their contacts allowing for both intra-grain and grain boundary breakage (fracturing). The grain boundaries are represented using smooth-joint contacts (Mas Ivars, 2010) (Fig. 4c-d). The cemented particle assemblies in the grains are represented using parallel bonds (Fig. 4e) (Potyondy and Cundall, 2004). Smooth-joint bond breakage is representative of grain boundary fracture, and parallel bond breakage of intra-grain fracture in the synthetic rock. Smooth-joints, schematically illustrated in Figure 4c remove the previous limitation in PFC2D where discontinuities or planar contacts were simulated as unrealistically rough and bumpy. Particles that are on adjacent sides of a smooth-joint can pass through each other during sliding, forcing the sliding path along the smooth-joint contact (Fig. 4c) opposed to riding over the particles along the sliding path (Fig. 4d). Parallel bonds, schematically illustrated in Figure 4e, can resist both a force and moment between individual particles. Once a smooth-joint or parallel bond breaks (in either shear or tension), the contact transitions to frictional behaviour depending on the assigned smooth-joint or particle (disk) residual coefficients of friction, respectively (see Bewick et al., 2014a Table 3 for the complete list of micro-parameters used in the PFC2D-GBM simulations).

The synthetic rock (Fig. 4a) was calibrated to the rupture characteristics of Lodève sandstone deformed in direct shear under constant normal stress boundary conditions reported by Petit (1988) and Wibberley et al. (2000) considering the following (as outlined in detail in Bewick et al., 2014a), which were successfully calibrated to:

peak shear strength envelope for normal stresses from 5 to 90 MPa; tensile strength; post-peak shear stress versus horizontal displacement response; rupture zone geometry change with increasing applied normal stress; and fracture angles (both tensile and shear) generated during rupture.

The synthetic rock generation and calibration methodology are described in Bewick et al. (2014a) and sensitivity to grain boundary and intra-grain strength parameters is discussed in Bewick et al. (2014b). The sandstone is a fine to medium grain brittle low porosity (<2%) rock consisting of feldspar, quartz, and calcite (cementation). The synthetic rock (Fig. 4a) used for direct shear testing is a simplified representation of Lodève sandstone and is 50mm x 50mm (length to height; aspect ratio 1:1). Bewick et al. (2014a) explored the influence of synthetic rock length to height ratio and found that the 1:1 aspect ratio was suitable for shear rupture investigation


purposes. The synthetic rock is composed of 41,388 particles and approximately 1405 mineral grains (composition of 50% feldspar, 30% calcite, and 20% quartz) with an overall average grain size of 1.4mm.

Figure 4 Synthetic rock specimen: (a) (white – calcite, light grey – feldspar, darkest grey – quartz) and elements forming a grain; (b) example grains showing grain boundaries, broken smooth-joint contacts along the grain boundaries, the internal parallel bonded particle assemblies (not

showing the parallel bonds for clarity), and broken parallel bonds in the grains; (c) schematic representation of the behaviour of a contact along a smooth-joint; (d) schematic representation of the behaviour of a contact without a smooth-joint; (e) schemaitc representation of a parallel bond.

From Bewick et al. (2014c).

The numerical set up used (Fig. 5; discussed in the following section) imposes the width of the shear zone. The purpose of the research was to investigate the influence of a constant normal stiffness boundary condition on some shear rupture zone characteristics (e.g., creation process, geometry, shear stress – versus horizontal displacement). Therefore, the width of the shear zone imposed by the numerical simulation was not considered as a variable. Taboada et al. (2005) have shown that the width of a granular shear zone is up to 10 disk diameters. As can be seen in Figure 5, there are approximately 3 to 4 mineral grains across the 5 mm shear box gap. While only 3 to 4 mineral grains are evident in Figure 5, there are more than 21 particles (disks) across the shear box gap. Each grain is composed of a large number of individual parallel bonded particles (as outlined previously and in Fig. 4).

By numerically reproducing, in direct shear, the strength, deformation, fracturing characteristics, and rupture zone geometry in a brittle rock, and simulating fracture nucleation, propagation, and evolution in a grain structure that allows both grain boundary (analogous to joint) and intra-grain (analogous to rock block) fracturing, insight can be gained into the fracturing process leading to shear rupture and its relationship to the shear stress versus horizontal displacement response and strength under different boundary conditions. The numerical simulations allow one to track the internal state-of-stress, fracturing (location, orientation, mechanism, and type – e.g., grain boundary and intra-grain), and particle displacement and velocity vectors showing the explicit mechanism of rupture for fracture systems. A fracture system is a larger fracture created by the coalescence of individual fractures (see Fig. 21 in Bewick et al., 2014a for examples of fracture system development).


Figure 5 (a) Constant normal stress boundary condition direct shear simulation schematic (b) Constant normal stiffness boundary condition direct shear simulation schematic. Both (a) and (b) showing central measurment circle internal to the syntehtic specimen. The syntheic specimen shows the polygonal grain assembly (white – calcite, light grey – feldspar, darkest grey – quartz). Not showing the individual particles in each grain or the ‘smooth-joint’ contats along the grain boundaries. For more details on the syntheitc specimen and specific terminology refer to Bewick et al.

(2014a and b) Sections 3.0 and 4.0.

2.2 Simulation procedure

Thenumericaldirect shear setup (Fig.5a) is a simplificationof the laboratorysetupusedbyPetit(1988). The shear box with constant normal stress boundary conditions created in PFC2D (shown in Fig. 5a) is composed of a 50 x50 mm synthetic rock bound by an upper portion of the shear box which has two separate fixed lateral walls (Walls 3 and 4, Fig. 5a) and a lower portion simulated as a single wall, of ‘U’ shape (Wall 1) which moves in the horizontal direction (Fig. 5a). The contacts between the walls and the synthetic rock are frictionless. A 5 mm gap is introduced to match the shear box gap reported by Petit (1988). Constant normal (vertical) stress is first applied to the synthetic rock through applied constant velocity to the top wall of the shear box (Wall 2, Fig. 5a). Six normal stress magnitudes were investigated (5, 15, 25, 40, 60, 90 MPa). When the desired constant normal stress magnitude is achieved throughout the synthetic rock, the velocity of the top wall is stopped. Shear displacement is then applied to the synthetic rock through a constant velocity movement of the lower wall (Wall 1, Fig. 5a) with rotation of the shear box restricted and a constant normal stress maintained at the top wall (Wall 2). Diederichs (1999) found that constant velocity loading influenced both the peak strength and post-peak load–displacement response of synthetic rocks in PFC2D. Therefore, the constant shear velocity of the lower wall was chosen to ensure that both the peak shear strength and post-peak shear stress versus horizontal displacement response were not influenced by slower constant shear velocities. A constant shear velocity of 0.04 m/s was selected which relates to a displacement of the lower wall of approximately 2.75e‐7 mm per time step. This velocity resulted in quasistatic loading conditions.

The constant normal stiffness simulation set up as shown in Figure 5b (Bewick et al., 2014c) is based on the constant normal stress boundary condition simulations (Fig. 5a) reported by Bewick et al. (2014a and b). The synthetic rock is created in the same manner but the constant normal stiffness simulation differs from the constant normal stress simulation as follows (Fig. 5b):

(1) A cap of material is located on top of the synthetic rock and is composed of a parallel bonded particle assembly. This cap is not breakable. Its deformability is controlled by the assigned modulus (a proxy for stiffness) of the particles and the bonds between them, and the cap’s geometry (50mm length and 40mm height). The cap is not bonded to the synthetic rock and the contact between the synthetic rock and cap is frictionless. The following three cap modulus values were investigated: 10GPa, 30GPa, and 100GPa (i.e., confining modulus magnitudes).


(2) Normal stress is applied to the synthetic rock using an applied velocity to the top wall of the cap (Wall 2, Fig. 5b). Initial applied normal stress magnitudes of 5MPa, 25MPa, and 40MPa for each assigned cap modulus (confining modulus) are simulated. Once the desired normal stress is achieved (defined as the ‘initial applied normal stress’) in the synthetic rock of the constant normal stiffness simulations, the applied velocity is stopped and the top wall locked followed by shear displacement of the lower wall (Wall 1, Fig. 5b) at a constant velocity. A constant shear velocity of 0.04m/s was selected resulting in quasistatic loading conditions. Due to the locked top wall (Wall 2, Fig. 5b) in the constant normal stiffness simulations normal stress develops during shearing as a function of the confining modulus. As elastic vertical deformation, dilatant fracturing, or shear induced dilation occurs in the synthetic rock, the movement is resisted by the confining modulus providing feedback normal stress. The confining modulus, therefore, simulates the influence of the deformability (stiffness) of a surrounding material, such as a rock or a rock mass surrounding a shear rupture zone. The confining modulus values represent equivalent spring stiffnesses; they are related to but do not represent actual values of rock mass modulus.

The mean shear stress is determined by dividing the reaction forces acting along Wall 4 by the synthetic rock length. The normal stress is determined by the reaction forces acting along Wall 2 divided by the synthetic rock length. The horizontal displacement is recorded as the movement of the lower wall (Wall 1). Principal stress magnitudes and the orientation of the major principal stress internal to the synthetic rock are determined using the measurement circle (10mm diameter) shown in Figure 5 in the center of the synthetic rock. The logic adopted by Cho et al. (2008) where each stress component (σxx, σyy, σxy) is monitored for every particle within the measurement circle was adopted.

3.0 Summary of simulation results

In the following sub-sections, the results for the constant normal stiffness simulations are summarized and later in Section 4.0 compared to the findings of the constant normal stress simulations summarized in Section 1.0 and in Bewick et al. (2014 a and b).

3.1 Normal-shear stress-path and shear strength envelopes – constant normal stiffness

The normal-shear stress-path is coupled (Fig. 6 showing the results for initial applied normal stress magnitudes of 5 and 40MPa, respectively) and depends on the confining modulus with higher normal stresses developing more rapidly for higher confining modulus values. Each suite of simulations for the different initial applied normal stresses (i.e., 5, 25, and 40MPa) produces a unique strength envelope with the coefficient of friction increasing and the cohesion (MPa) decreasing for increasing applied normal stress.

For an initial applied normal stress of 5MPa (Fig. 6a):

τ = 0.91σn+30, R2 0.99 (1)

For an initial applied normal stress of 25MPa (not graphically presented):

τ = 0.95σn+23, R2 0.99 (2)

For an initial applied normal stress of 40MPa (Fig. 6b):

τ = 0.96σn+16, R2 0.99 (3)

Once the strength envelope is reached in each suite of results for the different initial applied normal stresses, the stress-path follows a linear Coulomb strength envelope. The envelope is followed due to dilation acting against the cap of material causing an increase in normal stress during applied shear displacement. This in turn leads to an increase in shear resistance proportional to the normal stress. Eventually, the stress-path deviates, i.e., drops below the linear envelope when the maximum peak shear strength is reached, and eventually approaches the residual strength defined by a friction angle of approximately 28° (coefficient of friction, μ=0.52) (Fig. 6).

The peak shear strength for a given normal stress cannot be predicted by the linear Coulomb strength envelope under constant normal stiffness boundary conditions because the stress-path is dependent on the normal


stiffness (confining modulus) as illustrated by Figure 6. Under this boundary condition, the linear Coulomb strength envelope defines the initial yield point. Interestingly, as illustrated by Figure 7, it was found that both the yield point and failure point (maximum peak strength) can be represented by constant horizontal displacement criteria of approximately 0.17 and 0.35mm, respectively. This suggests that a strain or displacement based failure criterion is more appropriate for the description of rupture zone behaviour under constant normal stiffness boundary conditions.

Figure 6 Normal-shear stress-paths (for the indicated confining modulus values), and peak and residual linear Coulomb strength envelopes: (a) initial applied normal stress of 5MPa; and (b) initial applied normal stress of 40MPa.

Figure 7 Constant horizontal displacement criteria for yield and maximum peak shear strength. Normal stress is the magnitude at the point when yield and maximum peak strength are reached.


3.2 Shear stress versus applied horizontal displacement response – constant normal stiffness

Under constant normal stiffness (Fig. 8a, initial applied normal stress of 5MPa, and Fig. 8b, initial applied normal stress of 40MPa), the shear stress versus applied horizontal displacement response of the synthetic rock is initially elastic (up to approximately 0.17mm to 0.18mm of horizontal displacement, i.e., the yield point) and then becomes inelastic (suggestive of strain hardening or more appropriately strain strengthening). Shear stress oscillations occur in the curves prior to reaching the maximum peak shear strength when the confining modulus is <100GPa (Fig. 8a, b). These oscillations are related to internal fracture creation (as described in Sections 3.3 and 3.5) and are not a result of frictional stick-slip mechanisms. The post-peak load-displacement response is brittle with a ‘stepped’ or ‘staircase’ character and with larger stress drops compared to those occurring before peak strength.

Figure 8 Shear stress versus applied horizontal displacement response for (a) 5MPa and (b) 40MPa initial applied normal stresses (for confining moduli indicated) showing shear stress oscillations before peak and large brittle stress drops post-peak.

3.3 Shear rupture zone structure – constant normal stiffness

The fracturing processes leading to the shear rupture zone creation occur consistently in four stages (I to IV) that are independent of the initial applied normal stress. PFC2D-GBM rupture zone images were captured at selected applied horizontal displacements and are used to describe the fracturing process leading to the shear rupture zone creation for the 5MPa (Fig. 9) and 40MPa (Fig. 10) initial applied normal stress, respectively. The 25MPa initial applied normal stress rupture zones are created in the same manner and are thus not presented in detail. The stages leading to rupture zone creation are as follows:

Stage I is characterized by the occurrence of grain boundary tensile fractures (orange fractures in Fig. 9a, g, m and Fig. 10a, g, m) which are oriented in the direction of the internal major principal stress (Table 1, summary of fracture system angles at Stage I for a confining modulus of 30GPa and initial applied normal stresses of 5, 25, and 40MPa).

Stage II is characterized by the development of an array of en échelon tensile fracture systems (composed of both grain boundary and intra-grain tensile fractures, orange and black fractures respectively, Fig. 9b, h, n; Fig. 10b, h, n). Figure 11b shows the particle relative displacement vectors indicating predominant opening modes along the en échelon fracture systems at the time of creation. The tensile en échelon fracture systems induce


changing kinematics as stresses rotate in the synthetic rock (i.e., tensile opening transitions to shear) and begin to grow fractures from their tips (Fig. 9c, i, o and Fig. 10c, i, o, Stage IIa). The tip fractures are of opening mode (Fig. 11d-e). From Stage II to Stage III, the en échelon array becomes progressively more connected (Fig. 9d, j, p and Fig. 10d, j, p) with increasing participation of grain cracking.

Stage III is characterized by the maximum peak shear strength and the emergence of a connected, localized but irregularly shaped shear rupture (Fig. 9e, k, q and Fig. 10e, k, q) with both the top and bottom of the rupture surface indicating opposite sense of shear (Fig. 11c). From Stage III to IV, the non-continuous rupture zone evolves and becomes less irregular and progresses into an almost fully continuous rupture surface across the synthetic rock dominated by intra-granular tensile fractures (black fractures in Fig. 9f, l, r and Fig. 10f, l, r).

Stage IV is characterized by a gouge or cataclastic damage creation process during the formation of a continuous rupture surface across the synthetic rock with fracture systems having increasing angles at increasing confining modulus values (see Fig. 9f, l, r and Fig. 10f, l, r labeled increasing fracture angles).

In summary, the described process of rupture zone creation for constant normal stiffness boundary conditions is significantly different from the process of rupture zone creation for constant normal stress boundary conditions where the rupture zone creation process is dependent on the applied normal stress to UCS ratio as summarized in Section 1.0.

Table 1 Pre-peak (Stage I) fracture angles for a confining modulus of 30GPa.

Initial applied normal stress

(MPa)

Stage I fracture orientation (°)

(average/standard deviation)

Stage I σ1 orientation just

prior to yield point (°)

5 26/12 26

25 32/18 30

40 32/36 33


Figure 9 Images of shear rupture zone creation process with initial applied normal stress of 5MPa, at selected applied horizontal displacement magnitudes (δh) for confining (cap) modulus values of: (a)-(f) 10GPa; (g)-(l) 30GPa; and (m)-(r) 100GPa (orange – grain boundary (GB)

fractures; black – intra-grain tensile fractures).


Figure 10 Rupture zone creation images with initial applied normal stress of 40MPa, at selected applied horizontal displacement magnitudes (δh) for confining (cap) modulus values of: (a)-(f) 10GPa; (g)-(l) 30GPa; and (m)-(r) 100GPa (orange – grain boundary fractures; black – intra-grain

tensile fractures).


Figure 11 Rupture zone images for initial applied normal stress of 5MPa and 30 GPa confining modulus: (a) Stage II en échelon tension fracture systems showing opening mode in (b); (c) Stage IIa fractures propagating from en échelon fracture system tips showing shear along original en

échelon tension fracture systems in (d) and tip fracture opening in (e); (f) Stage III, maximum peak shear strength showing shear along the rupture zone in (g) and (h). (orange - grain boundary fractures; black - intra-grain tensile fractures). Applied horizontal displacement (δh) the rupture zone images are from is indicated. Vector fields show relative displacements from the related rupture zone image's applied horizontal

displacement capture point compared to an earlier 3e-5 mm of applied horizontal displacement.


3.4 Rupture zone evolution for various practically feasible stress-paths

The rupture zone creation stages outlined in Section 3.3 can be related to the shear stress versus applied horizontal displacement curves, the normal-shear stress-paths (tracked along the external boundaries of the synthetic rock described in Section 2.2), and the major-minor principal stress-paths tracked internal to the synthetic rock (using the measurement circle described in Section 2.2). The following interpretation is given with reference to Figure 12 and Figure 13 for initial applied normal stress magnitudes of 5 and 40MPa, respectively (other stress states are described in the thesis). These two sets of figures show shear stress versus applied horizontal displacement curves (Fig. 12a and Fig. 13a) for 10, 30 and 100 GPa confining modulus, normal-shear stress-paths (Fig. 12b and Fig. 13b), the development of minor principal stress internal to the synthetic rock versus the applied horizontal displacement (Fig. 12c and Fig. 13c), and the internal stress-paths in the major versus minor principal stress space (Fig. 12d and Fig. 13d). Three distinct deformation stages can be identified:

Stage I: grain boundary tensile fracturing (Fig. 9a, g, m and Fig. 10a, g, m) is characterized by a linear shear stress versus applied horizontal displacement response (Fig. 12a and Fig. 13a) up to an applied horizontal displacement of approximately 0.17mm to 0.18mm when the yield surface is reached (first point of yield but not peak strength for constant normal stiffness conditions).

Stage II: development of an array of en échelon tensile fracture systems (Fig. 9b, h, n and Fig. 10b, h, n) with characteristic non-linear shear stress versus applied horizontal displacement response (Fig. 12a and Fig. 13a). This occurs when the internal tensile strength of the synthetic rock is reached and exceeded (Fig. 12d and Fig. 13d showing the principal stress-path internal to the synthetic rock). Peak tensile stress development relative to the applied horizontal displacement is also shown in Figure 12c and Figure 13c. The point when the internal tensile strength of the synthetic rock is reached (at the “first” yield point) is at or near the linear Coulomb strength envelope in normal-shear stress space (Fig. 12b and Fig. 13b). The horizontal arrows indicating the shear stress magnitude when the linear Coulomb strength envelope is reached (corresponding to the “first” yield points) are shown in Figure 12a and Figure 13a on the shear stress versus applied horizontal displacement curves. This stage is reached long before the rupture zones peak strength.

From Stage II to Stage III (0.17-0.18mm to 0.35mm of applied horizontal displacement): the normal-shear stress-path follows the linear Coulomb strength envelope (Fig. 12b and Fig. 13b), with the en échelon array of fracture systems becoming progressively more connected. During this period, a rupture zone is created before the maximum peak shear strength (Fig. 9 and Fig. 10) is reached. During this continuous inelastic deformation process, while the shear stress generally increases, a distinct shear stress oscillatory behaviour is encountered (particularly visible in Fig. 12a and Fig. 13a for the 10 and 30GPa confining modulus values). Each oscillation is related to a drop below the linear strength envelope (Fig. 12b and 13b), indicating a sudden strength loss that is recovered during further displacement. This oscillation is not a stick-slip (frictional) process but a tensile fracture process that temporarily leads to a stress but not a strength drop. With increasing displacement, the rupture zone locks up until a next rock bridge fails (mostly by internal extension failure; see Section 3.5).

Stage III (Fig. 9e, k, q and Fig. 10e, k, q): starts with the characteristic conditions at the maximum peak shear strength and the emergence of a more connected but irregular rupture zone (Fig. 12a and Fig. 13a; and circles in Fig 12b and Fig. 13d). Past the peak, as displacements increase in Stage III to IV (Fig. 9f, l, r and Fig. 10f, l, r), the non-continuous rupture zone evolves and becomes less irregular and discontinuous with an almost fully continuous rupture surface being created with a strength state changing from peak with a cohesion (c) and friction (ϕ) to residual with predominate frictional strength (28° in this case; Fig. 12b and Fig. 13b).


Figure 12 Linked mechanical response of the synthetic rock with initial applied normal stress of 5 MPa: (a) shear stress versus applied horizontal displacement response; (b) normal-shear stress-path; (c) development of minor principal stress with applied horizontal displacement; and (d)

principal stress-path internal to the synthetic rock.


Figure 13 Linked mechanical response of the synthetic rock with initial applied normal stress of 40 MPa: (a) shear stress versus applied horizontal displacement response; (b) normal-shear stress-path; (c) development of minor principal stress with applied horizontal displacement;

and (d) principal stress-path internal to the synthetic rock.


3.5 Shear stress oscillatory during rupture evolution

A close up view of the shear stress versus applied horizontal displacement curve for one example, the simulation with a confining modulus of 30GPa and an initial applied normal stress of 25MPa, is shown in Figure 14a alongside the PFC2D-GBM rupture images (orange – grain boundary, black – intra-grain tensile fracture) at selected applied horizontal displacements (δh) (Fig. 14c-f; the locations of the images are referenced in Fig. 14a by arrows). The grain boundary and intra-grain fracturing history in this range of applied horizontal displacement are presented in Figure 14b as fracture rates and cumulative fracture counts for grain boundary and intra-grain fractures as well as for the combined, total fracture count.

The shear stress versus applied horizontal displacement curve (Fig. 14a) shows three distinct shear stress oscillations with nearly instantaneous shear stress drops as indicated in Figure 14a (I, II, and III). The fracturing events in these three zones show increasing fracture rates from the shear stress peaks of the oscillation to the subsequent low in the shear stress oscillation with both grain boundary and intra-grain fracturing occurring simultaneously (Fig. 14b). The cumulative fracture count curves show steps at each oscillation. Locations of some of the newly created fractures are indicated by the grey circled areas of the rupture zone images (Fig. 14c-f) and indicate (along with the stepped cumulative fracture count curves) a progressive evolution of a discontinuous rupture zone or network towards a more connected and eventually continuous state with fracturing occurring in less fractured areas. Much of the fracturing occurs in the damage zone outside the shear rupture. Hence, the oscillation is clearly related to tensile fracturing and a related loss of cohesion (rather than a stick-slip frictional shear process). Little fracturing occurs during the shear stress ‘re-charging’ phases (increasing in shear stress magnitudes during the oscillations). In some instances there are periods without fracturing during this strengthening as indicated by the term ‘quiescence’ in Figure 14b. Quiescence is used here to describe a period of applied horizontal displacement without fracturing. From a practical and modeling perspective, these observations are highly relevant. Shear rupture processes and related development seismicity do not occur on a planar structure but within a zone of limited extent. The source mechanism is thus not one of shear slip on a plane but shear deformation facilitated by extension cracking in the shear rupture zone.


Figure 14 Characteristics of shear stress oscillations in the shear stress (τ) versus applied horizontal displacement (δh) curve for 25MPa initial applied normal stress and 30 GPa confining modulus: (a) shear stress versus applied horizontal displacement curve; (b) fracture rates and

cumulative fracture counts for grain boundary (GB) and intra-grain (IG) fractures; (c)–(f) rupture zone images at indicated applied horizontal displacements as indicated using ‘arrows’ in (a) (orange - grain boundary, black - intra-grain tensile fracture).


4.0 Discussion of shear rupture simulations

4.1 Rupture mechanics under constant normal stiffness compared to constant normal stress

The mechanical response and fracturing of massive brittle rock deformed in direct shear under the boundary condition of constant normal stiffness (Bewick et al., 2014c) is different from that of constant normal stress (Bewick et al. 2014a and b):

For constant normal stiffness, the stress-path reaches and then follows the strength envelope and is dilation-controlled, increasing normal stress and eventually, after reaching the maximum peak strength (which occurs for a given rock at a constant applied horizontal displacement), approaches the residual frictional strength. This is opposed to reaching the strength envelope and then immediately falling to the residual or ultimate strength under constant normal stress boundary conditions;

The load-displacement response is typically brittle for constant stiffness opposed to changing from brittle to ductile under constant normal stress boundary conditions at increasing constant normal stress magnitudes; and

For constant stiffness, the rupture zone is created before the maximum peak strength is reached, opposed to post-peak strength under constant normal stress boundary conditions.

The pre-peak strength rupture zone creation in the constant normal stiffness simulations is generated as a result of the dilatant fracturing and shearing interacting with the cap material of assigned modulus. As dilation occurs in the synthetic rock, the dilatant opening of fractures is resisted by the constraining material around the rupture increasing the normal stress. Thus, an increase in shear stress is facilitated by the dilatational resistance evolution, i.e., an increase in normal stress results in an increase in shear strength as per the linear Coulomb strength criterion:

τ = c+ σntanϕ (4)

In other words, the cohesion intercept is maintained during yield until the peak resistance at a unique displacement is reached. The post-peak rupture zone creation under constant normal stress occurs because dilatant fracturing in the synthetic rock does not generate increases in normal stress and therefore, no increases in shear strength during rupture zone creation. Thus, shear strength is typically lost after peak during rupture zone creation.

In summary, these results highlight the importance of boundary conditions on various shear rupture characteristics. The characteristics of a shear rupture zone are not only a function of the rock or rock mass properties but the boundary conditions under which the rupture zone is created.

4.2 Rupture connectivity, smoothing, and “apparent stick-slip” behaviour

According to Byerlee (1970), stick-slip behaviour of a surface in brittle rock with a sliding contact (under dry and room temperature conditions) results from the abrupt brittle fracture of locked asperities. Asperity influence on stick-slip behaviour is also evident from the descriptions of rupture surfaces undergoing stick-slip behaviour in Brace and Byerlee (1966) which generated a thin powder of material and thus must have involved fracture of asperities. According to them, the force needed to overcome and fracture the asperities holding the system in a stable state is the shear force. If sliding occurs in this way along a brittle surface then the shear force will increase when the surfaces become locked and decrease when fractured (Byerlee, 1970). These initial findings of asperity control on stick-slip initiation have been corroborated by the more recent works of Lei et al. (2003) and Thompson et al. (2009) where it was found that stick-slip events in a fractured specimen loaded in triaxial compression initiated from geometric heterogeneities (asperities) along the rupture surfaces. In all cases, stick-slip instability was preceded by fracture of intact material in locations of geometric heterogeneity with the stick-slip event occurring because the loading system was not capable of responding fast enough to the rapid fracturing and resulting displacements. In the case of Thompson et al. (2009), the initiation points for the stick-slip were in the ‘locked’ regions along the rupture surface with minimal micro-fracture damage. While stick-slip behaviour related to energy release depends on the loading system stiffness (i.e., overall system response, Scholz, 2002) the initiation of the stick-slip instability based on the above described processes occurs due to a cohesion loss on the slip surface.


The results presented in this thesis show that the stress oscillation is not primarily caused by a cohesion loss processes on the rupture surface but by damage accumulation with related sudden shear resistance (apparent cohesion loss) in the surrounding damage zone. The shear oscillatory character of the shear stress versus applied horizontal displacement curves under constant normal stiffness boundary conditions (generally when the confining modulus is <100GPa) described in Sections 3.2, 3.4, and 3.5 is the result of local fracturing along an evolving, discontinuous rupture zone and a halo surrounding the eventual rupture surface. The fracturing events appear to be similar to those prior to stick-slip instability, e.g., as in the cases of Byerlee (1970) and Thompson et al. (2009). The simulation results show that the oscillatory shear stress drops under constant normal stiffness boundary conditions are related to the rupture-smoothing and fault zone damage process. The results are in general agreement with findings from other studies (e.g., King, 1986; King and Nabelek, 1985) where discontinuous rupture zones are recognized as a factor for controlling the initiation and termination of seismic events which highlights the “cohesional” nature of the rupture process. A sudden cohesion loss, through extension fracturing, is thus the initiator of stick-slip behaviour.

A practical implication of the extension-failure-driven cohesion loss process and associated shear stress oscillations is that the strain-strengthening effect outlined previously (Section 3.2 and 3.4) makes rupture zones stronger with increasing applied shear displacement (even though energy would be released at each pre-maximum peak strength stress oscillation). Therefore, rupture zones have the potential to store more energy which could be released as higher magnitude seismic events once the maximum peak strength is reached in a system with sufficient loading system stiffness. This interpretation is relevant for stick-slip behaviour because cohesion loss is not gradual but a sudden process leading to the shear stress oscillations each time a ‘locked’ area is starting to fail. Normal stiffness, therefore, allows rupture zones to ‘lock’ creating the potential for energy release during rupture zone propagation and larger energy releases may occur as the maximum peak strength is approached and the largest stress drops occur.

5.0 Application to Mining – Shear Rupture Interpretation

Two pillar case histories were analyzed and re-interpreted at a mining camp (Fig. 15). Each pillar underwent a failure process resulting in shear rupture. In the first case, at the Golden Giant mine (Fig. 16 and 18), the rupture zone was created in the pillar over a period of a few months and did not generate any sizable seismic events Mn ≥ 0 (Nuttli magnitude). In the second case, at the Williams mine (Fig. 17 and 19), the rupture zone was created in the pillar over the period of a few years and contributed to or was the direct cause of a number of seismic events with magnitudes exceeding Mn = 2.7. Space does not allow for the presentation of the details for this investigation but the field evidence clearly supported the interpretation of shear rupture evolution within the context of boundary condition control on shear rupture zone creation and seismic behaviour.

In general, each case was re-evaluated using a number of datasets and methods as summarized in Bewick (2013) as follows: micro-seismic data, Principal Component Analyses (PCA) (Urbancic et al., 1993; Saccorotti et al., 2002; used to determine the orientation of planes in micro-seismic data clouds), assessed stress-paths from three dimensional elastic numerical stress models, and where available, extensometer data and pillar geometry changes. A new directional Loading System Stiffness (LSS) methodology was developed and used to assess stiffness changes normal to and in the direction of the dip line of the rupture zones using a three dimension numerical stress modelling tool. Each case was analyzed and re-interpreted assuming constant stress and stiffness boundary conditions. The field monitoring data was used to test the boundary condition assumptions. It was found that the first pillar case ruptured under a boundary condition that essentially was one of constant stress while the second pillar case ruptured under a boundary condition changing from stiffness to stress control. These case histories highlighted the importance of properly testing the assumptions used to interpret field monitoring data and provided evidence that other boundary conditions exist in mines. Based on the knowledge gained from the numerical simulations it was possible to better interpret the observed rock mass response to shear rupture zone creation under different boundary conditions.


Figure 15 Longitudinal view of the Hemlo mining camp showing Williams, Golden Giant, and David Bell mines and inset geographical location of the mining camp in Ontario, Canada. (A) Golden Giant shaft pillar region where pillar Case 1 is located. (B) Williams sill pillar region where

pillar Case 2 is located (modified from Coulson, 2009).


Figure 16 Golden Giant mine shaft and Case 1 pillar region showing mining to the end of 2003. (a) Longsection view looking north showing mine development and stoping around the shaft and Case 1 pillar. (b) View looking west showing proximity of shaft to de-stress slot and ore

body.


Figure 17 Williams mine longsection showing mining to the end of 1999, separate mining Blocks 3 and 4 separated by a sill pillar, and mining directions. The Case 2 pillar is located in the sill pillar between Easting 9412E and 9462E and levels 9390L and 9415L.

Figure 18 Case 1 pillar rupture zone initiation and propagation. (a-b) Micro-seismic source locations for 2002 and 2003, respectively. (c-f) Contour of micro-seismic density (5 events per 125m3) showing progression of rupture plane east to west from 2002-02 to 2003-03 (modified

from Coulson, 2009).


Figure 19 Micro-seismic event source locations for the Case 2 pillar showing cross section (9430E ±12.5m) and the creation of a shear rupture zone. Dates indicated (modified from Coulson, 2009).

6.0 Conclusions

Constant normal stress and stiffness direct shear DEM simulations were conducted using a calibrated synthetic rock model with a polygonal grain structure that could fracture along both grain boundaries and in grains. The constant normal stiffness results were compared to those conducted under constant normal stress boundary conditions and rupture zone creation theories and provide insight into possible explanations of rock behaviour. It is shown that shear rupture zone characteristics differ when subjected to constant normal stiffness and normal stress boundary conditions. The two boundary conditions generate different fracturing processes leading to rupture zone creation and thus different ultimate rupture zone geometries as well as different shear stress versus horizontal displacement responses due to the coupling between the normal and shear stresses (stress-path) under constant normal stiffness boundary conditions.

It was conclusively demonstrated that the characteristics of a shear rupture zone are not only a function of the rock or rock mass properties but also the prevailing boundary conditions under which the rupture zone is created.

Under constant normal stiffness boundary conditions:

a discontinuous rupture zone is created before the maximum shear strength is reached (i.e., pre-peak);


the shear stress versus applied horizontal displacement response is always brittle (i.e., has near instantaneous large shear stress drops) during the transition from maximum peak strength to a linear residual frictional shear strength;

specimen rupture occurs via a shear mechanism with rupture zone creation occurring in a consistent process regardless of initial applied normal stress magnitude (i.e., unlike rupture under constant normal stress which is dependent on the applied normal stress magnitude);

shear stress oscillatory behaviour is evident in the shear stress versus applied horizontal displacement curves;

the shear stress oscillatory behaviour is caused by cohesion loss processes in the damage zone around the shear rupture zone and the creation of fracture networks which lead to stepped cumulative fracture curves;

the linear strength envelope obtained by constant stiffness tests is not a conventional peak strength criterion, it is a threshold at which the rupture process is initiated (when yield starts) and becomes an envelope that defines the evolution of strength with increasing applied displacement until shear rupture occurs; and

The shear rupture creation process is displacement rather than stress controlled.

When mining, both the stress and deformation boundaries are changed. In this thesis it is demonstrated, as an important conclusion, that shear rupture zones induced by mining are not only dependent on the rock mass properties in which they are created but are controlled, to a large extent, by the boundary conditions imposed by the mining system. That this is true is not new, Salamon (1970) amongst others has long shown that the mining system affects, for example, pillar energy release behaviour and McKinnon and Garrido (1998) have shown that boundary conditions have an influence on fracture orientation and sequence of development. What is new is that it is conclusively demonstrated that the evolution and final characteristics of a shear rupture zone depend on the boundary condition under which it is created.

The implication of this improved understanding of shear rupture zone creation is of greatest economic value in mining when localized failure occurs as in confined pillars, abutments, or when mining into discontinuous fault structures. However, it should also lead to reinterpretations of rupture processes during earthquake events.

While not demonstrated in this thesis, the findings of this work also assist in better understanding differences in fault-slip (double-couple source mechanism) rockbursting created by mining. As is shown in Chapters 2 and 3 of the thesis, the post-peak load-displacement characteristics of near identical rock masses changed drastically depending on the boundary condition. Under constant normal stress, rupture at low normal stress was more brittle (Thesis Chapter 2), whereas, it was more brittle under higher normal stresses when under high normal stiffness (Thesis Chapter 3). Since shear rupture at low normal stresses under constant normal stress boundary conditions will release less energy compared to higher normal stresses under constant normal stiffness boundary conditions (due to the stress drop increase), conditions of shear rupture in high normal stiffness zones are most critical for mining because they have the potential to release more energy.

This body of work provides an improved understanding of the shear rupture of brittle rock under both constant normal stress and normal stiffness boundary conditions through the use of calibrated numerical simulations. By applying this understanding to two field case histories, which also support the findings from the DEM simulations, it was possible to arrive at an improved interpretation of shear rupture zone creation in pillars and to provide evidence of boundary condition effects in the field.

7.0 References

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